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Recent posts

- Conventions are confusing. Unfortunately, it seems like we always need to choose one when we want to concretely write down a mathematical idea. Meanwhile, when taking limits of things, we need to be careful. The conventions and relationships between discrete Fourier transforms (DFT), Fourier series, and continuous Fourier transforms (FT), are confusing enough that I decided to write a reference on it. + The conventions and relationships between discrete Fourier transforms (DFT), Fourier series, and continuous Fourier transforms (FT), are confusing enough that I decided to write a reference on it.
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- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

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- Conventions are confusing. Unfortunately, it seems like we always need to choose one when we want to concretely write down a mathematical idea. Meanwhile, when taking limits of things, we need to be careful. The conventions and relationships between discrete Fourier transforms (DFT), Fourier series, and continuous Fourier transforms (FT), are confusing enough that I decided to write a reference on it. + The conventions and relationships between discrete Fourier transforms (DFT), Fourier series, and continuous Fourier transforms (FT), are confusing enough that I decided to write a reference on it.
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- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

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+ diff --git a/en/index.json b/en/index.json index 5d043ba7..a414d6d2 100644 --- a/en/index.json +++ b/en/index.json @@ -5,4 +5,4 @@ - [{"authors":null,"categories":null,"content":"Hello, my name is Sun Woo, a PhD student at King’s College London interested in theoretical condensed matter physics and machine learning, working with Prof. Joe Bhaseen.\nI served Republic of Korea’s national service as a ‘Skilled Industry Personnel’, at a medical startup AIRS Medical, where I developed novel machine-learning algorithms for MRI reconstruction and ultrasound vein detection combining MRI Physics, Compressed Sensing, and recent deep learning techniques. I was discharged on November 2021 and continued to work there until April 2023.\nDuring my service, I also did part-time research at Max Planck Institute of Complex Systems (mostly remotely) with Prof. Markus Heyl and Giuseppe de Tomasi on developing efficient ways of studying the dynamics of bosonic systems in the Many-Body Localized (MBL) regime.\nI completed the Masters of Advanced Studies in Physics at Cambridge University in 2019, and a Bachelors of Science in Physics with Theoretical Physics at Imperial College London in 2018. Please check my CV and Projects page for more details. My Google Scholar can be found here, and my Twitter here.\nIf you would like to get in touch, please feel free to contact me at swk34 [at] cantab [dot] ac [dot] uk.\nPlease note that this site is still under construction. So, some things may be incomplete.\n","date":1693440000,"expirydate":-62135596800,"kind":"term","lang":"en","lastmod":1693440000,"objectID":"68b6b3d93c9b3620d7854d6d088cfd84","permalink":"","publishdate":"0001-01-01T00:00:00Z","relpermalink":"","section":"authors","summary":"Hello, my name is Sun Woo, a PhD student at King’s College London interested in theoretical condensed matter physics and machine learning, working with Prof. Joe Bhaseen.\nI served Republic of Korea’s national service as a ‘Skilled Industry Personnel’, at a medical startup AIRS Medical, where I developed novel machine-learning algorithms for MRI reconstruction and ultrasound vein detection combining MRI Physics, Compressed Sensing, and recent deep learning techniques.","tags":null,"title":"Sun Woo Kim","type":"authors"},{"authors":["Sun Woo Kim"],"categories":null,"content":" To cite this page @misc{swkim2023continuous, author={Kim, Sun Woo}, title={Continuous and discrete Fourier transforms}, year={2023}, howpublished={\\url{https://sunwoo-kim.github.io/en/posts/fourier-transforms/}}, note={Accessed: 2023-08-31} } Conventions are confusing. Unfortunately, it seems like we always need to choose one when we want to concretely write down a mathematical idea. Meanwhile, when taking limits of things, we need to be careful. The conventions and relationships between discrete Fourier transforms (DFT), Fourier series, and continuous Fourier transforms (FT), are confusing enough that I decided to write a reference on it.\nThese notes closely follow Austen Lamacraft’s notes, but with some different choices in notation to explain each step as clearly as I can.\nDiscrete Fourier transform Consider a function $f_j$ defined for discrete inputs $j = {0, \\dots, N-1}$.\nThen we can define the discrete fourier transform (DFT) by\n$$ \\tilde{f}_n = C(N) \\sum_{j=0}^{N-1} f_j e^{-i 2 \\pi n j/N}, $$\nwhere $C(N)$ is a normalisation constant. Since $\\tilde{f}_n$ is periodic in $n$ with period $L$, there are many choices for the domain for $n$. We could choose $n \\in {0, \\dots, N-1}$, or choose the domain to be symmetric(ish),\n\\begin{equation} \\label{symmetric_domain} n \\in \\begin{dcases} \\left \\{ -\\frac{N}{2}, \\dots, \\frac{N}{2}-1 \\right\\} \u0026amp; N \\text{ even} \\\\ \\left \\{ -\\frac{N-1}{2}, \\dots, \\frac{N-1}{2} \\right\\} \u0026amp; N\\text{ odd}. \\end{dcases} \\end{equation}\nWhichever convention we choose, we can use the identity $\\sum_n e^{+i 2 \\pi n j /L} = N\\delta_{j,0 \\text{mod} N}$, to show that\n$$ f_j = \\tilde C(N) \\sum_{n} \\tilde{f}_n e^{+i 2 \\pi n j/N}, $$\nwhere the normalisation constants $C(N)$, $\\tilde C(N)$ can be chosen to be anything, as long as $C(N) \\tilde C(N) = 1/N$. The popular choice is $C, \\tilde C = 1/\\sqrt{N}$.\nLater, we will want to look at the limits of $N \\rightarrow \\infty$. There we will choose the appropriate normalisations $C(N), \\tilde C(N)$ such that the integrals have a well-defined limit.\n(Of course, the sign in the exponential is also a convention in defining the Fourier transforms, but the convention used here is used almost everywhere, so let’s not be worried about that.)\nAdding space We can consider the case where our function is a function on discrete space $x_j = aj$, where $a$ is the lattice spacing. Then the system size is $L=Na$, and $N$ the number of discrete sptial points. Then the Fourier transform becomes\n$$ \\tilde{f}_n = C(N) \\sum_{j=1}^N f(x_j) e^{-i 2 \\pi n x_j/L}. $$\nWe can also define the wavevector $k_n = 2 \\pi n /L$, and write\n$$ \\tilde{f}(k_n) = {C}(N) \\sum_{j=1}^N f(x_j) e^{-i k_n x_j}, $$\nwhilst the inverse transform becomes\n$$ f(x_j) = \\tilde{C}(N) \\sum_{n} \\tilde{f}(k_n) e^{+i k_n x_j}. $$\nNow, let’s make two choices from this point onwards:\nFirst, let’s choose the symmetric domain for $n$. And, let’s expand the domain for $j$ to be $\\mathbb Z$, and say that $f_j$ is periodic in $N$, $f_{j} = f_{j+N}$. This doesn’t do anything, as we can just look at the function in the domain we were interested in the end. But it does mean that we can also choose the symmetric domain Eq. \\eqref{symmetric_domain} for $j$ as well. $N \\rightarrow \\infty$ and $a \\rightarrow 0$ keeping $Na = L$ fixed Here, we are going to continuous real-space, but fix the length of the system. This will result in a countably infinite wavevector space. We can write the discrete Fourier transform as\n$$ \\tilde{f}(k_n) = {C}(N) \\sum_{j} f(x_j) e^{-i k_n x_j}. $$\nThen, the spacing between positions becomes $\\delta x_j = a$. We can choose $C(N) = a$, so we have the limit\n$$ \\tilde{f}(k_n) = \\int_{-L/2}^{L/2} dx f(x) e^{-i k_n x}, $$\nwhich then means that we require $\\tilde C(N) = 1/L$, so we have the inverse transform as\n$$ f(x) = \\frac{1}{L} \\sum_{n \\in \\mathbb{Z}} \\tilde{f}(k_n) e^{+i k_n x}. $$\nwhere now $n \\in \\mathbb{Z}$.\nThis then is just the Fourier series of a periodic function $f(x)$.\n$N \\rightarrow \\infty$ and $L \\rightarrow \\infty$, keeping $a$ fixed Here, we are going to keep the real-space discrete, but just make it infinitely long. This will result in a continuously varying wavevector space. We have\n$$ \\tilde{f}(k) = {C}(N) \\sum_{j \\in \\mathbb{Z}} f(x_j) e^{-i k x_j}, $$\nand\n$$ f(x_j) = \\tilde{C}(N) \\sum_{n} \\tilde{f}(k_n) e^{+i k_n x_j}. $$\nSince $k_n = 2 \\pi n /L = 2\\pi n /Na$, our spacing between wavevectors goes to zero. We have $\\delta k_n = 2 \\pi /N a$. The most popular convention is to choose $\\tilde C (N) = 1/Na$, so that we have\n$$ f(x_j) = \\int_{-\\pi/a}^{\\pi /a} \\frac{dk}{2 \\pi} \\tilde{f}(k) e^{+i k x_j}. $$\nThis means that the forward Fourier transform must be\n$$ \\tilde{f}(k) = a \\sum_{j \\in \\mathbb{Z}} f(x_j) e^{-i k x_j}. $$\nNormally, in this limit, people work with $a=1$ with $x_j =j$.\nSending $N \\rightarrow \\infty$, $L \\rightarrow \\infty$, $a \\rightarrow 0$ From the expressions for the $N \\rightarrow \\infty$, $L \\rightarrow \\infty$, and $a$ fixed, we just send $a \\rightarrow 0$. …","date":1693440000,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1693440000,"objectID":"c67c3a3d5a4b806ecee7c0566ec6a8e9","permalink":"https://sunwoo-kim.github.io/en/posts/fourier-transforms/","publishdate":"2023-08-31T00:00:00Z","relpermalink":"/en/posts/fourier-transforms/","section":"posts","summary":"Conventions are confusing. Unfortunately, it seems like we always need to choose one when we want to concretely write down a mathematical idea. Meanwhile, when taking limits of things, we need to be careful. The conventions and relationships between discrete Fourier transforms (DFT), Fourier series, and continuous Fourier transforms (FT), are confusing enough that I decided to write a reference on it.","tags":null,"title":"Continuous and discrete Fourier transforms","type":"posts"},{"authors":["Sun Woo Kim"],"categories":null,"content":" To cite this page @misc{swkim2023transverse, author={Kim, Sun Woo}, title={Transverse field Ising model}, year={2023}, howpublished={\\url{https://sunwoo-kim.github.io/en/posts/tfi-model/}}, note={Accessed: 2023-08-31} } In condensed matter physics, there are myriads of phenomenological models out there, with information about them scattered throughout different papers and resources. In the field of machine learning, there are efforts such as the Model Zoo which tries to organise these models in one place.\nIn this new series, I wanted to write down an overview of some of the models I looked at in my studies, with, where possible, the precise definition of what physicists mean when they say things like ’this model has a $U(1)$ symmetry’, or ’the FM case is dual to the AFM case’, etc., which were confusing to me when I was first reading about them. I will also try to support the discussion with some simulations from exact diagonalisation (ED) or Monté Carlo results. Since websites are interactive, I’ll try to nest the finer details in collapsibles.\nTable of Contents Introduction Hamiltonian Symmetries Phases of the ground state Quantum to classical mapping $d=1$ case Mapping to free fermions Wavefunction Entanglement $d=2$ case Introduction The transverse-field Ising (TFI) model is a prototypical model of a quantum magnet, where quantum fluctuations are used to generate paramagnetism instead of temperature. Another common name is ‘quantum Ising model’, or ’transverse Ising model’ (TIM). There is already a pretty good Wikipedia article about this model, but I wanted to just write it down in a more technical way, and fill in some missing gaps. Hamiltonian Broadly, I will use the following conventions for the TFI model,\n\\begin{equation} \\hat{H} = J\\sum_{\\langle i,j \\rangle} \\hat{\\sigma}^z_i \\hat{\\sigma}^z_j - g \\sum_i \\hat{\\sigma}^x_i - b \\sum_i \\hat{\\sigma}^z_i, \\end{equation} where $\\langle i, j\\rangle$ denotes nearest neighbours, and $\\sigma^\\alpha_i$ are Pauli matrices. We consider periodic boundary conditions (PBC). Without a loss of generalisation, we can set $J=\\pm 1$. Then $J=-1$ corresponds to ferrmomagnetic (FM) couplings, $J=1$ to antiferromagnetic couplings (AFM), respectively. $g$ is the transverse field strength, and $b$ the longitudinal field strength, which we will take as $b=0$ unless stated otherwise. Symmetries For $b=0$, the Hamiltonian is $\\mathbb{Z}_2$ symmetric, in the sense that we can globally send $\\hat{\\sigma}^z_i \\rightarrow - \\hat{\\sigma}^z_i$ and keep the Hamiltonian invariant, or concretely, $[\\hat{H}, \\hat{U}] = 0$ for $\\hat{U} = \\vec{\\prod_i} \\hat{\\sigma}^x_i$. Phases of the ground state The TFI model admits 3 phases: the ordered phase $g\u0026lt;g_c$, the critical/gapless phase $g=g_c$, and the disordered phase $g\u0026gt;g_c$. Quantum to classical mapping The TFI model in spatial dimensions $d$ can be explicitly mapped to a $d+1$ classical Ising model, in the sense that the zero-temperature (ground-state) density matrix can be explicitly mapped to the partition function of the classical Ising model. It is well known that $d$-dimensional quantum systems map to $(d+1)$-dimensional classical systems. Here, I’ll show an explicit mapping between $d$-dimensional quantum Ising model of length $L_Q$ with a $(d+1)$-dimensional Ising Hamiltonian, which is very similar to1 but is for general $d$ and with longitudinal field $b$.\nLet $\\hat{H}_0$ be the part of $\\hat{H}$ that only contain $\\hat{\\sigma}^z$’s, and $\\hat{H}_1$ be the rest of the Hamiltonian. Then the quantum partition function is given by\n\\begin{align} Z_Q = \\mathrm{tr} [e^{-\\beta_Q \\hat{H}_0 - \\beta_Q \\hat{H}_1}]. \\end{align}\nFrom Trotter’s theorem, for any two Hermitian operators bounded from below, $\\hat{A}$, $\\hat{B}$, we have $e^{\\hat{A} + \\hat{B}}= \\lim_{L \\rightarrow \\infty} \\left(e^{-\\hat{A}/L} e^{-\\hat{B}/L}\\right)^L$, therefore. defining $\\tau = \\beta_Q / L$,\n\\begin{align} Z_Q = \\sum_\\sigma \\langle \\sigma \\rvert \\lim_{L \\rightarrow \\infty} \\left(e^{-\\tau\\hat{H}_0} e^{-\\tau \\hat{H}_1}\\right)^L \\lvert \\sigma \\rangle. \\end{align}\nInserting identities, we have\n\\begin{align} Z_Q \u0026amp; = \\lim_{L \\rightarrow \\infty} \\sum_{\\sigma^1, \\dots, \\sigma^L} \\prod_{l=1}^{L} \\langle \\sigma^{l+1} \\rvert e^{-\\tau \\hat{H}_1} e^{-\\tau \\hat{H}_0} \\lvert \\sigma^l \\rangle \\end{align}\n\\begin{align} Z_Q \u0026amp; = \\lim_{L \\rightarrow \\infty} \\sum_{\\sigma^1, \\dots, \\sigma^L} \\prod_{l=1}^{L} e^{-\\tau H_0(\\sigma^l)} \\langle \\sigma^{l+1} \\rvert e^{\\tau g \\sum_i \\hat{\\sigma}^x_i } \\lvert \\sigma^l \\rangle, \\end{align}\nwhere $H_0(\\sigma^l) = - J \\sum_{\\langle i, j\\rangle} \\sigma^l_i \\sigma^l_j - b \\sum_i \\sigma_i^l$. Now we can use the identity that $\\langle \\sigma^{l+1}_i \\rvert e^{\\tau g \\hat{\\sigma}^x_i} \\lvert \\sigma^l_i \\rangle = \\Lambda e^{\\gamma \\sigma^{l+1}_i \\sigma^{l}_i}$, where $\\Lambda = \\sqrt{\\sinh(\\tau g)\\cosh(\\tau g)}$ and $\\gamma = -\\frac{1}{2} \\ln \\tanh(\\tau g)$. Therefore we have\n\\begin{align} Z_Q = \\lim_{L \\rightarrow \\infty} \\Lambda^{L_Q^d L} \\sum_{\\sigma^1, …","date":1691971200,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1691971200,"objectID":"815c9e667263186829c8628516d7a35e","permalink":"https://sunwoo-kim.github.io/en/posts/tfi-model/","publishdate":"2023-08-14T00:00:00Z","relpermalink":"/en/posts/tfi-model/","section":"posts","summary":"The transverse-field Ising (TFI) model is a prototypical model of a quantum magnet, where quantum fluctuations are used to generate paramagnetism instead of temperature.","tags":["Model Zoo"],"title":"Transverse field Ising model","type":"posts"},{"authors":["Sun Woo Kim"],"categories":null,"content":" To cite this page @misc{swkim2023hello, author={Kim, Sun Woo}, title={Hello World}, year={2021}, howpublished={\\url{https://sunwoo-kim.github.io/en/posts/hello-world/}}, note={Accessed: 2023-08-31} } Welcome to my new website! I hope to post some expository writeups on my research projects here.\nEDIT 2023-08-14\nNow $\\LaTeX$ should also work, complete with equation numbering,\n\\begin{equation} \\label{eq:gaussian-integral} \\int_{-\\infty}^{\\infty} dx \\frac{e^{-x^2/2\\sigma^2}}{\\sqrt{2 \\pi \\sigma^2}} = 1, \\end{equation} where Eq. \\eqref{eq:gaussian-integral} is the Gaussian integral identity.\nI also added a little snippet that automatically generates bibtex for my posts, so people can cite the posts if they want.\n","date":1618704000,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1691971200,"objectID":"6764dc15ea7b8ee22c2266fbedf1629e","permalink":"https://sunwoo-kim.github.io/en/posts/hello-world/","publishdate":"2021-04-18T00:00:00Z","relpermalink":"/en/posts/hello-world/","section":"posts","summary":"Welcome to my new website!","tags":null,"title":"Hello World","type":"posts"},{"authors":["Sun Woo Kim"],"categories":null,"content":" For the general public (click to open and close) When we put espresso and milk together, we expect the two types of particles to mix, eventually resulting in a homogeneous mixture that we call latte. This phenomenon is called ‘thermalization’. In usual circumstances, we expect systems to thermalize, where a ‘system’ is simply a set of governing rules and agents. In our case of coffee and milk, the governing rules would be the physics of fluids, and the agents would be the coffee and milk particles.\nThere are also systems that do not thermalize. Non-interacting systems are a common example of non-thermalizing systems, since the individual particles cannot talk to each other and therefore cannot mix. One important non-interacting system is an ‘Anderson Localized’ (AL) one, where non-interacting particles are ‘localized’, i.e. they do not venture far away from their initial positions. In other words, they retain a memory of their initial states.\nNaively, we expect that interacting systems to thermalize. What is somewhat unexpected, then, is that in certain situations, even interacting particles can stay localized and not thermalize. These are called ‘Many-Body Localized’ (MBL) systems and are a big part of condensed matter physics research today. Theorists usually study such phenomena on a lattice. A lattice is a simplification of space, so that instead of particles being in continuous positions, they occupy discrete ‘sites’, which are states that particles can be in.\nIn physics, there are two types of fundamental particles. The first are ‘fermions’, which are particles where only one particle can occupy a given site, much like the game of musical chairs. The second are ‘bosons’, where any number of particles can occupy a single site at a time. In the musical chairs analogy, it would be as if multiple players could sit on top of a single chair. Experimentalists have recently been studying MBL systems using bosons. However, both theoretically and computationally, bosons are much harder to solve compared to fermions. Why? Well, if you think about the possible states for fermions, there can either be a particle or no particle per site, so there are $2 \\times 2 \\times \\cdots \\times 2 = 2^L$ possible states, where $L$ is the number of sites. However, for bosons, if there are $N$ particles in total, there can be 0 to $N$ particles per site, so there are $(N+1)^L$ possible states. As you have more particles, there are simply more possible states for bosons compared to fermions.\nIn our work, we developed a way to study bosonic MBL systems efficiently. The method is quite abstract, but can be explained with the following analogy. A non-interacting quantum system can be thought of as a collection of pendulums, each located on a site in the lattice and each with their own frequency. Each pendulum only affects their own weights, each oscillating with their own frequency as they swing, without getting in the way of pendulums in other sites. In the roughest approximation, the interactions will only change the frequencies of these pendulums. This approximation is called ‘Poincaré-Lindstedt’ theory. By applying this theory/principle in our analyses of bosonic MBL systems, we found that even the most basic approximation is enough to show some of the hallmarks of MBL systems.\nOne of the hallmarks of MBL systems is the slow spreading of information due to the interactions of the particles. The above cover photo depicts a lattice. On the left is the non-interacting AL system, and on the right is the interacting MBL system. The x-axis is space, and the y-axis is ‘log-time’, where time increases exponentially as you go further up the graph, from 1 to 10 to 100 and so on. The areas where the lattice/graph is lit up indicates that information was transmitted. We can see that in the case of the AL system, the spread of information is stunted, but in the case of the MBL system, because the particles interact, information is transmitted, albeit very slowly, since you need to wait an exponentially longer time for information to spread to the next site.\nIf you’d like to know more about this work, please check out the PDF link, which links to an ArXiv page.\nAbstract (click to open and close) Recent experiments in quantum simulators have provided evidence for the Many-Body Localized (MBL) phase in 1D and 2D bosonic quantum matter. The theoretical study of such bosonic MBL, however, is a daunting task due to the unbounded nature of its Hilbert space. In this work, we introduce a method to compute the long-time real-time evolution of 1D and 2D bosonic systems in an MBL phase at strong disorder and weak interactions. We focus on local dynamical indicators that are able to distinguish an MBL phase from an Anderson localized phase. In particular, we consider the temporal fluctuations of local observables, the spatiotemporal behavior of two-time correlators and Out-Of-Time-Correlators (OTOCs). We show that these few-body observables can be …","date":1618704000,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1618704000,"objectID":"814db4367a531471bdb556b67639d72c","permalink":"https://sunwoo-kim.github.io/en/projects/mblbosons/","publishdate":"2021-04-18T00:00:00Z","relpermalink":"/en/projects/mblbosons/","section":"projects","summary":" ","tags":null,"title":"Many-body localization in bosons","type":"projects"},{"authors":["Sun Woo Kim"],"categories":null,"content":" For the general public (click to open and close) You are probably well aware of tiles on the sidewalk or wallpapers with patterns that repeat themselves, potentially forever. There is some kind of symmetry there, but how do we describe it? Mathematicians were able to codify these patterns using a field called ‘group theory’. In group theory, we define symmetries as transformations that leave the pattern the same, in other words, invariant. For example, assuming that the pattern goes on forever, we can shift the repeating pattern of wallpapers by certain directions and amount can retrieve the same exact pattern as before.\nIn physics, systems that repeat themselves forever are called crystals. What’s cool about crystals is that the symmetries of the crystals have a direct impact on how waves propagating in the crystals themselves behave, such as if we were to shine light (famously known as a wave) through it, or disturb it (resulting in a sound wave) and so on. Only knowing the symmetries of the crystal, we can predict whether a wave propagating in the crystal will spread out evenly, or propagate in a highly directed fashion, such as in the cover photo.\nIn wallpapers or sidewalks, there are two different kinds of symmetries. The first is symmorphic symmetry, where there exists two kinds of separable symmetries: translational, shifting the entire pattern, and rotational, only rotating the entire pattern around certain points. Nonsymmorphic symmetries are a bit more complicated, in that you cannot separate out the translational and rotational symmetries; there are certain symmetries that require partial translation, followed by a rotation.\nIn this project, we predicted how certain crystals would behave based on their symmetries, and used their properties to design ‘waveguides’, that are able to steer the waves around in particular directions efficiently.\nIf you like to know more about the work, please check out the PDF.\nAbstract (click to open and close) The study of wave propagation through structured periodic media depends critically upon the periodic lattice from which the medium is constructed. That is unsurprising, but perhaps what is slightly more surprising, is that pieces of pure mathematics play a key role - in particular, group and representation theory. Group theory is the natural language that encodes the symmetries of shape and form. Here we use it to consider a class of $2D$ periodic crystals whose lattice is encoded by nonsymmorphic space groups. These are often overlooked due to their relative complexity compared to the symmorphic space groups. We demonstrate that nonsymmorphic groups have possible practical interest in terms of coalescence of dispersion curves, Dirac points and band-sticking, using both theory and simulation. Once we’ve laid out the group theoretical framework in the context of the nonsymmorphic crystals, we use it to illustrate how accidental degeneracies can arise in symmorphic square lattices. We combine this phenomenon with topological valley effects to design highly-efficient topological waveguides and energy-splitters.\n","date":1534636800,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1534636800,"objectID":"f92f02b2c075a850d5cbd92b58eda4ea","permalink":"https://sunwoo-kim.github.io/en/projects/nonsymmorphic/","publishdate":"2018-08-19T00:00:00Z","relpermalink":"/en/projects/nonsymmorphic/","section":"projects","summary":" ","tags":null,"title":"Nonsymmorphic crystals and topological waveguiding","type":"projects"},{"authors":null,"categories":null,"content":"","date":-62135596800,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":-62135596800,"objectID":"ecaf0acadd75077acbe19596e39f5f60","permalink":"https://sunwoo-kim.github.io/en/activities/kclmanybodycircle/","publishdate":"0001-01-01T00:00:00Z","relpermalink":"/en/activities/kclmanybodycircle/","section":"activities","summary":"I run a reading group with KCL maths and physics PhD students, on topics regarding theories of many-body systems. Click heading for details.","tags":null,"title":"KCL Many Body Circle","type":"activities"}] \ No newline at end of file + [{"authors":null,"categories":null,"content":"Hello, my name is Sun Woo, a PhD student at King’s College London interested in theoretical condensed matter physics and machine learning, working with Prof. Joe Bhaseen.\nI served Republic of Korea’s national service as a ‘Skilled Industry Personnel’, at a medical startup AIRS Medical, where I developed novel machine-learning algorithms for MRI reconstruction and ultrasound vein detection combining MRI Physics, Compressed Sensing, and recent deep learning techniques. I was discharged on November 2021 and continued to work there until April 2023.\nDuring my service, I also did part-time research at Max Planck Institute of Complex Systems (mostly remotely) with Prof. Markus Heyl and Giuseppe de Tomasi on developing efficient ways of studying the dynamics of bosonic systems in the Many-Body Localized (MBL) regime.\nI completed the Masters of Advanced Studies in Physics at Cambridge University in 2019, and a Bachelors of Science in Physics with Theoretical Physics at Imperial College London in 2018. Please check my CV and Projects page for more details. My Google Scholar can be found here, and my Twitter here.\nIf you would like to get in touch, please feel free to contact me at swk34 [at] cantab [dot] ac [dot] uk.\nPlease note that this site is still under construction. So, some things may be incomplete.\n","date":1693440000,"expirydate":-62135596800,"kind":"term","lang":"en","lastmod":1693440000,"objectID":"68b6b3d93c9b3620d7854d6d088cfd84","permalink":"","publishdate":"0001-01-01T00:00:00Z","relpermalink":"","section":"authors","summary":"Hello, my name is Sun Woo, a PhD student at King’s College London interested in theoretical condensed matter physics and machine learning, working with Prof. Joe Bhaseen.\nI served Republic of Korea’s national service as a ‘Skilled Industry Personnel’, at a medical startup AIRS Medical, where I developed novel machine-learning algorithms for MRI reconstruction and ultrasound vein detection combining MRI Physics, Compressed Sensing, and recent deep learning techniques.","tags":null,"title":"Sun Woo Kim","type":"authors"},{"authors":["Sun Woo Kim"],"categories":null,"content":" To cite this page @misc{swkim2023continuous, author={Kim, Sun Woo}, title={Continuous and discrete Fourier transforms}, year={2023}, howpublished={\\url{https://sunwoo-kim.github.io/en/posts/fourier-transforms/}}, note={Accessed: 2023-09-02} } The conventions and relationships between discrete Fourier transforms (DFT), Fourier series, and continuous Fourier transforms (FT), are confusing enough that I decided to write a reference on it.\nThese notes closely follow Austen Lamacraft’s notes, but with some different choices in notation to explain each step as clearly as I can.\nDiscrete Fourier transform Consider a function $f_j$ defined for discrete inputs $j = {0, \\dots, N-1}$.\nThen we can define the discrete fourier transform (DFT) by\n$$ \\tilde{f}_n = C(N) \\sum_{j=0}^{N-1} f_j e^{-i 2 \\pi n j/N}, $$\nwhere $C(N)$ is a normalisation constant. Since $\\tilde{f}_n$ is periodic in $n$ with period $L$, there are many choices for the domain for $n$. We could choose $n \\in {0, \\dots, N-1}$, or choose the domain to be symmetric(ish),\n\\begin{equation} \\label{symmetric_domain} n \\in \\begin{dcases} \\left \\{ -\\frac{N}{2}, \\dots, \\frac{N}{2}-1 \\right\\} \u0026amp; N \\text{ even} \\\\ \\left \\{ -\\frac{N-1}{2}, \\dots, \\frac{N-1}{2} \\right\\} \u0026amp; N\\text{ odd}. \\end{dcases} \\end{equation}\nWhichever convention we choose, we can use the identity $\\sum_n e^{+i 2 \\pi n j /L} = N\\delta_{j,0 \\text{mod} N}$, to show that\n$$ f_j = \\tilde C(N) \\sum_{n} \\tilde{f}_n e^{+i 2 \\pi n j/N}, $$\nwhere the normalisation constants $C(N)$, $\\tilde C(N)$ can be chosen to be anything, as long as $C(N) \\tilde C(N) = 1/N$. The popular choice is $C, \\tilde C = 1/\\sqrt{N}$.\nLater, we will want to look at the limits of $N \\rightarrow \\infty$. There we will choose the appropriate normalisations $C(N), \\tilde C(N)$ such that the integrals have a well-defined limit.\n(Of course, the sign in the exponential is also a convention in defining the Fourier transforms, but the convention used here is used almost everywhere, so let’s not be worried about that.)\nAdding space We can consider the case where our function is a function on discrete space $x_j = aj$, where $a$ is the lattice spacing. Then the system size is $L=Na$, and $N$ the number of discrete sptial points. Then the Fourier transform becomes\n$$ \\tilde{f}_n = C(N) \\sum_{j=1}^N f(x_j) e^{-i 2 \\pi n x_j/L}. $$\nWe can also define the wavevector $k_n = 2 \\pi n /L$, and write\n$$ \\tilde{f}(k_n) = {C}(N) \\sum_{j=1}^N f(x_j) e^{-i k_n x_j}, $$\nwhilst the inverse transform becomes\n$$ f(x_j) = \\tilde{C}(N) \\sum_{n} \\tilde{f}(k_n) e^{+i k_n x_j}. $$\nNow, let’s make two choices from this point onwards:\nFirst, let’s choose the symmetric domain for $n$. And, let’s expand the domain for $j$ to be $\\mathbb Z$, and say that $f_j$ is periodic in $N$, $f_{j} = f_{j+N}$. This doesn’t do anything, as we can just look at the function in the domain we were interested in the end. But it does mean that we can also choose the symmetric domain Eq. \\eqref{symmetric_domain} for $j$ as well. $N \\rightarrow \\infty$ and $a \\rightarrow 0$ keeping $Na = L$ fixed Here, we are going to continuous real-space, but fix the length of the system. This will result in a countably infinite wavevector space. We can write the discrete Fourier transform as\n$$ \\tilde{f}(k_n) = {C}(N) \\sum_{j} f(x_j) e^{-i k_n x_j}. $$\nThen, the spacing between positions becomes $\\delta x_j = a$. We can choose $C(N) = a$, so we have the limit\n$$ \\tilde{f}(k_n) = \\int_{-L/2}^{L/2} dx f(x) e^{-i k_n x}, $$\nwhich then means that we require $\\tilde C(N) = 1/L$, so we have the inverse transform as\n$$ f(x) = \\frac{1}{L} \\sum_{n \\in \\mathbb{Z}} \\tilde{f}(k_n) e^{+i k_n x}. $$\nwhere now $n \\in \\mathbb{Z}$.\nThis then is just the Fourier series of a periodic function $f(x)$.\n$N \\rightarrow \\infty$ and $L \\rightarrow \\infty$, keeping $a$ fixed Here, we are going to keep the real-space discrete, but just make it infinitely long. This will result in a continuously varying wavevector space. We have\n$$ \\tilde{f}(k) = {C}(N) \\sum_{j \\in \\mathbb{Z}} f(x_j) e^{-i k x_j}, $$\nand\n$$ f(x_j) = \\tilde{C}(N) \\sum_{n} \\tilde{f}(k_n) e^{+i k_n x_j}. $$\nSince $k_n = 2 \\pi n /L = 2\\pi n /Na$, our spacing between wavevectors goes to zero. We have $\\delta k_n = 2 \\pi /N a$. The most popular convention is to choose $\\tilde C (N) = 1/Na$, so that we have\n$$ f(x_j) = \\int_{-\\pi/a}^{\\pi /a} \\frac{dk}{2 \\pi} \\tilde{f}(k) e^{+i k x_j}. $$\nThis means that the forward Fourier transform must be\n$$ \\tilde{f}(k) = a \\sum_{j \\in \\mathbb{Z}} f(x_j) e^{-i k x_j}. $$\nNormally, in this limit, people work with $a=1$ with $x_j =j$.\nSending $N \\rightarrow \\infty$, $L \\rightarrow \\infty$, $a \\rightarrow 0$ From the expressions for the $N \\rightarrow \\infty$, $L \\rightarrow \\infty$, and $a$ fixed, we just send $a \\rightarrow 0$. Since the spacing between spatial points is $\\delta x_j = a$, we can just write down\n$$ \\tilde{f}(k) = \\int_{-\\infty}^{\\infty} dxf(x) e^{-i k x}, $$\nand\n$$ f(x) = \\int_{-\\infty}^{\\infty} \\frac{dk}{2 \\pi} …","date":1693440000,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1693440000,"objectID":"c67c3a3d5a4b806ecee7c0566ec6a8e9","permalink":"https://sunwoo-kim.github.io/en/posts/fourier-transforms/","publishdate":"2023-08-31T00:00:00Z","relpermalink":"/en/posts/fourier-transforms/","section":"posts","summary":"The conventions and relationships between discrete Fourier transforms (DFT), Fourier series, and continuous Fourier transforms (FT), are confusing enough that I decided to write a reference on it.","tags":null,"title":"Continuous and discrete Fourier transforms","type":"posts"},{"authors":["Sun Woo Kim"],"categories":null,"content":" To cite this page @misc{swkim2023transverse, author={Kim, Sun Woo}, title={Transverse field Ising model}, year={2023}, howpublished={\\url{https://sunwoo-kim.github.io/en/posts/tfi-model/}}, note={Accessed: 2023-09-02} } In condensed matter physics, there are myriads of phenomenological models out there, with information about them scattered throughout different papers and resources. In the field of machine learning, there are efforts such as the Model Zoo which tries to organise these models in one place.\nIn this new series, I wanted to write down an overview of some of the models I looked at in my studies, with, where possible, the precise definition of what physicists mean when they say things like ’this model has a $U(1)$ symmetry’, or ’the FM case is dual to the AFM case’, etc., which were confusing to me when I was first reading about them. I will also try to support the discussion with some simulations from exact diagonalisation (ED) or Monté Carlo results. Since websites are interactive, I’ll try to nest the finer details in collapsibles.\nTable of Contents Introduction Hamiltonian Symmetries Phases of the ground state Quantum to classical mapping $d=1$ case Mapping to free fermions Wavefunction Entanglement $d=2$ case Introduction The transverse-field Ising (TFI) model is a prototypical model of a quantum magnet, where quantum fluctuations are used to generate paramagnetism instead of temperature. Another common name is ‘quantum Ising model’, or ’transverse Ising model’ (TIM). There is already a pretty good Wikipedia article about this model, but I wanted to just write it down in a more technical way, and fill in some missing gaps. Hamiltonian Broadly, I will use the following conventions for the TFI model,\n\\begin{equation} \\hat{H} = J\\sum_{\\langle i,j \\rangle} \\hat{\\sigma}^z_i \\hat{\\sigma}^z_j - g \\sum_i \\hat{\\sigma}^x_i - b \\sum_i \\hat{\\sigma}^z_i, \\end{equation} where $\\langle i, j\\rangle$ denotes nearest neighbours, and $\\sigma^\\alpha_i$ are Pauli matrices. We consider periodic boundary conditions (PBC). Without a loss of generalisation, we can set $J=\\pm 1$. Then $J=-1$ corresponds to ferrmomagnetic (FM) couplings, $J=1$ to antiferromagnetic couplings (AFM), respectively. $g$ is the transverse field strength, and $b$ the longitudinal field strength, which we will take as $b=0$ unless stated otherwise. Symmetries For $b=0$, the Hamiltonian is $\\mathbb{Z}_2$ symmetric, in the sense that we can globally send $\\hat{\\sigma}^z_i \\rightarrow - \\hat{\\sigma}^z_i$ and keep the Hamiltonian invariant, or concretely, $[\\hat{H}, \\hat{U}] = 0$ for $\\hat{U} = \\vec{\\prod_i} \\hat{\\sigma}^x_i$. Phases of the ground state The TFI model admits 3 phases: the ordered phase $g\u0026lt;g_c$, the critical/gapless phase $g=g_c$, and the disordered phase $g\u0026gt;g_c$. Quantum to classical mapping The TFI model in spatial dimensions $d$ can be explicitly mapped to a $d+1$ classical Ising model, in the sense that the zero-temperature (ground-state) density matrix can be explicitly mapped to the partition function of the classical Ising model. It is well known that $d$-dimensional quantum systems map to $(d+1)$-dimensional classical systems. Here, I’ll show an explicit mapping between $d$-dimensional quantum Ising model of length $L_Q$ with a $(d+1)$-dimensional Ising Hamiltonian, which is very similar to1 but is for general $d$ and with longitudinal field $b$.\nLet $\\hat{H}_0$ be the part of $\\hat{H}$ that only contain $\\hat{\\sigma}^z$’s, and $\\hat{H}_1$ be the rest of the Hamiltonian. Then the quantum partition function is given by\n\\begin{align} Z_Q = \\mathrm{tr} [e^{-\\beta_Q \\hat{H}_0 - \\beta_Q \\hat{H}_1}]. \\end{align}\nFrom Trotter’s theorem, for any two Hermitian operators bounded from below, $\\hat{A}$, $\\hat{B}$, we have $e^{\\hat{A} + \\hat{B}}= \\lim_{L \\rightarrow \\infty} \\left(e^{-\\hat{A}/L} e^{-\\hat{B}/L}\\right)^L$, therefore. defining $\\tau = \\beta_Q / L$,\n\\begin{align} Z_Q = \\sum_\\sigma \\langle \\sigma \\rvert \\lim_{L \\rightarrow \\infty} \\left(e^{-\\tau\\hat{H}_0} e^{-\\tau \\hat{H}_1}\\right)^L \\lvert \\sigma \\rangle. \\end{align}\nInserting identities, we have\n\\begin{align} Z_Q \u0026amp; = \\lim_{L \\rightarrow \\infty} \\sum_{\\sigma^1, \\dots, \\sigma^L} \\prod_{l=1}^{L} \\langle \\sigma^{l+1} \\rvert e^{-\\tau \\hat{H}_1} e^{-\\tau \\hat{H}_0} \\lvert \\sigma^l \\rangle \\end{align}\n\\begin{align} Z_Q \u0026amp; = \\lim_{L \\rightarrow \\infty} \\sum_{\\sigma^1, \\dots, \\sigma^L} \\prod_{l=1}^{L} e^{-\\tau H_0(\\sigma^l)} \\langle \\sigma^{l+1} \\rvert e^{\\tau g \\sum_i \\hat{\\sigma}^x_i } \\lvert \\sigma^l \\rangle, \\end{align}\nwhere $H_0(\\sigma^l) = - J \\sum_{\\langle i, j\\rangle} \\sigma^l_i \\sigma^l_j - b \\sum_i \\sigma_i^l$. Now we can use the identity that $\\langle \\sigma^{l+1}_i \\rvert e^{\\tau g \\hat{\\sigma}^x_i} \\lvert \\sigma^l_i \\rangle = \\Lambda e^{\\gamma \\sigma^{l+1}_i \\sigma^{l}_i}$, where $\\Lambda = \\sqrt{\\sinh(\\tau g)\\cosh(\\tau g)}$ and $\\gamma = -\\frac{1}{2} \\ln \\tanh(\\tau g)$. Therefore we have\n\\begin{align} Z_Q = \\lim_{L \\rightarrow \\infty} \\Lambda^{L_Q^d L} \\sum_{\\sigma^1, …","date":1691971200,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1691971200,"objectID":"815c9e667263186829c8628516d7a35e","permalink":"https://sunwoo-kim.github.io/en/posts/tfi-model/","publishdate":"2023-08-14T00:00:00Z","relpermalink":"/en/posts/tfi-model/","section":"posts","summary":"The transverse-field Ising (TFI) model is a prototypical model of a quantum magnet, where quantum fluctuations are used to generate paramagnetism instead of temperature.","tags":["Model Zoo"],"title":"Transverse field Ising model","type":"posts"},{"authors":["Sun Woo Kim"],"categories":null,"content":" To cite this page @misc{swkim2023hello, author={Kim, Sun Woo}, title={Hello World}, year={2021}, howpublished={\\url{https://sunwoo-kim.github.io/en/posts/hello-world/}}, note={Accessed: 2023-09-02} } Welcome to my new website! I hope to post some expository writeups on my research projects here.\nEDIT 2023-08-14\nNow $\\LaTeX$ should also work, complete with equation numbering,\n\\begin{equation} \\label{eq:gaussian-integral} \\int_{-\\infty}^{\\infty} dx \\frac{e^{-x^2/2\\sigma^2}}{\\sqrt{2 \\pi \\sigma^2}} = 1, \\end{equation} where Eq. \\eqref{eq:gaussian-integral} is the Gaussian integral identity.\nI also added a little snippet that automatically generates bibtex for my posts, so people can cite the posts if they want.\n","date":1618704000,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1691971200,"objectID":"6764dc15ea7b8ee22c2266fbedf1629e","permalink":"https://sunwoo-kim.github.io/en/posts/hello-world/","publishdate":"2021-04-18T00:00:00Z","relpermalink":"/en/posts/hello-world/","section":"posts","summary":"Welcome to my new website!","tags":null,"title":"Hello World","type":"posts"},{"authors":["Sun Woo Kim"],"categories":null,"content":" For the general public (click to open and close) When we put espresso and milk together, we expect the two types of particles to mix, eventually resulting in a homogeneous mixture that we call latte. This phenomenon is called ‘thermalization’. In usual circumstances, we expect systems to thermalize, where a ‘system’ is simply a set of governing rules and agents. In our case of coffee and milk, the governing rules would be the physics of fluids, and the agents would be the coffee and milk particles.\nThere are also systems that do not thermalize. Non-interacting systems are a common example of non-thermalizing systems, since the individual particles cannot talk to each other and therefore cannot mix. One important non-interacting system is an ‘Anderson Localized’ (AL) one, where non-interacting particles are ‘localized’, i.e. they do not venture far away from their initial positions. In other words, they retain a memory of their initial states.\nNaively, we expect that interacting systems to thermalize. What is somewhat unexpected, then, is that in certain situations, even interacting particles can stay localized and not thermalize. These are called ‘Many-Body Localized’ (MBL) systems and are a big part of condensed matter physics research today. Theorists usually study such phenomena on a lattice. A lattice is a simplification of space, so that instead of particles being in continuous positions, they occupy discrete ‘sites’, which are states that particles can be in.\nIn physics, there are two types of fundamental particles. The first are ‘fermions’, which are particles where only one particle can occupy a given site, much like the game of musical chairs. The second are ‘bosons’, where any number of particles can occupy a single site at a time. In the musical chairs analogy, it would be as if multiple players could sit on top of a single chair. Experimentalists have recently been studying MBL systems using bosons. However, both theoretically and computationally, bosons are much harder to solve compared to fermions. Why? Well, if you think about the possible states for fermions, there can either be a particle or no particle per site, so there are $2 \\times 2 \\times \\cdots \\times 2 = 2^L$ possible states, where $L$ is the number of sites. However, for bosons, if there are $N$ particles in total, there can be 0 to $N$ particles per site, so there are $(N+1)^L$ possible states. As you have more particles, there are simply more possible states for bosons compared to fermions.\nIn our work, we developed a way to study bosonic MBL systems efficiently. The method is quite abstract, but can be explained with the following analogy. A non-interacting quantum system can be thought of as a collection of pendulums, each located on a site in the lattice and each with their own frequency. Each pendulum only affects their own weights, each oscillating with their own frequency as they swing, without getting in the way of pendulums in other sites. In the roughest approximation, the interactions will only change the frequencies of these pendulums. This approximation is called ‘Poincaré-Lindstedt’ theory. By applying this theory/principle in our analyses of bosonic MBL systems, we found that even the most basic approximation is enough to show some of the hallmarks of MBL systems.\nOne of the hallmarks of MBL systems is the slow spreading of information due to the interactions of the particles. The above cover photo depicts a lattice. On the left is the non-interacting AL system, and on the right is the interacting MBL system. The x-axis is space, and the y-axis is ‘log-time’, where time increases exponentially as you go further up the graph, from 1 to 10 to 100 and so on. The areas where the lattice/graph is lit up indicates that information was transmitted. We can see that in the case of the AL system, the spread of information is stunted, but in the case of the MBL system, because the particles interact, information is transmitted, albeit very slowly, since you need to wait an exponentially longer time for information to spread to the next site.\nIf you’d like to know more about this work, please check out the PDF link, which links to an ArXiv page.\nAbstract (click to open and close) Recent experiments in quantum simulators have provided evidence for the Many-Body Localized (MBL) phase in 1D and 2D bosonic quantum matter. The theoretical study of such bosonic MBL, however, is a daunting task due to the unbounded nature of its Hilbert space. In this work, we introduce a method to compute the long-time real-time evolution of 1D and 2D bosonic systems in an MBL phase at strong disorder and weak interactions. We focus on local dynamical indicators that are able to distinguish an MBL phase from an Anderson localized phase. In particular, we consider the temporal fluctuations of local observables, the spatiotemporal behavior of two-time correlators and Out-Of-Time-Correlators (OTOCs). We show that these few-body observables can be …","date":1618704000,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1618704000,"objectID":"814db4367a531471bdb556b67639d72c","permalink":"https://sunwoo-kim.github.io/en/projects/mblbosons/","publishdate":"2021-04-18T00:00:00Z","relpermalink":"/en/projects/mblbosons/","section":"projects","summary":" ","tags":null,"title":"Many-body localization in bosons","type":"projects"},{"authors":["Sun Woo Kim"],"categories":null,"content":" For the general public (click to open and close) You are probably well aware of tiles on the sidewalk or wallpapers with patterns that repeat themselves, potentially forever. There is some kind of symmetry there, but how do we describe it? Mathematicians were able to codify these patterns using a field called ‘group theory’. In group theory, we define symmetries as transformations that leave the pattern the same, in other words, invariant. For example, assuming that the pattern goes on forever, we can shift the repeating pattern of wallpapers by certain directions and amount can retrieve the same exact pattern as before.\nIn physics, systems that repeat themselves forever are called crystals. What’s cool about crystals is that the symmetries of the crystals have a direct impact on how waves propagating in the crystals themselves behave, such as if we were to shine light (famously known as a wave) through it, or disturb it (resulting in a sound wave) and so on. Only knowing the symmetries of the crystal, we can predict whether a wave propagating in the crystal will spread out evenly, or propagate in a highly directed fashion, such as in the cover photo.\nIn wallpapers or sidewalks, there are two different kinds of symmetries. The first is symmorphic symmetry, where there exists two kinds of separable symmetries: translational, shifting the entire pattern, and rotational, only rotating the entire pattern around certain points. Nonsymmorphic symmetries are a bit more complicated, in that you cannot separate out the translational and rotational symmetries; there are certain symmetries that require partial translation, followed by a rotation.\nIn this project, we predicted how certain crystals would behave based on their symmetries, and used their properties to design ‘waveguides’, that are able to steer the waves around in particular directions efficiently.\nIf you like to know more about the work, please check out the PDF.\nAbstract (click to open and close) The study of wave propagation through structured periodic media depends critically upon the periodic lattice from which the medium is constructed. That is unsurprising, but perhaps what is slightly more surprising, is that pieces of pure mathematics play a key role - in particular, group and representation theory. Group theory is the natural language that encodes the symmetries of shape and form. Here we use it to consider a class of $2D$ periodic crystals whose lattice is encoded by nonsymmorphic space groups. These are often overlooked due to their relative complexity compared to the symmorphic space groups. We demonstrate that nonsymmorphic groups have possible practical interest in terms of coalescence of dispersion curves, Dirac points and band-sticking, using both theory and simulation. Once we’ve laid out the group theoretical framework in the context of the nonsymmorphic crystals, we use it to illustrate how accidental degeneracies can arise in symmorphic square lattices. We combine this phenomenon with topological valley effects to design highly-efficient topological waveguides and energy-splitters.\n","date":1534636800,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1534636800,"objectID":"f92f02b2c075a850d5cbd92b58eda4ea","permalink":"https://sunwoo-kim.github.io/en/projects/nonsymmorphic/","publishdate":"2018-08-19T00:00:00Z","relpermalink":"/en/projects/nonsymmorphic/","section":"projects","summary":" ","tags":null,"title":"Nonsymmorphic crystals and topological waveguiding","type":"projects"},{"authors":null,"categories":null,"content":"","date":-62135596800,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":-62135596800,"objectID":"ecaf0acadd75077acbe19596e39f5f60","permalink":"https://sunwoo-kim.github.io/en/activities/kclmanybodycircle/","publishdate":"0001-01-01T00:00:00Z","relpermalink":"/en/activities/kclmanybodycircle/","section":"activities","summary":"I run a reading group with KCL maths and physics PhD students, on topics regarding theories of many-body systems. Click heading for details.","tags":null,"title":"KCL Many Body Circle","type":"activities"}] \ No newline at end of file diff --git a/en/index.xml b/en/index.xml index 723ac321..41bab002 100644 --- a/en/index.xml +++ b/en/index.xml @@ -24,10 +24,10 @@ </span></span><span class="line"><span class="cl"> title=<span class="nb">{</span>Continuous and discrete Fourier transforms<span class="nb">}</span>, </span></span><span class="line"><span class="cl"> year=<span class="nb">{</span>2023<span class="nb">}</span>, </span></span><span class="line"><span class="cl"> howpublished=<span class="nb">{</span><span class="k">\url</span><span class="nb">{</span>https://sunwoo-kim.github.io/en/posts/fourier-transforms/<span class="nb">}}</span>, -</span></span><span class="line"><span class="cl"> note=<span class="nb">{</span>Accessed: 2023-08-31<span class="nb">}</span> +</span></span><span class="line"><span class="cl"> note=<span class="nb">{</span>Accessed: 2023-09-02<span class="nb">}</span> </span></span><span class="line"><span class="cl"><span class="nb">}</span> </span></span></code></pre></div></details> -<p>Conventions are confusing. Unfortunately, it seems like we always need to choose one when we want to concretely write down a mathematical idea. Meanwhile, when taking limits of things, we need to be careful. The conventions and relationships between discrete Fourier transforms (DFT), Fourier series, and continuous Fourier transforms (FT), are confusing enough that I decided to write a reference on it.</p> +<p>The conventions and relationships between discrete Fourier transforms (DFT), Fourier series, and continuous Fourier transforms (FT), are confusing enough that I decided to write a reference on it.</p> <p>These notes closely follow <a href="https://austen.uk/courses/tqm/elastic-chain/" target="_blank" rel="noopener">Austen Lamacraft</a>’s notes, but with some different choices in notation to explain each step as clearly as I can.</p> <h2 id="discrete-fourier-transform">Discrete Fourier transform</h2> <p> @@ -244,7 +244,7 @@ $$</p> </span></span><span class="line"><span class="cl"> title=<span class="nb">{</span>Transverse field Ising model<span class="nb">}</span>, </span></span><span class="line"><span class="cl"> year=<span class="nb">{</span>2023<span class="nb">}</span>, </span></span><span class="line"><span class="cl"> howpublished=<span class="nb">{</span><span class="k">\url</span><span class="nb">{</span>https://sunwoo-kim.github.io/en/posts/tfi-model/<span class="nb">}}</span>, -</span></span><span class="line"><span class="cl"> note=<span class="nb">{</span>Accessed: 2023-08-31<span class="nb">}</span> +</span></span><span class="line"><span class="cl"> note=<span class="nb">{</span>Accessed: 2023-09-02<span class="nb">}</span> </span></span><span class="line"><span class="cl"><span class="nb">}</span> </span></span></code></pre></div></details> <p>In condensed matter physics, there are myriads of phenomenological models out there, with information about them scattered throughout different papers and resources. In the field of machine learning, there are efforts such as the <a href="https://modelzoo.co" target="_blank" rel="noopener">Model Zoo</a> which tries to organise these models in one place.</p> @@ -475,7 +475,7 @@ where $B$ is some constant.</p> </span></span><span class="line"><span class="cl"> title=<span class="nb">{</span>Hello World<span class="nb">}</span>, </span></span><span class="line"><span class="cl"> year=<span class="nb">{</span>2021<span class="nb">}</span>, </span></span><span class="line"><span class="cl"> howpublished=<span class="nb">{</span><span class="k">\url</span><span class="nb">{</span>https://sunwoo-kim.github.io/en/posts/hello-world/<span class="nb">}}</span>, -</span></span><span class="line"><span class="cl"> note=<span class="nb">{</span>Accessed: 2023-08-31<span class="nb">}</span> +</span></span><span class="line"><span class="cl"> note=<span class="nb">{</span>Accessed: 2023-09-02<span class="nb">}</span> </span></span><span class="line"><span class="cl"><span class="nb">}</span> </span></span></code></pre></div></details> <p>Welcome to my new website! I hope to post some expository writeups on my research projects here.</p> diff --git a/en/posts/fourier-transforms/index.html b/en/posts/fourier-transforms/index.html index 0896e15d..8c3556a7 100644 --- a/en/posts/fourier-transforms/index.html +++ b/en/posts/fourier-transforms/index.html @@ -1,6 +1,6 @@ - + @@ -279,7 +279,7 @@ - + @@ -319,7 +319,7 @@ - + Continuous and discrete Fourier transforms title={Continuous and discrete Fourier transforms}, year={2023}, howpublished={\url{https://sunwoo-kim.github.io/en/posts/fourier-transforms/}}, - note={Accessed: 2023-08-31} + note={Accessed: 2023-09-02} } -

Conventions are confusing. Unfortunately, it seems like we always need to choose one when we want to concretely write down a mathematical idea. Meanwhile, when taking limits of things, we need to be careful. The conventions and relationships between discrete Fourier transforms (DFT), Fourier series, and continuous Fourier transforms (FT), are confusing enough that I decided to write a reference on it.

+

The conventions and relationships between discrete Fourier transforms (DFT), Fourier series, and continuous Fourier transforms (FT), are confusing enough that I decided to write a reference on it.

These notes closely follow Austen Lamacraft’s notes, but with some different choices in notation to explain each step as clearly as I can.

Discrete Fourier transform

@@ -1113,13 +1113,40 @@

Sun Woo Kim - Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

+

+ + + + + + + + + + +

+ @@ -1127,7 +1154,7 @@
Sun Woo Kim - Site last updated 2023-08-31 20:11:00.167111 + Site last updated 2023-09-02 12:03:10.971288

diff --git a/en/posts/fourier-transforms/index.html.bak b/en/posts/fourier-transforms/index.html.bak index 6ed8758d..c8bc4bd1 100644 --- a/en/posts/fourier-transforms/index.html.bak +++ b/en/posts/fourier-transforms/index.html.bak @@ -1,6 +1,6 @@ - + @@ -279,7 +279,7 @@ - + @@ -319,7 +319,7 @@ - + title={Continuous and discrete Fourier transforms}, year={2023}, howpublished={\url{https://sunwoo-kim.github.io/en/posts/fourier-transforms/}}, - note={Accessed: 2023-08-31} + note={Accessed: 2023-09-02} } -

Conventions are confusing. Unfortunately, it seems like we always need to choose one when we want to concretely write down a mathematical idea. Meanwhile, when taking limits of things, we need to be careful. The conventions and relationships between discrete Fourier transforms (DFT), Fourier series, and continuous Fourier transforms (FT), are confusing enough that I decided to write a reference on it.

+

The conventions and relationships between discrete Fourier transforms (DFT), Fourier series, and continuous Fourier transforms (FT), are confusing enough that I decided to write a reference on it.

These notes closely follow Austen Lamacraft’s notes, but with some different choices in notation to explain each step as clearly as I can.

Discrete Fourier transform

@@ -1113,13 +1113,40 @@ $$

+ + + + + + + + + + + + + +

- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

+

+ + + + + + + + + + +

+ diff --git a/en/posts/hello-world/index.html b/en/posts/hello-world/index.html index 4a83d854..06d75ae8 100644 --- a/en/posts/hello-world/index.html +++ b/en/posts/hello-world/index.html @@ -1,6 +1,6 @@ - + @@ -796,7 +796,7 @@

Hello World

title={Hello World}, year={2021}, howpublished={\url{https://sunwoo-kim.github.io/en/posts/hello-world/}}, - note={Accessed: 2023-08-31} + note={Accessed: 2023-09-02} }

Welcome to my new website! I hope to post some expository writeups on my research projects here.

@@ -939,13 +939,40 @@
Sun Woo Kim - Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

+

+ + + + + + + + + + +

+ @@ -953,7 +980,7 @@
Sun Woo Kim - Site last updated 2023-08-31 20:11:00.167111 + Site last updated 2023-09-02 12:03:10.971288

diff --git a/en/posts/hello-world/index.html.bak b/en/posts/hello-world/index.html.bak index f3107d70..8690463b 100644 --- a/en/posts/hello-world/index.html.bak +++ b/en/posts/hello-world/index.html.bak @@ -1,6 +1,6 @@ - + @@ -796,7 +796,7 @@ title={Hello World}, year={2021}, howpublished={\url{https://sunwoo-kim.github.io/en/posts/hello-world/}}, - note={Accessed: 2023-08-31} + note={Accessed: 2023-09-02} }

Welcome to my new website! I hope to post some expository writeups on my research projects here.

@@ -939,13 +939,40 @@ where Eq. \eqref{eq:gaussian-integral} is the Gaussian integral identity.

+ + + + + + + + + + + + + +

- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

+

+ + + + + + + + + + +

+ diff --git a/en/posts/index.html b/en/posts/index.html index 665bef80..646c7f9e 100644 --- a/en/posts/index.html +++ b/en/posts/index.html @@ -1,6 +1,6 @@ - + @@ -757,7 +757,7 @@

Posts

- Conventions are confusing. Unfortunately, it seems like we always need to choose one when we want to concretely write down a mathematical idea. Meanwhile, when taking limits of things, we need to be careful. The conventions and relationships between discrete Fourier transforms (DFT), Fourier series, and continuous Fourier transforms (FT), are confusing enough that I decided to write a reference on it. + The conventions and relationships between discrete Fourier transforms (DFT), Fourier series, and continuous Fourier transforms (FT), are confusing enough that I decided to write a reference on it.
@@ -1073,13 +1073,40 @@

Posts

+ + + + + + + + + + + + + +

- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

+

+ + + + + + + + + + +

+ @@ -1087,7 +1114,7 @@

Posts

- Site last updated 2023-08-31 20:11:00.167111 + Site last updated 2023-09-02 12:03:10.971288

diff --git a/en/posts/index.html.bak b/en/posts/index.html.bak index 4a7ecfe7..05f4c8fc 100644 --- a/en/posts/index.html.bak +++ b/en/posts/index.html.bak @@ -1,6 +1,6 @@ - + @@ -757,7 +757,7 @@
- Conventions are confusing. Unfortunately, it seems like we always need to choose one when we want to concretely write down a mathematical idea. Meanwhile, when taking limits of things, we need to be careful. The conventions and relationships between discrete Fourier transforms (DFT), Fourier series, and continuous Fourier transforms (FT), are confusing enough that I decided to write a reference on it. + The conventions and relationships between discrete Fourier transforms (DFT), Fourier series, and continuous Fourier transforms (FT), are confusing enough that I decided to write a reference on it.
@@ -1073,13 +1073,40 @@ + + + + + + + + + + + + + +

- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

+

+ + + + + + + + + + +

+ diff --git a/en/posts/index.xml b/en/posts/index.xml index b0df6e11..95830bf9 100644 --- a/en/posts/index.xml +++ b/en/posts/index.xml @@ -24,10 +24,10 @@ </span></span><span class="line"><span class="cl"> title=<span class="nb">{</span>Continuous and discrete Fourier transforms<span class="nb">}</span>, </span></span><span class="line"><span class="cl"> year=<span class="nb">{</span>2023<span class="nb">}</span>, </span></span><span class="line"><span class="cl"> howpublished=<span class="nb">{</span><span class="k">\url</span><span class="nb">{</span>https://sunwoo-kim.github.io/en/posts/fourier-transforms/<span class="nb">}}</span>, -</span></span><span class="line"><span class="cl"> note=<span class="nb">{</span>Accessed: 2023-08-31<span class="nb">}</span> +</span></span><span class="line"><span class="cl"> note=<span class="nb">{</span>Accessed: 2023-09-02<span class="nb">}</span> </span></span><span class="line"><span class="cl"><span class="nb">}</span> </span></span></code></pre></div></details> -<p>Conventions are confusing. Unfortunately, it seems like we always need to choose one when we want to concretely write down a mathematical idea. Meanwhile, when taking limits of things, we need to be careful. The conventions and relationships between discrete Fourier transforms (DFT), Fourier series, and continuous Fourier transforms (FT), are confusing enough that I decided to write a reference on it.</p> +<p>The conventions and relationships between discrete Fourier transforms (DFT), Fourier series, and continuous Fourier transforms (FT), are confusing enough that I decided to write a reference on it.</p> <p>These notes closely follow <a href="https://austen.uk/courses/tqm/elastic-chain/" target="_blank" rel="noopener">Austen Lamacraft</a>’s notes, but with some different choices in notation to explain each step as clearly as I can.</p> <h2 id="discrete-fourier-transform">Discrete Fourier transform</h2> <p> @@ -244,7 +244,7 @@ $$</p> </span></span><span class="line"><span class="cl"> title=<span class="nb">{</span>Transverse field Ising model<span class="nb">}</span>, </span></span><span class="line"><span class="cl"> year=<span class="nb">{</span>2023<span class="nb">}</span>, </span></span><span class="line"><span class="cl"> howpublished=<span class="nb">{</span><span class="k">\url</span><span class="nb">{</span>https://sunwoo-kim.github.io/en/posts/tfi-model/<span class="nb">}}</span>, -</span></span><span class="line"><span class="cl"> note=<span class="nb">{</span>Accessed: 2023-08-31<span class="nb">}</span> +</span></span><span class="line"><span class="cl"> note=<span class="nb">{</span>Accessed: 2023-09-02<span class="nb">}</span> </span></span><span class="line"><span class="cl"><span class="nb">}</span> </span></span></code></pre></div></details> <p>In condensed matter physics, there are myriads of phenomenological models out there, with information about them scattered throughout different papers and resources. In the field of machine learning, there are efforts such as the <a href="https://modelzoo.co" target="_blank" rel="noopener">Model Zoo</a> which tries to organise these models in one place.</p> @@ -475,7 +475,7 @@ where $B$ is some constant.</p> </span></span><span class="line"><span class="cl"> title=<span class="nb">{</span>Hello World<span class="nb">}</span>, </span></span><span class="line"><span class="cl"> year=<span class="nb">{</span>2021<span class="nb">}</span>, </span></span><span class="line"><span class="cl"> howpublished=<span class="nb">{</span><span class="k">\url</span><span class="nb">{</span>https://sunwoo-kim.github.io/en/posts/hello-world/<span class="nb">}}</span>, -</span></span><span class="line"><span class="cl"> note=<span class="nb">{</span>Accessed: 2023-08-31<span class="nb">}</span> +</span></span><span class="line"><span class="cl"> note=<span class="nb">{</span>Accessed: 2023-09-02<span class="nb">}</span> </span></span><span class="line"><span class="cl"><span class="nb">}</span> </span></span></code></pre></div></details> <p>Welcome to my new website! I hope to post some expository writeups on my research projects here.</p> diff --git a/en/posts/tfi-model/index.html b/en/posts/tfi-model/index.html index 4c3f4088..41cbcbc1 100644 --- a/en/posts/tfi-model/index.html +++ b/en/posts/tfi-model/index.html @@ -1,6 +1,6 @@ - + @@ -776,7 +776,7 @@

Transverse field Ising model

title={Transverse field Ising model}, year={2023}, howpublished={\url{https://sunwoo-kim.github.io/en/posts/tfi-model/}}, - note={Accessed: 2023-08-31} + note={Accessed: 2023-09-02} }

In condensed matter physics, there are myriads of phenomenological models out there, with information about them scattered throughout different papers and resources. In the field of machine learning, there are efforts such as the Model Zoo which tries to organise these models in one place.

@@ -1130,13 +1130,40 @@
Sun Woo Kim - Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

+

+ + + + + + + + + + +

+ @@ -1144,7 +1171,7 @@
Sun Woo Kim - Site last updated 2023-08-31 20:11:00.167111 + Site last updated 2023-09-02 12:03:10.971288

diff --git a/en/posts/tfi-model/index.html.bak b/en/posts/tfi-model/index.html.bak index dc26fcb4..59ff5e1e 100644 --- a/en/posts/tfi-model/index.html.bak +++ b/en/posts/tfi-model/index.html.bak @@ -1,6 +1,6 @@ - + @@ -776,7 +776,7 @@ title={Transverse field Ising model}, year={2023}, howpublished={\url{https://sunwoo-kim.github.io/en/posts/tfi-model/}}, - note={Accessed: 2023-08-31} + note={Accessed: 2023-09-02} }

In condensed matter physics, there are myriads of phenomenological models out there, with information about them scattered throughout different papers and resources. In the field of machine learning, there are efforts such as the Model Zoo which tries to organise these models in one place.

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- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

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Projects

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- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

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Sun Woo Kim - Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

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Sun Woo Kim - Site last updated 2023-08-31 20:11:00.167111 + Site last updated 2023-09-02 12:03:10.971288

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- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

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Sun Woo Kim - Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

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Sun Woo Kim - Site last updated 2023-08-31 20:11:00.167111 + Site last updated 2023-09-02 12:03:10.971288

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Publication_types

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- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

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+ diff --git a/en/tags/model-zoo/index.html b/en/tags/model-zoo/index.html index 75893a9c..ded9a121 100644 --- a/en/tags/model-zoo/index.html +++ b/en/tags/model-zoo/index.html @@ -1,6 +1,6 @@ - + @@ -855,13 +855,40 @@

Model Zoo

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Model Zoo

- Site last updated 2023-08-31 20:11:00.167111 + Site last updated 2023-09-02 12:03:10.971288

diff --git a/en/tags/model-zoo/index.html.bak b/en/tags/model-zoo/index.html.bak index 543d572a..06e38de7 100644 --- a/en/tags/model-zoo/index.html.bak +++ b/en/tags/model-zoo/index.html.bak @@ -1,6 +1,6 @@ - + @@ -855,13 +855,40 @@ + + + + + + + + + + + + + +

- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

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+ diff --git a/en/tags/model-zoo/index.xml b/en/tags/model-zoo/index.xml index 1156d3fd..eddb04e6 100644 --- a/en/tags/model-zoo/index.xml +++ b/en/tags/model-zoo/index.xml @@ -24,7 +24,7 @@ </span></span><span class="line"><span class="cl"> title=<span class="nb">{</span>Transverse field Ising model<span class="nb">}</span>, </span></span><span class="line"><span class="cl"> year=<span class="nb">{</span>2023<span class="nb">}</span>, </span></span><span class="line"><span class="cl"> howpublished=<span class="nb">{</span><span class="k">\url</span><span class="nb">{</span>https://sunwoo-kim.github.io/en/posts/tfi-model/<span class="nb">}}</span>, -</span></span><span class="line"><span class="cl"> note=<span class="nb">{</span>Accessed: 2023-08-31<span class="nb">}</span> +</span></span><span class="line"><span class="cl"> note=<span class="nb">{</span>Accessed: 2023-09-02<span class="nb">}</span> </span></span><span class="line"><span class="cl"><span class="nb">}</span> </span></span></code></pre></div></details> <p>In condensed matter physics, there are myriads of phenomenological models out there, with information about them scattered throughout different papers and resources. In the field of machine learning, there are efforts such as the <a href="https://modelzoo.co" target="_blank" rel="noopener">Model Zoo</a> which tries to organise these models in one place.</p> diff --git a/ko/.DS_Store b/ko/.DS_Store index 971e9819d12c858b67001616e6a4ff628a69492b..e473075bc32a5f2f6b89d84bd14575913762b596 100644 GIT binary patch delta 30 mcmZp1XmQx!Ak21iqS&IofXQ;gVv}Qpd6~i=Y|a**$pZkXg$jxQ delta 23 ecmZp1XmQx!AUs(@SY~pJFfWq_!{%(^nLGeh-3FBa diff --git a/ko/404.html b/ko/404.html index 28f29269..a51b2459 100644 --- a/ko/404.html +++ b/ko/404.html @@ -1,6 +1,6 @@ - + @@ -756,13 +756,40 @@

최신

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활동

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최신

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최신

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diff --git a/ko/authors/sunwoo/index.html.bak b/ko/authors/sunwoo/index.html.bak index 3d8c06e4..c00062c8 100644 --- a/ko/authors/sunwoo/index.html.bak +++ b/ko/authors/sunwoo/index.html.bak @@ -1,6 +1,6 @@ - + @@ -835,13 +835,40 @@ + + + + + + + + + + + + + +

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+ diff --git a/ko/authors/sunwoo/index.xml b/ko/authors/sunwoo/index.xml index 44aa7d2d..30efef82 100644 --- a/ko/authors/sunwoo/index.xml +++ b/ko/authors/sunwoo/index.xml @@ -24,7 +24,7 @@ </span></span><span class="line"><span class="cl"> title=<span class="nb">{</span>Hello World<span class="nb">}</span>, </span></span><span class="line"><span class="cl"> year=<span class="nb">{</span>2021<span class="nb">}</span>, </span></span><span class="line"><span class="cl"> howpublished=<span class="nb">{</span><span class="k">\url</span><span class="nb">{</span>https://sunwoo-kim.github.io/ko/posts/hello-world/<span class="nb">}}</span>, -</span></span><span class="line"><span class="cl"> note=<span class="nb">{</span>Accessed: 2023-08-31<span class="nb">}</span> +</span></span><span class="line"><span class="cl"> note=<span class="nb">{</span>Accessed: 2023-09-02<span class="nb">}</span> </span></span><span class="line"><span class="cl"><span class="nb">}</span> </span></span></code></pre></div></details> <!-- Welcome to my new website! I hope to post some expository writeups on my research projects here. Stay tuned! --> diff --git a/ko/categories/index.html b/ko/categories/index.html index 298ba617..5ade0d98 100644 --- a/ko/categories/index.html +++ b/ko/categories/index.html @@ -1,6 +1,6 @@ - + @@ -784,13 +784,40 @@

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+ diff --git a/ko/index.json b/ko/index.json index e445f3c6..d122516f 100644 --- a/ko/index.json +++ b/ko/index.json @@ -2,4 +2,4 @@ - [{"authors":null,"categories":null,"content":"안녕하세요, 김선우라고 합니다. 이론 응집물질물리학과 머신 러닝에 관심이 있고, 킹스 칼리지 런던에서 Joe Bhaseen 교수님 밑에서 박사과정을 밟고 있습니다.\n저는 산업기능요원으로서 에어스메디컬이라는 메디컬 스타트업에서 복무했으며 MRI 물리, Compressed Sensing, 그리고 현대 딥 러닝 테크닉을 동원해 MRI 복원과 초음파 정맥 감지를 위한 머신러닝 알고리즘을 개발했습니다. 2021년 11월에 전역했고, 2023년 4월까지 근무했습니다.\n또, 저는 복무중 파트타임 (대부분 비대면)으로 막스플랭크협회-복잡계연구소에서 Markus Heyl 교수와 Giuseppe de Tomasi 와 Many-Body Localized (MBL) bosonic systems의 dynamics를 study 할수 있는 방법을 연구했습니다.\n2019년 캠브릿지 대학에서 물리 석사 (Masters of Advanced Studies in physics)를 수료하였고, 2018년 임페리얼 칼리지에서 이론물리로 학사를 취득했습니다. 추가적인 항목은 제 CV와 프로젝트 페이지들을 참고해주시기 바랍니다. 제 Google Scholar는 여기서 확인하실수 있고, 트위터는 여기서 확인하실수 있습니다.\n저와 연락을 하고 싶으시다면, swk34 [at] cantab [dot] ac [dot] uk로 연락주시기 바랍니다.\n사이트가 준비중입니다. 미완성된 부분들이 있을수 있는점 양해 부탁 드립니다.\n","date":1618704000,"expirydate":-62135596800,"kind":"term","lang":"ko","lastmod":1618704000,"objectID":"b4184399a2f095c25b2bc76f9e06368f","permalink":"https://sunwoo-kim.github.io/ko/authors/sunwoo/","publishdate":"0001-01-01T00:00:00Z","relpermalink":"/ko/authors/sunwoo/","section":"authors","summary":"안녕하세요, 김선우라고 합니다. 이론 응집물질물리학과 머신 러닝에","tags":null,"title":"김선우","type":"authors"},{"authors":["김선우"],"categories":null,"content":" 이 페이지를 인용하려면 @misc{swkim2023hello, author={Kim, Sun Woo}, title={Hello World}, year={2021}, howpublished={\\url{https://sunwoo-kim.github.io/ko/posts/hello-world/}}, note={Accessed: 2023-08-31} } 뜨끈뜨끈한 제 사이트에 오신걸 환영합니다. 이 사이트엔 제 연구 프로젝트에 관한 소개문서를 올릴 예정이고, 가능하면 한국어로도 번역하려고 합니다. 기대해주세요!\n","date":1618704000,"expirydate":-62135596800,"kind":"page","lang":"ko","lastmod":1618704000,"objectID":"90736d2a0756bf49333db738c06c5ee6","permalink":"https://sunwoo-kim.github.io/ko/posts/hello-world/","publishdate":"2021-04-18T00:00:00Z","relpermalink":"/ko/posts/hello-world/","section":"posts","summary":"뜨끈뜨끈한 제 사이트에 오신걸 환영합니다.","tags":null,"title":"Hello World","type":"posts"},{"authors":["김선우"],"categories":null,"content":" 일반적인 사람들을 위해 (클릭해 열고 닫을수 있습니다) 커피믹스와 우유를 컵에 넣으면, 우린 보통 두가지의 입자들이 섞여, 라떼라는 균질한 혼합상태가 될거라고 생각합니다. 물리학자들은 이런 현상을 보고 시스템이 ‘열평형화’(thermalization) 된다고 표현합니다. 컵과 우유 뿐만아니라, 많은 일반적인 상황에서, 우린 감각적으로 시스템들이 열평형상태로 도달할거라고 예상합니다. 여기서 ‘시스템’ 이란, 위의 예의 유체 물리같은 어떤 ‘법칙’과, 커피 믹스와 우유 입자들 같이 그 법칙을 따르는 대상들의 집합을 뜻합니다.\n하지만 열평형화가 되지 않는 시스템들도 존재합니다. 입자간 아무런 소통을 하지 않는 ‘비상호작용’ 시스템들은 입자간 섞일수 없기 때문입니다. 물리학자들이 중요하게 생각하는 비상호작용 시스템중, 중요하게 여겨지는 ‘앤더슨 국소화’ (Anderson Localized)되어 있는 시스템들이 있습니다. 입자들이 초기 상태에서 멀리 퍼지지 않고 한곳에 모여있기때문에 ‘국소화’ 되어 있다고 표현합니다. 이 현상을 다른 관점으로 생각해 보면 해석해 보면, 초기 상태의 기억을 잃지 않는다고 해석할수 있겠죠.\n순진하게 생각해 보면, 우리는 상호작용을 하는 시스템들은 모조리 다 열평형화 될꺼라고 생각할수 있습니다. 하지만, 예상과 다르게 특정 상황에선 상호작용을 하는 시스템마저 국소화되어, 열평형상태가 되지 않는 경우가 있습니다. 이런 시스템들을 ‘다체 국소화’ (Many-Body Localized, MBL) 시스템이라고 하고, 다체 국소화(MBL)는 오늘날 응집물질물리 연구에 중요한 분야중 하나입니다. 이론 물리학자들은 보통 이런 현상을 ‘격자’ (lattice)에서 공부합니다. 격자란 공간을 단순화 한것으로, 입자들이 연속적인 위치에 존재하는것이 아니라, 이산된 ‘자리’에 위치 한다고 단순화 하며, 각 위치가 분리되어 있는 상태라고 생각하는 것입니다.\n물리학에는, 두가지의 기초적 입자들이 존재합니다. 첫번째는 ‘페르미온’ 이라는 입자 인데, 한 자리에 최대 한개의 입자가 자리잡을수 있다는것이 특징입니다. 두번째론 ‘보손’ 이라는 입자가 있습니다. 보손은 한 자리에 자리잡을수 있는 입자의 갯수가 제한되어 있지 않습니다. 최근, 실험물리학자들은 다체 국소화 (MBL) 시스템들을 보손입자를 이용하여 공부하고 있습니다. 그런데, 이론적으로나 연산적으로나, 페르미온에 비해, 보손을 공부하는것이 훨씬 더 어렵습니다. 왜냐고요? 우선 페르미온이 가질수 있는 상태를 모두 고려해봅시다. 각 자리에 페르미온이 있을수도, 없을수도 있기때문에, $2 \\times 2 \\times \\cdots \\times 2 = 2^L$ 가지의 경우의 수가 있습니다 (여기서 $L$는 자리의 총 갯수를 뜻합니다). 하지만 보손의 경우엔, 이론적으론 한 자리에 무한개의 보손이 존재할수 있고, 만약 보손입자의 총 갯수가 $N$개라고 가정해도 각 자리에 0부터 $N$까지의 입자들이 있을수 있기 때문에 $(N+1)^L$ 개의 경우의 수가 있습니다. 자리와 입자의 갯수가 늘어날수록 보손 시스템의 상태 갯수가 페르미온에 비해 훨씬 많다는걸 볼수 있죠.\n저희 연구에선 보손 다체 국소화 (bosonic MBL)시스템을 효율적으로 계산할수 있는 방법을 구안했습니다. 방법은 꽤나 추상적이지만, 은유적으로 설명할수 있습니다. 상호작용이 없는 양자 시스템은 각 자리에 자기만의 주기를 가지고 있는 진자들을 모아놓은것이라고 생각할수 있는데요, 이때 각 진자는 자신의 추에게만 영향을 미치고, 자기만의 주기를 가지고 흔들거리며, 다른 자리에 있는 진자들에게 영향을 미치지 않습니다. 이 시스템에 입자간 상호작용을 추가했을때, 가장 대략적 근사로 상호작용은 각 진자의 주기만 변화를 주고, 나머지 요소는 바뀌지 않는다고 가정할수 있는데, 이 근사 방법을 물리학자들은 ‘푸엥카레-린드스테드 섭동 이론’ (Poincaré-Lindstedt perturbation theory) 이라고 합니다. 아주 간단한 근사임에도 불구하고, 저희는 이 방법이 다체 국소화(MBL) 시스템 고유의 특징적 현상을 보여주기에 충분하다는걸 확인했습니다.\n다체 국소화 (MBL)시스템의 고유적 특성중 하나는 바로 상호작용으로 인해 정보가 느리게나마 전파된다는 점입니다. 위에 있는 커버 사진은 격자를 보여줍니다. 왼쪽은 상호작용이 없는 앤더슨 국소화 시스템의 격자이고, 오른쪽은 상호작용이 있는 다체 국소화 시스템의 격자입니다. x축은 공간이고, y축은 ‘로그 시간’ 인데, 위로 올라갈수록 간격이 급수함수적으로 증가하는 (예를들어 1, 10, 100, 등등) 시간 축 입니다. 밝은색은 정보가 전송되었다는것을 의미하고, 어두운색은 정보가 미치지 못했다는것을 의미합니다. 왼쪽은 상호작용이 없는 앤더슨 국소화(AL) 시스템, 오른쪽은 상호작용이 있는 다체 국소화(MBL) 시스템 입니다. 입자간 상호작용이 존재하기 때문에, 다체 국소화 시스템의 경우 아주 느리게나마 정보가 확산된다는것을 알수 있습니다.\n이 연구에 대해서 더 궁금하시다면, PDF 버튼을 눌러 ArXiv에 게시된 논문을 확인해 주시기 바랍니다.\n초록 (클릭해 열고 닫을수 있습니다) Recent experiments in quantum simulators have provided evidence for the Many-Body Localized (MBL) phase in 1D and 2D bosonic quantum matter. The theoretical study of such bosonic MBL, however, is a daunting task due to the unbounded nature of its Hilbert space. In this work, we introduce a method to compute the long-time real-time evolution of 1D and 2D bosonic systems in an MBL phase at strong disorder and weak interactions. We focus on local dynamical indicators that are able to distinguish an MBL phase from an Anderson localized phase. In particular, we consider the temporal fluctuations of local observables, the spatiotemporal behavior of two-time correlators and Out-Of-Time-Correlators (OTOCs). We show that these few-body observables can be computed with a computational effort that depends only polynomially on system size but is independent of the target time, by extending a recently proposed numerical method [Phys. Rev. B 99, 241114 (2019)] to mixed states and bosons. Our method also allows us to surrogate our numerical study with analytical considerations of the time-dependent behavior of the studied quantities.\n","date":1618704000,"expirydate":-62135596800,"kind":"page","lang":"ko","lastmod":1618704000,"objectID":"04bb235cd251b640eb9176edb6c0a2ef","permalink":"https://sunwoo-kim.github.io/ko/projects/mblbosons/","publishdate":"2021-04-18T00:00:00Z","relpermalink":"/ko/projects/mblbosons/","section":"projects","summary":" ","tags":null,"title":"보손입자의 다체 국소화 (Many-body localization in bosons)","type":"projects"},{"authors":["김선우"],"categories":null,"content":" For the general public (click to open and close) You are probably well aware of tiles on the sidewalk or wallpapers with patterns that repeat themselves, potentially forever. There is some kind of symmetry there, but how do we describe it? Mathematicians were able to codify these patterns using a field called ‘group theory’. In group theory, we define symmetries as transformations that leave the pattern the same, in other words, invariant. For example, assuming that the pattern goes on forever, we can shift the repeating pattern of wallpapers by certain directions and amount can retrieve the same exact pattern as before.\nIn physics, systems that repeat themselves forever are called crystals. What’s cool about crystals is that the symmetries of the crystals have a direct impact on how waves propagating in the crystals themselves behave, such as if we were to shine light (famously known as a wave) through it, or disturb it (resulting in a sound wave) and so on. Only knowing the symmetries of the crystal, we can predict whether a wave propagating in the crystal will spread out evenly, or propagate in a highly directed fashion, such as in the cover photo.\nIn wallpapers or sidewalks, there are two different kinds of symmetries. The first is symmorphic symmetry, where there exists two kinds of separable symmetries: translational, shifting the entire pattern, and rotational, only rotating the entire pattern around certain points. Nonsymmorphic symmetries are a bit more complicated, in that you cannot separate out the translational and rotational symmetries; there are certain symmetries that require partial translation, followed by a rotation.\nIn this project, we predicted how certain crystals would behave based on their symmetries, and used their properties to design ‘waveguides’, that are able to steer the waves around in particular directions efficiently.\nIf you like to know more about the work, please check out the PDF.\nAbstract (click to open and close) The study of wave propagation through structured periodic media depends critically upon the periodic lattice from which the medium is constructed. That is unsurprising, but perhaps what is slightly more surprising, is that pieces of pure mathematics play a key role - in particular, group and representation theory. Group theory is the natural language that encodes the symmetries of shape and form. Here we use it to consider a class of $2D$ periodic crystals whose lattice is encoded by nonsymmorphic space groups. These are often overlooked due to their relative complexity compared to the symmorphic space groups. We demonstrate that nonsymmorphic groups have possible practical interest in terms of coalescence of dispersion curves, Dirac points and band-sticking, using both theory and simulation. Once we’ve laid out the group theoretical framework in the context of the nonsymmorphic crystals, we use it to illustrate how accidental degeneracies can arise in symmorphic square lattices. We combine this phenomenon with topological valley effects to design highly-efficient topological waveguides and energy-splitters.\n","date":1534636800,"expirydate":-62135596800,"kind":"page","lang":"ko","lastmod":1534636800,"objectID":"ef65a6f62d2f386aa87f247b595a33dc","permalink":"https://sunwoo-kim.github.io/ko/projects/nonsymmorphic/","publishdate":"2018-08-19T00:00:00Z","relpermalink":"/ko/projects/nonsymmorphic/","section":"projects","summary":" ","tags":null,"title":"Nonsymmorphic crystals and topological waveguiding","type":"projects"}] \ No newline at end of file + [{"authors":null,"categories":null,"content":"안녕하세요, 김선우라고 합니다. 이론 응집물질물리학과 머신 러닝에 관심이 있고, 킹스 칼리지 런던에서 Joe Bhaseen 교수님 밑에서 박사과정을 밟고 있습니다.\n저는 산업기능요원으로서 에어스메디컬이라는 메디컬 스타트업에서 복무했으며 MRI 물리, Compressed Sensing, 그리고 현대 딥 러닝 테크닉을 동원해 MRI 복원과 초음파 정맥 감지를 위한 머신러닝 알고리즘을 개발했습니다. 2021년 11월에 전역했고, 2023년 4월까지 근무했습니다.\n또, 저는 복무중 파트타임 (대부분 비대면)으로 막스플랭크협회-복잡계연구소에서 Markus Heyl 교수와 Giuseppe de Tomasi 와 Many-Body Localized (MBL) bosonic systems의 dynamics를 study 할수 있는 방법을 연구했습니다.\n2019년 캠브릿지 대학에서 물리 석사 (Masters of Advanced Studies in physics)를 수료하였고, 2018년 임페리얼 칼리지에서 이론물리로 학사를 취득했습니다. 추가적인 항목은 제 CV와 프로젝트 페이지들을 참고해주시기 바랍니다. 제 Google Scholar는 여기서 확인하실수 있고, 트위터는 여기서 확인하실수 있습니다.\n저와 연락을 하고 싶으시다면, swk34 [at] cantab [dot] ac [dot] uk로 연락주시기 바랍니다.\n사이트가 준비중입니다. 미완성된 부분들이 있을수 있는점 양해 부탁 드립니다.\n","date":1618704000,"expirydate":-62135596800,"kind":"term","lang":"ko","lastmod":1618704000,"objectID":"b4184399a2f095c25b2bc76f9e06368f","permalink":"https://sunwoo-kim.github.io/ko/authors/sunwoo/","publishdate":"0001-01-01T00:00:00Z","relpermalink":"/ko/authors/sunwoo/","section":"authors","summary":"안녕하세요, 김선우라고 합니다. 이론 응집물질물리학과 머신 러닝에","tags":null,"title":"김선우","type":"authors"},{"authors":["김선우"],"categories":null,"content":" 이 페이지를 인용하려면 @misc{swkim2023hello, author={Kim, Sun Woo}, title={Hello World}, year={2021}, howpublished={\\url{https://sunwoo-kim.github.io/ko/posts/hello-world/}}, note={Accessed: 2023-09-02} } 뜨끈뜨끈한 제 사이트에 오신걸 환영합니다. 이 사이트엔 제 연구 프로젝트에 관한 소개문서를 올릴 예정이고, 가능하면 한국어로도 번역하려고 합니다. 기대해주세요!\n","date":1618704000,"expirydate":-62135596800,"kind":"page","lang":"ko","lastmod":1618704000,"objectID":"90736d2a0756bf49333db738c06c5ee6","permalink":"https://sunwoo-kim.github.io/ko/posts/hello-world/","publishdate":"2021-04-18T00:00:00Z","relpermalink":"/ko/posts/hello-world/","section":"posts","summary":"뜨끈뜨끈한 제 사이트에 오신걸 환영합니다.","tags":null,"title":"Hello World","type":"posts"},{"authors":["김선우"],"categories":null,"content":" 일반적인 사람들을 위해 (클릭해 열고 닫을수 있습니다) 커피믹스와 우유를 컵에 넣으면, 우린 보통 두가지의 입자들이 섞여, 라떼라는 균질한 혼합상태가 될거라고 생각합니다. 물리학자들은 이런 현상을 보고 시스템이 ‘열평형화’(thermalization) 된다고 표현합니다. 컵과 우유 뿐만아니라, 많은 일반적인 상황에서, 우린 감각적으로 시스템들이 열평형상태로 도달할거라고 예상합니다. 여기서 ‘시스템’ 이란, 위의 예의 유체 물리같은 어떤 ‘법칙’과, 커피 믹스와 우유 입자들 같이 그 법칙을 따르는 대상들의 집합을 뜻합니다.\n하지만 열평형화가 되지 않는 시스템들도 존재합니다. 입자간 아무런 소통을 하지 않는 ‘비상호작용’ 시스템들은 입자간 섞일수 없기 때문입니다. 물리학자들이 중요하게 생각하는 비상호작용 시스템중, 중요하게 여겨지는 ‘앤더슨 국소화’ (Anderson Localized)되어 있는 시스템들이 있습니다. 입자들이 초기 상태에서 멀리 퍼지지 않고 한곳에 모여있기때문에 ‘국소화’ 되어 있다고 표현합니다. 이 현상을 다른 관점으로 생각해 보면 해석해 보면, 초기 상태의 기억을 잃지 않는다고 해석할수 있겠죠.\n순진하게 생각해 보면, 우리는 상호작용을 하는 시스템들은 모조리 다 열평형화 될꺼라고 생각할수 있습니다. 하지만, 예상과 다르게 특정 상황에선 상호작용을 하는 시스템마저 국소화되어, 열평형상태가 되지 않는 경우가 있습니다. 이런 시스템들을 ‘다체 국소화’ (Many-Body Localized, MBL) 시스템이라고 하고, 다체 국소화(MBL)는 오늘날 응집물질물리 연구에 중요한 분야중 하나입니다. 이론 물리학자들은 보통 이런 현상을 ‘격자’ (lattice)에서 공부합니다. 격자란 공간을 단순화 한것으로, 입자들이 연속적인 위치에 존재하는것이 아니라, 이산된 ‘자리’에 위치 한다고 단순화 하며, 각 위치가 분리되어 있는 상태라고 생각하는 것입니다.\n물리학에는, 두가지의 기초적 입자들이 존재합니다. 첫번째는 ‘페르미온’ 이라는 입자 인데, 한 자리에 최대 한개의 입자가 자리잡을수 있다는것이 특징입니다. 두번째론 ‘보손’ 이라는 입자가 있습니다. 보손은 한 자리에 자리잡을수 있는 입자의 갯수가 제한되어 있지 않습니다. 최근, 실험물리학자들은 다체 국소화 (MBL) 시스템들을 보손입자를 이용하여 공부하고 있습니다. 그런데, 이론적으로나 연산적으로나, 페르미온에 비해, 보손을 공부하는것이 훨씬 더 어렵습니다. 왜냐고요? 우선 페르미온이 가질수 있는 상태를 모두 고려해봅시다. 각 자리에 페르미온이 있을수도, 없을수도 있기때문에, $2 \\times 2 \\times \\cdots \\times 2 = 2^L$ 가지의 경우의 수가 있습니다 (여기서 $L$는 자리의 총 갯수를 뜻합니다). 하지만 보손의 경우엔, 이론적으론 한 자리에 무한개의 보손이 존재할수 있고, 만약 보손입자의 총 갯수가 $N$개라고 가정해도 각 자리에 0부터 $N$까지의 입자들이 있을수 있기 때문에 $(N+1)^L$ 개의 경우의 수가 있습니다. 자리와 입자의 갯수가 늘어날수록 보손 시스템의 상태 갯수가 페르미온에 비해 훨씬 많다는걸 볼수 있죠.\n저희 연구에선 보손 다체 국소화 (bosonic MBL)시스템을 효율적으로 계산할수 있는 방법을 구안했습니다. 방법은 꽤나 추상적이지만, 은유적으로 설명할수 있습니다. 상호작용이 없는 양자 시스템은 각 자리에 자기만의 주기를 가지고 있는 진자들을 모아놓은것이라고 생각할수 있는데요, 이때 각 진자는 자신의 추에게만 영향을 미치고, 자기만의 주기를 가지고 흔들거리며, 다른 자리에 있는 진자들에게 영향을 미치지 않습니다. 이 시스템에 입자간 상호작용을 추가했을때, 가장 대략적 근사로 상호작용은 각 진자의 주기만 변화를 주고, 나머지 요소는 바뀌지 않는다고 가정할수 있는데, 이 근사 방법을 물리학자들은 ‘푸엥카레-린드스테드 섭동 이론’ (Poincaré-Lindstedt perturbation theory) 이라고 합니다. 아주 간단한 근사임에도 불구하고, 저희는 이 방법이 다체 국소화(MBL) 시스템 고유의 특징적 현상을 보여주기에 충분하다는걸 확인했습니다.\n다체 국소화 (MBL)시스템의 고유적 특성중 하나는 바로 상호작용으로 인해 정보가 느리게나마 전파된다는 점입니다. 위에 있는 커버 사진은 격자를 보여줍니다. 왼쪽은 상호작용이 없는 앤더슨 국소화 시스템의 격자이고, 오른쪽은 상호작용이 있는 다체 국소화 시스템의 격자입니다. x축은 공간이고, y축은 ‘로그 시간’ 인데, 위로 올라갈수록 간격이 급수함수적으로 증가하는 (예를들어 1, 10, 100, 등등) 시간 축 입니다. 밝은색은 정보가 전송되었다는것을 의미하고, 어두운색은 정보가 미치지 못했다는것을 의미합니다. 왼쪽은 상호작용이 없는 앤더슨 국소화(AL) 시스템, 오른쪽은 상호작용이 있는 다체 국소화(MBL) 시스템 입니다. 입자간 상호작용이 존재하기 때문에, 다체 국소화 시스템의 경우 아주 느리게나마 정보가 확산된다는것을 알수 있습니다.\n이 연구에 대해서 더 궁금하시다면, PDF 버튼을 눌러 ArXiv에 게시된 논문을 확인해 주시기 바랍니다.\n초록 (클릭해 열고 닫을수 있습니다) Recent experiments in quantum simulators have provided evidence for the Many-Body Localized (MBL) phase in 1D and 2D bosonic quantum matter. The theoretical study of such bosonic MBL, however, is a daunting task due to the unbounded nature of its Hilbert space. In this work, we introduce a method to compute the long-time real-time evolution of 1D and 2D bosonic systems in an MBL phase at strong disorder and weak interactions. We focus on local dynamical indicators that are able to distinguish an MBL phase from an Anderson localized phase. In particular, we consider the temporal fluctuations of local observables, the spatiotemporal behavior of two-time correlators and Out-Of-Time-Correlators (OTOCs). We show that these few-body observables can be computed with a computational effort that depends only polynomially on system size but is independent of the target time, by extending a recently proposed numerical method [Phys. Rev. B 99, 241114 (2019)] to mixed states and bosons. Our method also allows us to surrogate our numerical study with analytical considerations of the time-dependent behavior of the studied quantities.\n","date":1618704000,"expirydate":-62135596800,"kind":"page","lang":"ko","lastmod":1618704000,"objectID":"04bb235cd251b640eb9176edb6c0a2ef","permalink":"https://sunwoo-kim.github.io/ko/projects/mblbosons/","publishdate":"2021-04-18T00:00:00Z","relpermalink":"/ko/projects/mblbosons/","section":"projects","summary":" ","tags":null,"title":"보손입자의 다체 국소화 (Many-body localization in bosons)","type":"projects"},{"authors":["김선우"],"categories":null,"content":" For the general public (click to open and close) You are probably well aware of tiles on the sidewalk or wallpapers with patterns that repeat themselves, potentially forever. There is some kind of symmetry there, but how do we describe it? Mathematicians were able to codify these patterns using a field called ‘group theory’. In group theory, we define symmetries as transformations that leave the pattern the same, in other words, invariant. For example, assuming that the pattern goes on forever, we can shift the repeating pattern of wallpapers by certain directions and amount can retrieve the same exact pattern as before.\nIn physics, systems that repeat themselves forever are called crystals. What’s cool about crystals is that the symmetries of the crystals have a direct impact on how waves propagating in the crystals themselves behave, such as if we were to shine light (famously known as a wave) through it, or disturb it (resulting in a sound wave) and so on. Only knowing the symmetries of the crystal, we can predict whether a wave propagating in the crystal will spread out evenly, or propagate in a highly directed fashion, such as in the cover photo.\nIn wallpapers or sidewalks, there are two different kinds of symmetries. The first is symmorphic symmetry, where there exists two kinds of separable symmetries: translational, shifting the entire pattern, and rotational, only rotating the entire pattern around certain points. Nonsymmorphic symmetries are a bit more complicated, in that you cannot separate out the translational and rotational symmetries; there are certain symmetries that require partial translation, followed by a rotation.\nIn this project, we predicted how certain crystals would behave based on their symmetries, and used their properties to design ‘waveguides’, that are able to steer the waves around in particular directions efficiently.\nIf you like to know more about the work, please check out the PDF.\nAbstract (click to open and close) The study of wave propagation through structured periodic media depends critically upon the periodic lattice from which the medium is constructed. That is unsurprising, but perhaps what is slightly more surprising, is that pieces of pure mathematics play a key role - in particular, group and representation theory. Group theory is the natural language that encodes the symmetries of shape and form. Here we use it to consider a class of $2D$ periodic crystals whose lattice is encoded by nonsymmorphic space groups. These are often overlooked due to their relative complexity compared to the symmorphic space groups. We demonstrate that nonsymmorphic groups have possible practical interest in terms of coalescence of dispersion curves, Dirac points and band-sticking, using both theory and simulation. Once we’ve laid out the group theoretical framework in the context of the nonsymmorphic crystals, we use it to illustrate how accidental degeneracies can arise in symmorphic square lattices. We combine this phenomenon with topological valley effects to design highly-efficient topological waveguides and energy-splitters.\n","date":1534636800,"expirydate":-62135596800,"kind":"page","lang":"ko","lastmod":1534636800,"objectID":"ef65a6f62d2f386aa87f247b595a33dc","permalink":"https://sunwoo-kim.github.io/ko/projects/nonsymmorphic/","publishdate":"2018-08-19T00:00:00Z","relpermalink":"/ko/projects/nonsymmorphic/","section":"projects","summary":" ","tags":null,"title":"Nonsymmorphic crystals and topological waveguiding","type":"projects"}] \ No newline at end of file diff --git a/ko/index.xml b/ko/index.xml index 1643560e..5274def7 100644 --- a/ko/index.xml +++ b/ko/index.xml @@ -24,7 +24,7 @@ </span></span><span class="line"><span class="cl"> title=<span class="nb">{</span>Hello World<span class="nb">}</span>, </span></span><span class="line"><span class="cl"> year=<span class="nb">{</span>2021<span class="nb">}</span>, </span></span><span class="line"><span class="cl"> howpublished=<span class="nb">{</span><span class="k">\url</span><span class="nb">{</span>https://sunwoo-kim.github.io/ko/posts/hello-world/<span class="nb">}}</span>, -</span></span><span class="line"><span class="cl"> note=<span class="nb">{</span>Accessed: 2023-08-31<span class="nb">}</span> +</span></span><span class="line"><span class="cl"> note=<span class="nb">{</span>Accessed: 2023-09-02<span class="nb">}</span> </span></span><span class="line"><span class="cl"><span class="nb">}</span> </span></span></code></pre></div></details> <!-- Welcome to my new website! I hope to post some expository writeups on my research projects here. Stay tuned! --> diff --git a/ko/posts/hello-world/index.html b/ko/posts/hello-world/index.html index fc06ef01..2c9cfaa6 100644 --- a/ko/posts/hello-world/index.html +++ b/ko/posts/hello-world/index.html @@ -1,6 +1,6 @@ - + @@ -806,7 +806,7 @@

Hello World

title={Hello World}, year={2021}, howpublished={\url{https://sunwoo-kim.github.io/ko/posts/hello-world/}}, - note={Accessed: 2023-08-31} + note={Accessed: 2023-09-02} } @@ -943,13 +943,40 @@
김선우
+ + + + + + + + + + + + + +

- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

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+ @@ -957,7 +984,7 @@
김선우

- Site last updated 2023-08-31 20:11:00.167111 + Site last updated 2023-09-02 12:03:10.971288

diff --git a/ko/posts/hello-world/index.html.bak b/ko/posts/hello-world/index.html.bak index 56bbd15c..26e6864b 100644 --- a/ko/posts/hello-world/index.html.bak +++ b/ko/posts/hello-world/index.html.bak @@ -1,6 +1,6 @@ - + @@ -806,7 +806,7 @@ title={Hello World}, year={2021}, howpublished={\url{https://sunwoo-kim.github.io/ko/posts/hello-world/}}, - note={Accessed: 2023-08-31} + note={Accessed: 2023-09-02} } @@ -943,13 +943,40 @@ + + + + + + + + + + + + + +

- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

+

+ + + + + + + + + + +

+ diff --git a/ko/posts/index.html b/ko/posts/index.html index 12ef6c91..b804d4d9 100644 --- a/ko/posts/index.html +++ b/ko/posts/index.html @@ -1,6 +1,6 @@ - + @@ -885,13 +885,40 @@

게시물

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- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

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게시물

- Site last updated 2023-08-31 20:11:00.167111 + Site last updated 2023-09-02 12:03:10.971288

diff --git a/ko/posts/index.html.bak b/ko/posts/index.html.bak index 9906c364..b6d3a3a6 100644 --- a/ko/posts/index.html.bak +++ b/ko/posts/index.html.bak @@ -1,6 +1,6 @@ - + @@ -885,13 +885,40 @@ + + + + + + + + + + + + + +

- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

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+ diff --git a/ko/posts/index.xml b/ko/posts/index.xml index d8e42097..9e3522e0 100644 --- a/ko/posts/index.xml +++ b/ko/posts/index.xml @@ -24,7 +24,7 @@ </span></span><span class="line"><span class="cl"> title=<span class="nb">{</span>Hello World<span class="nb">}</span>, </span></span><span class="line"><span class="cl"> year=<span class="nb">{</span>2021<span class="nb">}</span>, </span></span><span class="line"><span class="cl"> howpublished=<span class="nb">{</span><span class="k">\url</span><span class="nb">{</span>https://sunwoo-kim.github.io/ko/posts/hello-world/<span class="nb">}}</span>, -</span></span><span class="line"><span class="cl"> note=<span class="nb">{</span>Accessed: 2023-08-31<span class="nb">}</span> +</span></span><span class="line"><span class="cl"> note=<span class="nb">{</span>Accessed: 2023-09-02<span class="nb">}</span> </span></span><span class="line"><span class="cl"><span class="nb">}</span> </span></span></code></pre></div></details> <!-- Welcome to my new website! I hope to post some expository writeups on my research projects here. Stay tuned! --> diff --git a/ko/projects/index.html b/ko/projects/index.html index f62e2cf5..6320edd3 100644 --- a/ko/projects/index.html +++ b/ko/projects/index.html @@ -1,6 +1,6 @@ - + @@ -1055,13 +1055,40 @@

프로젝트

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- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

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+ + + + + + + + + + +

+ @@ -1069,7 +1096,7 @@

프로젝트

- Site last updated 2023-08-31 20:11:00.167111 + Site last updated 2023-09-02 12:03:10.971288

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- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

+

+ + + + + + + + + + +

+ diff --git a/ko/projects/mblbosons/index.html b/ko/projects/mblbosons/index.html index 469bbda2..0e2e0593 100644 --- a/ko/projects/mblbosons/index.html +++ b/ko/projects/mblbosons/index.html @@ -1,6 +1,6 @@ - + @@ -1004,13 +1004,40 @@
김선우
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- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

+

+ + + + + + + + + + +

+ @@ -1018,7 +1045,7 @@
김선우

- Site last updated 2023-08-31 20:11:00.167111 + Site last updated 2023-09-02 12:03:10.971288

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- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

+

+ + + + + + + + + + +

+ diff --git a/ko/projects/nonsymmorphic/index.html b/ko/projects/nonsymmorphic/index.html index 3b2aa286..42c42bca 100644 --- a/ko/projects/nonsymmorphic/index.html +++ b/ko/projects/nonsymmorphic/index.html @@ -1,6 +1,6 @@ - + @@ -996,13 +996,40 @@
김선우
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- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

+

+ + + + + + + + + + +

+ @@ -1010,7 +1037,7 @@
김선우

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- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

+

+ + + + + + + + + + +

+ diff --git a/ko/publication_types/index.html b/ko/publication_types/index.html index 9ab96061..b3f8db1c 100644 --- a/ko/publication_types/index.html +++ b/ko/publication_types/index.html @@ -1,6 +1,6 @@ - + @@ -784,13 +784,40 @@

Publication_types

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- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

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+ @@ -798,7 +825,7 @@

Publication_types

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- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

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+ diff --git a/ko/tags/index.html b/ko/tags/index.html index ed85bea1..85781646 100644 --- a/ko/tags/index.html +++ b/ko/tags/index.html @@ -1,6 +1,6 @@ - + @@ -784,13 +784,40 @@

Tags

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- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

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+ @@ -798,7 +825,7 @@

Tags

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diff --git a/ko/tags/index.html.bak b/ko/tags/index.html.bak index 85100dcf..a48ea628 100644 --- a/ko/tags/index.html.bak +++ b/ko/tags/index.html.bak @@ -1,6 +1,6 @@ - + @@ -784,13 +784,40 @@ + + + + + + + + + + + + + +

- Email: swk34 [at] cantab [dot] ac [dot] uk + © 2023 Sun Woo Kim. Email: swk34 [at] cantab [dot] ac [dot] uk

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