sign structure leads to non hermitian Ham

[dambuck]

Hello,

so this used to work in the older netket versions. I take some lattice and set a arbitrary sign struture. Here I take a 1d chain and make the spins to rotate 90° from one site to the next. This wont run because of an non hermitian hamiltonian, which should not be the case here but I might be misunderstanding something.

import netket as nk
import numpy as np


#some helper functions
def super_index(site=1,N=16):
    if site<N:
        return site
    else:
        return site%N

# the sign structure function
def phase_shift(Q,eta, eta_,N):
    eta_v=np.repeat([[eta,eta_]],N/2, axis=0).flatten()
    R=np.array([[x,x] for x in range(int(N/2))]).flatten()

    return np.exp((Q*R + eta_v)*1j, dtype='complex')


Sx = np.array([[0, 1], [1, 0]])*0.5
Sy = np.array([[0, -1j], [1j, 0]])*0.5
Sz = np.array([[1, 0], [0, -1]])*0.5

diag=np.kron(Sz, Sz)
offdiag=np.kron(Sx, Sx)+np.kron(Sy, Sy)



#############################################################
#############################################################
#HILBERT

# 1D Lattice
L = 4
# Hilbert space of spins on the graph
hi = nk.hilbert.Spin(s=1 / 2, N=L, total_sz=0)



edge_colors = []
color=1
for i in range(L):
    edge_colors.append([i, (i+1)%L, color])
    color+=1
graph = nk.graph.Graph(n_nodes=L, edges=edge_colors)



#############################################################
#############################################################
#HAMILTONIAN 


bond_operator=[]
bond_color=[]


#setting the sign structure
Q=1
eta=0.5
J=1
J_=1
signs=phase_shift(Q*np.pi, 0, eta*np.pi, L)
color=1
for site in range(L):
    #diag contribution NN
    bond_operator.append((J* diag).tolist())
    bond_color.append(color)
    #OFF diag NN
    bond_operator.append(signs[site]*signs[super_index(site+1, L)]\
                              *J * offdiag)
    bond_color.append(color)
    color+=1
        
        


ha = nk.operator.GraphOperator(hi,
       bond_ops=bond_operator, bond_ops_colors=bond_color, graph=graph, dtype=complex)



#############################################################
#############################################################
#MACHINE

# RBM Spin Machine
ma = nk.models.RBM(
    alpha=4,
    use_visible_bias=False,
    use_hidden_bias=True,
    dtype=float,
)

# Metropolis Local Sampling
sa = nk.sampler.MetropolisExchange(hi, graph=graph, n_chains=16)

# Optimizer
op = nk.optimizer.Sgd(learning_rate=0.01)
sr = nk.optimizer.SR(diag_shift=0.1)

# Variational State
vs = nk.vqs.MCState(sa, ma, n_samples=1000, n_discard_per_chain=100)

# Variational monte carlo driver
gs = nk.VMC(ha, op, variational_state=vs, preconditioner=sr)


# Run the optimization for 300 iterations
gs.run(n_iter=100, out="test")

[PhilipVinc]

The standard formula we use for VMC, the one you find in textbooks, is derived assuming hermitian hamiltonian.
If your hamiltonian is non hermitian, it gives wrong results. Are you sure you want to find the ground state of a non hermitian hamiltonian?

[dambuck]

Yeah sorry realized I was doing rubbish here. Please close