Variational Monte Carlo investigation of SU(N) Heisenberg chains

Motivated by recent experimental progress in the context of ultracold multicolor fermionic atoms in optical lattices, we have investigated the properties of the SU (N) Heisenberg chain with totally antisymmetric irreducible representations, the effective model of Mott phases with m < N particles per site. These models have been studied for arbitrary N and m with non-Abelian bosonization [I. Affleck, Nucl. Phys. B 265, 409 (1986); 305, 582 (1988)], leading to predictions about the nature of the ground state (gapped or critical) in most but not all cases. Using exact diagonalization and variational Monte Carlo based on Gutzwiller projected fermionic wave functions, we have been able to verify these predictions for a representative number of cases with N

A variational investigation of the SU(N) Heisenberg model in antisymmetric irreducible representations

Jérôme Dufour

Motivated by recent experimental progress in the context of ultra-cold multi-colour fermionic atoms in optical lattices, this thesis investigates the properties of the antiferromagnetic SU(N) Heisenberg models with fully antisymmetric irreducible representations labelled by a Young tableau with

m

boxes on each site of various lattices. Thanks to an intensive use of variational Monte Carlo method based on Gutzwiller projected fermionic wave functions, we aim to determine the ground state and its related properties of systems which are beyond the capabilities of other numerical methods. This typically means that we will study large lattices with many particles per site for

N>2

. The first part of this thesis will introduce the basic theoretical tools connected to the SU(N) Heisenberg model. A very short review of group theory introducing Young tableaux, will clarify the concept of fully antisymmetric irreducible representation of the SU(N) Lie algebra. Some properties of the Heisenberg model will be discussed and a mean-field solution will be given. The numerical methods used during this thesis and their limitations wil also be introduced. We will investigate the properties of 1-dimensional systems in the second part of this work. In particular, the chains have been studied for arbitrary

N

and

m

with non-abelian bosonisation leading to predictions about the nature of the ground state (gapped or critical) in most but not all cases. We have been able to verify these predictions for a representative number of cases with

N\leq10

and

m\leq N/2

, and we have shown that the opening of a gap is associated with a spontaneous dimerisation or trimerisation depending on the value of

m

and

N

. We have also investigated the marginal cases, where non-abelian bosonisation did not lead to any prediction. In these cases, variational Monte Carlo predicts that the ground state is critical with exponents that are consistent with conformal field theory. The other 1-dimensional system that has been studied is the ladder for which field-theory gives some predictions. For instance, the ground state of the SU(4) case in the fundamental irreducible representation is believed to consist of a 4-site plaquette state for any non-zero inter-leg coupling. We have confirmed the presence of this state, even if in the weak coupling regime, the optimal variational state is gapless. The study has been extended to various

N

and

m

values providing interesting results. The last part will present the results for 2-dimensional lattices. The square lattice will first be investigated. A mean-field analysis predicts a plaquette ground state for

3\leq N/m < 5

and of a chiral ground state for

N/m\geq 5

. By systematically studying the SU(3m) and SU(6m) Heisenberg model, we determined that the mean-field solution is valid even at small

m

(

m>1

for SU(3m) and

m>2

for SU(6m)). Then, for the SU(6m) Heisenberg model with

m

particles per site on a honeycomb lattice, we make the connection between the

m=1

case, for which the ground state has recently been shown to be in a plaquette singlet state, and the

m\rightarrow \infty

limit, where a mean-field approach has established that the ground state has chiral order. We were able to show that this system has a clear tendency toward the chiral order as soon as

m\geq 2

. This demonstrates that the chiral phase can be stabilised for not too large values of

m

, opening the way to its experimental realisations in other lattices. We will finally present results for the SU(3m) on the triangular lattice for which we find a plaquette state, when

m>1

EPFL2016

Variational Quantum Monte Carlo Method with a Neural-Network Ansatz for Open Quantum Systems

Alexandra Nagy, Vincenzo Savona

The possibility to simulate the properties of many-body open quantum systems with a large number of degrees of freedom (d.o.f.) is the premise to the solution of several outstanding problems in quantum science and quantum information. The challenge posed by this task lies in the complexity of the density matrix increasing exponentially with the system size. Here, we develop a variational method to efficiently simulate the nonequilibrium steady state of Markovian open quantum systems based on variational Monte Carlo methods and on a neural network representation of the density matrix. Thanks to the stochastic reconfiguration scheme, the application of the variational principle is translated into the actual integration of the quantum master equation. We test the effectiveness of the method by modeling the two-dimensional dissipative XYZ spin model on a lattice.

AMER PHYSICAL SOC2019