Ansatz Wavefunctions Research Articles

Liouville-space neural network representation of density matrices

Neural network quantum states such as ansatz wave functions have shown a great deal of promise for finding the ground state of spin models. Recently, work has focused on extending this idea to mixed states for simulating the dynamics of open systems. Most approaches so far have used a purification ansatz where a copy of the system Hilbert space is added, which when traced out gives the correct density matrix. Here we instead present an extension of the restricted Boltzmann machine which directly represents the density matrix in Liouville space. This allows the compact representation of states which appear in mean-field theory. We benchmark our approach on two different versions of the dissipative transverse-field Ising model, which show our ansatz is able to compete with other state-of-the-art approaches. Published by the American Physical Society 2024

Efficient neural-network based variational Monte Carlo scheme for direct optimization of excited energy states in frustrated quantum systems

We examine applicability of the valence bond basis correlator product state ansatz, equivalent to the restricted Boltzmann machine quantum artificial neural network ansatz, and variational Monte Carlo method for direct optimization of excited energy states to study properties of strongly correlated and frustrated quantum systems. The energy eigenstates are found by stochastic minimization of the variational function for the energy eigenstates which allows direct optimization of particular energy state without knowledge of the lower energy states. This approach combined with numerous tensor network or artificial neural network ansatz wavefunctions then allows further insight into quantum phases and phase transitions in various strongly correlated models by considering properties of these systems beyond the ground state properties. Also, the method is in general applicable to any dimension and has no sign instability. An example that we consider is the square lattice J1-J2 antiferromagnetic Heisenberg model. The model is one of the most studied models in frustrated quantum magnetism since it is closely related to the disappearance of the antiferromagnetic order in the high-Tc superconducting materials and there is still no agreement about the properties of the system in the highly frustrated regime near J2/J1 = 0.5. For J1-J2 model we write the variational ansatz in terms of the two site correlators and in the valence bond basis and calculate lowest energy eigenstates in the highly frustrated regime near J2/J1 = 0.5 where the system has a paramagnetic phase. We find that our results are in good agreement with previously obtained results which confirms applicability of the method to study frustrated spin systems.

Izergin–Korepin analysis on the wavefunctions of the $U_q(\mathrm {sl}_2)$ six-vertex model with reflecting end

We extend the recently developed Izergin–Korepin analysis on the wavefunctions of the U_q(\mathrm {sl}_2) six-vertex model to the reflecting boundary conditions. Based on the Izergin–Korepin analysis, we determine the exact forms of the symmetric functions which represent the wavefunctions and its dual. Comparison of the symmetric functions with the coordinate Bethe ansatz wavefunctions for the open XXZ chain by Alcaraz, Barber, Batchelor, Baxter, and Quispel is also made. As an application, we derive algebraic identities for the symmetric functions by combining the results with the determinant formula of the domain wall boundary partition function of the six-vertex model with reflecting end.

Probing the stability of the spin-liquid phases in the Kitaev-Heisenberg model using tensor network algorithms

We study the extent of the spin liquid phases in the Kitaev-Heisenberg model using infinite Projected Entangled-Pair States tensor network ansatz wave functions directly in the thermodynamic limit. To assess the accuracy of the ansatz wave functions we perform benchmarks against exact results for the Kitaev model and find very good agreement for various observables. In the case of the Kitaev-Heisenberg model we confirm the existence of 6 different phases: N\'eel, stripy, ferromagnetic, zigzag and two spin liquid phases. We find finite extents for both spin liquid phases and discontinuous phase transitions connecting them to symmetry-broken phases.

Wave-function vortex attachment via matrix products: Application to atomic Fermi gases in flat spin-orbit bands

Variational wave functions that introduce zeros (vortices) to screen repulsive interactions are typically difficult to verify in unbiased microscopic calculations. An approach is constructed to insert vortices into ansatz wave functions using a matrix-product representation. This approach opens the door to validation of a broad class of Jastrow-based wave functions. The formalism is applied to a model motivated by experiments on ultracold atomic gases in the presence of synthetic spin-orbit coupling. Validated wave functions show that vortices in atomic Fermi gases with flat Rashba spin-orbit bands cluster near the system center and should therefore be directly visible in time-of-flight imaging.

Bipartite Composite Fermion States

We study a class of ansatz wave functions in which composite fermions form two correlated "partitions." These "bipartite" composite fermion states are demonstrated to be very accurate for electrons in a strong magnetic field interacting via a short-range 3-body interaction potential over a broad range of filling factors. Furthermore, this approach gives accurate approximations for the exact Coulomb ground state at 2+3/5 and 2+4/7 and is thus a promising candidate for the observed fractional quantum Hall states at the hole conjugate fractions at 2+2/5 and 2+3/7.

Jastrow-correlated wave functions for flat-band lattices

The electronic band structure of many compounds, e.g., carbon-based structures, can exhibit essentially no dispersion. Models of electrons in flat-band lattices define non-perturbative strongly correlated problems by default. We construct a set of Jastrow-correlated ansatz wavefunctions to capture the low energy physics of interacting particles in flat bands. We test the ansatz in a simple Coulomb model of spinless electrons in a honeycomb ribbon. We find that the wavefunction accurately captures the ground state in a transition from a crystal to a uniform quantum liquid.

Distinguishing phases with ansatz wave functions

We propose an indistinguishability measure for assessment of ansatz wavefunctions with numerically determined wavefunctions. The measure efficiently compares all correlation functions of two states and can therefore be used to distinguish phases by defining correlator classes for ansatz wavefunctions. It also allows identification of quantum critical points. We demonstrate the approach for the transverse Ising model, using the matrix product state formalism with the time evolving block decimation algorithm.

Distinct solutions of infinite U Hubbard model through nested Bethe ansatz and Gutzwiller projection operator approach

Abstract The exact nested Bethe ansatz solution for the one dimensional (1-D) U infinity Hubbard model show that the state vectors are a product of spin-less fermion and spin wavefunctions, or an appropriate superposition of such factorized wavefunctions. The spin-less fermion component of the wavefunctions ensures no double occupancy at any site. It had been demonstrated that the nested Bethe ansatz wavefunctions in the U infinity limit obey orthofermi statistics. Gutzwiller projection operator formalism is the another well known technique employed to handle U infinity Hubbard model. In general, this approach does not lead to spin-less fermion wavefunctions. Therefore, the nested Bethe ansatz and Gutzwiller projection operator approach give rise to different kinds of the wavefunctions for the U infinity limit of 1-D Hubbard Hamiltonian. To compare the consequences of this dissimilarity in the wavefunctions, we have obtained the ground state energy of a finite system consisting of three particles on a four site closed chain. It is shown that in the nested Bethe ansatz implemented through orthofermion algebra, all the permissible 2 3 spin configurations are degenerate in the ground state. This eight fold degeneracy of the ground state is absent in the Gutzwiller projection operator approach. This finding becomes relevant in the context of known exact U infinity results, which require that all the energy levels are 2 N -fold degenerate for an N particle system.

Orbital Landau level dependence of the fractional quantum Hall effect in quasi-two-dimensional electron layers: Finite-thickness effects

The fractional quantum Hall effect (FQHE) in the second orbital Landau level at even-denominator filling factor of 5/2 remains enigmatic and motivates our work. We theoretically consider the effect of the quasi-two-dimensional (2D) nature of the experimental fractional quantum Hall system on a number of FQH states (filling factors: 1/3, 1/5, and 1/2) in the lowest, second, and third orbital Landau levels (LLL, SLL, and TLL, respectively) by calculating the wave-function overlap, as a function of quasi-2D layer thickness, between the exact ground state of a model Hamiltonian and the consensus variational ansatz wave functions, i.e., the Laughlin wave function for 1/3 and 1/5 and the Moore-Read Pfaffian wave function for 1/2. Using large numerical overlap as a stability (or FQHE robustness) criterion, we find that the FQHE does not occur in the TLL (for any quasi-2D layer thickness), is the most robust for zero thickness (strict 2D limit) in the LLL for 1/3 and 1/5 and for 11/5 in the SLL, and is the most robust at finite thickness (4--5 magnetic lengths) in the SLL for the mysterious even-denominator 5/2 state and the presumably more conventional 7/3 state. We do not find any FQHE at 1/2 in the LLL for any thickness for the quasi-2D models considered in our work. Furthermore, we examine the orbital effects of a nonzero in-plane (parallel) magnetic field, finding that its application effectively reduces the quasi-2D layer thickness and, therefore, could destroy the FQHE at 5/2 and 7/3, while it enhances the FQHE at 11/5, in the SLL. The in-plane field also enhances the LLL FQHE states by making the quasi-2D system more purely 2D. The in-plane field effects could thus be qualitatively different in the LLL and the SLL by virtue of magneto-orbital coupling through the finite-thickness effect. Using exact diagonalization on the torus geometry, we show the appearance of the characteristic threefold topological degeneracy expected for the Pfaffian state. This signature is enhanced by nonzero thickness, corroborating our findings from overlap calculations. Our results have ramifications for wave-function engineering, opening the possibility of creating an optimal experimental system where the 5/2 FQHE state is more likely described by the Moore-Read Pfaffian state with obvious applications in the burgeoning field of fault-tolerant topological quantum computing.

Non-regular eigenstate of theXXXmodel as some limit of the Bethe state

For the one-dimensional XXX model under the periodic boundary conditions, we discuss two types of eigenvectors, regular eigenvectors which have finite-valued rapidities satisfying the Bethe ansatz equations and non-regular eigenvectors which are descendants of some regular eigenvectors under the action of the SU(2) spin-lowering operator. It has been pointed out by many authors that the non-regular eigenvectors should correspond to the Bethe ansatz wavefunctions which have multiple infinite rapidities. However, it has not been explicitly shown whether such a delicate limiting procedure is possible. In this paper, we discuss it explicitly at the level of wavefunctions: we prove that any non-regular eigenvector of the XXX model is derived from the Bethe ansatz wavefunctions through some limit of infinite rapidities. We formulate the regularization also in terms of the algebraic Bethe ansatz method. As an application of infinite rapidity, we discuss the period of the spectral flow under the twisted periodic boundary conditions.

Origin of the singular Bethe ansatz solutions for the Heisenberg XXZ spin chain

We investigate symmetry properties of the Bethe ansatz wave functions for the Heisenberg XXZ spin chain. The XXZ Hamiltonian commutes simultaneously with the shift operator T and the lattice inversion operator V in the space of Ω=±1 with Ω the eigenvalue of T. We show that the Bethe ansatz solutions with normalizable wave functions cannot be the eigenstates of T and V with quantum number (Ω,ϒ)=(±1,∓1) where ϒ is the eigenvalue of V. Therefore, the Bethe ansatz wave functions should be singular for nondegenerate eigenstates of the Hamiltonian with quantum number (Ω,ϒ)=(±1,∓1). It is also shown that such states exist in any nontrivial down-spin number sector and that the number of them diverges exponentially with the chain length.

On the use of analytic ansatz three–body wave functions in the study of (e, 2e) band related processes

We present a critical analysis of the analytic ansatz wave functions that have been proposed for use in the calculation of triple–differential cross–sections for the electron–impact ionization of atoms. We will first discuss, in detail, the mathematical background to these approximations, and we will give, for the first time, cross–sections for the electron–impact ionization of hydrogen using the most elaborate of these. We will show that they give generally unsatisfactory agreement with experiment. We will argue that this failure comes from an inability to model the crucial region where the three particles are close together. The analysis presented should be of value not only in the case of electron–impact ionization but also for a wide range of Coulombic three–body problems in atomic physics.

Two-component one-dimensional gas with integrable open boundary conditions

The open nonlinear Schrodinger model with spin degrees of freedom is considered. We find two types of integrable open boundary conditions. We study the Bethe ansatz states of the model at the infinitely strong coupling limit. For some boundary conditions, the spins of the Bethe ansatz wavefunctions are all aligned up or down. For the other type of boundary conditions all the spin configurations are degenerate.

Class of ansatz wave functions for one-dimensional spin systems and their relation to the density matrix renormalization group

We investigate the density matrix renormalization group (DMRG) discovered by White and show that in the case where the renormalization eventually converges to a fixed point the DMRG ground state can be simply written as a ``matrix product'' form. This ground state can also be rederived through a simple variational ansatz making no reference to the DMRG construction. We also show how to construct the ``matrix product'' states and how to calculate their properties, including the excitation spectrum. This paper provides details of many results announced in an earlier letter.

Bethe ansatz wavefunctions and an infinite number of conserved operators of the Luttinger model

We find a transformation of the field operators which casts the Hamiltonian of the Luttinger model into a quadratic form. The transformed field operators satisfy interesting commutation relations, fermion anti-commutation relations between the same kinds of fields and anyon-like commutation relations between different kinds. Using the transformation, Bethe ansatz wavefunctions are constructed algebraically. In addition, an infinite number of conserved quantities are given.

Massless Thirring Model and Bethe Ansatz Wavefunction

The quantum one-dimensional massless Thirring model is studied. A transformation which casts the massless Thirring model into a quadratic form is found. The transformed field operators satisfy interesting commutation relations, fermion anti-commutation relations among the same kinds of fields and anyon-like commutation relations between different kinds. Bethe Ansatz wavefunctions are constructed algebraically using the transformation and the commutation relations in coordinate space.

Collective excitations in the fractional quantum Hall effect

The authors study the dispersion of collective excitations for fractional quantum Hall states using Feynman-Bijl ansatz wavefunctions. They take the ground-state pair correlation function from direct diagonalisations of the Hamiltonian for a small number of electrons confined to the surface of a sphere, instead of from the simulation of large systems described by a Laughlin-like wavefunction. The results seem remarkably well converged considering the small systems used. They also treat systems with partially filled N=1 Landau levels.

The three-colouring problem as a special eight-vertex model

The equations giving the free energy per site and those governing the Bethe ansatz wavefunctions for a 'solid-on-solid' (SOS) model equivalent to the eight-vertex model are shown to be identical to those arising in the three-colouring problem for the particular choice of parameters eta =-iK'/3, v=0, using the original Baxter notation.

Multi-configurational complex molecular orbital method

A simple version of the multi-configurational complex molecular orbital method is developed. Built-in assumptions limit its applicability to closed-shell states of molecules which belong by symmetry to a certain non-degenerate point group. Only closed-shell type configurations appear explicitly in the ansatz wavefunctions. Complex MO's are optimized separately for each configuration (“floating” CMO's). A set of quasi-independent secular equations of pseudo-eigenvalue form is derived, one equation for each configuration. This method could reliably treat election correlations in molecules close to their equilibrium geometry.