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arXiv:2001.06510

Variational-Correlations Approach to Quantum Many-body Problems

Arbel Haim, Richard Kueng, Gil Refael

Published Jan 17, 2020• 4 citations
arXivHTMLPDF

Abstract

We investigate an approach for studying the ground state of a quantum many-body Hamiltonian that is based on treating the correlation functions as variational parameters. In this approach, the challenge set by the exponentially-large Hilbert space is circumvented by approximating the positivity of the density matrix, order-by-order, in a way that keeps track of a limited set of correlation functions. In particular, the density-matrix description is replaced by a correlation matrix whose dimension is kept linear in system size, to all orders of the approximation. Unlike the conventional variational principle which provides an upper bound on the ground-state energy, in this approach one obtains a lower bound instead. By treating several one-dimensional spin $1/2$ Hamiltonians, we demonstrate the ability of this approach to produce long-range correlations, and a ground-state energy that converges to the exact result. Possible extensions, including to higher-excited states are discussed.

DOI
10.48550/arxiv.2001.06510

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•Jan 1, 2023•0 citations
We investigate an approach for studying the ground state of a quantum many-body Hamiltonian that is based on treating the correlation functions as variational parameters. In this approach, the challenge set by the exponentially-large Hilbert space is circumvented by approximating the positivity of the density matrix, order-by-order, in a way that keeps track of a limited set of correlation functions. In particular, the density-matrix description is replaced by a correlation matrix whose dimension is kept linear in system size, to all orders of the approximation. Unlike the conventional variational principle which provides an upper bound on the ground-state energy, in this approach one obtains a lower bound instead. By treating several one-dimensional spin 1/21/21/2 Hamiltonians, we demonstrate the ability of this approach to produce long-range correlations, and a ground-state energy that converges to the exact result. Possible extensions, including to higher-excited states are discussed.
Standard variational methods tend to obtain upper bounds on the ground state energy of quantum many-body systems. Here we study a complementary method that determines lower bounds on the ground state energy in a systematic fashion, scales polynomially in the system size and gives direct access to correlation functions. This is achieved by relaxing the positivity constraint on the density matrix and replacing it by positivity constraints on moment matrices, thus yielding a semi-definite programme. Further, the number of free parameters in the optimization problem can be reduced dramatically under the assumption of translational invariance. A novel numerical approach, principally a combination of a projected gradient algorithm with Dykstra's algorithm, for solving the optimization problem in a memory-efficient manner is presented and a proof of convergence for this iterative method is given. Numerical experiments that determine lower bounds on the ground state energies for the Ising and Heisenberg Hamiltonians confirm that the approach can be applied to large systems, especially under the assumption of translational invariance.
A variational upper bound on the ground state energy E gs of a quantum system, E gs ⩽ 〈Ψ| H |Ψ〉, is well-known (here H is the Hamiltonian of the system and Ψ is an arbitrary wave function). Much less known are variational lower bounds on the ground state. We consider one such bound which is valid for a many-body translation-invariant lattice system. Such a lattice can be divided into clusters which are identical up to translations. The Hamiltonian of such a system can be written as , where a term H i is supported on the i 'th cluster. The bound reads , where is some wisely chosen set of reduced density matrices of a single cluster. The implementation of this latter variational principle can be hampered by the difficulty of parameterizing the set , which is a necessary prerequisite for a variational procedure. The root cause of this difficulty is the nonlinear positivity constraint ρ > 0 which is to be satisfied by a density matrix. The squaring parametrization of the density matrix, ρ = τ 2 /tr τ 2 , where τ is an arbitrary (not necessarily positive) Hermitian operator, accounts for positivity automatically. We discuss how the squaring parametrization can be utilized to find variational lower bounds on ground states of translation-invariant many-body systems. As an example, we consider a one-dimensional Heisenberg antiferromagnet.
A ubiquitous problem in quantum physics is to understand the ground-state properties of many-body systems. Confronted with the fact that exact diagonalization quickly becomes impossible when increasing the system size, variational approaches are typically employed as a scalable alternative: Energy is minimized over a subset of all possible states and then different physical quantities are computed over the solution state. Despite remarkable success, rigorously speaking, all that variational methods offer are upper bounds on the ground-state energy. On the other hand, so-called relaxations of the ground-state problem based on semidefinite programming represent a complementary approach, providing lower bounds to the ground-state energy. However, in their current implementation, neither variational nor relaxation methods offer provable bound on other observables in the ground state beyond the energy. In this work, we show that the combination of the two classes of approaches can be used to derive certifiable bounds on the value of any observable in the ground state, such as correlation functions of arbitrary order, structure factors, or order parameters. We illustrate the power of this approach in paradigmatic examples of 1D and 2D spin- 1 / 2 Heisenberg models. To improve the scalability of the method, we exploit the symmetries and sparsity of the considered systems to reach sizes of hundreds of particles at much higher precision than previous works. Our analysis therefore shows how to obtain certifiable bounds on many-body ground-state properties beyond energy in a scalable way. Published by the American Physical Society 2024
Solving for the lowest energy eigenstate of the many-body Schrodinger equation is a cornerstone problem that hinders understanding of a variety of quantum phenomena. The difficulty arises from the exponential nature of the Hilbert space which casts the governing equations as an eigenvalue problem of exponentially large, structured matrices. Variational methods approach this problem by searching for the best approximation within a lower-dimensional variational manifold. In this work we use graph neural networks to define a structured variational manifold and optimize its parameters to find high quality approximations of the lowest energy solutions on a diverse set of Heisenberg Hamiltonians. Using graph networks we learn distributed representations that by construction respect underlying physical symmetries of the problem and generalize to problems of larger size. Our approach achieves state-of-the-art results on a set of quantum many-body benchmark problems and works well on problems whose solutions are not positive-definite. The discussed techniques hold promise of being a useful tool for studying quantum many-body systems and providing insights into optimization and implicit modeling of exponentially-sized objects.
Variational methods are of fundamental importance and widely used in theoretical physics, especially for strongly interacting systems. In this work, we present a set of variational equations of state (VES) for pure states of an interacting quantum Hamiltonian. The VES can be expressed in terms of the variation of the density operators or static correlation functions. We derive the algebraic relationship between a known pure state density matrix and its variation, and obtain the VES by applying this relation to the averaged Heisenberg-equations-of-motion for the exact density matrix. Additionally, we provide a direct expression of the VES in terms of correlation functions to make it computable. We present three nontrivial applications of the VES: a perturbation calculation of correlation functions of the transverse field Ising model in arbitrary spatial dimensions, a study of a longitudinal field perturbation to the one-dimensional transverse field Ising model at the critical point and variational calculation of magnetization and ground state energy of the two-dimensional spin-1/2 Heisenberg model on a square lattice. For the second one, our results not only recover the scaling limit, but also indicate the possibility of continuous tuning of the critical exponents by adjusting the longitudinal fields differently from the scaling limit. For the Heisenberg model, we obtained results numerically comparable to established results with simple calculations. The VES approach provides a powerful and versatile tool for studying interacting quantum systems.
Understanding strongly correlated quantum many-body systems requires dealing with a largeconfiguration space comprised of a set of all possible measurable configurations. The total number of configurations grows exponentially with the size of the system, which makes exact numerical calculations impossible for even moderately sized systems. Since the advent of high-temperature superconductivity and the subsequent interest in strongly correlated phases of matter, progress in the field has been marked by ingenuity of the field to overcome these computational limitations. The first techniques to emerge were inspired by quantum chemistry and based on exact diagonalization[1], which is limited to small clusters, and different variants of quantum Monte Carlo[2], that suffers from the sign problem in fermionic and frustrated systems. Circa 1992, the density matrix renormalization group (DMRG) [3, 4, 5, 6, 7] became one of the major breakthroughs of the past few decades. Despite the success of DMRG for one-dimensional(1D) and quasi-one-dimensional geometries, extensions to actual two-dimensional systems remain challenging and applications are constrained to long cylinders and strips. One promising alternative is to combine one the oldest methods, variational wave functions (VWFs), with neural networks, which have been used to recently with great success in machine learning. The focus of this dissertation is on developing novel variational Monte Carlo algorithms to make neural network variational wave functions applicable to many problems in quantum many-body physics. These algorithms draw from existing variational wave-function, tensor network, and machine learning methods. We calculate dynamical spectral functions using the vector correction and Chebyshev expansion method. We do finite temperature calculations with minimally entangled thermal states. Finally, we do ground state calculations for the frustrated J1−J2 Heisenberg model. We benchmark these results using restricted Boltzmann machines as the neural network for the the variational wave-functions.--Author's abstract