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arXiv:2111.13007

Bootstrapping Lattice Vacua

Scott Lawrence

Published Nov 25, 2021• 2 citations
arXivHTMLPDF

Abstract

This paper demonstrates the application of semidefinite programming to lattice field theories, showcasing spin chains and lattice scalar field theory. Requiring expectation values of manifestly positive semi-definite operators to be non-negative results in a lower bound on the ground-state energy of any quantum mechanical system, which can be made arbitrarily tight for systems described by finite-dimensional Hilbert spaces. Such bounds can be obtained directly in the infinite-volume limit. The process of optimizing these lower bounds also yields estimates for a chosen set of expectation values in the ground state.

DOI
10.48550/arxiv.2111.13007

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This paper demonstrates the application of semidefinite programming to lattice field theories, showcasing spin chains and lattice scalar field theory. Requiring expectation values of manifestly positive semi-definite operators to be non-negative results in a lower bound on the ground-state energy of any quantum mechanical system, which can be made arbitrarily tight for systems described by finite-dimensional Hilbert spaces. Such bounds can be obtained directly in the infinite-volume limit. The process of optimizing these lower bounds also yields estimates for a chosen set of expectation values in the ground state.
A novel bootstrap method is exemplified by lattice theories and quantum mechanics. Positivity of expectation values combined with sets of non-trivial constraint equations and semidefinite optimization rigorously and precisely bound correlation functions. Starting with the example of a one-dimensional integral, the technique is built upon to extract limit cycles in a two-dimensional classical dynamical system and spin correlation functions in the Statistical Ising Model. Further positivity constraints and simplifications arising in a large rank limit are implemented to bound the energy expectation value as a function of temperature in Matrix Quantum Mechanics. Along the way, the S-matrix of Chern-Simons-matter theory is computed as an example of computational simplification at large gauge group rank.
Semidefinite programs can be constructed to provide a nonperturbative view of the zero-temperature behavior of quantum systems. Past work has found that small semidefinite programs (relative to the dimension of Hilbert space) yield quantitatively accurate estimates for the ground-state energy and expectation values. This paper examines the properties of these semidefinite programs when applied to lattice-regulated field theories exhibiting fermion sign problems. Specifically on the finite-density Thirring model, small semidefinite programs still yield quantitatively accurate results, and there is no indication that these methods encounter exponential costs (analogous to the fermion sign problem) at finite chemical potential.
A numerical bootstrap method is proposed to provide rigorous and nontrivial bounds in general quantum many-body systems with locality. In particular, lower bounds on ground state energies of local lattice systems are obtained by imposing positivity constraints on certain operator expectation values. Complemented with variational upper bounds, ground state observables are constrained to be within a narrow range. The method is demonstrated with the Hubbard model in one and two dimensions, and bounds on ground state double occupancy and magnetization are discussed.
Gauge theories describe fundamental interactions in the standard model of particle physics and build effective theories in condensed matter physics. Being notoriously hard to simulate, these theories are commonly regularized as lattice gauge theories which can be evaluated with high precision through Monte Carlo algorithms or variational methods. However, Monte Carlo algorithms are not applicable in all regimes due to the sign problem; while variational methods, quantum and classical alike, only give upper-bounds on the ground-state energy. Their precision crucially depends on the chosen ansatz. Here, we present a framework based on a hierarchy of semidefinite programs to yield increasingly good lower-bounds on the ground-state energy and propagate these bounds to extended observables like Wilson loops and mesonic strings. We demonstrate the numerical capabilities by applying the algorithm to a (1+1)-dimensional Z2\mathbb{Z}_2Z2​ theory with dynamic fermionic matter and its pure-gauge version in (2+1)-dimensions. In both cases, we obtain certified intervals for the ground-state energy. For a one-dimensional system with 129 sites, the certified interval has a relative spread of 0.04%0.04\%0.04%. For a two-dimensional system with 97 sites (corresponding to a 7×87 \times 87×8 lattice), the relative spread is 2.08%2.08\%2.08%. Beyond energy estimates, the proposed method provides rigorous bounds on both short- and long-range observables such as Wilson loops over multiple plaquettes and the magnitude of the mesonic string. The method can be both used as a computational microscope into many-body physics and a way to certify results of quantum simulators.
We present a semidefinite programming algorithm to find eigenvalues of Schrödinger operators within the bootstrap approach to quantum mechanics. The bootstrap approach involves two ingredients: a nonlinear set of constraints on the variables (expectation values of operators in an energy eigenstate), plus positivity constraints (unitarity) that need to be satisfied. By fixing the energy we linearize all the constraints and show that the feasibility problem can be presented as an optimization problem for the variables that are not fixed by the constraints and one additional slack variable that measures the failure of positivity. To illustrate the method we are able to obtain high-precision, sharp bounds on eigenenergies for arbitrary confining polynomial potentials in one dimension.
A fundamental challenge in quantum physics is determining the ground-state properties of many-body systems. Whereas standard variational approaches posit a wave-function ansatz and minimize over the possible states expressible by that ansatz, the problem can alternatively be formulated as a noncommutative polynomial optimization problem and treated through a hierarchy of semidefinite programming relaxations. In contrast to variational calculations, these relaxations provide lower bounds on ground-state energies and both lower and upper bounds on observable expectation values. However, this approach typically suffers from severe scalability issues, limiting its applicability to small-to-medium-scale systems. In this article, we demonstrate that systematically leveraging the inherent structures of the system can substantially mitigate these scalability challenges and thus permits computing meaningful bounds for quantum spin systems on square lattices of size up to 16×1616\times1616×16.