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arXiv:1106.4966 · Physical Review Letters

Solving Condensed-Matter Ground-State Problems by Semidefinite Relaxations

Thomas Barthel, R. Hübener

Published May 17, 2012• 42 citations
arXivHTMLPDF

Abstract

We present a generic approach to the condensed-matter ground-state problem which is complementary to variational techniques and works directly in the thermodynamic limit. Relaxing the ground-state problem, we obtain semidefinite programs (SDP). These can be solved efficiently, yielding strict lower bounds to the ground-state energy and approximations to the few-particle Green's functions. As the method is applicable for all particle statistics, it represents, in particular, a novel route for the study of strongly correlated fermionic and frustrated spin systems in D>1 spatial dimensions. It is demonstrated for the XXZ model and the Hubbard model of spinless fermions. The results are compared against exact solutions, quantum Monte Carlo calculations, and Anderson bounds, showing the competitiveness of the SDP method.

DOI
10.1103/physrevlett.108.200404

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We present a generic approach to the condensed-matter ground-state problem which is complementary to variational techniques and works directly in the thermodynamic limit. Relaxing the ground-state problem, we obtain semidefinite programs (SDP). These can be solved efficiently, yielding strict lower bounds to the ground-state energy and approximations to the few-particle Green's functions. As the method is applicable for all particle statistics, it represents, in particular, a novel route for the study of strongly correlated fermionic and frustrated spin systems in D>1 spatial dimensions. It is demonstrated for the XXZ model and the Hubbard model of spinless fermions. The results are compared against exact solutions, quantum Monte Carlo calculations, and Anderson bounds, showing the competitiveness of the SDP method.
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
We study semidefinite programming (SDP) methods to analyze spin chain Hamiltonians. We examine the ground state energy, the first excited charged states and ground state correlators in two simple models: the Ising model in a transverse magnetic field and the closely related 3-state Potts model. Our goal is to understand precisely what the SDP program is doing and when it works well, why it does so. We focus on the following novel ingredients: using charge constraints to obtain excited states and to see if additional constraints from integrable models are effective at improving the method. At criticality we also explore to what extent we can use approximate Virasoro correlators to extract conformal data: the central charge and some critical exponents of charged states. We also use these to identify the location of the phase transition. In the special case where the system is made of free fermions we prove that the SDP finds the exact energy of the ground state and produces the correct two point functions of the fermions. Away from free fermion theories, the SDP gets progressively worse at estimating data beyond the value of the ground state energy (like correlation functions), although it qualitatively matches these. In order to be effective, the SDP seems to run into scaling issues where the amount of input needed scales poorly with the lattice volume.
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.
Identifying quantum phase transitions poses a significant challenge in condensed matter physics, as this requires methods that both provide accurate results and scale well with system size. In this work, we demonstrate how relaxation methods can be used to generate the phase diagram for one- and two-dimensional quantum systems. To do so, we formulate a relaxed version of the ground-state problem as a semidefinite program, which we can solve efficiently. Then, by taking the resulting vector of moments for different model parameters, we identify all phase transitions based on their cosine similarity. Furthermore, we show how spontaneous symmetry breaking is naturally captured by bounding the corresponding observable. Using these methods, we reproduce the phase transitions for the one-dimensional transverse field Ising model and the two-dimensional frustrated bilayer Heisenberg model. We also illustrate how the phase diagram of the latter changes when a next-nearest-neighbor interaction is introduced. Overall, our work demonstrates how relaxation methods provide a novel framework for studying and understanding quantum phase transitions.
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.
Sometimes a mathematical tool emerges as uniquely useful and even a defining feature within some branch of physics such as Feynman diagrams. In quantum information theory, the semidefinite program (SDP) has emerged as such a tool. SDP is an optimization task in which a linear objective function is maximized over a set of Hermitian matrices with positive eigenvalues. This review discusses the highly efficient algorithms available for the SDP, and shows how comprehensively the SDP has been deployed in problems of entanglement characterization, quantum nonlocality, quantum channel capacities, and the bounding of ground-state energies.