Post-matrix product state methods: To tangent space and beyond

Haegeman, Jutho; Osborne, Tobias J.; Verstraete, Frank

doi:10.1103/physrevb.88.075133

Cited by 203 publications

(239 citation statements)

References 92 publications

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“…Finally, it seems that tangent-space methods that have proven successful in the context of matrix product states 39 are now within reach for PEPS simulations for generic two-dimensional quantum spin models. In particular, this paper opens up the prospect of simulating real-time evolution according to the time-dependent variational principle 38 and/or computing the low-energy spectrum on top of a generic PEPS with the quasiparticle excitation ansatz 75,76 .…”

Section: Discussionmentioning

confidence: 99%

“…In the case of imaginary time, this tangent vector is exactly the gradient, which shows that different optimization algorithms can be compared within this unifying manifold interpretation 41 . Moreover, whereas imaginary-time evolution more or less corresponds to a steepest-descent method 39 , more advanced optimization methods such as conjugate-gradient or quasi-Newton algorithms can find an optimal matrix product state much more efficiently 42,43 .…”

Section: Introductionmentioning

confidence: 99%

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Gradient methods for variational optimization of projected entangled-pair states

et al. 2016

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We present a conjugate-gradient method for the ground-state optimization of projected entangledpair states (PEPS) in the thermodynamic limit, as a direct implementation of the variational principle within the PEPS manifold. Our optimization is based on an efficient and accurate evaluation of the gradient of the global energy functional by using effective corner environments, and is robust with respect to the initial starting points. It has the additional advantage that physical and virtual symmetries can be straightforwardly implemented. We provide the tools to compute static structure factors directly in momentum space, as well as the variance of the Hamiltonian. We benchmark our method on Ising and Heisenberg models, and show a significant improvement on the energies and order parameters as compared to algorithms based on imaginary-time evolution.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Gradient methods for variational optimization of projected entangled-pair states

et al. 2016

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“…This is a data-sparse tensor format introduced in [20] in the mathematical literature. It has previously been used in physics under the name of matrix product states; see, e.g., [22,25] and, in a time-dependent context, [6,7].…”

Section: 2)ẏ (T) = P (Y (T))f (T Y (T))mentioning

confidence: 99%

Discretized Dynamical Low-Rank Approximation in the Presence of Small Singular Values

Kieri¹,

Lubich²,

Walach³

2016

SIAM J. Numer. Anal.

145

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Abstract. Low-rank approximations to large time-dependent matrices and tensors are the subject of this paper. These matrices and tensors either are given explicitly or are the unknown solutions of matrix and tensor differential equations. Based on splitting the orthogonal projection onto the tangent space of the low-rank manifold, novel time integrators for obtaining approximations by low-rank matrices and low-rank tensor trains were recently proposed. By standard theory, the Lie-Trotter and Strang projector-splitting methods are first and second order accurate, respectively, but the usual error bounds break down when the low-rank approximation has small singular values. This happens when the singular values of the solution decay without a distinct gap or when the effective rank of the solution is overestimated. On the other hand, the integrators are exact when given time-dependent matrices or tensors are already of the prescribed rank. We provide an error analysis which unifies these properties. We show that in cases where the exact solution is an ε-perturbation of a low-rank matrix or tensor train, the error of the projector-splitting integrator is favorably bounded in terms of ε and the stepsize, independently of the smallness of the singular values. Such a result does not hold for any standard integrator. Numerical experiments illustrate the theory.Key words. tensor train, low-rank approximation, tensor differential equations, splitting integrator AMS subject classifications. 15A18, 65L05, 65L70 DOI. 10.1137/15M10267911. Introduction. Low-rank approximations to matrices and tensors are a basic tool in data and model reduction; see, e.g., [4,5]. In this paper we are concerned with the low-rank approximation of time-dependent matrices and tensors, which either are given explicitly by their increments or are the unknown solutions of differential equations. In [11] such time-dependent problems and their numerical treatment were first studied for matrices. Differential equations for the factors of a low-rank factorization similar to the singular value decomposition were derived and their approximation properties were studied. Extensions to time-dependent tensors in various tensor formats were given in [2,12,18,19]; see also [15] for a review of dynamical low-rank approximation.The approach yields differential equations on low-rank matrix and tensor manifolds, which need to be solved numerically. Recently, very efficient integrators based on splitting the projection onto the tangent space of the low-rank manifold have been proposed and studied for matrices and for tensors in the tensor-train format in [16] and [17], respectively. The objective of the present paper is to show that these projector-splitting integrators are insensitive to the presence of small singular values

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“…Once we have a good approximation for the ground state, we can use the method of [17,18] to obtain the one-particle excited states. The excitations are labelled by their (physical) momentum k ∈ [−π/2a, π/2a[ and their CT quantum number γ = ±1.…”

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confidence: 99%

“…A state in this form is also invariant under any gauge transformation. Indeed, according to (18), we need to find unitary matrices U l and V l such that…”

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confidence: 99%

Matrix Product States for Gauge Field Theories

et al. 2014

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The matrix product state formalism is used to simulate Hamiltonian lattice gauge theories. To this end, we define matrix product state manifolds which are manifestly gauge invariant. As an application, we study 1+1 dimensional one flavor quantum electrodynamics, also known as the massive Schwinger model, and are able to determine very accurately the ground state properties and elementary one-particle excitations in the continuum limit. In particular, a novel particle excitation in the form of a heavy vector boson is uncovered, compatible with the strong coupling expansion in the continuum. We also study full quantum non-equilibrium dynamics by simulating the real-time evolution of the system induced by a quench in the form of a uniform background electric field. PACS numbers:Gauge theories hold a most prominent place in physics. They appear as effective low energy descriptions at different instances in condensed matter physics and nuclear physics. But far and foremost they lie at the root of our understanding of the four fundamental interactions that are each mediated by the gauge fields corresponding to a particular gauge symmetry. At the perturbative quantum level, this picture translates to the Feynman diagrammatic approach that has produced physical predictions with unlevelled precision, most famously in quantum electrodynamics (QED). However the perturbative approach miserably fails once the interactions become strong. This problem is most pressing for quantum chromodynamics (QCD), where all low energy features like quark confinement, chiral symmetry breaking and mass generation are essentially non-perturbative.Lattice QCD, which is based on Monte Carlo sampling of Wilson's Euclidean lattice version of gauge theories, has historically been by far the most successful method in tackling this strongly coupled regime. Using up a sizable fraction of the global supercomputer time, state of the art calculations have now reached impressive accuracy, for instance in the ab initio determination of the light hadron masses [1]. But in spite of its clear superiority, the lattice Monte Carlo sampling also suffers from a few drawbacks. There is the infamous sign problem that prevents application to systems with large fermionic densities. In addition, the use of Euclidean time, as opposed to real time, presents a serious barrier for the understanding of dynamical non-equilibrium phenomena. Over the last few years there has been a growing experimental and theoretical interest in precisely these elusive regimes, e.g. in the study of heavy ion collisions or early time cosmology.In this letter we study the application of tensor network states (TNS) as a possible complementary approach to the numerical simulation of gauge theories. This is highly relevant as this Hamiltonian method is free from the sign problem and allows for real-time dynamics. As a first application we focus on the massive Schwinger model. For this model the TNS approach has been studied before by Byrnes et al [2] and Bañuls et al [3]. By integrating out the ...

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Post-matrix product state methods: To tangent space and beyond

Cited by 203 publications

References 92 publications

Gradient methods for variational optimization of projected entangled-pair states

Gradient methods for variational optimization of projected entangled-pair states

Discretized Dynamical Low-Rank Approximation in the Presence of Small Singular Values

Matrix Product States for Gauge Field Theories

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