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arXiv:1901.06124

Continuum Limits of Quantum Lattice Systems

Tobias J. Osborne

Published Jan 18, 2019• 9 citations
arXivHTMLPDF

Abstract

We describe a general procedure to give effective continuous descriptions of quantum lattice systems in terms of quantum fields. There are two key novelties of our method: firstly, it is framed in the hamiltonian setting and applies equally to distinguishable quantum spins, bosons, and fermions and, secondly, it works for arbitrary variational tensor network states and can easily produce computable non-gaussian quantum field states. Our construction extends the mean-field fluctuation formalism of Hepp and Lieb (developed later by Verbeure and coworkers) to identify emergent continuous large-scale degrees of freedom - the continuous degrees of freedom are not identified beforehand. We apply the construction to to tensor network states, including, matrix product states and projected entangled-pair states, where we recover their recently introduced continuous counterparts, and also for tree tensor networks and the multi-scale entanglement renormalisation ansatz. Finally, extending the continuum limit to include dynamics we obtain a strict light cone for the propagation of information.

DOI
10.48550/arxiv.1901.06124

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We describe a general procedure to give effective continuous descriptions of quantum lattice systems in terms of quantum fields. There are two key novelties of our method: firstly, it is framed in the hamiltonian setting and applies equally to distinguishable quantum spins, bosons, and fermions and, secondly, it works for arbitrary variational tensor network states and can easily produce computable non-gaussian quantum field states. Our construction extends the mean-field fluctuation formalism of Hepp and Lieb (developed later by Verbeure and coworkers) to identify emergent continuous large-scale degrees of freedom - the continuous degrees of freedom are not identified beforehand. We apply the construction to to tensor network states, including, matrix product states and projected entangled-pair states, where we recover their recently introduced continuous counterparts, and also for tree tensor networks and the multi-scale entanglement renormalisation ansatz. Finally, extending the continuum limit to include dynamics we obtain a strict light cone for the propagation of information.
We introduce a new class of states for bosonic quantum fields which extend tensor network states to the continuum and generalize continuous matrix product states (cMPS) to spatial dimensions d≥2d\geq 2d≥2. By construction, they are Euclidean invariant, and are genuine continuum limits of discrete tensor network states. Admitting both a functional integral and an operator representation, they share the important properties of their discrete counterparts: expressiveness, invariance under gauge transformations, simple rescaling flow, and compact expressions for the NNN-point functions of local observables. While we discuss mostly the continuous tensor network states extending Projected Entangled Pair States (PEPS), we propose a generalization bearing similarities with the continuum Multi-scale Entanglement Renormalization Ansatz (cMERA).
We introduce a new class of states for bosonic quantum fields which extend tensor network states to the continuum and generalize continuous matrix product states to spatial dimensions d ≥ 2. By construction, they are Euclidean invariant and are genuine continuum limits of discrete tensor network states.Admitting both a functional integral and an operator representation, they share the important properties of their discrete counterparts: expressiveness, invariance under gauge transformations, simple rescaling flow, and compact expressions for the N-point functions of local observables.While we discuss mostly the continuous tensor network states extending projected entangled-pair states, we propose a generalization bearing similarities with the continuum multiscale entanglement renormalization ansatz.
We study Gaussian continuous tensor network states (GCTNS) - a finitely parameterized subclass of Gaussian states admitting an interpretation as continuum limits of discrete tensor network states. We show that, at short distance, GCTNS correspond to vacua of free Lifshitz theories with even dynamical exponent, establishing a connection between certain entanglement properties of the two. Two schemes for approximating ground states of free bosonic field theories using GCTNS are presented: rational approximants to the exact dispersion relation and Trotterized imaginary-time evolution. We apply them to Klein-Gordon theory and characterize the resulting GCTNS, identifying the scale at which deviations from its conformal short-distance behavior appear. In particular, we construct an optimal rational approximation for which, at fixed accuracy, the bond dimension grows only logarithmically with the UV cutoff. These results provide a simple and analytically controlled setting to assess the strengths and limitations of GCTNS as variational ansätze for relativistic quantum fields.
Determining the properties of quantum many-body systems is a central challenge in modern physics. Being able to determine the macroscopic properties of a system from its microscopic description would hasten progress in many fields of science and technology. However, we currently lack the tools to solve such problems generally, or even to develop a theoretical intuition about how many systems might behave. From a simple Hamiltonian description of the system, one may obtain complex, highly correlated collective behaviour. Computational techniques have played a major part in the effort to determine properties of quantum many-body systems. However, as the total degrees of freedom in the system scales exponentially in the system size, numerical diagonalization of the Hamiltonian quickly becomes computationally intractable and one must develop more efficient approximate techniques to explore the system. Present numerical methods such as quantum Monte Carlo and series expansion have provided insight into many systems of interest, but are also held back by fundamental difficulties. In this thesis, we focus on tensor networks, a relatively new ansatz for representing quantum many-body states. Tensor networks are motivated by two ideas from quantum information: firstly, that quantum entanglement is the source of the immense difficulty of simulating quantum systems classically, and secondly that the ground states of certain Hamiltonians exist in a low-entanglement region of the entire Hilbert space. The strength of tensor networks is that they provide a systematic way of representing this class of low-entanglement quantum states. In particular, this thesis describes the iPEPS algorithm for computing the ground states of infinite, two-dimensional quantum lattice systems based on the Projected Entangled Pair States (PEPS) ansatz. We then benchmark the algorithm by computing the phase diagrams of several systems that have been studied with other techniques. Lastly, we apply our algorithm to problems that are not well solved by current approaches, such as frustrated spin systems.
Tree tensor network descriptions of critical quantum spin chains are empirically known to reproduce correlation functions matching conformal field theory (CFT) predictions in the continuum limit. It is natural to seek a more complete correspondence, additionally incorporating dynamics. On the CFT side, this is determined by a representation of the diffeomorphism group of the circle. In a remarkable series of papers, Jones outlined a research program where the Thompson group T takes the role of the latter in the discrete setting, and representations of T are constructed from certain elements of a subfactor planar algebra. He also showed that, for a particular example of such a construction, this approach only yields-in the continuum limit-a representation which is highly discontinuous and hence unphysical. Here we show that the same issue arises generically when considering tree tensor networks: the set of coarse-graining maps yielding discontinuous representations has full measure in the set of all isometries. This extends Jones's no-go example to typical elements of the so-called tensor planar algebra. We also identify an easily verified necessary condition for a continuous limit to exist. This singles out a particular class of tree tensor networks. Our considerations apply to recent approaches for introducing dynamics in holographic codes.
Gauge theories form the basis of our understanding of modern physics—ranging from the description of quarks and gluons to effective models in condensed matter physics. In the nonperturbative regime, gauge theories are conventionally treated discretely as lattice gauge theories. The resulting systems are evaluated with path-integral based Monte Carlo methods. These methods, however, can suffer from the sign problem and do not allow for a direct evaluation of real-time dynamics. In this work, we present a unified and comprehensive framework for gauged Gaussian projected entangled pair states (PEPS), a variational ansatz based on tensor networks. We review the construction of Hamiltonian lattice gauge theories, explain their similarities to PEPS, and detail the construction of the ansatz state. The estimation of ground states is based on a variational Monte Carlo procedure with the PEPS as an ansatz state. This sign-problem-free ansatz can be efficiently evaluated in any dimension with arbitrary gauge groups, and can include dynamical fermionic matter, suggesting new options for the simulation of nonperturbative regimes of gauge theories, including quantum chromodynamics. Published by the American Physical Society 2024