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Probing universality with entanglement entropy via quantum Monte Carlo

Bohdan Kulchytskyy

Published Aug 30, 2019• 9 citations
OpenAlex

Abstract

Our understanding of physical phenomena hinges on finding universal core mechanisms that unite them. The concept of universality is deeply ingrained in the study of quantum many-body systems. At zero temperature, microscopically different systems with long-range order collapse into their universal state described by a handful of universal parameters. Establishing those parameters implies identification of the universal theory that is effectively describing the system and ultimately providing the desired understanding. We take up this task with the help of Renyi entanglement entropy. Defined with respect to a system bipartition, this measure quantifies information shared between the subsystems. Understanding what insights into universality are encoded within this information-theoretic quantity as well as developing numerical tools to efficiently estimate the Renyi entanglement entropy are the subjects of this thesis. \n \nOn the computational end of this far-reaching goal, we develop a novel theoretical framework for constructing improved Renyi entanglement entropy estimators in the context of d+1 quantum Monte Carlo methods. The discovery of a connection of this methodology to the well-established Kandel-Domany formalism provides a clear path towards generalization. Additionally, we embrace a data-driven approach towards learning the ground state wavefunction. We demonstrate how a restricted Boltzmann machine can be used to reconstruct the Renyi entanglement entropy from projective measurements of a quantum ground state. Furthermore, we extend this classical generative architecture to a quantum analogue that we call the quantum Boltzmann machine. \n \nOn the theoretical side of this endeavour, we study the Renyi entanglement entropy scaling terms for two quantum lattice models embedded in two dimensional space via extensive quantum Monte Carlo simulations. For the ground state of the XY model, we provide conclusive numerical evidence for a logarithmic contribution that uniquely characterizes the continuous symmetry of the emerging order parameter. Moreover, we confirm the form of the subleading universal geometric contribution arising due to the bosonic nature of low-energy degrees of freedom in this model. For the critical ground state of the transverse field Ising model, we develop a novel scaling procedure to extract a universal number κ revealed via a cylindrical entangling bipartition in the thin-slice limit. The combined product of our work sheds new light on the entanglement-based classification of universality and brings a suite of new powerful numerical tools to continue illuminating this theoretical program in the future.

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Our understanding of physical phenomena hinges on finding universal core mechanisms that unite them. The concept of universality is deeply ingrained in the study of quantum many-body systems. At zero temperature, microscopically different systems with long-range order collapse into their universal state described by a handful of universal parameters. Establishing those parameters implies identification of the universal theory that is effectively describing the system and ultimately providing the desired understanding. We take up this task with the help of Renyi entanglement entropy. Defined with respect to a system bipartition, this measure quantifies information shared between the subsystems. Understanding what insights into universality are encoded within this information-theoretic quantity as well as developing numerical tools to efficiently estimate the Renyi entanglement entropy are the subjects of this thesis. \n \nOn the computational end of this far-reaching goal, we develop a novel theoretical framework for constructing improved Renyi entanglement entropy estimators in the context of d+1 quantum Monte Carlo methods. The discovery of a connection of this methodology to the well-established Kandel-Domany formalism provides a clear path towards generalization. Additionally, we embrace a data-driven approach towards learning the ground state wavefunction. We demonstrate how a restricted Boltzmann machine can be used to reconstruct the Renyi entanglement entropy from projective measurements of a quantum ground state. Furthermore, we extend this classical generative architecture to a quantum analogue that we call the quantum Boltzmann machine. \n \nOn the theoretical side of this endeavour, we study the Renyi entanglement entropy scaling terms for two quantum lattice models embedded in two dimensional space via extensive quantum Monte Carlo simulations. For the ground state of the XY model, we provide conclusive numerical evidence for a logarithmic contribution that uniquely characterizes the continuous symmetry of the emerging order parameter. Moreover, we confirm the form of the subleading universal geometric contribution arising due to the bosonic nature of low-energy degrees of freedom in this model. For the critical ground state of the transverse field Ising model, we develop a novel scaling procedure to extract a universal number κ revealed via a cylindrical entangling bipartition in the thin-slice limit. The combined product of our work sheds new light on the entanglement-based classification of universality and brings a suite of new powerful numerical tools to continue illuminating this theoretical program in the future.
At the interface of quantum information and condensed matter physics, the study of entanglement in quantum many-body systems requires a new toolset which combines concepts from each. This thesis introduces a set of computational methods to study phases and phase transitions in lattice models of quantum systems, using the Renyi entropies as a means of quantifying entanglement. The scaling of entanglement entropy can give valuable insight into the phase of a condensed matter system. It can be used to detect exotic types of phases, to pinpoint transitions between phases, and can give us universal information about a system. \nThe first approach in this thesis is a technique to measure entanglement in finite size \nlattice systems using zero-temperature quantum Monte Carlo simulations. The algorithm \nis developed, implemented, and used to explore anomalous entanglement scaling terms in \nthe spin-1/2 Heisenberg antiferromagnet. \nIn the second part of this thesis, a new and complementary numerical technique is introduced to study entanglement not just in finite size systems, but as we approach the thermodynamic limit. This “numerical linked-cluster expansion” is used to study two different systems at their quantum critical points — continuous phase transitions occurring at zero temperature, at which these systems exhibit universal properties. Remarkably, these universal properties can be reflected in the scaling of entanglement. \nEntanglement offers a new perspective on condensed matter systems, one which takes us closer to genuinely understanding what goes on in these materials at the quantum mechanical level. This thesis demonstrates the first steps in developing an extensive list of computational tools that can be used to study entanglement over a wide range of interacting quantum many-body systems. With the ever increasing computational power available, it may be only a matter of time before these tools are used to create a comprehensive framework for the characterization of condensed matter phases and phase transitions.
Computing the subleading logarithmic term in the entanglement entropy (EE) of (2+1)d quantum many-body systems remains a significant challenge, despite its central role in revealing universal information about quantum states and quantum critical points (QCPs). Building on recent algorithmic advances that enable the stable calculation of EE as an exponential observable~\cite{zhouIncremental2024,zhangIntegral2024,liaoExtracting2024}, we develop a {\it bubble basis} projector quantum Monte Carlo (QMC) algorithm to precisely and efficiently compute the universal corner of EE at QCPs in a (2+1)d square-lattice transverse-field Ising model augmented with a four-body interaction. Turning on this interaction allows us to trace an Ising critical line, reaching the tricritical point, and then a line of first-order phase transition. In (2+1)d, the tricritical point is described by the Gaussian theory, where a theoretical calculation of the corner logarithmic term in the 2nd Rényi entropy term is available~\cite{UniversalCasini2007}. Our QMC results are in quantitative agreement with this theoretical value, providing a highly nontrivial benchmark of the algorithm. Furthermore, we also study the Rényi EE at the Ising critical line and on the first-order transition line, obtaining results consistent with theoretical expectations. These findings establish the long-sought connection between the universal values of an exactly solvable limit and those of a strongly correlated regime at (2+1)d.
A ground state path integral quantum Monte Carlo algorithm is introduced that allows for the study of entanglement in lattice bosons at zero temperature. The Rényi entanglement entropy between spatial subregions is explored across the phase diagram of the one dimensional Bose-Hubbard model for systems consisting of up to L=256 L = 256 sites at unit-filling without any restrictions on site occupancy, far beyond the reach of exact diagonalization. The favorable scaling of the algorithm is demonstrated through a further measurement of the Rényi entanglement entropy at the two dimensional superfluid-insulator critical point for large system sizes, confirming the existence of the expected entanglement boundary law in the ground state. The Rényi estimator is extended to measure the symmetry resolved entanglement that is operationally accessible as a resource for experimentally relevant lattice gases with fixed total particle number.
The interplay of the constituents of interacting many-body systems may reveal emergent properties on the macroscopic scale which are not inherent to the individual constituents. These properties are expressed in macroscopic observables \ndescribing the state — denoted as the phase of the system. Continuous phase transition between phases are generically manifested in critical behavior, for example, a divergence of a macroscopic observable. The identification of the present phase of a system and the classification of critical phenomena into universality classes are exciting challenges of condensed-matter physics. \nIn this thesis, we use entanglement entropies as macroscopic quantities for the characterization of phases of quantum matter and critical theories. For groundstates of quantum many-body systems the entanglement entropy is a measure of \nthe amount of entanglement between two subsystems. The generic dependence of the entanglement entropy on the size and shape of the subsystems is contained in the well-known boundary law — stating a scaling of the entanglement entropy with the boundary between the two subsystems. We numerically investigate how the coefficient of this dependence reflects quantum phase transitions in simple spin-half bilayer models. \nSubleading terms to the boundary law such as a logarithmic contribution provide universal numbers for the criticality of field theories. We examine free and interacting theories from an entanglement entropy perspective in order to assess the role of the coefficient of the logarithmic correction induced by corners in the subsystems. Beyond its universality, this coefficient also quantifies degrees of freedom of low-lying excitations in the conformal field theory describing a critical point. \nA constant contribution to the boundary law indicates the presence of so-called topological order in the ground state of a many-body system. This extremely useful property can also be identified in classical counterparts of entanglement entropy which we study at the example of various toric code models. To this endeavor, we have designed Monte Carlo techniques which allow for an efficient numerical computation of the constant contribution. In particular, we analyze via entanglement entropies under which conditions remnants of topological order are present in the quantum system at finite temperature and at perturbations from a magnetic field. The major motivation behind this effort is to use topological order for the robust storage of a quantum information — a basic need for the construction of quantum \ncomputers.
The Shannon-Rényi (SR) entropy characterizes the probability distribution of many-body wave functions in a chosen basis and is directly accessible from single-shot measurement snapshots in cold-atom experiments, yet its theoretical evaluation in interacting systems carries exponential cost. Here we construct a field-theoretic spacetime manifold for the SR entropy in which qqq replicas are glued at the imaginary-time boundaries, recasting it as a ratio of generalized partition functions amenable to large-scale quantum Monte Carlo simulation. On this basis we develop a reweight-annealing framework with two complementary strategies: a virtual-parameter scheme interpolating between unglued and fully glued limits via binomial-weighted partial gluing, and a real-parameter scheme using a physical coupling constant as the annealing variable within stochastic series expansion. Both reduce the computational cost from exponential to polynomial. Most importantly, we establish for the first time a finite-size scaling theory for the SR entropy near a quantum critical point, Sq=a(g)L2+Sq0((g−gc)L1/ν)S_q=a(g)L^2+S_q^0\left((g-g_c)L^{1/ν}\right)Sq​=a(g)L2+Sq0​((g−gc​)L1/ν); data collapse of its first derivative yields the correlation-length exponent ννν for both the O(3) and Ising universality classes, realized in the spin-12\frac{1}{2}21​ columnar dimerized Heisenberg model and the bilayer Heisenberg-Ising model. The same simulations extract the Goldstone-mode count NG=2N_G=2NG​=2 and the Z2Z_2Z2​ ground-state degeneracy ln⁡2\ln 2ln2 from subleading terms. These results elevate the SR entropy to a standalone, order-parameter-free diagnostic of quantum criticality with the same scaling status as the entanglement entropy, and close the loop between theory and quantum simulation by enabling quantitative benchmarking of experimental snapshots from state-of-the-art quantum simulators.
In the face of mounting numerical evidence, Metlitski and Grover (arXiv:1112.5166) have given compelling analytical arguments that systems with spontaneous broken continuous symmetry contain a subleading contribution to the entanglement entropy that diverges logarithmically with system size. They predict that the coefficient of this log is a universal quantity that depends on the number of Goldstone modes. In this paper, we confirm the presence of this log term through quantum Monte Carlo calculations of the second R\'enyi entropy on the spin-1/2 XYXYXY model. Devising an algorithm to facilitate convergence of entropy data at extremely low temperatures, we demonstrate that the single Goldstone mode in the ground state can be identified through the coefficient of the log term. Furthermore, our simulation accuracy allows us to obtain an additional geometric constant additive to the R\'enyi entropy, that matches a predicted fully universal form obtained from a free bosonic field theory with no adjustable parameters.