Entanglement entropies can conceal novel universal features that underlie a quantum phase transition. The universal content is determined by the geometry of the boundary between two entangled regions. In the quantum Monte Carlo simulations here, the authors numerically extract a universal number that arises at a quantum phase transition in the two-dimensional quantum Ising model. This universal number, measured when a two-dimensional torus is partitioned with a thin cylindrical strip, is so far lacking a comprehensive theoretical interpretation.
Entanglement entropies can conceal novel universal features that underlie a quantum phase transition. The universal content is determined by the geometry of the boundary between two entangled regions. In the quantum Monte Carlo simulations here, the authors numerically extract a universal number that arises at a quantum phase transition in the two-dimensional quantum Ising model. This universal number, measured when a two-dimensional torus is partitioned with a thin cylindrical strip, is so far lacking a comprehensive theoretical interpretation.
Although the leading-order scaling of entanglement entropy is non-universal at a quantum critical point (QCP), sub-leading scaling can contain universal behaviour. Such universal quantities are commonly studied in non-interacting field theories, however it typically requires numerical calculation to access them in interacting theories. In this paper, we use large-scale T = 0 quantum Monte Carlo simulations to examine in detail the second Rényi entropy of entangled regions at the QCP in the transverse-field Ising model in 2 + 1 space–time dimensions—a fixed point for which there is no exact result for the scaling of entanglement entropy. We calculate a universal coefficient of a vertex-induced logarithmic scaling for a polygonal entangled subregion, and compare the result to interacting and non-interacting theories. We also examine the shape-dependence of the Rényi entropy for finite-size toroidal lattices divided into two entangled cylinders by smooth boundaries. Remarkably, we find that the dependence on cylinder length follows a shape-dependent function calculated previously by Stephan et al (2013 New J. Phys. 15 015004) at the QCP corresponding to the 2 + 1 dimensional quantum Lifshitz free scalar field theory. The quality of the fit of our data to this scaling function, as well as the apparent cutoff-independent coefficient that results, presents tantalizing evidence that this function may reflect universal behaviour across these and other very disparate QCPs in 2 + 1 dimensional systems.
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.
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.
The entanglement entropy is a unique probe to reveal universal features of strongly interacting many-body systems. In two or more dimensions these features are subtle, and detecting them numerically requires extreme precision, a notoriously difficult task. This is especially challenging in models of interacting fermions, where many such universal features have yet to be observed. In this Letter we tackle this challenge by introducing a new method to compute the Rényi entanglement entropy in auxiliary-field quantum Monte Carlo simulations, where we treat the entangling region itself as a stochastic variable. We demonstrate the efficiency of this method by extracting, for the first time, universal subleading logarithmic terms in a two-dimensional model of interacting fermions, focusing on the half-filled honeycomb Hubbard model at T=0. We detect the universal corner contribution due to gapless fermions throughout the Dirac semi-metal phase and at the Gross-Neveu-Yukawa critical point, where the latter shows a pronounced enhancement depending on the type of entangling cut. Finally, we observe the universal Goldstone mode contribution in the antiferromagnetic Mott insulating phase.
Although entanglement is a well studied property in the context of quantum systems, the ability to measure it in Monte Carlo methods is relatively new. \nThrough measures of the Renyi entanglement entropy and mutual information one is able to examine and characterize criticality, pinpoint phase transitions, and probe universality. \nWe describe the most basic algorithms for calculating these quantities in straightforward Monte Carlo methods and state of the art techniques used in high performance computing. \nThis description emphasizes the core principal of these measurements and allows one to both build an intuition for these quantities and how they are useful in numerical studies. \n \nUsing the Renyi entanglement entropy we demonstrate the ability to detect thermal phase transitions in the Ising model and XY model without use of an order parameter. \nThe scaling near the critical point also shows signatures identifying the universality class of the model. \nImproved methods are explored using extended ensemble techniques that can increase calculation efficiency, and show good agreement with the standard approach. \nWe explore the "ratio trick" at finite temperature and use it to explore the quantum critical fan of the one dimensional transverse field Ising model, showing agreement with finite temperature and finite size scaling from field theory. \nThis same technique is used at zero temperature to explore the geometric dependence of the entanglement entropy and examine the universal scaling functions in the two dimensional transverse field Ising model. \nAll of this shows the multitude of ways in which the study of the Renyi entanglement entropy can be efficiently and practically used in conventional and exotic condensed matter systems, and should serve as a reference for those wishing to use it as a tool.
Criticality on Rényi Defects at (2+1)D O(3) Quantum Critical Points
At a quantum critical point, the universal scaling behavior of Rényi entanglement entropy is controlled by the universality class of the codimension-two Rényi (or conical) defects in the infrared theory. In this work we perform a systematic study of critical correlations along Rényi defect lines in (2+1)D quantum spin models realizing quantum phase transitions described by the O(3) Wilson-Fisher universality class, using large-scale quantum Monte Carlo simulations. We present numerical evidence that, for a fixed Rényi index n, there exist multiple Rényi defect universality classes, with distinct critical exponents for the O(3) order parameter on the defect. These universality classes are realized by choosing microscopically different entanglement cuts in lattice models, which we classify as ordinary, special and extraordinary according to their relation to surface criticality. For the extraordinary entanglement cut, we further find evidence for a phase transition on the defect as a function of the Rényi index. Our results highlight the key role of defect universality classes in determining the universal scaling of Rényi entropy, and provide a framework for understanding the previously observed dependence of Rényi entropy scaling on microscopic lattice details.