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The scaling behavior of the entanglement entropy in 2+1 dimensional critical systems

Benjamin Hsu

Published Aug 25, 2011• 0 citations
OpenAlex

Abstract

The entanglement entropy in 1+1 dimensional critical system has been well studied and known to have a universal scaling form that gives information about the underlying critical system. While the case for 1+1 dimensional critical systems has been well studied, the situation in higher dimensional systems is less clear. In this work, I show that the entanglement entropy in a certain class of 2+1 dimensional critical systems, the conformal quantum critical theories, has a universal subleading correction that likewise gives information about the underlying critical system. In addition, I address the issue of whether the dynamical entanglement entropy, the entanglement generated by a local quantum quench, is an experimentally measurable quantity. This turns out not to be the case, but to arrive at this conclusion we studied quenching in a theory of strongly interacting electrons and applied methods of boundary conformal field theory to solve a hitherto open problem: the noise from an instantaneous switching of a quantum point contact separating two fractional quantum Hall fluids.

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The entanglement entropy in 1+1 dimensional critical system has been well studied and known to have a universal scaling form that gives information about the underlying critical system. While the case for 1+1 dimensional critical systems has been well studied, the situation in higher dimensional systems is less clear. In this work, I show that the entanglement entropy in a certain class of 2+1 dimensional critical systems, the conformal quantum critical theories, has a universal subleading correction that likewise gives information about the underlying critical system. In addition, I address the issue of whether the dynamical entanglement entropy, the entanglement generated by a local quantum quench, is an experimentally measurable quantity. This turns out not to be the case, but to arrive at this conclusion we studied quenching in a theory of strongly interacting electrons and applied methods of boundary conformal field theory to solve a hitherto open problem: the noise from an instantaneous switching of a quantum point contact separating two fractional quantum Hall fluids.
A formidable perspective in understanding quantum criticality of a given many-body system is through its entanglement contents. Until now, most progress are only limited to the disorder-free case. Here, we develop an efficient scheme to compute the entanglement entropy of (2+1)(2+1)(2+1)-dimensional quantum critical points with randomness, from a conceptual angle where the quenched disorder can be considered as dimensionally reducible interactions. As a concrete example, we reveal novel entanglement signatures of (2+1)(2+1)(2+1)-dimensional Dirac fermion exposed to a random magnetic field, which hosts a class of emergent disordered quantum critical points. We demonstrate that the entanglement entropy satisfies the area-law scaling, and observe a modification of the area-law coefficient that points to the emergent disordered quantum criticality. Moreover, we also obtain the subleading correction to the entanglement entropy due to a finite correlation length. This subleading correction is found to be a universal function of the correlation length and disorder strength. We discuss its connection to the renormalization group flows of underlying theories.
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The entanglement entropy of a pure quantum state of a bipartite system is defined as the von Neumann entropy of the reduced density matrix obtained by tracing over one of the two parts. Critical ground states of local Hamiltonians in one dimension have an entanglement entropy that diverges logarithmically in the subsystem size, with a universal coefficient that is is related to the central charge of the associated conformal field theory. Here I will discuss recent extensions of these ideas to a class of quantum critical points with dynamic critical exponent z = 2 in two space dimensions and to 2D systems in a topological phase. The application of these ideas to quantum dimer models and fractional quantum Hall states will be discussed.
We examine the growth of entanglement under a quantum quench at point contacts of simple fractional quantum Hall fluids and its relation with the measurement of local observables. Recently Klich and Levitov proposed that the noise generated from a local quantum quench provides a measure of the entanglement entropy. Their methods were specific to noninteracting electrons and the generalization to interacting systems was left as an open question. In this work, we generalize their result to the Laughlin states. We investigate the noise generated in the current along the edge of a fractional quantum Hall state at filling factors $\ensuremath{\nu}=1/m$, when a quantum point contact, initially closed, is fully opened at some initial time t0=0{t}_{0}=0t0​=0. We find that local quenching in these systems gives time dependent correlation functions that have universal behavior on sufficiently long time and length scales. We calculate the noise and full counting statistics for $\ensuremath{\nu}=1/m$ and find that in general, the entanglement entropy and noise generated are unrelated quantities. We also discuss a generalization of this problem to the critical quantum Ising spin chain.
Characterizing universal entanglement features in higher-dimensional quantum matter is a central goal of quantum information science and condensed matter physics. While the subleading corner terms in two-dimensional quantum systems encapsulate essential universal information of the underlying conformal field theory, our understanding of these features remains remarkably limited compared to their one-dimensional counterparts. We address this challenge by investigating the entanglement dynamics of fermionic systems along the imaginary-time evolution. We uncover a pioneering non-equilibrium scaling law where the corner entanglement entropy grows linearly with the logarithm of imaginary time, dictated solely by the universality class of the quantum critical point. Through unbiased Quantum Monte Carlo simulations, we verify this scaling in the interacting Gross-Neveu-Yukawa model, demonstrating that universal data can be accurately recovered from the early stages of relaxation. Our findings significantly circumvent the computational bottlenecks inherent in reaching full equilibrium convergence. This work establishes a direct link between the fundamental theory of non-equilibrium critical phenomena and the high-precision determination of universal entanglement properties on both classical and quantum platforms, paving the way for probing the rich entanglement structure of quantum critical systems.
The entanglement entropy in one-dimensional critical systems with boundaries has been associated with the noninteger ground-state degeneracy. This quantity, being a characteristic of boundary fixed points, decreases under renormalization group flow, as predicted by the g theorem. Here, using conformal field theory methods, we exactly calculate the entanglement entropy in the boundary Ising universality class. Our expression can be separated into the well-known bulk term and a boundary entanglement term, displaying a universal flow between two boundary conditions, in accordance with the g theorem. These results are obtained within the replica trick approach, where we show that the associated twist field, a central object generating the geometry of an nnn-sheeted Riemann surface, can be bosonized, giving simple analytic access to multiple quantities of interest. We argue that our result applies to other models falling into the same universality class. This includes the vicinity of the quantum critical point of the two-channel Kondo model, allowing one to track in real space the presence of a region containing one-half of a qubit with entropy 12log(2)\frac{1}{2}log(2)21​log(2), associated with a free local Majorana fermion.