Critical Scaling Behaviors of Entanglement Spectra^*

Abstract

We investigate the evolution of entanglement spectra under a global quantum quench from a short-range correlated state to the quantum critical point. Motivated by the conformal mapping, we find that the dynamical entanglement spectra demonstrates distinct finite-size scaling behaviors from the static case. As a prototypical example, we compute real-time dynamics of the entanglement spectra of a one-dimensional transverse-field Ising chain. Numerical simulation confirms that, the entanglement spectra scales with the subsystem size l as ∼ l −1 for the dynamical equilibrium state, much faster than ∝ log −1 l for the critical ground state. In particular, as a byproduct, the entanglement spectra at the long time limit faithfully gives universal tower structure of underlying Ising criticality, which shows the emergence of operator-state correspondence in the quantum dynamics.PACS numbers: 03.65. Ud, 11.25.Hf Conformal field theory (CFT) [1] has become a profitable tool as a diagnosis of critical phenomena in two dimensional statistical models. In the equilibrium case, the conformal invariance at the critical point sets rigid constrains on physical properties by a set of conformal data including the central charge, conformal dimensions and operator product expansion coefficients. In the past decades, great success has been achieved in condensed matter physics, especially for minimal models with a finite number of primary scaling operators (irreducible representations of the Virasoro algebra) [2][3][4][5][6].In general, due to gaplessness nature massive entanglement should play a vital role at or around the critical point. One remarkable achievement is [7-34], CFT provides a novel way to connect the quantum entanglement and critical phenomena. It is found that the conformal invariance in critical ground states results in a universal scaling of the entanglement entropy depending on the central charge c [8-10, 13, 16]. Interestingly, by extending this idea, the entropy can be applied to identify quantum critical points in higher dimensions [13,16,[35][36][37]. Besides the entropy, other entanglement-based measures also attract lots of attention. The eigenvalues of reduced density matrix, called entanglement spectrum (ES), is such an example, which contains much richer information than the entropy [38,39]. In addition to the evidences in topological gapped systems [40][41][42][43], the ES is also proposed to describe the quantum critical point [15,[44][45][46][47][48][49][50][51]. However, compared to the well-established boundary law for gapped states, much less is known about the critical behavior of the ES [52,53], which casts doubt on direct application of the ES in the critical systems.Beyond equilibrium, quantum dynamics attracts considerable attention recently, particular in approaching to steadiness and thermalization. Universal entanglement structures are expected to leave some marks in the dynamic process, e.g., central charge c controls the growth of entropy [11,17]. However, novel example [8] is still rare, a...

Thermal correction to entanglement spectrum for conformal field theories

“…In this context, it is interesting to consider whether the ES -as a more precise measure of quantum entanglement than the EE -can be determinated by using the conformal symmetry. In (1 + 1)D critical systems, it is found that the ES can be indeed solved through CFT techniques [24], and the conformal data can be resolved by the ES [25]. Moreover, at the continuum limit, the gap of entanglement spectrum closes and the continuous distribution of the ES is found to be dependent only on the central charge [34].…”

“…In particular, the study of quantum entanglement brings novel insights into understanding critical phenomena in many-body systems. Especially for (1+1)-dimensional [(1+1)D] critical systems in equilibration [11][12][13][14][15][16] and out-of-equilibrium [17][18][19][20][21][22][23][24][25], the intrinsic conformal invariance leads to rigid constrains on physical proprieties, including the structure of quantum entanglement. One of the most important measures of quantum entanglement is the entanglement entropy (EE), defined as the von Neumann entropy of the reduced density matrix of the subsystem ρ A :…”

Quenches to the critical point of the three-state Potts model: Matrix product state simulations and conformal field theory

“…The hope is substantiated by the previous results obtained simulating the long-time behavior of quantum systems out-of-equilibrium [13][14][15] and comparing them with the CFT predictions. In previous works [13,[15][16][17], we have identified the low-energy part of the entanglement Hamiltonian as the relevant quantity to focus on in this context. We have shown how the CFT predictions are recovered by focusing on the low-energy part of the entanglement Hamiltonian.…”

Coexistence of Charge Order and Antiferromagnetic Order in an Extended Periodic Anderson Model

“…Moreover, the critical behaviors of the entangled spectra is worth studying, as it would provide significant information on the quantum phase transitions. [31]…”