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arXiv:2203.10272 · Physical Review A

Scaling of the finite-size effect of the α -Rényi entropy in disjointed intervals under dilation

Long Xiong, Shunyao Zhang, Guang‐Can Guo, Ming Gong

Published Apr 21, 2022• 2 citations
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Abstract

The $\ensuremath{\alpha}$-R\'enyi entropy in the gapless models has been obtained by the conformal field theory, which is exact in the thermodynamic limit. However, the calculation of its finite-size effect (FSE) is challenging. So far only the FSE in a single interval in the $XX$ model has been understood and the FSE in the other models and in the other conditions is totally unknown. Here we report the FSE of this entropy in disjointed intervals $A={\ensuremath{\cup}}_{i}{A}_{i}$ under a uniform dilation $\ensuremath{\lambda}A$ in the $XY$ model, showing a universal scaling law as ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\lambda}A}^{\ensuremath{\alpha}}={\mathrm{\ensuremath{\Delta}}}_{A}^{\ensuremath{\alpha}}{\ensuremath{\lambda}}^{\ensuremath{-}\ensuremath{\eta}}\mathcal{B}(A,\ensuremath{\lambda})$, where $|\mathcal{B}(A,\ensuremath{\lambda})|\ensuremath{\le}1$ is a bounded function and $\ensuremath{\eta}=\text{min}(2,2/\ensuremath{\alpha})$ when $\ensuremath{\alpha}<10$. We verify this relation in the phase boundaries of the $XY$ model, in which the different central charges correspond to the physics of free fermion and free boson models. We find that in the disjointed intervals two FSEs, termed as extrinsic FSE and intrinsic FSE, are required to fully account for the FSE of the entropy. Physically, we find that only the edge modes of the correlation matrix localized at the open ends $\ensuremath{\partial}A$ have contribution to the total entropy and its FSE. Our results provide some incisive insight into the entanglement entropy in the many-body systems.

DOI
10.1103/physreva.105.042429

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The \ensuremathα\ensuremath{\alpha}\ensuremathα-R'enyi entropy in the gapless models has been obtained by the conformal field theory, which is exact in the thermodynamic limit. However, the calculation of its finite-size effect (FSE) is challenging. So far only the FSE in a single interval in the XXXXXX model has been understood and the FSE in the other models and in the other conditions is totally unknown. Here we report the FSE of this entropy in disjointed intervals A=\ensuremath∪iAiA={\ensuremath{\cup}}_{i}{A}_{i}A=\ensuremath∪i​Ai​ under a uniform dilation \ensuremathλA\ensuremath{\lambda}A\ensuremathλA in the XYXYXY model, showing a universal scaling law as \ensuremathΔ\ensuremathλA\ensuremathα=\ensuremathΔA\ensuremathα\ensuremathλ\ensuremath−\ensuremathηB(A,\ensuremathλ){\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\lambda}A}^{\ensuremath{\alpha}}={\mathrm{\ensuremath{\Delta}}}_{A}^{\ensuremath{\alpha}}{\ensuremath{\lambda}}^{\ensuremath{-}\ensuremath{\eta}}\mathcal{B}(A,\ensuremath{\lambda})\ensuremathΔ\ensuremathλA\ensuremathα​=\ensuremathΔA\ensuremathα​\ensuremathλ\ensuremath−\ensuremathηB(A,\ensuremathλ), where ∣B(A,\ensuremathλ)∣\ensuremath≤1|\mathcal{B}(A,\ensuremath{\lambda})|\ensuremath{\le}1∣B(A,\ensuremathλ)∣\ensuremath≤1 is a bounded function and \ensuremathη=min(2,2/\ensuremathα)\ensuremath{\eta}=\text{min}(2,2/\ensuremath{\alpha})\ensuremathη=min(2,2/\ensuremathα) when \ensuremathα<10\ensuremath{\alpha}<10\ensuremathα<10. We verify this relation in the phase boundaries of the XYXYXY model, in which the different central charges correspond to the physics of free fermion and free boson models. We find that in the disjointed intervals two FSEs, termed as extrinsic FSE and intrinsic FSE, are required to fully account for the FSE of the entropy. Physically, we find that only the edge modes of the correlation matrix localized at the open ends \ensuremath∂A\ensuremath{\partial}A\ensuremath∂A have contribution to the total entropy and its FSE. Our results provide some incisive insight into the entanglement entropy in the many-body systems.
We study the scaling of the Renyi entanglement entropy of two disjoint blocks of critical lattice models described by conformal field theories with central charge c=1. We provide the analytic conformal field theory result for the second order Renyi entropy for a free boson compactified on an orbifold describing the scaling limit of the Ashkin-Teller (AT) model on the self-dual line. We have checked this prediction in cluster Monte Carlo simulations of the classical two dimensional AT model. We have also performed extensive numerical simulations of the anisotropic Heisenberg quantum spin-chain with tree-tensor network techniques that allowed to obtain the reduced density matrices of disjoint blocks of the spin-chain and to check the correctness of the predictions for Renyi and entanglement entropies from conformal field theory. In order to match these predictions, we have extrapolated the numerical results by properly taking into account the corrections induced by the finite length of the blocks to the leading scaling behavior.
The subsystem-size dependence of ground-state entanglement entropy and its crossover to thermal entropy as a function of temperature are well understood for one-dimensional (1D) gapless systems described by conformal field theory (CFT), and for free fermions with a Fermi surface in any dimension. However, little is known about the entanglement entropy for gapless fermionic systems without quasi-particles, such as a strange metal. Here we study the entanglement entropy of fermions in a solvable large-NNN 1D lattice model akin to the Yukawa-Sachdev-Ye-Kitaev (Yukawa SYK) model. In this model, two Fermi points are coupled to scalar bosons via spatially random Yukawa interactions, providing a solvable model of a strange metal when the bosons become critical at a quantum critical point. We exactly compute the second Rényi entropy of fermions in a spatial subregion in this model. Our results unravel crucial role of intra-subregion entanglement between fermionic and bosonic degrees of freedom along with the inter-subregion entanglement in understanding the ground states of such strongly coupled fermion-boson systems. We show that the crossover from thermal entropy to entanglement entropy, is captured by a single scaling ansatz, that collapses the second Rényi entropy of fermions for different subregion sizes and temperatures into a single universal curve. We further show that the universal scaling curve for the critical strange metal is well described by the standard CFT formula, albeit with an effective central charge substantially larger than the non-interacting value.
Entanglement entropy in even-dimensional conformal field theories (CFTs) contains well-known universal terms arising from the conformal anomaly. R\'enyi entropies are natural generalizations of the entanglement entropy that are much less understood. Above two spacetime dimensions, the universal terms in the R\'enyi entropies are unknown for general entangling geometries. We conjecture a new structure in the dependence of the four-dimensional R\'enyi entropies on the intrinsic and extrinsic geometry of the entangling surface. We provide evidence for this conjecture by direct numerical computations in the free scalar and fermion field theories. The computation involves relating the four-dimensional free massless R\'enyi entropies across cylindrical entangling surfaces to corresponding three-dimensional massive R\'enyi entropies across circular entangling surfaces. Our numerical technique also allows us to directly probe other interesting aspects of three-dimensional R\'enyi entropy, including the massless renormalized R\'enyi entropy and calculable contributions to the perimeter law.
We study the Rényi entanglement entropies of two disjoint intervals in XY chains. We exploit the exact solution of the model in terms of free Majorana fermions and we show how to construct the reduced density matrix in the spin variables by taking the Jordan–Wigner string between the two blocks properly into account. From this we can evaluate any Rényi entropy of finite integer order. We study in detail critical XX and Ising chains and we show that the asymptotic results for large blocks agree with recent conformal field theory predictions if corrections to the scaling are included in the analysis correctly. We also report results for the gapped phase and after a quantum quench.
We study the Rényi entropies of N disjoint intervals in the conformal field theories given by the free compactified boson and the Ising model. They are computed as the 2N point function of twist fields, by employing the partition function of the model on a particular class of Riemann surfaces. The results are written in terms of Riemann theta functions. The prediction for the free boson in the decompactification regime is checked against exact results for the harmonic chain. For the Ising model, matrix product states computations agree with the conformal field theory result once the finite size corrections have been taken into account.
Using the density matrix renormalization group, we calculated the finite-size corrections of the entanglement $\ensuremath{\alpha}$-R\'enyi entropy of a single interval for several critical quantum chains. We considered models with U(1)U(1)U(1) symmetry such as the spin-1/2 XXZXXZXXZ and spin-1 Fateev-Zamolodchikov models, as well as models with discrete symmetries such as the Ising, the Blume-Capel, and the three-state Potts models. These corrections contain physically relevant information. Their amplitudes, which depend on the value of $\ensuremath{\alpha}$, are related to the dimensions of operators in the conformal field theory governing the long-distance correlations of the critical quantum chains. The obtained results together with earlier exact and numerical ones allow us to formulate some general conjectures about the operator responsible for the leading finite-size correction of the $\ensuremath{\alpha}$-R\'enyi entropies. We conjecture that the exponent of the leading finite-size correction of the $\ensuremath{\alpha}$-R\'enyi entropies is ${p}_{\ensuremath{\alpha}}=2{X}_{\ensuremath{\epsilon}}/\ensuremath{\alpha}$ for $\ensuremath{\alpha}>1$ and ${p}_{1}=\ensuremath{\nu}$, where ${X}_{\ensuremath{\epsilon}}$ denotes the dimensions of the energy operator of the model and $\ensuremath{\nu}=2$ for all the models.