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arXiv:1611.01847 · Journal of Statistical Mechanics Theory and Experiment

Two-cylinder entanglement entropy under a twist

Xiao Chen, William Witczak‐Krempa, Thomas Faulkner, Eduardo Fradkin

Published Apr 1, 2017• 33 citations
arXivHTMLPDF

Abstract

Abstract We study the von Neumann and Rényi entanglement entropy (EE) of the scale-invariant theories defined on the tori in 2 + 1 and 3 + 1 spacetime dimensions. We focus on the spatial bi-partitions of the torus into two cylinders, and allow for twisted boundary conditions along the non-contractible cycles. Various analytical and numerical results are obtained for the universal EE of the relativistic boson and Dirac fermion conformal field theories (CFTs), the fermionic quadratic band touching and the boson with z = 2 Lifshitz scaling. The shape dependence of the EE clearly distinguishes these theories, although intriguing similarities are found in certain limits. We also study the evolution of the EE when a mass is introduced to detune the system from its scale-invariant point, by employing a renormalized EE that goes beyond a naive subtraction of the area law. In certain cases we find the non-monotonic behavior of the torus EE under RG flow, which distinguishes it from the EE of a disk.

DOI
10.1088/1742-5468/aa668a

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Abstract We study the von Neumann and Rényi entanglement entropy (EE) of the scale-invariant theories defined on the tori in 2 + 1 and 3 + 1 spacetime dimensions. We focus on the spatial bi-partitions of the torus into two cylinders, and allow for twisted boundary conditions along the non-contractible cycles. Various analytical and numerical results are obtained for the universal EE of the relativistic boson and Dirac fermion conformal field theories (CFTs), the fermionic quadratic band touching and the boson with z = 2 Lifshitz scaling. The shape dependence of the EE clearly distinguishes these theories, although intriguing similarities are found in certain limits. We also study the evolution of the EE when a mass is introduced to detune the system from its scale-invariant point, by employing a renormalized EE that goes beyond a naive subtraction of the area law. In certain cases we find the non-monotonic behavior of the torus EE under RG flow, which distinguishes it from the EE of a disk.
We study the universal scaling behavior of the entanglement entropy of critical theories in 2 + 1 dimensions. We specially consider two fermionic scale-invariant models, free massless Dirac fermions and a model of fermions with quadratic band touching, and numerically study the two-cylinder entanglement entropy of the models on the torus. We find that in both cases the entanglement entropy satisfies the area law and has the subleading term which is a scaling function of the aspect ratios of the cylindrical regions. We test the scaling of entanglement in both the free fermion models using three possible scaling functions for the subleading term derived from (a) the quasi-1D conformal field theory, (b) the bosonic quantum Lifshitz model and (c) the holographic AdS/CFT correspondence. For the later case we construct an analytic scaling function using holography, appropriate for critical theories with a gravitational dual description. We find that the subleading term in the fermionic models is well described, for a range of aspect ratios, by the scaling form derived from the quantum Lifshitz model as well as that derived using the AdS/CFT correspondence (in this case only for the Dirac model). For the case where the fermionic models are placed on a square torus we find the fit to the different scaling forms is in agreement to surprisingly high precision.
We study the universal contributions to the entanglement entropy (EE) of 2+12+12+1-dimensional and 3+13+13+1-dimensional holographic conformal field theories (CFTs) on topologically nontrivial manifolds, focusing on tori. The holographic bulk corresponds to anti--de Sitter-soliton geometries. We characterize the properties of these regulator-independent EE terms as a function of both the size of the cylindrical entangling region, and the shape of the torus. In 2+12+12+1 dimensions, in the simple limit where the torus becomes a thin one-dimensional ring, the EE reduces to a shape-independent constant $2\ensuremath{\gamma}$. This is twice the EE obtained by bipartitioning an infinite cylinder into equal halves. We study the renormalization group flow of $\ensuremath{\gamma}$ by defining a renormalized EE that (1) is applicable to general QFTs, (2) resolves the failure of the area law subtraction, and (3) is inspired by the F-theorem. We find that the renormalized $\ensuremath{\gamma}$ decreases monotonically at small coupling when the holographic CFT is deformed by a relevant operator for all allowed scaling dimensions. We also discuss the question of nonuniqueness of such renormalized EEs both in 2+12+12+1 dimensions and 3+13+13+1 dimensions.
The entanglement entropy (EE) has emerged as an important window into the structure of complex quantum states of matter. We analyze the universal part of the EE for gapless systems on tori in 2D and 3D, denoted by χ. Focusing on scale-invariant systems, we derive general nonperturbative properties for the shape dependence of χ and reveal surprising relations to the EE associated with corners in the entangling surface. We obtain closed-form expressions for χ in 2D and 3D within a model that arises in the study of conformal field theories (CFTs), and we use them to obtain Ansätze without fitting parameters for the 2D and 3D free boson CFTs. Our numerical lattice calculations show that the Ansätze are highly accurate. Finally, we discuss how the torus EE can act as a fingerprint of exotic states such as gapless quantum spin liquids, e.g., Kitaev's honeycomb model.
Abstract In this paper we study the properties of two-dimensional CFTs defined by cyclic and symmetric orbifolds of free Dirac fermions, especially by focusing on the partition function and entanglement entropy. Via the bosonization, we construct the twist operators which glue two complex planes to calculate the partition function of ℤ2 orbifold CFT on a torus. We also find an expression of ℤN cyclic orbifold in terms of Hecke operators, which provides an explicit relation between the partition functions of cyclic orbifolds and those of symmetric ones. We compute the entanglement entropy and Renyi entropy in cyclic orbifolds on a circle both for finite temperature states and for time-dependent states under quantum quenches. We find that the replica method calculation is highly non-trivial and new because of the contributions from replicas with different boundary conditions. We find the full expression for the ℤ2 orbifold and show that the periodicity gets doubled. Finally, we discuss extensions of our results on entanglement entropy to symmetric orbifold CFTs and make a heuristic argument towards holographic CFTs.
Abstract We compute universal finite corrections to entanglement entropy for generalised quantum Lifshitz models in arbitrary odd spacetime dimensions. These are generalised free field theories with Lifshitz scaling symmetry, where the dynamical critical exponent z equals the number of spatial dimensions d, and which generalise the 2+1-dimensional quantum Lifshitz model to higher dimensions. We analyse two cases: one where the spatial manifold is a d-dimensional sphere and the entanglement entropy is evaluated for a hemisphere, and another where a d-dimensional flat torus is divided into two cylinders. In both examples the finite universal terms in the entanglement entropy are scale invariant and depend on the compactification radius of the scalar field.
We use holographic duality to study the entanglement entropy (EE) of Conformal Field Theories (CFTs) in various spacetime dimensions d, in the presence of various deformations: a relevant Lorentz scalar operator with constant source, a temperature T , a chemical potential μ, a marginal Lorentz scalar operator with source linear in a spatial coordinate, and a circle-compactified spatial direction. We consider EE between a strip or sphere sub-region and the rest of the system, and define the “entanglement density” (ED) as the change in EE due to the deformation, divided by the sub-region’s volume. Using the deformed CFTs above, we show how the ED’s dependence on the strip width or sphere radius, L, is useful for characterizing states of matter. For example, the ED’s small-L behavior is determined either by the dimension of the perturbing operator or by the first law of EE. For Lorentz-invariant renormalization group (RG) flows between CFTs, the “area theorem” states that the coefficient of the EE’s area law term must be larger in the UV than in the IR. In these cases the ED must therefore approach zero from below as L→∞. However, when Lorentz symmetry is broken and the IR fixed point has different scaling from the UV, we find that the ED often approaches the thermal entropy density from above, indicating area theorem violation.