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arXiv:1605.00444 · Physics Letters B

Entanglement entropy from the truncated conformal space

Tamás Pálmai

Published Jun 11, 2016• 19 citations
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Abstract

A new numerical approach to entanglement entropies of the Rényi type is proposed for one-dimensional quantum field theories. The method extends the truncated conformal spectrum approach and we will demonstrate that it is especially suited to study the crossover from massless to massive behavior when the subsystem size is comparable to the correlation length. We apply it to different deformations of massless free fermions, corresponding to the scaling limit of the Ising model in transverse and longitudinal fields. For massive free fermions the exactly known crossover function is reproduced already in very small system sizes. The new method treats ground states and excited states on the same footing, and the applicability for excited states is illustrated by reproducing Rényi entropies of low-lying states in the transverse field Ising model.

DOI
10.1016/j.physletb.2016.06.012

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A new numerical approach to entanglement entropies of the Rényi type is proposed for one-dimensional quantum field theories. The method extends the truncated conformal spectrum approach and we will demonstrate that it is especially suited to study the crossover from massless to massive behavior when the subsystem size is comparable to the correlation length. We apply it to different deformations of massless free fermions, corresponding to the scaling limit of the Ising model in transverse and longitudinal fields. For massive free fermions the exactly known crossover function is reproduced already in very small system sizes. The new method treats ground states and excited states on the same footing, and the applicability for excited states is illustrated by reproducing Rényi entropies of low-lying states in the transverse field Ising model.
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.
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.
We compute the Renyi entropy in a one-dimensional transverse-field quantum Ising model by employing a swapping operator acting on the states which are prepared from the neural network methods. In the static ground state, Renyi entropy can uncover the critical point of the quantum phase transition from paramagnetic to ferromagnetic. At the critical point, the relation between the Renyi entropy and the subsystem size satisfies the predictions from conformal field theory. In the dynamical case, we find coherent oscillations of the Renyi entropy after the end of the linear quench. These oscillations have universal frequencies which may come from the superpositions of excited states. The asymptotic form of the Renyi entropy implies a new length scale away from the critical point. This length scale is also verified by the overlap of the reduced Renyi entropy against the dimensionless subsystem size.
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In this paper we study the scaling of the correction of the Renyi entropy of the excited states in systems described, in the continuum limit, by a conformal field theory (CFT). These corrections scale as L−2ΔnL^{-\frac{2Δ}{n}}L−n2Δ​, where LLL is the system size and ΔΔΔ is the scaling dimension of a relevant bulk operator located around the singularities of the Riemann surface Rn\mathcal{R}_nRn​. Their name is due to their explicit dependence on the Riemann surface Rn\mathcal{R}_nRn​. Their presence has been detected in several works on the entanglement entropy in finite size systems, both in the ground and the excited states. Here, we present a general study of these corrections based on the perturbation expansion on Rn\mathcal{R}_nRn​. Some of the terms in this expansion are divergent and they will be cured with addition cut-offs. These cut-offs will determine how these corrections scale with the system size LLL. Exact numerical computations of the Renyi entropy of the excited states of the XX model are provided and they confirm our theoretical prediction on the scaling of corrections. They allow also a comparison with the other works present in the literature finding that the corrections, for the excited states, have the exact same form of the ones of the ground state case multiplied by a model dependend function of nnn and l/Ll/Ll/L.
Here we show that the Rényi entanglement entropy of a region of large size ℓ in a one-dimensional critical model whose ground state breaks conformal invariance (such as in those described by non-unitary conformal field theories), behaves as , where is the effective central charge, c (which may be negative) is the central charge of the conformal field theory and is the lowest holomorphic conformal dimension in the theory. We also obtain results for models with boundaries, and with a large but finite correlation length, and we show that if the lowest conformal eigenspace is logarithmic ( with N nilpotent), then there is an additional term proportional to . These results generalize the well known expressions for unitary models. We provide a general proof, and report on numerical evidence for a non-unitary spin chain and an analytical computation using the corner transfer matrix method for a non-unitary lattice model. We use a new algebraic technique for studying the branching that arises within the replica approach, and find a new expression for the entanglement entropy in terms of correlation functions of twist fields for non-unitary models.