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arXiv:1205.6426 · Physical Review B

Approaching the restricted solid-on-solid critical points through entanglement: One model for many universalities

Andrea De Luca, Fabio Franchini

Published Jan 22, 2013• 2 citations
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Abstract

We analytically compute the Renyi entropies for the RSOS models, representing a wide class of exactly solvable models with multicritical conformal points described by unitary minimal models and ${\mathbb{Z}}_{n}$ parafermions. The exact expressions allow for an explicit comparison of the expansions around the critical points with the predictions coming from field theory. In this way, it is possible to point out the nature of the so-called ``unusual corrections,'' clarifying the link with the operator content, the role of the symmetries and the boundary conditions. By choosing different boundary conditions, we can single out the ground states as well as certain combinations of high-energy states. We find that the entanglement spectrum is given by operators different from those responsible for the off-critical perturbation, although they belong to the same representation of a Virasoro algebra. In the parafermionic case, we observe unexpected logarithmic corrections.

DOI
10.1103/physrevb.87.045118

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We analytically compute the Renyi entropies for the RSOS models, representing a wide class of exactly solvable models with multicritical conformal points described by unitary minimal models and Zn{\mathbb{Z}}_{n}Zn​ parafermions. The exact expressions allow for an explicit comparison of the expansions around the critical points with the predictions coming from field theory. In this way, it is possible to point out the nature of the so-called ``unusual corrections,'' clarifying the link with the operator content, the role of the symmetries and the boundary conditions. By choosing different boundary conditions, we can single out the ground states as well as certain combinations of high-energy states. We find that the entanglement spectrum is given by operators different from those responsible for the off-critical perturbation, although they belong to the same representation of a Virasoro algebra. In the parafermionic case, we observe unexpected logarithmic corrections.
We analytically compute the Renyi entropies for the RSOS models, representing a wide class of exactly solvable models with multicritical conformal points described by unitary minimal models and Zn\mathbb{Z}_nZn​ parafermions. The exact expressions allow for an explicit comparison of the expansions around the critical points with the predictions coming from field theory. In this way it is possible to point out the nature of the so-called "unusual corrections", clarifying the link with the operator content, the role of the symmetries and the boundary conditions. By choosing different boundary conditions, we can single out the ground states as well as certain combinations of high energy states. We find that the {\it entanglement spectrum} is given by operators that are not present in the bulk Hamiltonian, although they belong to the same representation of a Virasoro Algebra. In the parafermionic case we observe unexpected logarithmic corrections.
Abstract Using a corner transfer matrix approach, we compute the bipartite entanglement Rényi entropy in the off-critical perturbations of non-unitary conformal minimal models realised by lattice spin chains Hamiltonians related to the Forrester–Baxter RSOS models (Bianchini et al 2015 J. Stat. Mech. P03010 ) in regime III. This allows to show on a set of explicit examples that the Rényi entropies for non-unitary theories rescale near criticality as the logarithm of the correlation length with a coefficient proportional to the effective central charge. This complements a similar result, recently established for the size rescaling at the critical point (Bianchini et al 2015 J. Phys. A: Math. Theor. 48 04FT01 ), showing the expected agreement of the two behaviours. We also compute the first subleading unusual correction to the scaling behaviour, showing that it is expressible in terms of expansions of various fractional powers of the correlation length, related to the differences between the conformal dimensions of fields in the theory and the minimal conformal dimension. Finally, a few observations on the limit leading to the off-critical logarithmic minimal models of Pearce and Seaton (2012 J. Stat. Mech. P09014 ) are put forward.
At a quantum critical point, bipartite entanglement entropies have universal quantities which are subleading to the ubiquitous area law. For Renyi entropies, these terms are known to be similar to the von Neumann entropy, while being much more amenable to numerical and even experimental measurement. We show here that when calculating universal properties of Renyi entropies, it is important to account for unusual corrections to scaling that arise from relevant local operators present at the conical singularity in the multisheeted Riemann surface. These corrections grow in importance with increasing Renyi index. We present studies of Renyi correlation functions in the 1+11+11+1 transverse-field Ising model (TFIM) using conformal field theory, mapping to free fermions, and series expansions, and the logarithmic entropy singularity at a corner in 2+12+12+1 for both free bosonic field theory and the TFIM, using numerical linked cluster expansions. In all numerical studies, accurate results are only obtained when unusual corrections to scaling are taken into account. In the worst case, an analysis ignoring these corrections can get qualitatively incorrect answers, such as predicting a decrease in critical exponents with the Renyi index, when they are actually increasing. We discuss a two-step extrapolation procedure that can be used to account for the unusual corrections to scaling.
We investigate the entanglement properties of the Quantum Six-Vertex Model on a cylinder, focusing on the Shannon-Renyi entropy in the limit of Renyi order n=∞n = \inftyn=∞. This entropy, calculated from the ground state amplitudes of the equivalent XXZ spin-1/2 chain, allows us to determine the Renyi entanglement entropy of the corresponding Rokhsar-Kivelson wavefunctions, which describe the ground states of certain conformal quantum critical points. Our analysis reveals a novel logarithmic correction to the expected entanglement scaling when the system size is odd. This anomaly arises from the geometric frustration of spin configurations imposed by periodic boundary conditions on odd-sized chains. We demonstrate that the scaling prefactor of this logarithmic term is directly related to the compactification radius of the low-energy bosonic field theory description, or equivalently, the Luttinger parameter. Thus, this correction provides a direct probe of the underlying Conformal Field Theory (CFT) describing the critical point. Our findings highlight the crucial role of system size parity in determining the entanglement properties of this model and offer insights into the interplay between geometry, frustration, and criticality.
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.
Two-dimensional conformal field theories with a large central charge and a small number of low-dimension operators are studied using the conformal block expansion. A universal formula is derived for the Renyi entropies of N disjoint intervals in the ground state, valid to all orders in a series expansion. This is possible because the full perturbative answer in this regime comes from the exchange of the stress tensor and other descendants of the vacuum state. Therefore, the Renyi entropy is related to the Virasoro vacuum block at large central charge. The entanglement entropy, computed from the Renyi entropy by an analytic continuation, decouples into a sum of single-interval entanglements. This field theory result agrees with the Ryu-Takayanagi formula for the holographic entanglement entropy of a 2d CFT, applied to any number of intervals, and thus can be interpreted as a microscopic calculation of the area of minimal surfaces in 3d gravity.