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arXiv:1006.1605 · Physical Review B

Rényi entropy of a line in two-dimensional Ising models

Jean-Marie Stéphan, Grégoire Misguich, Vincent Pasquier

Published Sep 30, 2010• 86 citations
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Abstract

We consider the two-dimensional Ising model on an infinitely long cylinder and study the probabilities ${p}_{i}$ to observe a given spin configuration $i$ along a circular section of the cylinder. These probabilities also occur as eigenvalues of reduced density matrices in some Rokhsar-Kivelson wave functions. We analyze the subleading constant to the R\'enyi entropy ${R}_{n}=1/(1\ensuremath{-}n)\text{ln}({\ensuremath{\sum}}_{i}{p}_{i}^{n})$ and discuss its scaling properties at the critical point. Studying three different microscopic realizations, we provide numerical evidence that it is universal and behaves in a steplike fashion as a function of $n$ with a discontinuity at the Shannon point $n=1$. As a consequence, a field theoretical argument based on the replica trick would fail to give the correct value at this point. We nevertheless compute it numerically with high precision. Two other values of the R\'enyi parameter are of special interest: $n=1/2$ and $n=\ensuremath{\infty}$ are related in a simple way to the Affleck-Ludwig boundary entropies associated to free and fixed boundary conditions, respectively.

DOI
10.1103/physrevb.82.125455

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We consider the two-dimensional Ising model on an infinitely long cylinder and study the probabilities pi{p}_{i}pi​ to observe a given spin configuration iii along a circular section of the cylinder. These probabilities also occur as eigenvalues of reduced density matrices in some Rokhsar-Kivelson wave functions. We analyze the subleading constant to the R'enyi entropy Rn=1/(1\ensuremath−n)ln(\ensuremath∑ipin){R}_{n}=1/(1\ensuremath{-}n)\text{ln}({\ensuremath{\sum}}_{i}{p}_{i}^{n})Rn​=1/(1\ensuremath−n)ln(\ensuremath∑i​pin​) and discuss its scaling properties at the critical point. Studying three different microscopic realizations, we provide numerical evidence that it is universal and behaves in a steplike fashion as a function of nnn with a discontinuity at the Shannon point n=1n=1n=1. As a consequence, a field theoretical argument based on the replica trick would fail to give the correct value at this point. We nevertheless compute it numerically with high precision. Two other values of the R'enyi parameter are of special interest: n=1/2n=1/2n=1/2 and n=\ensuremath∞n=\ensuremath{\infty}n=\ensuremath∞ are related in a simple way to the Affleck-Ludwig boundary entropies associated to free and fixed boundary conditions, respectively.
The R\'enyi-Shannon entropy allows extraction of some universal information about many-body wave functions. For a critical spin chain with central charge c=1c=1c=1, we show that it exhibits a phase transition at some value nc{n}_{c}nc​ of the R\'enyi parameter nnn which depends on the Luttinger parameter RRR. A replica-free formulation establishes a connection to boundary entropies in conformal field theory and reveals that the transition is triggered by a vertex operator which becomes relevant at the boundary. Our numerical results (XXZ and ${J}_{1}\ensuremath{-}{J}_{2}$ spin chains) match the continuum limit prediction, confirming its universal character. The replica approach used in previous works turns out to be correct only for n<ncn<{n}_{c}n<nc​. From the point of view of two-dimensional Rokhsar-Kivelson states, this transition reveals a singularity in the entanglement spectra.
Thanks to Pfaffian techniques, we study the Rényi entanglement entropies and the entanglement spectrum of large subsystems for two-dimensional Rokhsar–Kivelson wavefunctions constructed from a dimer model on the triangular lattice. By including a fugacity t on some suitable bonds, one interpolates between the triangular lattice ( t = 1) and the square lattice ( t = 0). The wavefunction is known to be a massive topological liquid for t > 0 whereas it is a gapless critical state at t = 0. We mainly consider two geometries for the subsystem: that of a semi-infinite cylinder and the disc-like set-up proposed by Kitaev and Preskill (2006 Phys. Rev. Lett. 96 110404). In the cylinder case, the entropies contain an extensive term—proportional to the length of the boundary—and a universal subleading constant s n ( t ). Fitting these cylinder data (up to a perimeter of L = 32 sites) provides s n with a very high numerical accuracy (10 − 9 at t = 1 and 10 − 6 at t = 0.5). In the topological liquid phase we find s n ( t > 0) = − ln2, independent of the fugacity t and the Rényi parameter n . At t = 0 we recover a previously known result, for n < 1 and s n ( t = 0) = − ln(2)/( n − 1) for n > 1. In the disc-like geometry—designed to get rid of the boundary contributions—we find an entropy s n KP ( t 0) = − ln2 in the whole massive phase whatever n > 0, in agreement with the result of Flammia et al (2009 Phys. Rev. Lett. 103 261601). Some results for the gapless limit are discussed.
We study the Shannon entropy of the probability distribution resulting from the ground-state wave function of a one-dimensional quantum model. This entropy is related to the entanglement entropy of a Rokhsar-Kivelson-type wave function built from the corresponding two-dimensional classical model. In both critical and massive cases, we observe that it is composed of an extensive part proportional to the length of the system and a subleading universal constant S0{S}_{0}S0​. In c=1c=1c=1 critical systems (Tomonaga-Luttinger liquids), we find that S0{S}_{0}S0​ is a simple function of the boson compactification radius. This finding is based on a field-theoretical analysis of the Dyson-Gaudin gas related to dimer and Calogero-Sutherland models. We also performed numerical demonstrations in the dimer models and the spin-1/2 XXZXXZXXZ chain. In a massive (crystal) phase, S0{S}_{0}S0​ is related to the ground-state degeneracy. We also examine this entropy in the Ising chain in a transverse field as an example showing a c=1/2c=1/2c=1/2 critical point.
It is generally believed that in spatial dimension d>1d>1d>1, the leading contribution to the entanglement entropy $S=\ensuremath{-}\text{tr}\text{ }{\ensuremath{\rho}}_{A}\text{ }\text{log}\text{ }{\ensuremath{\rho}}_{A}$ scales as the area of the boundary of subsystem AAA. The coefficient of this ``area law'' is nonuniversal. However, in the neighborhood of a quantum critical point SSS is believed to possess subleading universal corrections. In the present work, we study the entanglement entropy in the quantum O(N)O(N)O(N) model in 1<d<31<d<31<d<3. We use an expansion in $ϵ=3\ensuremath{-}d$ to evaluate (i) the universal geometric correction to SSS for an infinite cylinder divided along a circular boundary; (ii) the universal correction to SSS due to a finite correlation length. Both corrections are different at the Wilson-Fisher and Gaussian fixed points, and the $ϵ\ensuremath{\rightarrow}0$ limit of the Wilson-Fisher fixed point is distinct from the Gaussian fixed point. In addition, we compute the correlation length correction to the Renyi entropy ${S}_{n}=\frac{1}{1\ensuremath{-}n}\text{log}\text{ }\text{tr}\text{ }{\ensuremath{\rho}}_{A}^{n}$ in ϵϵϵ and large-NNN expansions. For $N\ensuremath{\rightarrow}\ensuremath{\infty}$, this correction generally scales as N2{N}^{2}N2 rather than the naively expected NNN. Moreover, the Renyi entropy has a phase transition as a function of nnn for ddd close to 3.
We study the scaling of the (basis dependent) Shannon entropy for two-dimensional quantum antiferromagnets with N\'eel long-range order. We use a massless free-field description of the gapless spin wave modes and phase space arguments to treat the fact that the finite-size ground state is rotationally symmetric, while there are degenerate physical ground states which break the symmetry. Our results show that the Shannon entropy (and its R\'enyi generalizations) possesses some universal logarithmic term proportional to the number NNGN_\text{NG}NNG​ of Nambu-Goldstone modes. In the case of a torus, we show that Sn>1≃O(N)+NNG4nn−1log⁡NS_{n>1} \simeq \mathcal{O}(N)+ \frac{N_\text{NG}}{4}\frac{n}{n-1} \log{N}Sn>1​≃O(N)+4NNG​​n−1n​logN and S1≃O(N)−NNG4log⁡NS_1 \simeq \mathcal{O}(N) - \frac{N_\text{NG}}{4} \log{N}S1​≃O(N)−4NNG​​logN, where NNN is the total number of sites and nnn the R\'enyi index. The result for n>1n>1n>1 is in reasonable agreement with the quantum Monte Carlo results of Luitz et al. [Phys. Rev. Lett. 112, 057203 (2014)], and qualitatively similar to those obtained previously for the entanglement entropy. The Shannon entropy of a line subsystem (embedded in the two-dimensional system) is also considered. Finally, we present some density-matrix renormalization group (DMRG) calculations for a spin12\frac{1}{2}21​ XY model on the square lattice in a cylinder geometry. These numerical data confirm our findings for logarithmic terms in the n=∞n=\inftyn=∞ R\'enyi entropy (also called −log⁡pmax-\log{p_{\rm max}}−logpmax​). They also reveal some universal dependence on the cylinder aspect ratio, in good agreement with the fact that, in that case, pmaxp_{\rm max}pmax​ is related to a non-compact free-boson partition function in dimension 1+1.
We study the scaling of the (basis-dependent) Shannon entropy for two-dimensional quantum antiferromagnets with N\'eel long-range order. We use a massless free-field description of the gapless spin wave modes and phase space arguments to treat the fact that the finite-size ground state is rotationally symmetric, while there are degenerate physical ground states which break the symmetry. Our results show that the Shannon entropy (and its R\'enyi generalizations) possesses some universal logarithmic term proportional to the number NNG{N}_{\text{NG}}NNG​ of Nambu-Goldstone modes. In the case of a torus, we show that ${S}_{n>1}\ensuremath{\simeq}\mathrm{const}.\ifmmode\times\else\texttimes\fi{}N+\frac{{N}_{\text{NG}}}{4}\frac{n}{n\ensuremath{-}1}lnN$ and ${S}_{1}\ensuremath{\simeq}\mathrm{const}.\ifmmode\times\else\texttimes\fi{}N\ensuremath{-}\frac{{N}_{\text{NG}}}{4}lnN$, where NNN is the total number of sites and nnn the R\'enyi index. The result for n>1n>1n>1 is in reasonable agreement with the quantum Monte Carlo results of Luitz et al. [Phys. Rev. Lett. 112, 057203 (2014)], and qualitatively similar to those obtained previously for the entanglement entropy. The Shannon entropy of a line subsystem (embedded in the two-dimensional system) is also considered. Finally, we present some density-matrix renormalization group (DMRG) calculations for a spin-12\frac{1}{2}21​ XY model on the square lattice in a cylinder geometry. These numerical data confirm our findings for logarithmic terms in the $n=\ensuremath{\infty}$ R\'enyi entropy (also called $\ensuremath{-}ln{p}_{\mathrm{max}})$. They also reveal some universal dependence on the cylinder aspect ratio, in good agreement with the fact that, in that case, pmax{p}_{\mathrm{max}}pmax​ is related to a noncompact free-boson partition function in dimension 1+1.