Detecting classical phase transitions with Renyi mutual information

Iaconis, Jason; Inglis, Stephen; Kallin, Ann B.; Melko, Roger G.

doi:10.1103/physrevb.87.195134

Cited by 56 publications

(71 citation statements)

References 40 publications

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“…Let us finally mention that universal scaling forms proportional to n/(n − 1) have been found in the 2d classical Rényi mutual information 24 for Ising, as well as in the Rényi entropy of the 2d quantum transverse field Ising model 27 . In both cases the underlying ordering assumption is easier to justify: for the former the critical system is coupled to a bulk in the ordered phase 42 , while for the latter the higher dimensionality makes an extraordinary transition more likely.…”

Section: Path-integral and Replicasmentioning

confidence: 99%

Shannon and Rényi mutual information in quantum critical spin chains

Stéphan

2014

Phys. Rev. B

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We study the Shannon mutual information in one-dimensional critical spin chains, following a recent conjecture (Phys. Rev. Lett. 111, 017201 (2013)[1]), as well as Rényi generalizations of it. We combine conformal field theory arguments with numerical computations in lattice discretizations with central charge c = 1 and c = 1/2. For a periodic system of length L cut into two parts of length ℓ and L − ℓ, all our results agree with the general shape-dependence In(ℓ,, where bn is a universal coefficient. For the free boson CFT we show from general arguments that bn = c = 1. At c = 1/2 we conjecture a result for n > 1. We perform extensive numerical computations in Ising chains to confirm this, and also find b1 ≃ 0.4801629(2), a nontrivial number which we do not understand analytically. Open chains at c = 1/2 and n = 1 are even more intriguing, with a shape-dependent logarithmic divergence of the Shannon mutual information.

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Section: Path-integral and Replicasmentioning

confidence: 99%

Shannon and Rényi mutual information in quantum critical spin chains

Stéphan

2014

Phys. Rev. B

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“…This is a striking example of the success of information-theoretic concepts applied to condensed-matter problems. Since a CFT also describes the large-distance limit of a two-dimensional classical critical model with rotational invariance, it is natural to expect that informationtheoretic concepts can be used to analyze classical critical systems [7][8][9][10]. The aim of this letter is to define and compute the geometric mutual information G n , a quantity quite analogous to the quantum result in eq.…”

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confidence: 99%

“…It can be used for example to detect phase transitions, and extract the critical temperature accurately [9]. Because the leading bulk contributions cancel, the RMI obeys a boundary law [9]:…”

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Geometric Mutual Information at Classical Critical Points

et al. 2014

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A practical use of the entanglement entropy in a 1d quantum system is to identify the conformal field theory describing its critical behavior. It is exactly (c/3) ln for an interval of length in an infinite system, where c is the central charge of the conformal field theory. Here we define the geometric mutual information, an analogous quantity for classical critical points. We compute this for 2d conformal field theories in an arbitrary geometry, and show in particular that for a rectangle cut into two rectangles, it is proportional to c. This makes it possible to extract c in classical simulations, which we demonstrate for the critical Ising and 3-state Potts models. Introduction.-In studies of new and exotic phases of quantum matter, the entanglement entropy has established itself as an important resource [1]. It is particularly useful in the many 1d quantum critical systems governed by a conformal field theory (CFT) [2] in the large-distance limit. Here the Rényi entanglement entropy S n of the ground-state is universal [3][4][5][6], and the leading piece is proportional to the central charge c of the CFT characterizing the universality class. Namely, for a periodic system of length L cut into two open segments of respective sizes L A and L B = (L − L A ),

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“…The mutual information peaks at temperatures above the Kosterlitz-Thouless transition of (T /J) KT = 0.343 (which can be detected by the crossing of the finite-size curves; see Ref. [38]). For T /J < (T /J) KT , the mutual information reaches a minimum (at J/T ≡ β ≈ 4 in Fig.…”

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Detecting Goldstone modes with entanglement entropy

et al. 2015

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In the face of mounting numerical evidence, Metlitski and Grover [arXiv:1112.5166] have given compelling analytical arguments that systems with spontaneous broken continuous symmetry contain a sub-leading contribution to the entanglement entropy that diverges logarithmically with system size. They predict that the coefficient of this log is a universal quantity that depends on the number of Goldstone modes. In this paper, we confirm the presence of this log term through quantum Monte Carlo calculations of the second Rényi entropy on the spin 1/2 XY model. Devising an algorithm to facilitate convergence of entropy data at extremely low temperatures, we demonstrate that the single Goldstone mode in the ground state can be identified through the coefficient of the log term. Furthermore, our simulation accuracy allows us to obtain an additional geometric constant additive to the Rényi entropy, that matches a predicted fully-universal form obtained from a free bosonic field theory with no adjustable parameters.Introduction -In condensed matter, the entanglement entropy of a bipartition contains an incredible amount of information about the correlations in a system. In spatial dimensions d ≥ 2, quantum spins or bosons display an entanglement entropy that, to leading order, scales as the boundary of the bipartition [1][2][3]. Subleading to this "area-law" are various constants and -particularly in gapless phases -functions that depend non-trivially on length and energy scales. Some of these subleading terms are known to act as informatic "order parameters" which can detect non-trivial correlations, such as the topological entanglement entropy in a gapped spin liquid phase [4][5][6][7]. At a quantum critical point, subleading terms contain novel quantities that identify the universality class, and potentially can provide constraints on renormalization group flows to other nearby fixed points [8][9][10][11][12][13][14].In systems with a continuous broken symmetry, evidence is mounting that the entanglement entropy between two subsystems with a smooth spatial bipartition contains a term, subleading to the area law, that diverges logarithmically with the subsystem size. First observed in spin wave [15] and finite-size lattice numerics [16], the apparently anomalous logarithm had no rigorous explanation until 2011, when Metlitski and Grover developed a comprehensive theory [17]. They argued that, for a finite-size subsystem with length scale L, the term is a manifestation of the two long-wavelength energy scales corresponding to the spin wave gap, and the "tower of states" arising from the restoration of symmetry in a finite volume [18][19][20][21]. Remarkably, their theory not only explains the subleading logarithm, but predicts that the * bkulchyt@uwaterloo.ca FIG. 1. Schematic energy level structure of the low energy tower of states for finite-size systems with spontaneous breaking of a continuous symmetry. The correction to the entanglement entropy may be approximated by the log of the number of quantum rotor states bel...

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Detecting classical phase transitions with Renyi mutual information

Cited by 56 publications

References 40 publications

Shannon and Rényi mutual information in quantum critical spin chains

Shannon and Rényi mutual information in quantum critical spin chains

Geometric Mutual Information at Classical Critical Points

Detecting Goldstone modes with entanglement entropy

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