Quantum Monte Carlo calculation of entanglement Rényi entropies for generic quantum systems

Humeniuk, Stephan; Roscilde, Tommaso

doi:10.1103/physrevb.86.235116

Cited by 113 publications

(184 citation statements)

References 24 publications

Supporting

Mentioning

174

Contrasting

Order By: Relevance

“…Such terms have been calculated previously using methods known as "ratio tricks" [23,24]. This is typically done by generating a valid state from partition function Z[A i , n, β] and looking at the weight of the same configuration as a state from the partition function Z[A i+1 , n, β].…”

Section: (C)mentioning

confidence: 99%

Geometric Mutual Information at Classical Critical Points

et al. 2014

View full text Add to dashboard Cite

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 ),

show abstract

Section: (C)mentioning

confidence: 99%

Geometric Mutual Information at Classical Critical Points

et al. 2014

View full text Add to dashboard Cite

show abstract

“…By employing an extended-ensemble generalization of a ratio method [22][23][24], we are able to carefully converge the mutual information of the second Rényi entropy to its low-temperature value. There, finite-size scaling reveals the coefficient of the subleading logarithmic term to precisely match the prediction of Metlitski and Grover, identifying the lone Goldstone mode in the theory.…”

mentioning

confidence: 99%

Detecting Goldstone modes with entanglement entropy

et al. 2015

View full text Add to dashboard Cite

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...

show abstract

“…Rather, one needs a calculation of the quantity in an interacting theory -the lack of which we hope will motivate future field-theoretic calculations of a α at the Wilson-Fisher fixed point. The coefficient −aα from fitting cα to the function aα log √ O + bα (purple dots) plotted along with other numerical estimates of −aα from free scalar field theory [4] (green diamonds), tensor tree network [22] (red star), finite-T QMC [20] (yellow square), and series expansion [18,19] (black circle). Standard error from the fit is shown for integer values of α. Inset: cα for α = 1, 2, 10 along with fits plotted vs 1/ √ O on a logarithmic scale.…”

mentioning

confidence: 99%

Entanglement at a Two-Dimensional Quantum Critical Point: A Numerical Linked-Cluster Expansion Study

et al. 2013

View full text Add to dashboard Cite

We develop a method to calculate the bipartite entanglement entropy of quantum models, in the thermodynamic limit, using a Numerical Linked Cluster Expansion (NLCE) involving only rectangular clusters. It is based on exact diagonalization of all n × m rectangular clusters at the interface between entangled subsystems A and B. We use it to obtain the Renyi entanglement entropy of the two-dimensional transverse field Ising model, for arbitrary real Renyi index α. Extrapolating these results as a function of the order of the calculation, we obtain universal pieces of the entanglement entropy associated with lines and corners at the quantum critical point. They show NLCE to be one of the few methods capable of accurately calculating universal properties of arbitrary Renyi entropies at higher dimensional critical points.

show abstract

Quantum Monte Carlo calculation of entanglement Rényi entropies for generic quantum systems

Cited by 113 publications

References 24 publications

Geometric Mutual Information at Classical Critical Points

Geometric Mutual Information at Classical Critical Points

Detecting Goldstone modes with entanglement entropy

Entanglement at a Two-Dimensional Quantum Critical Point: A Numerical Linked-Cluster Expansion Study

Contact Info

Product

Resources

About