Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1108.3942 · Europhysics Letters (EPL)

Universal logarithmic terms in the entanglement entropy of 2d, 3d and 4d random transverse-field Ising models

István A. Kovács, Ferenc Iglói

Published Mar 1, 2012• 43 citations
arXivHTMLPDF

Abstract

The entanglement entropy of the random transverse-field Ising model is calculated by a numerical implementation of the asymptotically exact strong disorder renormalization group method in 2 d , 3 d and 4 d hypercubic lattices for different shapes of the subregion. We find that the area law is always satisfied, but there are analytic corrections due to E -dimensional edges (1⩽ E ⩽ d − 2). More interesting is the contribution arising from corners, which is logarithmically divergent at the critical point and its prefactor in a given dimension is universal, i.e. , independent of the form of disorder.

DOI
10.1209/0295-5075/97/67009

Related papers

Quantum entanglement in the multicritical disordered Ising model

István A. Kovács

Quantum entanglement at critical points is often marked by universal characteristics. Here, the entanglement entropy is calculated at the quantum multicritical point of the random transverse-field Ising model (RTIM). We use an efficient implementation of the strong disorder renormalization group method in two and three dimensions for two types of disorder. For cubic subsystems we find a universal logarithmic corner contribution to the area law $bln(\ensuremath{\ell})$ that is independent of the form of disorder. Our results agree qualitatively with those at the quantum critical points of the RTIM, but with new $b$ prefactors due to having both geometric and quantum fluctuations at play. By studying the vicinity of the multicritical point, we demonstrate that the corner contribution serves as an ``entanglement susceptibility,'' a useful tool to locate the phase transition and to measure the correlation length critical exponents.

arXiv:2404.12990•Physical review. B./Physical review. B•Jun 6, 2024•2 citations

Entanglement entropy in the two-dimensional random transverse field Ising model

Rong Yu, Hubert Saleur, Stephan Haas

The scaling behavior of the entanglement entropy in the two-dimensional random transverse field Ising model is numerically studied through the strong disordered renormalization group method. We find that the leading term of the entanglement entropy always linearly scales with the block size. However, besides this area law contribution, we find a subleading logarithmic correction at the quantum critical point. This correction is discussed from the point of view of an underlying percolation transition, both at finite and at zero temperature.

arXiv:0709.3840•Physical Review B•Apr 25, 2008•74 citations

Entanglement Entropy at Infinite-Randomness Fixed Points in Higher Dimensions

Yu‐Cheng Lin, Ferenc Iglói, Heiko Rieger

The entanglement entropy of the two-dimensional random transverse Ising model is studied with a numerical implementation of the strong-disorder renormalization group. The asymptotic behavior of the entropy per surface area diverges at, and only at, the quantum phase transition that is governed by an infinite-randomness fixed point. Here we identify a double-logarithmic multiplicative correction to the area law for the entanglement entropy. This contrasts with the pure area law valid at the infinite-randomness fixed point in the diluted transverse Ising model in higher dimensions.

arXiv:0704.0418•Physical Review Letters•Oct 2, 2007•78 citations

Universal shape-dependence of quantum entanglement in disordered magnets

Natalie Love, István A. Kovács

Abstract Disordered quantum magnets are not only experimentally relevant, but offer efficient computational methodologies to calculate the low energy states as well as various measures of quantum correlations. We present a systematic analysis of quantum entanglement in the paradigmatic random transverse-field Ising model in two dimensions, using an efficient implementation of the asymptotically exact strong disorder renormalization group method. The phase diagram is known to be governed by three distinct infinitely disordered fixed points (IDFPs) that we study here. For square subsystems, it has been recently established that quantum entanglement has a universal logarithmic correction due to the corners of the subsystem at all three IDFPs. This corner contribution has been proposed as an ‘entanglement susceptibility’, a useful tool to locate the phase transition and measure the correlation length critical exponent. Towards a deeper understanding, we quantify how the corner contribution depends on the shape of the subsystem. While the corner contribution remains universal, the shape-dependence is qualitatively different in each universality class, also confirmed by line segment subsystems, a special case of skeletal entanglement. Therefore, unlike in conformally invariant systems, in general different subsystem shapes are versatile probes to unveil new universal information on phase transitions in disordered quantum systems.

arXiv:2507.04557•New Journal of Physics•Jan 1, 2026•0 citations

Entanglement across a cubic interface in 3 + 1 dimensions

Trithep Devakul, Rajiv R. P. Singh

We calculate the area, edge, and corner Renyi entanglement entropies in the ground state of the transverse-field Ising model, on a simple-cubic lattice, by high-field and low-field series expansions. We find that while the area term is positive and the line term is negative as required by strong subadditivity, the corner contributions are positive in three dimensions. Analysis of the series suggests that the expansions converge up to the physical critical point from both sides. The leading area-law Renyi entropies match nicely from the high- and low-field expansions at the critical point, forming a sharp cusp there. We calculate the coefficients of the logarithmic divergence associated with the corner entropy and compare them with conformal field theory results with smooth interfaces and find a striking correspondence.

arXiv:1407.0084•Physical Review B•Aug 20, 2014•15 citations

Precise computation of universal corner entanglement entropy at 2+1 dimension: From Ising to Gaussian quantum critical points

Ben Lee-Yeung Ngai, Justin Tim-Lok Chau, Junchen Rong +3 more

Computing the subleading logarithmic term in the entanglement entropy (EE) of (2+1)d quantum many-body systems remains a significant challenge, despite its central role in revealing universal information about quantum states and quantum critical points (QCPs). Building on recent algorithmic advances that enable the stable calculation of EE as an exponential observable~\cite{zhouIncremental2024,zhangIntegral2024,liaoExtracting2024}, we develop a {\it bubble basis} projector quantum Monte Carlo (QMC) algorithm to precisely and efficiently compute the universal corner of EE at QCPs in a (2+1)d square-lattice transverse-field Ising model augmented with a four-body interaction. Turning on this interaction allows us to trace an Ising critical line, reaching the tricritical point, and then a line of first-order phase transition. In (2+1)d, the tricritical point is described by the Gaussian theory, where a theoretical calculation of the corner logarithmic term in the 2nd Rényi entropy term is available~\cite{UniversalCasini2007}. Our QMC results are in quantitative agreement with this theoretical value, providing a highly nontrivial benchmark of the algorithm. Furthermore, we also study the Rényi EE at the Ising critical line and on the first-order transition line, obtaining results consistent with theoretical expectations. These findings establish the long-sought connection between the universal values of an exactly solvable limit and those of a strongly correlated regime at (2+1)d.

cond-mat.str-elCross-listedarXiv:2512.00382•Reports on Progress in Physics•Dec 2, 2025•0 citations
The entanglement entropy of the random transverse-field Ising model is calculated by a numerical implementation of the asymptotically exact strong disorder renormalization group method in 2 d , 3 d and 4 d hypercubic lattices for different shapes of the subregion. We find that the area law is always satisfied, but there are analytic corrections due to E -dimensional edges (1⩽ E ⩽ d − 2). More interesting is the contribution arising from corners, which is logarithmically divergent at the critical point and its prefactor in a given dimension is universal, i.e. , independent of the form of disorder.
Quantum entanglement at critical points is often marked by universal characteristics. Here, the entanglement entropy is calculated at the quantum multicritical point of the random transverse-field Ising model (RTIM). We use an efficient implementation of the strong disorder renormalization group method in two and three dimensions for two types of disorder. For cubic subsystems we find a universal logarithmic corner contribution to the area law $bln(\ensuremath{\ell})$ that is independent of the form of disorder. Our results agree qualitatively with those at the quantum critical points of the RTIM, but with new bbb prefactors due to having both geometric and quantum fluctuations at play. By studying the vicinity of the multicritical point, we demonstrate that the corner contribution serves as an ``entanglement susceptibility,'' a useful tool to locate the phase transition and to measure the correlation length critical exponents.
The scaling behavior of the entanglement entropy in the two-dimensional random transverse field Ising model is numerically studied through the strong disordered renormalization group method. We find that the leading term of the entanglement entropy always linearly scales with the block size. However, besides this area law contribution, we find a subleading logarithmic correction at the quantum critical point. This correction is discussed from the point of view of an underlying percolation transition, both at finite and at zero temperature.
The entanglement entropy of the two-dimensional random transverse Ising model is studied with a numerical implementation of the strong-disorder renormalization group. The asymptotic behavior of the entropy per surface area diverges at, and only at, the quantum phase transition that is governed by an infinite-randomness fixed point. Here we identify a double-logarithmic multiplicative correction to the area law for the entanglement entropy. This contrasts with the pure area law valid at the infinite-randomness fixed point in the diluted transverse Ising model in higher dimensions.
Abstract Disordered quantum magnets are not only experimentally relevant, but offer efficient computational methodologies to calculate the low energy states as well as various measures of quantum correlations. We present a systematic analysis of quantum entanglement in the paradigmatic random transverse-field Ising model in two dimensions, using an efficient implementation of the asymptotically exact strong disorder renormalization group method. The phase diagram is known to be governed by three distinct infinitely disordered fixed points (IDFPs) that we study here. For square subsystems, it has been recently established that quantum entanglement has a universal logarithmic correction due to the corners of the subsystem at all three IDFPs. This corner contribution has been proposed as an ‘entanglement susceptibility’, a useful tool to locate the phase transition and measure the correlation length critical exponent. Towards a deeper understanding, we quantify how the corner contribution depends on the shape of the subsystem. While the corner contribution remains universal, the shape-dependence is qualitatively different in each universality class, also confirmed by line segment subsystems, a special case of skeletal entanglement. Therefore, unlike in conformally invariant systems, in general different subsystem shapes are versatile probes to unveil new universal information on phase transitions in disordered quantum systems.
We calculate the area, edge, and corner Renyi entanglement entropies in the ground state of the transverse-field Ising model, on a simple-cubic lattice, by high-field and low-field series expansions. We find that while the area term is positive and the line term is negative as required by strong subadditivity, the corner contributions are positive in three dimensions. Analysis of the series suggests that the expansions converge up to the physical critical point from both sides. The leading area-law Renyi entropies match nicely from the high- and low-field expansions at the critical point, forming a sharp cusp there. We calculate the coefficients of the logarithmic divergence associated with the corner entropy and compare them with conformal field theory results with smooth interfaces and find a striking correspondence.
Computing the subleading logarithmic term in the entanglement entropy (EE) of (2+1)d quantum many-body systems remains a significant challenge, despite its central role in revealing universal information about quantum states and quantum critical points (QCPs). Building on recent algorithmic advances that enable the stable calculation of EE as an exponential observable~\cite{zhouIncremental2024,zhangIntegral2024,liaoExtracting2024}, we develop a {\it bubble basis} projector quantum Monte Carlo (QMC) algorithm to precisely and efficiently compute the universal corner of EE at QCPs in a (2+1)d square-lattice transverse-field Ising model augmented with a four-body interaction. Turning on this interaction allows us to trace an Ising critical line, reaching the tricritical point, and then a line of first-order phase transition. In (2+1)d, the tricritical point is described by the Gaussian theory, where a theoretical calculation of the corner logarithmic term in the 2nd Rényi entropy term is available~\cite{UniversalCasini2007}. Our QMC results are in quantitative agreement with this theoretical value, providing a highly nontrivial benchmark of the algorithm. Furthermore, we also study the Rényi EE at the Ising critical line and on the first-order transition line, obtaining results consistent with theoretical expectations. These findings establish the long-sought connection between the universal values of an exactly solvable limit and those of a strongly correlated regime at (2+1)d.