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:1407.4911 · Physical Review B

Identifying topological order in the Shastry-Sutherland model via entanglement entropy

David C. Ronquillo, Michael R. Peterson

Published Nov 14, 2014• 10 citations
arXivHTMLPDF

Abstract

It is known that for a topologically ordered state the area law for the entanglement entropy shows a negative universal additive constant contribution, $\ensuremath{-}\ensuremath{\gamma}$, called the topological entanglement entropy. We theoretically study the entanglement entropy of the two-dimensional Shastry-Sutherland quantum antiferromagnet using exact diagonalization on clusters of 16 and 24 spins. By utilizing the Kitaev-Preskill construction [A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006)] we extract a finite topological term, $\ensuremath{-}\ensuremath{\gamma}$, in the region of bond-strength parameter space corresponding to high geometrical frustration. Thus, we provide strong evidence for the existence of an exotic topologically ordered state and shed light on the nature of this model's strongly frustrated, and long controversial, intermediate phase.

DOI
10.1103/physrevb.90.201108

Related papers

Chiral spin liquid on a Shastry-Sutherland Heisenberg antiferromagnet

Jian-Wei Yang, Weiwei Luo, W. Zhu +3 more

We demonstrate the existence of a topological chiral spin liquid in the frustrated Shastry-Sutherland Heisenberg model with an additional spin chirality interaction, using numerically unbiased exact diagonalization and density matrix renormalization group methods. We establish a quantum phase diagram where conventional phases, including dimer singlet, plaquette singlet, N\'eel, and collinear phase, can be clearly identified by suitable local order parameters. Among them, an $\text{SU}{(2)}_{1}$ chiral spin liquid emerges in the highly frustrated region, which is unambiguously identified by two topologically degenerate ground states, a modular matrix, and characteristic level counting in an entanglement spectrum, featuring the same topological order of the $\ensuremath{\nu}=1/2$ bosonic Laughlin state. The phase boundaries among the different orders are determined by the energy level crossing analysis and wave-function fidelity susceptibility.

arXiv:2403.00597•Physical review. B./Physical review. B•May 3, 2024•1 citation

Entanglement entropy at generalized Rokhsar-Kivelson points of quantum dimer models

Alexander Selem, C. M. Herdman, K. Birgitta Whaley

We study the $n=2$ R\'enyi entanglement entropy of the triangular quantum dimer model via Monte Carlo sampling of Rokhsar-Kivelson- (RK-) like ground-state wave functions. Using the construction proposed by Kitaev and Preskill [Phys. Rev. Lett. 96, 110404 (2006)] and an adaptation of the Monte Carlo algorithm described by Hastings et al. [Phys. Rev. Lett. 104, 157201 (2010)], we compute the topological entanglement entropy (TEE) at the RK point $\ensuremath{\gamma}=(1.001\ifmmode\pm\else\textpm\fi{}0.003)\mathrm{ln}2$, confirming earlier results. Additionally, we compute the TEE of the ground state of a generalized RK-like Hamiltonian and demonstrate the universality of TEE over a wide range of parameter values within a topologically ordered phase approaching a quantum phase transition. For system sizes that are accessible numerically, we find that the quantization of TEE depends sensitively on correlations. We characterize corner contributions to the entanglement entropy and show that these are well described by shifts proportional to the number and types of corners in the bipartition.

arXiv:1211.0321•Physical Review B•Mar 6, 2013•11 citations

Identifying the ground state phases by spin-patterns in the Shastry-Sutherland model

Yun-Tong Yang, Fuzhou Chen, Hong‐Gang Luo

Exploring the influence of frustration on the phases and related phase transitions in condensed matter physics is of fundamental importance in uncovering the role played by frustration. In the two-dimensional square lattice, a minimal frustration has been formulated in 1981 as the Shastry-Sutherland (SS) model described by competitions between the nearest-neighbor bond ($J_1$) and the next-nearest-neighbor one ($J_2$). In the two limits of $α=J_2/J_1$, i.e. $α\ll 1$ and $α\gg 1$, the corresponding phases are the N{é}el antiferromagnet (AFM) and the dimer-singlet(DS). Unfortunately, the intermediate regime remains controversial, and the nature of transition from the N{é}el AFM to the intermediate state is also unclear. Here we provide a pattern language to explore the SS model and take the lattice size $L=4 \times4$ with periodic boundary condition. We firstly diagonalize the Hamiltonian in an operator space to obtain all fundamental spin-patterns and then analyze their energy and occupancy evolutions with the frustration parameter $κ=α/ (1+α)$. Our results indicate that the intermediate regime is characterized by diagonal two-domain spin-pattern while the N{é}el AFM state has a diagonal single-domain and the DS has mixings of diagonal single- and four-domain. While the transition from the DS to the intermediate phase occurred around $α_c = 1.5$ is the first-order in nature, consistent with that in literature, the one from the intermediate phase to the AFM is clearly seen around $α_c = 1.277$, where it has a reversal of the contributions from the single- and two-domain patterns to the ground state. The result indicates that the pattern language is powerful in identifying the possible phases in frustrated models.

arXiv:2404.16330•Apr 25, 2024•0 citations

Negativity and topological order in the toric code

Claudio Castelnovo

In this paper we study the behavior of the entanglement measure dubbed negativity in the context of the toric code model. Using a replica method introduced recently by Calabrese, Cardy, and Tonni [Phys. Rev. Lett. 109, 130502 (2012)], we obtain an exact expression which illustrates how the nonlocal correlations present in a topologically ordered state reflect in the behavior of the negativity of the system. We find that the negativity has a leading area-law contribution if the subsystems are in direct contact with one another (as expected in a zero-range correlated model). We also find a topological contribution directly related to the topological entropy, provided that the partitions are topologically nontrivial in both directions on a torus. We further confirm by explicit calculation that the negativity captures only quantum contributions to the entanglement. Indeed, we show that the negativity vanishes identically for the classical topologically ordered eight-vertex model, which on the contrary exhibits a finite von Neumann entropy, inclusive of topological correction.

arXiv:1306.4990•Physical Review A•Oct 16, 2013•123 citations

Geometric entanglement in topologically ordered states

Román Orús, Tzu-Chieh Wei, Oliver Buerschaper +1 more

Here we investigate the connection between topological order and the geometric entanglement, as measured by the logarithm of the overlap between a given state and its closest product state of blocks. We do this for a variety of topologically ordered systems such as the toric code, double semion, colour code and quantum double models. As happens for the entanglement entropy, we find that for sufficiently large block sizes the geometric entanglement is, up to possible sub-leading corrections, the sum of two contributions: a bulk contribution obeying a boundary law times the number of blocks and a contribution quantifying the underlying pattern of long-range entanglement of the topologically ordered state. This topological contribution is also present in the case of single-spin blocks in most cases, and constitutes an alternative characterization of topological order for these quantum states based on a multipartite entanglement measure. In particular, we see that the topological term for the two-dimensional colour code is twice as much as the one for the toric code, in accordance with recent renormalization group arguments (Bombin et al 2012 New J. Phys. 14 073048). Motivated by these results, we also derive a general formalism to obtain upper- and lower-bounds to the geometric entanglement of states with a non-Abelian group symmetry, and which we explicitly use to analyse quantum double models. Furthermore, we also provide an analysis of the robustness of the topological contribution in terms of renormalization and perturbation theory arguments, as well as a numerical estimation for small systems. Some of the results in this paper rely on the ability to disentangle single sites from the quantum state, which is always possible for the systems that we consider. Additionally we relate our results to the behaviour of the relative entropy of entanglement in topologically ordered systems, and discuss a number of numerical approaches based on tensor networks that could be employed to extract this topological contribution for large systems beyond exactly solvable models.

arXiv:1304.1339•New Journal of Physics•Jan 13, 2014•26 citations

Long-range entanglement and topological excitations

Gianpaolo Torre, Jovan Odavić, Pierre Fromholz +2 more

Topological order comes in different forms, and its classification and detection is an important field of modern research. In this work, we show that the Disconnected Entanglement Entropy, a measure originally introduced to identify topological phases, is also able to unveil the long-range entanglement (LRE) carried by a single, fractionalized excitation. We show this by considering a quantum, delocalized domain wall excitation that can be introduced into a system by inducing geometric frustration in an antiferromagnetic spin chain. Furthermore, we show that the LRE of such systems is resilient against a quantum quench and the introduction of disorder, as it happens in traditional symmetry-protected topological phases. All these evidences establish the existence of a new phase induced by frustration with topological features despite not being of the usual type.

arXiv:2310.16091•SciPost Physics Core•Aug 5, 2024•13 citations
It is known that for a topologically ordered state the area law for the entanglement entropy shows a negative universal additive constant contribution, \ensuremath−\ensuremathγ\ensuremath{-}\ensuremath{\gamma}\ensuremath−\ensuremathγ, called the topological entanglement entropy. We theoretically study the entanglement entropy of the two-dimensional Shastry-Sutherland quantum antiferromagnet using exact diagonalization on clusters of 16 and 24 spins. By utilizing the Kitaev-Preskill construction [A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006)] we extract a finite topological term, \ensuremath−\ensuremathγ\ensuremath{-}\ensuremath{\gamma}\ensuremath−\ensuremathγ, in the region of bond-strength parameter space corresponding to high geometrical frustration. Thus, we provide strong evidence for the existence of an exotic topologically ordered state and shed light on the nature of this model's strongly frustrated, and long controversial, intermediate phase.
We demonstrate the existence of a topological chiral spin liquid in the frustrated Shastry-Sutherland Heisenberg model with an additional spin chirality interaction, using numerically unbiased exact diagonalization and density matrix renormalization group methods. We establish a quantum phase diagram where conventional phases, including dimer singlet, plaquette singlet, N\'eel, and collinear phase, can be clearly identified by suitable local order parameters. Among them, an SU(2)1\text{SU}{(2)}_{1}SU(2)1​ chiral spin liquid emerges in the highly frustrated region, which is unambiguously identified by two topologically degenerate ground states, a modular matrix, and characteristic level counting in an entanglement spectrum, featuring the same topological order of the $\ensuremath{\nu}=1/2$ bosonic Laughlin state. The phase boundaries among the different orders are determined by the energy level crossing analysis and wave-function fidelity susceptibility.
We study the n=2n=2n=2 R\'enyi entanglement entropy of the triangular quantum dimer model via Monte Carlo sampling of Rokhsar-Kivelson- (RK-) like ground-state wave functions. Using the construction proposed by Kitaev and Preskill [Phys. Rev. Lett. 96, 110404 (2006)] and an adaptation of the Monte Carlo algorithm described by Hastings et al. [Phys. Rev. Lett. 104, 157201 (2010)], we compute the topological entanglement entropy (TEE) at the RK point $\ensuremath{\gamma}=(1.001\ifmmode\pm\else\textpm\fi{}0.003)\mathrm{ln}2$, confirming earlier results. Additionally, we compute the TEE of the ground state of a generalized RK-like Hamiltonian and demonstrate the universality of TEE over a wide range of parameter values within a topologically ordered phase approaching a quantum phase transition. For system sizes that are accessible numerically, we find that the quantization of TEE depends sensitively on correlations. We characterize corner contributions to the entanglement entropy and show that these are well described by shifts proportional to the number and types of corners in the bipartition.
Exploring the influence of frustration on the phases and related phase transitions in condensed matter physics is of fundamental importance in uncovering the role played by frustration. In the two-dimensional square lattice, a minimal frustration has been formulated in 1981 as the Shastry-Sutherland (SS) model described by competitions between the nearest-neighbor bond (J1J_1J1​) and the next-nearest-neighbor one (J2J_2J2​). In the two limits of α=J2/J1α=J_2/J_1α=J2​/J1​, i.e. α≪1α\ll 1α≪1 and α≫1α\gg 1α≫1, the corresponding phases are the N{é}el antiferromagnet (AFM) and the dimer-singlet(DS). Unfortunately, the intermediate regime remains controversial, and the nature of transition from the N{é}el AFM to the intermediate state is also unclear. Here we provide a pattern language to explore the SS model and take the lattice size L=4×4L=4 \times4L=4×4 with periodic boundary condition. We firstly diagonalize the Hamiltonian in an operator space to obtain all fundamental spin-patterns and then analyze their energy and occupancy evolutions with the frustration parameter κ=α/(1+α)κ=α/ (1+α)κ=α/(1+α). Our results indicate that the intermediate regime is characterized by diagonal two-domain spin-pattern while the N{é}el AFM state has a diagonal single-domain and the DS has mixings of diagonal single- and four-domain. While the transition from the DS to the intermediate phase occurred around αc=1.5α_c = 1.5αc​=1.5 is the first-order in nature, consistent with that in literature, the one from the intermediate phase to the AFM is clearly seen around αc=1.277α_c = 1.277αc​=1.277, where it has a reversal of the contributions from the single- and two-domain patterns to the ground state. The result indicates that the pattern language is powerful in identifying the possible phases in frustrated models.
In this paper we study the behavior of the entanglement measure dubbed negativity in the context of the toric code model. Using a replica method introduced recently by Calabrese, Cardy, and Tonni [Phys. Rev. Lett. 109, 130502 (2012)], we obtain an exact expression which illustrates how the nonlocal correlations present in a topologically ordered state reflect in the behavior of the negativity of the system. We find that the negativity has a leading area-law contribution if the subsystems are in direct contact with one another (as expected in a zero-range correlated model). We also find a topological contribution directly related to the topological entropy, provided that the partitions are topologically nontrivial in both directions on a torus. We further confirm by explicit calculation that the negativity captures only quantum contributions to the entanglement. Indeed, we show that the negativity vanishes identically for the classical topologically ordered eight-vertex model, which on the contrary exhibits a finite von Neumann entropy, inclusive of topological correction.
Here we investigate the connection between topological order and the geometric entanglement, as measured by the logarithm of the overlap between a given state and its closest product state of blocks. We do this for a variety of topologically ordered systems such as the toric code, double semion, colour code and quantum double models. As happens for the entanglement entropy, we find that for sufficiently large block sizes the geometric entanglement is, up to possible sub-leading corrections, the sum of two contributions: a bulk contribution obeying a boundary law times the number of blocks and a contribution quantifying the underlying pattern of long-range entanglement of the topologically ordered state. This topological contribution is also present in the case of single-spin blocks in most cases, and constitutes an alternative characterization of topological order for these quantum states based on a multipartite entanglement measure. In particular, we see that the topological term for the two-dimensional colour code is twice as much as the one for the toric code, in accordance with recent renormalization group arguments (Bombin et al 2012 New J. Phys. 14 073048). Motivated by these results, we also derive a general formalism to obtain upper- and lower-bounds to the geometric entanglement of states with a non-Abelian group symmetry, and which we explicitly use to analyse quantum double models. Furthermore, we also provide an analysis of the robustness of the topological contribution in terms of renormalization and perturbation theory arguments, as well as a numerical estimation for small systems. Some of the results in this paper rely on the ability to disentangle single sites from the quantum state, which is always possible for the systems that we consider. Additionally we relate our results to the behaviour of the relative entropy of entanglement in topologically ordered systems, and discuss a number of numerical approaches based on tensor networks that could be employed to extract this topological contribution for large systems beyond exactly solvable models.
Topological order comes in different forms, and its classification and detection is an important field of modern research. In this work, we show that the Disconnected Entanglement Entropy, a measure originally introduced to identify topological phases, is also able to unveil the long-range entanglement (LRE) carried by a single, fractionalized excitation. We show this by considering a quantum, delocalized domain wall excitation that can be introduced into a system by inducing geometric frustration in an antiferromagnetic spin chain. Furthermore, we show that the LRE of such systems is resilient against a quantum quench and the introduction of disorder, as it happens in traditional symmetry-protected topological phases. All these evidences establish the existence of a new phase induced by frustration with topological features despite not being of the usual type.