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:1012.1740 · Physical Review Letters

Robustness of a Perturbed Topological Phase

Sébastien Dusuel, Michael Kamfor, Román Orús, Kai Phillip Schmidt, Julien Vidal

Published Mar 8, 2011• 207 citations
arXivHTMLPDF

Abstract

We investigate the stability of the topological phase of the toric code model in the presence of a uniform magnetic field by means of variational and high-order series expansion approaches. We find that when this perturbation is strong enough, the system undergoes a topological phase transition whose first- or second-order nature depends on the field orientation. When this transition is of second order, it is in the Ising universality class except for a special line on which the critical exponent driving the closure of the gap varies continuously, unveiling a new topological universality class.

DOI
10.1103/physrevlett.106.107203

Related papers

Breakdown of a perturbed $\boldsymbol{\mathbbm{Z}}_N$ topological phase

Marc Daniel Schulz, Sébastien Dusuel, Román Orús +2 more

We studied the robustness of a generalized Kitaev's toric code with degrees of freedom in the presence of local perturbations. For N = 2, this model reduces to the conventional toric code in a uniform magnetic field. A quantitative analysis was performed for the perturbed toric code by applying a combination of high-order series expansions and variational techniques. We found strong evidence for first- and second-order phase transitions between topologically ordered and polarized phases. Most interestingly, our results also indicate the existence of topological multi-critical points in the phase diagram.

arXiv:1110.3632•New Journal of Physics•Feb 17, 2012•74 citations

Robustness and spectroscopy of the toric code in a magnetic field

Michael Kamfor

We study the robustness of topologically ordered phases under perturbation by considering the specific examples of the toric code and Kitaev's honeycomb models.More precisely, by means of high-order series expansions (pCUT) combined with a variational method (iPEPS) we analyze transitions between the topological and polarized phases of the toric code in a uniform magnetic field.We first describe the phase diagram of the perturbed toric code, and develop a quasi-particle picture of its elementary anyonic excitations.The effective model of interacting quasi-particles we derive allows us to analyze spectral properties of the low-energy physics of the perturbed toric code and unveil the presence of bound states.In a second step, we compute dynamical correlation functions (spectral densities) that are relevant for potential scattering experiments.Finally, we draw a connection between Kitaev's honeycomb model in a vortex superlattice and graphene in a strongly-modulated magnetic field, and present analytical results on metal-insulator transitions in these systems.

•Jun 4, 2013•1 citation

Robustness of a topological phase: Topological color code in a parallel magnetic field

Saeed S. Jahromi, Mehdi Kargarian, Seyed Farhad Masoudi +1 more

The robustness of the topological color code, which is a class of error-correcting quantum codes, is investigated under the influence of a uniform magnetic field on the honeycomb lattice. Our study relies on two high-order series expansions using perturbative continuous unitary transformations in the limit of low and high fields, exact diagonalization, and a classical approximation. We show that the topological color code in a single parallel field is isospectral to the Baxter-Wu model in a transverse field on the triangular lattice. It is found that the topological phase is stable up to a critical field beyond which it breaks down to the polarized phase by a first-order phase transition. The results also suggest that the topological color code is more robust than the toric code in the parallel magnetic field.

arXiv:1211.1687•Physical Review B•Mar 13, 2013•35 citations

Quantum phase transitions out of a Z2×Z2 topological phase

Saeed S. Jahromi, Seyed Farhad Masoudi, Mehdi Kargarian +1 more

We investigate the low-energy spectral properties and robustness of the topological phase of color code, which is a quantum spin model for the aim of fault-tolerant quantum computation, in the presence of a uniform magnetic field or Ising interactions, using high-order series expansion and exact diagonalization. In a uniform magnetic field, we find first-order phase transitions in all field directions. In contrast, our results for the Ising interactions unveil that for strong enough Ising couplings, the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological phase of color code breaks down to symmetry broken phases by first- or second-order phase transitions.

arXiv:1308.1407•Physical Review B•Dec 12, 2013•20 citations

Quantum robustness of the toric code in a parallel field on the honeycomb and triangular lattice

Viktor Kott, Matthias Mühlhauser, Jan Alexander Koziol +1 more

We investigate the quantum robustness of the topological order in the toric code on the honeycomb lattice in the presence of a uniform parallel field. For a field in z z -direction, the low-energy physics is in the flux-free sector and can be mapped to the transverse-field Ising model on the honeycomb lattice. One finds a second-order quantum phase transition in the 3D Ising ^\star ⋆ universality class for both signs of the field. The same is true for a postive field in x x -direction where an analogue mapping in the charge-free sector yields a ferromagnetic transverse-field Ising model on the triangular lattice and the phase transition is still 3D Ising ^\star ⋆ . In contrast, for negative x x -field, the charge-free sector is mapped to the highly frustrated antiferromagnetic transverse-field Ising model on the triangular lattice which is known to host a quantum phase transition in the 3D XY ^\star ⋆ universality class. Further, the charge-free sector does not always contain the low-energy physics for negative x x -fields and a first-order phase transition to the polarized phase in the charge-full sector takes place at larger negative field values. We quantify the location of this transition by comparing quantum Monte Carlo simulations and high-field series expansions. The full extension of the topological phase in the presence of x x - and z z -fields is determined by perturbative linked-cluster expansions using a full graph decomposition. Extrapolating the high-order series of the charge and the flux gap allows to estimate critical exponents of the gap closing. This analysis indicates that the topological order breaks down by critical lines of 3D Ising ^\star ⋆ and 3D XY ^\star ⋆ type with interesting potential multi-critical crossing points. All findings for the toric code on the honeycomb lattice can be transferred exactly to the toric code on the triangular lattice.

arXiv:2402.15389•SciPost Physics•Aug 15, 2024•4 citations

Topological line in frustrated toric code models

Mohammad Hossein Zarei, Jahanfar Abouie

Typical topological systems undergo a topological phase transition in the presence of a strong enough perturbation. We propose an adjustable frustrated toric code with a ``topological line'' at which no phase transition happens in the system and the topological order is robust against a nonlinear perturbation of arbitrary strength. This important result is a consequence of the interplay between frustration and nonlinearity in our system, which also causes the emergence of other interesting phenomena such as reentrant topological phases and survival of the topological order under local projection operations. Our study opens a window towards more robust topological quantum codes which are the cornerstones of large-scale quantum computing.

arXiv:2101.03156•Physical review. B./Physical review. B•Sep 21, 2021•9 citations
We investigate the stability of the topological phase of the toric code model in the presence of a uniform magnetic field by means of variational and high-order series expansion approaches. We find that when this perturbation is strong enough, the system undergoes a topological phase transition whose first- or second-order nature depends on the field orientation. When this transition is of second order, it is in the Ising universality class except for a special line on which the critical exponent driving the closure of the gap varies continuously, unveiling a new topological universality class.
We studied the robustness of a generalized Kitaev's toric code with degrees of freedom in the presence of local perturbations. For N = 2, this model reduces to the conventional toric code in a uniform magnetic field. A quantitative analysis was performed for the perturbed toric code by applying a combination of high-order series expansions and variational techniques. We found strong evidence for first- and second-order phase transitions between topologically ordered and polarized phases. Most interestingly, our results also indicate the existence of topological multi-critical points in the phase diagram.
We study the robustness of topologically ordered phases under perturbation by considering the specific examples of the toric code and Kitaev's honeycomb models.More precisely, by means of high-order series expansions (pCUT) combined with a variational method (iPEPS) we analyze transitions between the topological and polarized phases of the toric code in a uniform magnetic field.We first describe the phase diagram of the perturbed toric code, and develop a quasi-particle picture of its elementary anyonic excitations.The effective model of interacting quasi-particles we derive allows us to analyze spectral properties of the low-energy physics of the perturbed toric code and unveil the presence of bound states.In a second step, we compute dynamical correlation functions (spectral densities) that are relevant for potential scattering experiments.Finally, we draw a connection between Kitaev's honeycomb model in a vortex superlattice and graphene in a strongly-modulated magnetic field, and present analytical results on metal-insulator transitions in these systems.
The robustness of the topological color code, which is a class of error-correcting quantum codes, is investigated under the influence of a uniform magnetic field on the honeycomb lattice. Our study relies on two high-order series expansions using perturbative continuous unitary transformations in the limit of low and high fields, exact diagonalization, and a classical approximation. We show that the topological color code in a single parallel field is isospectral to the Baxter-Wu model in a transverse field on the triangular lattice. It is found that the topological phase is stable up to a critical field beyond which it breaks down to the polarized phase by a first-order phase transition. The results also suggest that the topological color code is more robust than the toric code in the parallel magnetic field.
We investigate the low-energy spectral properties and robustness of the topological phase of color code, which is a quantum spin model for the aim of fault-tolerant quantum computation, in the presence of a uniform magnetic field or Ising interactions, using high-order series expansion and exact diagonalization. In a uniform magnetic field, we find first-order phase transitions in all field directions. In contrast, our results for the Ising interactions unveil that for strong enough Ising couplings, the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological phase of color code breaks down to symmetry broken phases by first- or second-order phase transitions.
We investigate the quantum robustness of the topological order in the toric code on the honeycomb lattice in the presence of a uniform parallel field. For a field in z z -direction, the low-energy physics is in the flux-free sector and can be mapped to the transverse-field Ising model on the honeycomb lattice. One finds a second-order quantum phase transition in the 3D Ising ^\star ⋆ universality class for both signs of the field. The same is true for a postive field in x x -direction where an analogue mapping in the charge-free sector yields a ferromagnetic transverse-field Ising model on the triangular lattice and the phase transition is still 3D Ising ^\star ⋆ . In contrast, for negative x x -field, the charge-free sector is mapped to the highly frustrated antiferromagnetic transverse-field Ising model on the triangular lattice which is known to host a quantum phase transition in the 3D XY ^\star ⋆ universality class. Further, the charge-free sector does not always contain the low-energy physics for negative x x -fields and a first-order phase transition to the polarized phase in the charge-full sector takes place at larger negative field values. We quantify the location of this transition by comparing quantum Monte Carlo simulations and high-field series expansions. The full extension of the topological phase in the presence of x x - and z z -fields is determined by perturbative linked-cluster expansions using a full graph decomposition. Extrapolating the high-order series of the charge and the flux gap allows to estimate critical exponents of the gap closing. This analysis indicates that the topological order breaks down by critical lines of 3D Ising ^\star ⋆ and 3D XY ^\star ⋆ type with interesting potential multi-critical crossing points. All findings for the toric code on the honeycomb lattice can be transferred exactly to the toric code on the triangular lattice.
Typical topological systems undergo a topological phase transition in the presence of a strong enough perturbation. We propose an adjustable frustrated toric code with a ``topological line'' at which no phase transition happens in the system and the topological order is robust against a nonlinear perturbation of arbitrary strength. This important result is a consequence of the interplay between frustration and nonlinearity in our system, which also causes the emergence of other interesting phenomena such as reentrant topological phases and survival of the topological order under local projection operations. Our study opens a window towards more robust topological quantum codes which are the cornerstones of large-scale quantum computing.