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Phase transitions in topological lattice models via topological symmetry breaking

Burnell, F.J., Steven H. Simon, Johannes K. Slingerland

Published Jan 1, 2012• 64 citations
OpenAlex

Abstract

We study transitions between phases of matter with topological order. By studying these transitions in exactly solvable lattice models we show how universality classes may be identified and critical properties described. As a familiar example to elucidate our results concretely, we describe in detail a transition between a fully gapped achiral 2D p -wave superconductor ( p +i p for pseudo-spin up/ p − i p for pseudo-spin down) to an s-wave superconductor. We show in particular that this transition is of the 2D transverse field Ising universality class.

Related papers

Phase transitions in topological lattice models via topological symmetry breaking

F J Burnell, Steven H. Simon, Johannes K. Slingerland

We study transitions between phases of matter with topological order. By studying these transitions in exactly solvable lattice models we show how universality classes may be identified and critical properties described. As a familiar example to elucidate our results concretely, we describe in detail a transition between a fully gapped achiral 2D p -wave superconductor ( p + i p for pseudo-spin up/ p − i p for pseudo-spin down) to an s-wave superconductor. We show in particular that this transition is of the 2D transverse field Ising universality class.

arXiv:1012.0317•New Journal of Physics•Jan 20, 2012•89 citations

Topological quantum phase transitions and topological quantum criticality in superfluids and superconductors

Fan Yang

Superfluids and superconductors can be either fully gapped or gapless with nodal structures in momentum space. Both the gapped and nodal phases can be topologically protected and possess nontrivial topological invariants. Topological quantum phase transitions exist at zero temperature between different gapped phases or between gapped and nodal phases. These phase transitions can be driven by chemical potential and/or spin exchange fields. The two phases separated by the phase transition can have the same local order but differ in global topology. Thus, these topological quantum phase transitions cannot be described by the Landau paradigm of symmetry breaking. Although some aspects of these transitions, such as the change of topological invariants and gapless boundary states, have been discussed before, a complete theory of these transitions has yet to be developed. In this dissertation, we construct effective field theories to study the universality and thermodynamic signatures of these transitions. We find four different universality classes in superfluids and superconductors. Certain thermodynamic quantities, such as compressibility or spin susceptibility, change non-analytically across the transitions. For certain time-reversal symmetry breaking fields that lead to bulk phase transitions, there also exist topological phase transitions on the surface. All the topological phase transitions studied in this dissertation only exist at zero temperature. At finite temperature, different states are connected by smooth crossovers. There exists a quantum critical region at finite temperature near the quantum critical point (QCP). In this quantum critical region, thermodynamic quantities have universal scaling dependence on temperature dictated by the universality class of the QCP. We argue that these scaling properties can be used to probe and differentiate these QCPs. These bulk and surface topological quantum phase transitions are discussed in various concrete models, such as chiral and time-reversal invariant p-wave superfluids, topological superconductors of emergent Dirac fermions, and topological superconducting model of CuₓBi₂Se₃.

•Jan 1, 2021•0 citations

Topological transitions in Ising models

Somenath Jalal, Rishabh Khare, Siddhartha Lal

The thermal dynamics of the two-dimensional Ising model and quantum dynamics of the one-dimensional transverse-field Ising model (TFIM) are mapped to one another through the transfer-matrix formalism. We show that the fermionised TFIM undergoes a Fermi-surface topology-changing Lifshitz transition at its critical point. We identify the degree of freedom which tracks the Lifshitz transition via changes in topological quantum numbers (e.g., Chern number, Berry phase etc.). An emergent $SU(2)$ symmetry at criticality is observed to lead to a topological quantum number different from that which characterises the ordered phase. The topological transition is also understood via a spectral flow thought-experiment in a Thouless charge pump, revealing the bulk-boundary correspondence across the transition. The duality property of the phases and their entanglement content are studied, revealing a holographic relation with the entanglement at criticality. The effects of a non-zero longitudinal field and interactions that scatter across the singular Fermi surface are treated within the renormalisation group (RG) formalism. The analysis reveals that the critical point of the 1D TFIM and the 1D spin-1/2 Heisenberg chain are connected via a line of $SU(2)$-symmetric theories. We extend our analysis to show that the classical to quantum correspondence links the critical theories of Ising models in various dimensions holographically through the universal effective Hamiltonian that describes the Lifshitz transition of the 1D TFIM. We obtain in this way a unified perspective of transitions in Ising models that lie beyond the traditional Ginzburg-Landau-Wilson paradigm. We discuss the consequences of our results for similar topological transitions observed in classical spin models, topological insulators, superconductors and lattice gauge-field theories which are related to the Ising universality class.

arXiv:1610.09845•Oct 31, 2016•2 citations

Ising order parameter and topological phase transitions: Toric code in a uniform magnetic field

Mohammad Hossein Zarei

A quantum Ising model in a transverse field is of the simplest quantum many-body systems used for studying universal properties of quantum phase transitions. Interestingly, it is well known that such phase transitions can be mapped to topological phase transitions in toric code models. Therefore, one can expect that well-known properties of the transverse Ising model are used for characterizing topological phase transitions in toric code models. In this paper, we consider the magnetization of the Ising model and show, while it is a local order parameter, it is mapped to a nonlocal order parameter on the topological side. Consequently, we introduce a string order parameter for topological phase transitions in toric code models defined on different lattices with different dimensions in the presence of a uniform magnetic field. Since such a string order parameter is dual of the Ising order parameter with well-known properties, it leads to a topological (nonlocal) characterization of phase transition in the toric code model in uniform field. Our results show that although topological phase transitions do not follow a symmetry-breaking mechanism, it might still be possible to use concepts of symmetry-breaking phase transitions for topological ones by finding suitable mappings.

arXiv:1907.06219•Physical review. B./Physical review. B•Sep 25, 2019•18 citations

Condensation of achiral simple currents in topological lattice models: Hamiltonian study of topological symmetry breaking

F. J. Burnell, Steven H. Simon, Johannes K. Slingerland

We describe a family of phase transitions connecting phases of differing nontrivial topological order by explicitly constructing Hamiltonians of the Levin-Wen [Levin and Wen, Phys. Rev. B 71, 045110 (2005)] type which can be tuned between two solvable points, each of which realizes a different topologically ordered phase. We show that the low-energy degrees of freedom near the phase transition can be mapped onto those of a Potts model, and we discuss the stability of the resulting phase diagram to small perturbations about the model. We further explain how the excitations in the condensed phase are formed from those in the original topological theory, some of which are split into multiple components by condensation, and we discuss the implications of our results for understanding the nature of general achiral topological phases in 2 $+$ 1 dimensions in terms of doubled Chern-Simons theories.

arXiv:1104.1701•Physical Review B•Sep 16, 2011•103 citations

Topological order in three-dimensional systems and 2-gauge symmetry

Ricardo Costa de Almeida

Topological order is a new paradigm for quantum phases of matter developed to explain phase transitions which do not fit the symmetry breaking scheme for classifying phases of matter. They are characterized by patterns of entanglement that lead to topologically depended ground state degeneracy and anyonic excitations. One common approach for studying such phases in two-dimensional systems is through exactly solvable lattice Hamiltonian models such as quantum double models and String-Net models. The former can be understood as the Hamiltonian formulation of lattice gauge theories and, as such, it is defined by a finite gauge group. However, not much is known about topological phases in tridimensional systems. Motivated by this we develop a new class of three-dimensional exactly solvable models which go beyond quantum double models by using finite crossed modules instead of gauge groups. This approach relies on a lattice implementation of 2-gauge theory to obtain models with a richer topological structure. We construct the Hamiltonian model explicitly and provide a rigorous proof that the ground state degeneracy is a topological invariant and that the ground states can only be characterized with nonlocal order parameters.

•Dec 5, 2017•1 citation
We study transitions between phases of matter with topological order. By studying these transitions in exactly solvable lattice models we show how universality classes may be identified and critical properties described. As a familiar example to elucidate our results concretely, we describe in detail a transition between a fully gapped achiral 2D p -wave superconductor ( p +i p for pseudo-spin up/ p − i p for pseudo-spin down) to an s-wave superconductor. We show in particular that this transition is of the 2D transverse field Ising universality class.
We study transitions between phases of matter with topological order. By studying these transitions in exactly solvable lattice models we show how universality classes may be identified and critical properties described. As a familiar example to elucidate our results concretely, we describe in detail a transition between a fully gapped achiral 2D p -wave superconductor ( p + i p for pseudo-spin up/ p − i p for pseudo-spin down) to an s-wave superconductor. We show in particular that this transition is of the 2D transverse field Ising universality class.
Superfluids and superconductors can be either fully gapped or gapless with nodal structures in momentum space. Both the gapped and nodal phases can be topologically protected and possess nontrivial topological invariants. Topological quantum phase transitions exist at zero temperature between different gapped phases or between gapped and nodal phases. These phase transitions can be driven by chemical potential and/or spin exchange fields. The two phases separated by the phase transition can have the same local order but differ in global topology. Thus, these topological quantum phase transitions cannot be described by the Landau paradigm of symmetry breaking. Although some aspects of these transitions, such as the change of topological invariants and gapless boundary states, have been discussed before, a complete theory of these transitions has yet to be developed. In this dissertation, we construct effective field theories to study the universality and thermodynamic signatures of these transitions. We find four different universality classes in superfluids and superconductors. Certain thermodynamic quantities, such as compressibility or spin susceptibility, change non-analytically across the transitions. For certain time-reversal symmetry breaking fields that lead to bulk phase transitions, there also exist topological phase transitions on the surface. All the topological phase transitions studied in this dissertation only exist at zero temperature. At finite temperature, different states are connected by smooth crossovers. There exists a quantum critical region at finite temperature near the quantum critical point (QCP). In this quantum critical region, thermodynamic quantities have universal scaling dependence on temperature dictated by the universality class of the QCP. We argue that these scaling properties can be used to probe and differentiate these QCPs. These bulk and surface topological quantum phase transitions are discussed in various concrete models, such as chiral and time-reversal invariant p-wave superfluids, topological superconductors of emergent Dirac fermions, and topological superconducting model of CuₓBi₂Se₃.
The thermal dynamics of the two-dimensional Ising model and quantum dynamics of the one-dimensional transverse-field Ising model (TFIM) are mapped to one another through the transfer-matrix formalism. We show that the fermionised TFIM undergoes a Fermi-surface topology-changing Lifshitz transition at its critical point. We identify the degree of freedom which tracks the Lifshitz transition via changes in topological quantum numbers (e.g., Chern number, Berry phase etc.). An emergent SU(2)SU(2)SU(2) symmetry at criticality is observed to lead to a topological quantum number different from that which characterises the ordered phase. The topological transition is also understood via a spectral flow thought-experiment in a Thouless charge pump, revealing the bulk-boundary correspondence across the transition. The duality property of the phases and their entanglement content are studied, revealing a holographic relation with the entanglement at criticality. The effects of a non-zero longitudinal field and interactions that scatter across the singular Fermi surface are treated within the renormalisation group (RG) formalism. The analysis reveals that the critical point of the 1D TFIM and the 1D spin-1/2 Heisenberg chain are connected via a line of SU(2)SU(2)SU(2)-symmetric theories. We extend our analysis to show that the classical to quantum correspondence links the critical theories of Ising models in various dimensions holographically through the universal effective Hamiltonian that describes the Lifshitz transition of the 1D TFIM. We obtain in this way a unified perspective of transitions in Ising models that lie beyond the traditional Ginzburg-Landau-Wilson paradigm. We discuss the consequences of our results for similar topological transitions observed in classical spin models, topological insulators, superconductors and lattice gauge-field theories which are related to the Ising universality class.
A quantum Ising model in a transverse field is of the simplest quantum many-body systems used for studying universal properties of quantum phase transitions. Interestingly, it is well known that such phase transitions can be mapped to topological phase transitions in toric code models. Therefore, one can expect that well-known properties of the transverse Ising model are used for characterizing topological phase transitions in toric code models. In this paper, we consider the magnetization of the Ising model and show, while it is a local order parameter, it is mapped to a nonlocal order parameter on the topological side. Consequently, we introduce a string order parameter for topological phase transitions in toric code models defined on different lattices with different dimensions in the presence of a uniform magnetic field. Since such a string order parameter is dual of the Ising order parameter with well-known properties, it leads to a topological (nonlocal) characterization of phase transition in the toric code model in uniform field. Our results show that although topological phase transitions do not follow a symmetry-breaking mechanism, it might still be possible to use concepts of symmetry-breaking phase transitions for topological ones by finding suitable mappings.
We describe a family of phase transitions connecting phases of differing nontrivial topological order by explicitly constructing Hamiltonians of the Levin-Wen [Levin and Wen, Phys. Rev. B 71, 045110 (2005)] type which can be tuned between two solvable points, each of which realizes a different topologically ordered phase. We show that the low-energy degrees of freedom near the phase transition can be mapped onto those of a Potts model, and we discuss the stability of the resulting phase diagram to small perturbations about the model. We further explain how the excitations in the condensed phase are formed from those in the original topological theory, some of which are split into multiple components by condensation, and we discuss the implications of our results for understanding the nature of general achiral topological phases in 2 +++ 1 dimensions in terms of doubled Chern-Simons theories.
Topological order is a new paradigm for quantum phases of matter developed to explain phase transitions which do not fit the symmetry breaking scheme for classifying phases of matter. They are characterized by patterns of entanglement that lead to topologically depended ground state degeneracy and anyonic excitations. One common approach for studying such phases in two-dimensional systems is through exactly solvable lattice Hamiltonian models such as quantum double models and String-Net models. The former can be understood as the Hamiltonian formulation of lattice gauge theories and, as such, it is defined by a finite gauge group. However, not much is known about topological phases in tridimensional systems. Motivated by this we develop a new class of three-dimensional exactly solvable models which go beyond quantum double models by using finite crossed modules instead of gauge groups. This approach relies on a lattice implementation of 2-gauge theory to obtain models with a richer topological structure. We construct the Hamiltonian model explicitly and provide a rigorous proof that the ground state degeneracy is a topological invariant and that the ground states can only be characterized with nonlocal order parameters.