We investigate the neighborhood of topological lattice field theories (TLFT's) in the parameter space of general lattice field theories in D\ensuremath{\ge}2 dimensions, and discuss the phase structures associated with them. We first discuss the decomposition of a TLFT into a direct sum of irreducible TLFT's, which cannot be decomposed anymore. Using this decomposed form, we discuss the phase structure and the renormalization group flow near a TLFT in a certain restricted parameter space. We find that a TLFT is on a multiple first-order phase transition point as well as on the fixed point of the flow. The phase structure is controlled by the physical states on a (D-1)-sphere of the TLFT. The flow agrees with the Nienhuis-Nauenberg criterion. We also discuss the neighborhood of a TLFT in general directions by a perturbative method, the so-called cluster expansion. We investigate especially the ${\mathit{Z}}_{\mathit{p}}$ analogue of the Turaev-Viro model, and find that the TLFT is in general on a higher order discrete phase transition point. The phase structures depend on the topology of the base manifold and are controlled by the physical states on topologically nontrivial surfaces. We also discuss the correlation lengths of local fluctuations, and find some long-range modes propagating along topological defects. Thus various discrete phase transitions are associated with TLFT's.We investigate the neighborhood of topological lattice field theories (TLFT's) in the parameter space of general lattice field theories in D\ensuremath{\ge}2 dimensions, and discuss the phase structures associated with them. We first discuss the decomposition of a TLFT into a direct sum of irreducible TLFT's, which cannot be decomposed anymore. Using this decomposed form, we discuss the phase structure and the renormalization group flow near a TLFT in a certain restricted parameter space. We find that a TLFT is on a multiple first-order phase transition point as well as on the fixed point of the flow. The phase structure is controlled by the physical states on a (D-1)-sphere of the TLFT. The flow agrees with the Nienhuis-Nauenberg criterion. We also discuss the neighborhood of a TLFT in general directions by a perturbative method, the so-called cluster expansion. We investigate especially the ${\mathit{Z}}_{\mathit{p}}$ analogue of the Turaev-Viro model, and find that the TLFT is in general on a higher order discrete phase transition point. The phase structures depend on the topology of the base manifold and are controlled by the physical states on topologically nontrivial surfaces. We also discuss the correlation lengths of local fluctuations, and find some long-range modes propagating along topological defects. Thus various discrete phase transitions are associated with TLFT's.

arXiv:hep-th/9401094Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsJan 15, 19951 citation

TOPOLOGICAL QUANTUM NUMBERS AND PHASE TRANSITIONS IN MATTER

D. J. Thouless

Conference in Honor of C N Yang's 85th BirthdayDec 1, 20080 citations

Tricritical point with fractional supersymmetry from a Fibonacci topological state

Wen-Tao Xu, Guang-Ming Zhang

We consider a generic Fibonacci topological wave function on a square lattice, and the norm of this wave function can be mapped into the partition function of two-coupled $ϕ^{2}$ -state Potts models with $ϕ=(\sqrt{5}+1)/2$ as the golden ratio. A global phase diagram is thus established to display non-abelian topological phase transitions. The Fibonacci topological phase corresponds to an emergent new phase of the two-coupled Potts models, and continuously change into two non-topological phases separately, which are dual each other and divided by a first-order phase transition line. Under the self-duality, the Fibonacci topological state enters into the first-order transition state at a quantum tricritical point, where two continuous quantum phase transitions bifurcate. All the topological phase transitions are driven by condensation of anyonic bosons consisting of Fibonacci anyon and its conjugate. However, a fractional supersymmetry is displayed at the quantum tricritical point, characterized by a coset conformal field theory.We consider a generic Fibonacci topological wave function on a square lattice, and the norm of this wave function can be mapped into the partition function of two-coupled $ϕ^{2}$-state Potts models with $ϕ=(\sqrt{5}+1)/2$ as the golden ratio. A global phase diagram is thus established to display non-abelian topological phase transitions. The Fibonacci topological phase corresponds to an emergent new phase of the two-coupled Potts models, and continuously change into two non-topological phases separately, which are dual each other and divided by a first-order phase transition line. Under the self-duality, the Fibonacci topological state enters into the first-order transition state at a quantum tricritical point, where two continuous quantum phase transitions bifurcate. All the topological phase transitions are driven by condensation of anyonic bosons consisting of Fibonacci anyon and its conjugate. However, a fractional supersymmetry is displayed at the quantum tricritical point, characterized by a coset conformal field theory.

arXiv:1905.09960May 23, 20193 citations

Charge response of the Majorana toric code

Ananda Roy, Fabian Hassler

At zero temperature, a two-dimensional lattice of Majorana zero modes on mesoscopic superconducting islands has a topologically ordered toric-code phase. Recently, a Landau field theory has been proposed for the system that captures its different phases and the associated phase transitions. It was shown that with the increase of Josephson tunneling between the islands, a continuous symmetry-breaking three-dimensional (3D)-XY transition gets transformed into a discrete symmetry-breaking 3D-Ising transition through a couple of tricritical points and first-order transitions. Using the proposed field theory, we analyze the charge response of the system at the different continuous phase transitions. We calculate the universal conductivity at the 3D-XY transitions and the change in the superconducting density at the Ising transition using 1/ $N$ expansion. Furthermore, by computing a one-loop correction to the field theory, we show that an additional tricritical point is likely to be present in the phase diagram. Finally, we provide a mean-field calculation that supports the earlier proposed field theory.At zero temperature, a two-dimensional lattice of Majorana zero modes on mesoscopic superconducting islands has a topologically ordered toric-code phase. Recently, a Landau field theory has been proposed for the system that captures its different phases and the associated phase transitions. It was shown that with the increase of Josephson tunneling between the islands, a continuous symmetry-breaking three-dimensional (3D)-XY transition gets transformed into a discrete symmetry-breaking 3D-Ising transition through a couple of tricritical points and first-order transitions. Using the proposed field theory, we analyze the charge response of the system at the different continuous phase transitions. We calculate the universal conductivity at the 3D-XY transitions and the change in the superconducting density at the Ising transition using 1/$N$ expansion. Furthermore, by computing a one-loop correction to the field theory, we show that an additional tricritical point is likely to be present in the phase diagram. Finally, we provide a mean-field calculation that supports the earlier proposed field theory.

arXiv:1710.04168Physical review. B./Physical review. BJan 22, 20184 citations

Topological Phase Transition Theory of Strongly Correlated Many Body System

Tieyan Si

Jan 2, 20060 citations

{\mathit{Z}}_{\mathit{p}}

analogue of the Turaev-Viro model, and find that the TLFT is in general on a higher order discrete phase transition point. The phase structures depend on the topology of the base manifold and are controlled by the physical states on topologically nontrivial surfaces. We also discuss the correlation lengths of local fluctuations, and find some long-range modes propagating along topological defects. Thus various discrete phase transitions are associated with TLFT's.

We consider a generic Fibonacci topological wave function on a square lattice, and the norm of this wave function can be mapped into the partition function of two-coupled

ϕ^{2}

-state Potts models with

ϕ=(\sqrt{5}+1)/2

as the golden ratio. A global phase diagram is thus established to display non-abelian topological phase transitions. The Fibonacci topological phase corresponds to an emergent new phase of the two-coupled Potts models, and continuously change into two non-topological phases separately, which are dual each other and divided by a first-order phase transition line. Under the self-duality, the Fibonacci topological state enters into the first-order transition state at a quantum tricritical point, where two continuous quantum phase transitions bifurcate. All the topological phase transitions are driven by condensation of anyonic bosons consisting of Fibonacci anyon and its conjugate. However, a fractional supersymmetry is displayed at the quantum tricritical point, characterized by a coset conformal field theory.

N

expansion. Furthermore, by computing a one-loop correction to the field theory, we show that an additional tricritical point is likely to be present in the phase diagram. Finally, we provide a mean-field calculation that supports the earlier proposed field theory.