The same bulk two-dimensional topological phase can have multiple distinct, fully chiral edge phases. We show that this can occur in the integer quantum Hall states at $\ensuremath{\nu}=8$ and 12, with experimentally testable consequences. We show that this can occur in Abelian fractional quantum Hall states as well, with the simplest examples being at $\ensuremath{\nu}=\frac{8}{7},\frac{12}{11},\frac{8}{15},\frac{16}{5}$. We give a general criterion for the existence of multiple distinct chiral edge phases for the same bulk phase and discuss experimental consequences. Edge phases correspond to lattices while bulk phases correspond to genera of lattices. Since there are typically multiple lattices in a genus, the bulk-edge correspondence is typically one-to-many; there are usually many stable fully chiral edge phases corresponding to the same bulk. We explain these correspondences using the theory of integral quadratic forms. We show that fermionic systems can have edge phases with only bosonic low-energy excitations and discuss a fermionic generalization of the relation between bulk topological spins and the central charge. The latter follows from our demonstration that every fermionic topological phase can be represented as a bosonic topological phase, together with some number of filled Landau levels. Our analysis shows that every Abelian topological phase can be decomposed into a tensor product of theories associated with prime numbers $p$ in which every quasiparticle has a topological spin that is a ${p}^{n}$ th root of unity for some $n$ . It also leads to a simple demonstration that all Abelian topological phases can be represented by $\mathrm{U}{(1)}^{N}$ Chern-Simons theory parameterized by a $K$ matrix.The same bulk two-dimensional topological phase can have multiple distinct, fully chiral edge phases. We show that this can occur in the integer quantum Hall states at $\ensuremath{\nu}=8$ and 12, with experimentally testable consequences. We show that this can occur in Abelian fractional quantum Hall states as well, with the simplest examples being at $\ensuremath{\nu}=\frac{8}{7},\frac{12}{11},\frac{8}{15},\frac{16}{5}$. We give a general criterion for the existence of multiple distinct chiral edge phases for the same bulk phase and discuss experimental consequences. Edge phases correspond to lattices while bulk phases correspond to genera of lattices. Since there are typically multiple lattices in a genus, the bulk-edge correspondence is typically one-to-many; there are usually many stable fully chiral edge phases corresponding to the same bulk. We explain these correspondences using the theory of integral quadratic forms. We show that fermionic systems can have edge phases with only bosonic low-energy excitations and discuss a fermionic generalization of the relation between bulk topological spins and the central charge. The latter follows from our demonstration that every fermionic topological phase can be represented as a bosonic topological phase, together with some number of filled Landau levels. Our analysis shows that every Abelian topological phase can be decomposed into a tensor product of theories associated with prime numbers $p$ in which every quasiparticle has a topological spin that is a ${p}^{n}$th root of unity for some $n$. It also leads to a simple demonstration that all Abelian topological phases can be represented by $\mathrm{U}{(1)}^{N}$ Chern-Simons theory parameterized by a $K$ matrix.

arXiv:1310.5708Physical Review BMar 14, 2014123 citations

Edge-state critical behavior of the integer quantum Hall transition

Martin Puschmann, Philipp Cain, Michael Schreiber +1 more

arXiv:2004.01611The European Physical Journal Special TopicsApr 7, 20213 citations

Edge physics at the deconfined transition between a quantum spin Hall insulator and a superconductor

Ruochen Ma, Liujun Zou, Chong Wang

We study the edge physics of the deconfined quantum phase transition (DQCP) between a spontaneous quantum spin Hall (QSH) insulator and a spin-singlet superconductor (SC). Although the bulk of this transition is in the same universality class as the paradigmatic deconfined Neel to valence-bond-solid transition, the boundary physics has a richer structure due to proximity to a quantum spin Hall state. We use the parton trick to write down an effective field theory for the QSH-SC transition in the presence of a boundary. We calculate various edge properties in an N\to\infty N → ∞ limit. We show that the boundary Luttinger liquid in the QSH state survives at the phase transition, but only as ``fractional’’ degrees of freedom that carry charge but not spin. The physical fermion remains gapless on the edge at the critical point, with a universal jump in the fermion scaling dimension as the system approaches the transition from the QSH side. The critical point could be viewed as a gapless analogue of the quantum spin Hall state but with the full SU(2) S U ( 2 ) spin rotation symmetry, which cannot be realized if the bulk is gapped.We study the edge physics of the deconfined quantum phase transition (DQCP) between a spontaneous quantum spin Hall (QSH) insulator and a spin-singlet superconductor (SC). Although the bulk of this transition is in the same universality class as the paradigmatic deconfined Neel to valence-bond-solid transition, the boundary physics has a richer structure due to proximity to a quantum spin Hall state. We use the parton trick to write down an effective field theory for the QSH-SC transition in the presence of a boundary. We calculate various edge properties in an N\to\infty N → ∞ limit. We show that the boundary Luttinger liquid in the QSH state survives at the phase transition, but only as ``fractional’’ degrees of freedom that carry charge but not spin. The physical fermion remains gapless on the edge at the critical point, with a universal jump in the fermion scaling dimension as the system approaches the transition from the QSH side. The critical point could be viewed as a gapless analogue of the quantum spin Hall state but with the full SU(2) S U ( 2 ) spin rotation symmetry, which cannot be realized if the bulk is gapped.

arXiv:2110.08280SciPost PhysicsJun 14, 202225 citations

Phase transitions on the edge of the ν=5/2 Pfaffian and anti-Pfaffian quantum Hall state

B. J. Overbosch, Xiao-Gang Wen

Starting from the edge reconstructed Pfaffian state and the anti-Pfaffian state for the filling fraction $ν={5/2}$ fractional quantum Hall (FQH) state with the filled Landau levels included, we find that interactions between counterpropagating edge modes can induce phase transitions on the edge. In the new `Majorana-gapped' phases, a pair of counter propagating neutral Majorana modes becomes gapped. The quasiparticle tunneling exponent changes from $g=1/4$ to $g=1/2$ for the edge reconstructed Pfaffian state, and changes from $g=1/2$ to $g=0.55-0.75$ for the anti-Pfaffian state, in the new Majorana-gapped phases. The new phases are candidate states for the observed $ν={5/2}$ state. Furthermore, Majorana-gapped phases provide examples that non-trivial quantum phase transitions can happen on the edge of a FQH state without any change in bulk topological order.Starting from the edge reconstructed Pfaffian state and the anti-Pfaffian state for the filling fraction $ν={5/2}$ fractional quantum Hall (FQH) state with the filled Landau levels included, we find that interactions between counterpropagating edge modes can induce phase transitions on the edge. In the new `Majorana-gapped' phases, a pair of counter propagating neutral Majorana modes becomes gapped. The quasiparticle tunneling exponent changes from $g=1/4$ to $g=1/2$ for the edge reconstructed Pfaffian state, and changes from $g=1/2$ to $g=0.55-0.75$ for the anti-Pfaffian state, in the new Majorana-gapped phases. The new phases are candidate states for the observed $ν={5/2}$ state. Furthermore, Majorana-gapped phases provide examples that non-trivial quantum phase transitions can happen on the edge of a FQH state without any change in bulk topological order.

arXiv:0804.2087Apr 13, 20086 citations

Phases and Phase Transitions in the Quantum Hall Effect

D. H. Lee

Jan 1, 20000 citations

p

in which every quasiparticle has a topological spin that is a

{p}^{n}

th root of unity for some

n

. It also leads to a simple demonstration that all Abelian topological phases can be represented by

\mathrm{U}{(1)}^{N}

Chern-Simons theory parameterized by a

K

matrix.

Starting from the edge reconstructed Pfaffian state and the anti-Pfaffian state for the filling fraction

ν={5/2}

fractional quantum Hall (FQH) state with the filled Landau levels included, we find that interactions between counterpropagating edge modes can induce phase transitions on the edge. In the new `Majorana-gapped' phases, a pair of counter propagating neutral Majorana modes becomes gapped. The quasiparticle tunneling exponent changes from

g=1/4

g=1/2

for the edge reconstructed Pfaffian state, and changes from

g=1/2

g=0.55-0.75

for the anti-Pfaffian state, in the new Majorana-gapped phases. The new phases are candidate states for the observed

ν={5/2}

state. Furthermore, Majorana-gapped phases provide examples that non-trivial quantum phase transitions can happen on the edge of a FQH state without any change in bulk topological order.