Boundary Structure and Module Decomposition of the Bosonic $Z_2$ Orbifold Models with $R^2=1/2k$ Boundary Structure and Module Decomposition of the Bosonic $Z_2$ Orbifold Models with $R^2=1/2k$

Hatzinikitas, A, Smyrnakis, J M

The $Z_2$ bosonic orbifold models with compactification radius $R^2=1/2k$ are examined in the presence of boundaries. Demanding the extended algebra characters to have definite conformal dimension and to consist of an integer sum of Virasoro characters, we arrive at the right splitting of the partition function. This is used to derive a free field representation of a complete, consistent set of boundary states, without resorting to a basis of the extended algebra Ishibashi states. Finally the modules of the extended symmetry algebra that correspond to the finitely many characters are identified inside the direct sum of Fock modules that constitute the space of states of the theory.The $Z_2$ bosonic orbifold models with compactification radius $R^2=1/2k$ are examined in the presence of boundaries. Demanding the extended algebra characters to have definite conformal dimension and to consist of an integer sum of Virasoro characters, we arrive at the right splitting of the partition function. This is used to derive a free field representation of a complete, consistent set of boundary states, without resorting to a basis of the extended algebra Ishibashi states. Finally the modules of the extended symmetry algebra that correspond to the finitely many characters are identified inside the direct sum of Fock modules that constitute the space of states of the theory.

Mar 14, 20020 citations

Multiscale entanglement renormalization ansatz for spin chains with continuously varying criticality

Jacob C. Bridgeman, Aroon O’Brien, Stephen D. Bartlett +1 more

We use the multiscale entanglement renormalization ansatz (MERA) to numerically investigate three critical quantum spin chains with ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ on-site symmetry: a staggered XXZ model, a transverse field cluster model, and the quantum Ashkin-Teller model. All three models possess a continuous one-parameter family of critical points. Along this critical line, the thermodynamic limit of these models is expected to be described by classes of $c=1$ conformal field theories (CFTs) of two possible types: the ${S}^{1}$ free boson and its ${Z}_{2}$ orbifold. Our numerics using MERA with explicitly enforced ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry allow us to extract conformal data for each model, with strong evidence supporting the identification of the staggered XXZ model and critical transverse field cluster model with the ${S}^{1}$ boson CFT, and the Ashkin-Teller model with the ${Z}_{2}$ -orbifold boson CFT. Our first two models describe the phase transitions between symmetry-protected topologically ordered phases and trivial phases, which lie outside the usual Landau-Ginsburg-Wilson paradigm of symmetry breaking. Our results show that a range of critical theories can arise at the boundary of a single symmetry protected phase.We use the multiscale entanglement renormalization ansatz (MERA) to numerically investigate three critical quantum spin chains with ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ on-site symmetry: a staggered XXZ model, a transverse field cluster model, and the quantum Ashkin-Teller model. All three models possess a continuous one-parameter family of critical points. Along this critical line, the thermodynamic limit of these models is expected to be described by classes of $c=1$ conformal field theories (CFTs) of two possible types: the ${S}^{1}$ free boson and its ${Z}_{2}$ orbifold. Our numerics using MERA with explicitly enforced ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry allow us to extract conformal data for each model, with strong evidence supporting the identification of the staggered XXZ model and critical transverse field cluster model with the ${S}^{1}$ boson CFT, and the Ashkin-Teller model with the ${Z}_{2}$-orbifold boson CFT. Our first two models describe the phase transitions between symmetry-protected topologically ordered phases and trivial phases, which lie outside the usual Landau-Ginsburg-Wilson paradigm of symmetry breaking. Our results show that a range of critical theories can arise at the boundary of a single symmetry protected phase.

arXiv:1501.02817Physical Review BApr 24, 201535 citations

TOWARDS THE CLASSIFICATION OF Z2×Z2 FERMIONIC MODELS

Alon E. Faraggi, Costas Kounnas, S.E.M. Nooij +1 more

String Phenomenology ConferenceAug 1, 20045 citations

parafermionic current algebras of torus/orbifold models

A. Fujitsu

Physics Letters BApr 1, 19900 citations

Coupled wire model of Z2×Z2 orbifold quantum Hall states

Pok Man Tam, Yichen Hu, C. L. Kane

We construct a coupled wire model for a sequence of non-Abelian quantum Hall states occurring at filling factors $\ensuremath{\nu}=2/(2M+q)$ with integers $M$ and even (odd) integers $q$ for fermionic (bosonic) states. They are termed ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ orbifold states, which have a topological order with a neutral sector described by the $c=1$ orbifold conformal field theory (CFT) at radius ${R}_{\mathrm{orbifold}}=\sqrt{p/2}$ with even integers $p$ . When $p=2$ , the state can be viewed as two decoupled layers of Moore-Read states, whose neutral sector is described by the Ising $\ifmmode\times\else\texttimes\fi{}$ Ising CFT and contains a ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ fusion subalgebra. We demonstrate that orbifold states with $p>2$ , also containing a ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ fusion algebra, can be obtained by coupling together an array of $\mathrm{MR}\ifmmode\times\else\texttimes\fi{}\mathrm{MR}$ wires through local interactions. The corresponding charge spectrum of quasiparticles is also examined. The orbifold states constructed here are complementary to the ${Z}_{4}$ orbifold states, whose neutral edge theory is described by orbifold CFT with odd integer $p$ and contains a ${Z}_{4}$ fusion algebra.We construct a coupled wire model for a sequence of non-Abelian quantum Hall states occurring at filling factors $\ensuremath{\nu}=2/(2M+q)$ with integers $M$ and even (odd) integers $q$ for fermionic (bosonic) states. They are termed ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ orbifold states, which have a topological order with a neutral sector described by the $c=1$ orbifold conformal field theory (CFT) at radius ${R}_{\mathrm{orbifold}}=\sqrt{p/2}$ with even integers $p$. When $p=2$, the state can be viewed as two decoupled layers of Moore-Read states, whose neutral sector is described by the Ising $\ifmmode\times\else\texttimes\fi{}$ Ising CFT and contains a ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ fusion subalgebra. We demonstrate that orbifold states with $p>2$, also containing a ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ fusion algebra, can be obtained by coupling together an array of $\mathrm{MR}\ifmmode\times\else\texttimes\fi{}\mathrm{MR}$ wires through local interactions. The corresponding charge spectrum of quasiparticles is also examined. The orbifold states constructed here are complementary to the ${Z}_{4}$ orbifold states, whose neutral edge theory is described by orbifold CFT with odd integer $p$ and contains a ${Z}_{4}$ fusion algebra.

arXiv:1912.01034Physical review. B./Physical review. BMar 6, 20209 citations

Boundary Structure and Module Decomposition of the Bosonic

Z_2

Orbifold Models with

R^2=1/2k

The

Z_2

bosonic orbifold models with compactification radius

R^2=1/2k

are examined in the presence of boundaries. Demanding the extended algebra characters to have definite conformal dimension and to consist of an integer sum of Virasoro characters, we arrive at the right splitting of the partition function. This is used to derive a free field representation of a complete, consistent set of boundary states, without resorting to a basis of the extended algebra Ishibashi states. Finally the modules of the extended symmetry algebra that correspond to the finitely many characters are identified inside the direct sum of Fock modules that constitute the space of states of the theory.

c=1

conformal field theories (CFTs) of two possible types: the

{S}^{1}

free boson and its

{Z}_{2}

orbifold. Our numerics using MERA with explicitly enforced ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry allow us to extract conformal data for each model, with strong evidence supporting the identification of the staggered XXZ model and critical transverse field cluster model with the

{S}^{1}

boson CFT, and the Ashkin-Teller model with the

{Z}_{2}

-orbifold boson CFT. Our first two models describe the phase transitions between symmetry-protected topologically ordered phases and trivial phases, which lie outside the usual Landau-Ginsburg-Wilson paradigm of symmetry breaking. Our results show that a range of critical theories can arise at the boundary of a single symmetry protected phase.

We construct a coupled wire model for a sequence of non-Abelian quantum Hall states occurring at filling factors $\ensuremath{\nu}=2/(2M+q)$ with integers

M

and even (odd) integers

q

for fermionic (bosonic) states. They are termed ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ orbifold states, which have a topological order with a neutral sector described by the

c=1

orbifold conformal field theory (CFT) at radius

{R}_{\mathrm{orbifold}}=\sqrt{p/2}

with even integers

p

. When

p=2

, the state can be viewed as two decoupled layers of Moore-Read states, whose neutral sector is described by the Ising $\ifmmode\times\else\texttimes\fi{}$ Ising CFT and contains a ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ fusion subalgebra. We demonstrate that orbifold states with

p>2

, also containing a ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ fusion algebra, can be obtained by coupling together an array of $\mathrm{MR}\ifmmode\times\else\texttimes\fi{}\mathrm{MR}$ wires through local interactions. The corresponding charge spectrum of quasiparticles is also examined. The orbifold states constructed here are complementary to the

{Z}_{4}

orbifold states, whose neutral edge theory is described by orbifold CFT with odd integer

p

and contains a

{Z}_{4}

fusion algebra.