We propose a systematic approach to constructing microscopic models with fractional excitations in three-dimensional (3D) space. Building blocks are quantum wires described by the ( $1+1$ )-dimensional conformal field theory (CFT) associated with a current algebra $\mathfrak{g}$ . The wires are coupled with each other to form a 3D network through the current-current interactions of ${\mathfrak{g}}_{1}$ and ${\mathfrak{g}}_{2}$ CFTs that are related to the $\mathfrak{g}$ CFT by a nontrivial conformal embedding $\mathfrak{g}\ensuremath{\supset}{\mathfrak{g}}_{1}\ifmmode\times\else\texttimes\fi{}{\mathfrak{g}}_{2}$. The resulting model can be viewed as a layer construction of a 3D topologically ordered state, in which the conformal embedding in each wire implements the anyon condensation between adjacent layers. Local operators acting on the ground state create pointlike or looplike deconfined excitations depending on the branching rule. We demonstrate our construction for a simple solvable model based on the conformal embedding $SU{(2)}_{1}\ifmmode\times\else\texttimes\fi{}SU{(2)}_{1}\ensuremath{\supset}U{(1)}_{4}\ifmmode\times\else\texttimes\fi{}U{(1)}_{4}$. We show that the model possesses extensively degenerate ground states on a torus with deconfined quasiparticles, and that appropriate local perturbations lift the degeneracy and yield a 3D ${Z}_{2}$ gauge theory with a fermionic ${Z}_{2}$ charge.We propose a systematic approach to constructing microscopic models with fractional excitations in three-dimensional (3D) space. Building blocks are quantum wires described by the ($1+1$)-dimensional conformal field theory (CFT) associated with a current algebra $\mathfrak{g}$. The wires are coupled with each other to form a 3D network through the current-current interactions of ${\mathfrak{g}}_{1}$ and ${\mathfrak{g}}_{2}$ CFTs that are related to the $\mathfrak{g}$ CFT by a nontrivial conformal embedding $\mathfrak{g}\ensuremath{\supset}{\mathfrak{g}}_{1}\ifmmode\times\else\texttimes\fi{}{\mathfrak{g}}_{2}$. The resulting model can be viewed as a layer construction of a 3D topologically ordered state, in which the conformal embedding in each wire implements the anyon condensation between adjacent layers. Local operators acting on the ground state create pointlike or looplike deconfined excitations depending on the branching rule. We demonstrate our construction for a simple solvable model based on the conformal embedding $SU{(2)}_{1}\ifmmode\times\else\texttimes\fi{}SU{(2)}_{1}\ensuremath{\supset}U{(1)}_{4}\ifmmode\times\else\texttimes\fi{}U{(1)}_{4}$. We show that the model possesses extensively degenerate ground states on a torus with deconfined quasiparticles, and that appropriate local perturbations lift the degeneracy and yield a 3D ${Z}_{2}$ gauge theory with a fermionic ${Z}_{2}$ charge.

arXiv:1901.05918Physical review. B./Physical review. BJun 12, 201914 citations

Physics of conformal field theories

Yevgeny Kats

Jan 1, 20100 citations

Quantum field theories in finite volume: Excited state energies

Vladimir V. Bazhanov, Sergei L. Lukyanov, Alexander B. Zamolodchikov

We develop a method of computing the excited state energies in Integrable Quantum Field Theories (IQFT) in finite geometry, with the spatial coordinate compactified on a circle of circumference R. The IQFT “commuting transfer matrices” introduced earlier [Commun. Math. Phys. 177 (1996) 381] for Conformal Field Theories (CFT) are generalized to non-conformal IQFT obtained by perturbing CFT with the operator Φ1,3. We study the models in which the fusion relations for these “transfer matrices” truncate and provide closed integral equations which generalize the equations of the thermodynamic Bethe ansatz to excited states. The explicit calculations are done for the first excited state in the “scaling Lee-Yang model”.We develop a method of computing the excited state energies in Integrable Quantum Field Theories (IQFT) in finite geometry, with the spatial coordinate compactified on a circle of circumference R. The IQFT “commuting transfer matrices” introduced earlier [Commun. Math. Phys. 177 (1996) 381] for Conformal Field Theories (CFT) are generalized to non-conformal IQFT obtained by perturbing CFT with the operator Φ1,3. We study the models in which the fusion relations for these “transfer matrices” truncate and provide closed integral equations which generalize the equations of the thermodynamic Bethe ansatz to excited states. The explicit calculations are done for the first excited state in the “scaling Lee-Yang model”.

arXiv:hep-th/9607099Nuclear Physics BApr 1, 1997216 citations

Entanglement entropy of excited states in conformal perturbation theory and the Einstein equation

Antony J. Speranza

For a conformal field theory (CFT) deformed by a relevant operator, the entanglement entropy of a ball-shaped region may be computed as a perturbative expansion in the coupling. A similar perturbative expansion exists for excited states near the vacuum. Using these expansions, this work investigates the behavior of excited state entanglement entropies of small, ball-shaped regions. The motivation for these calculations is Jacobson’s recent work on the equivalence of the Einstein equation and the hypothesis of maximal vacuum entropy [ arXiv:1505.04753 ], which relies on a conjecture stating that the behavior of these entropies is sufficiently similar to a CFT. In addition to the expected type of terms which scale with the ball radius as R d , the entanglement entropy calculation gives rise to terms scaling as R 2Δ, where Δ is the dimension of the deforming operator. When $\varDelta \le \frac{d}{2}$ , the latter terms dominate the former, and suggest that a modification to the conjecture is needed.For a conformal field theory (CFT) deformed by a relevant operator, the entanglement entropy of a ball-shaped region may be computed as a perturbative expansion in the coupling. A similar perturbative expansion exists for excited states near the vacuum. Using these expansions, this work investigates the behavior of excited state entanglement entropies of small, ball-shaped regions. The motivation for these calculations is Jacobson’s recent work on the equivalence of the Einstein equation and the hypothesis of maximal vacuum entropy [ arXiv:1505.04753 ], which relies on a conjecture stating that the behavior of these entropies is sufficiently similar to a CFT. In addition to the expected type of terms which scale with the ball radius as R d , the entanglement entropy calculation gives rise to terms scaling as R 2Δ, where Δ is the dimension of the deforming operator. When $$ \varDelta \le \frac{d}{2} $$ , the latter terms dominate the former, and suggest that a modification to the conjecture is needed.

arXiv:1602.01380Journal of High Energy PhysicsApr 1, 201622 citations

We propose a systematic approach to constructing microscopic models with fractional excitations in three-dimensional (3D) space. Building blocks are quantum wires described by the (

1+1

)-dimensional conformal field theory (CFT) associated with a current algebra

\mathfrak{g}

. The wires are coupled with each other to form a 3D network through the current-current interactions of

{\mathfrak{g}}_{1}

and

{\mathfrak{g}}_{2}

CFTs that are related to the

\mathfrak{g}

CFT by a nontrivial conformal embedding $\mathfrak{g}\ensuremath{\supset}{\mathfrak{g}}_{1}\ifmmode\times\else\texttimes\fi{}{\mathfrak{g}}_{2}$. The resulting model can be viewed as a layer construction of a 3D topologically ordered state, in which the conformal embedding in each wire implements the anyon condensation between adjacent layers. Local operators acting on the ground state create pointlike or looplike deconfined excitations depending on the branching rule. We demonstrate our construction for a simple solvable model based on the conformal embedding $SU{(2)}_{1}\ifmmode\times\else\texttimes\fi{}SU{(2)}_{1}\ensuremath{\supset}U{(1)}_{4}\ifmmode\times\else\texttimes\fi{}U{(1)}_{4}$. We show that the model possesses extensively degenerate ground states on a torus with deconfined quasiparticles, and that appropriate local perturbations lift the degeneracy and yield a 3D

{Z}_{2}

gauge theory with a fermionic

{Z}_{2}

charge.

\varDelta \le \frac{d}{2}

, the latter terms dominate the former, and suggest that a modification to the conjecture is needed.