Dynamic Duality and the Phase Structure of Some Non-Abelian Spin Systems.

We introduce a family of quantum spin Hamiltonians on Z2 that can be regarded as perturbations of Kitaev’s Abelian quantum double models that preserve the gauge and duality symmetries of these models. We analyze in detail the sector with one electric charge and one magnetic flux and show that the spectrum in this sector consists of both bound states and scattering states of Abelian anyons. Concretely, we have defined a family of lattice models in which Abelian anyons arise naturally as finite-size quasi-particles with non-trivial dynamics that consist of a charge-flux pair. In particular, the anyons exhibit a non-trivial holonomy with a quantized phase, consistent with the gauge and duality symmetries of the Hamiltonian.We introduce a family of quantum spin Hamiltonians on Z2 that can be regarded as perturbations of Kitaev’s Abelian quantum double models that preserve the gauge and duality symmetries of these models. We analyze in detail the sector with one electric charge and one magnetic flux and show that the spectrum in this sector consists of both bound states and scattering states of Abelian anyons. Concretely, we have defined a family of lattice models in which Abelian anyons arise naturally as finite-size quasi-particles with non-trivial dynamics that consist of a charge-flux pair. In particular, the anyons exhibit a non-trivial holonomy with a quantized phase, consistent with the gauge and duality symmetries of the Hamiltonian.

arXiv:2303.07379Journal of Mathematical PhysicsJul 1, 20233 citations

Realization and stability of non-Abelian chiral quantum spin liquids via dimensional reduction

Rodrigo Corso B. Santos, Carlos A. Hernaski

A bstract This work is concerned with the realization and stability of a non-Abelian chiral quantum spin liquid phase. To do so, we cast the problem in a quantum wires framework, which is a dimensional deconstruction framework that allows us to study the (2+1) dimensional spin liquids phase from a series of coupled (1+1) dimensional theories. The lower dimension grants us the ability to perform a bosonization procedure, which yields two different partition functions connected by a strong-weak duality transformation. This bosonization procedure is illuminating in that it makes the fixed point structure of the model unequivocal. Then, we proceed by studying the RG flow through the β -functions, which we use to determine the phase structure. We find that the quantum spin liquid phase is realized and stable in the deep IR limit.A bstract This work is concerned with the realization and stability of a non-Abelian chiral quantum spin liquid phase. To do so, we cast the problem in a quantum wires framework, which is a dimensional deconstruction framework that allows us to study the (2+1) dimensional spin liquids phase from a series of coupled (1+1) dimensional theories. The lower dimension grants us the ability to perform a bosonization procedure, which yields two different partition functions connected by a strong-weak duality transformation. This bosonization procedure is illuminating in that it makes the fixed point structure of the model unequivocal. Then, we proceed by studying the RG flow through the β -functions, which we use to determine the phase structure. We find that the quantum spin liquid phase is realized and stable in the deep IR limit.

arXiv:2408.05327Journal of High Energy PhysicsMar 28, 20250 citations

On the phase transitions in quantum spin systems

R. Micnas, K. A. Chao

Physics Letters AJul 1, 19853 citations

Non-Abelian fermionization and the landscape of quantum Hall phases

Hart Goldman, Ramanjit Sohal, Eduardo Fradkin

The recent proposal of non-Abelian boson-fermion dualities in $2+1$ dimensions, which morally relate $U{(k)}_{N}$ to $SU{(N)}_{\ensuremath{-}k}$ Chern-Simons-matter theories, presents a new platform for exploring the landscape of non-Abelian quantum Hall states accessible from theories of Abelian composite particles. Here we focus on dualities relating theories of Abelian quantum Hall states of bosons or fermions to theories of non-Abelian ``composite fermions'' partially filling Landau levels. We show that these dualities predict special filling fractions where both Abelian and non-Abelian composite fermion theories appear capable of hosting distinct topologically ordered ground states, one Abelian and the other a non-Abelian, $U{(k)}_{2}$ Blok-Wen state. Rather than being in conflict with the duality, we argue that these results indicate unexpected dynamics in which the infrared and lowest Landau level limits fail to commute across the duality. In such a scenario, the non-Abelian topological order can be destabilized in favor of the Abelian ground state, suggesting the presence of a phase transition between the Abelian and non-Abelian states that is likely to be first order. We also generalize these constructions to other non-Abelian fermion-fermion dualities, in the process obtaining new derivations of a variety of paired composite fermion phases using duality, including the anti-Pfaffian state. Finally, we describe how, in multilayer constructions, excitonic pairing of the composite fermions across $N$ layers can also generate the family of Blok-Wen states with $U{(k)}_{2}$ topological order.The recent proposal of non-Abelian boson-fermion dualities in $2+1$ dimensions, which morally relate $U{(k)}_{N}$ to $SU{(N)}_{\ensuremath{-}k}$ Chern-Simons-matter theories, presents a new platform for exploring the landscape of non-Abelian quantum Hall states accessible from theories of Abelian composite particles. Here we focus on dualities relating theories of Abelian quantum Hall states of bosons or fermions to theories of non-Abelian ``composite fermions'' partially filling Landau levels. We show that these dualities predict special filling fractions where both Abelian and non-Abelian composite fermion theories appear capable of hosting distinct topologically ordered ground states, one Abelian and the other a non-Abelian, $U{(k)}_{2}$ Blok-Wen state. Rather than being in conflict with the duality, we argue that these results indicate unexpected dynamics in which the infrared and lowest Landau level limits fail to commute across the duality. In such a scenario, the non-Abelian topological order can be destabilized in favor of the Abelian ground state, suggesting the presence of a phase transition between the Abelian and non-Abelian states that is likely to be first order. We also generalize these constructions to other non-Abelian fermion-fermion dualities, in the process obtaining new derivations of a variety of paired composite fermion phases using duality, including the anti-Pfaffian state. Finally, we describe how, in multilayer constructions, excitonic pairing of the composite fermions across $N$ layers can also generate the family of Blok-Wen states with $U{(k)}_{2}$ topological order.

arXiv:2009.00011Physical review. B./Physical review. BNov 30, 20208 citations

Dynamical phase transitions in spin models and automata

Bernard Derrida, Henk van Beijeren

Jan 1, 19906 citations

The recent proposal of non-Abelian boson-fermion dualities in

2+1

dimensions, which morally relate

U{(k)}_{N}

to $SU{(N)}_{\ensuremath{-}k}$ Chern-Simons-matter theories, presents a new platform for exploring the landscape of non-Abelian quantum Hall states accessible from theories of Abelian composite particles. Here we focus on dualities relating theories of Abelian quantum Hall states of bosons or fermions to theories of non-Abelian ``composite fermions'' partially filling Landau levels. We show that these dualities predict special filling fractions where both Abelian and non-Abelian composite fermion theories appear capable of hosting distinct topologically ordered ground states, one Abelian and the other a non-Abelian,

U{(k)}_{2}

Blok-Wen state. Rather than being in conflict with the duality, we argue that these results indicate unexpected dynamics in which the infrared and lowest Landau level limits fail to commute across the duality. In such a scenario, the non-Abelian topological order can be destabilized in favor of the Abelian ground state, suggesting the presence of a phase transition between the Abelian and non-Abelian states that is likely to be first order. We also generalize these constructions to other non-Abelian fermion-fermion dualities, in the process obtaining new derivations of a variety of paired composite fermion phases using duality, including the anti-Pfaffian state. Finally, we describe how, in multilayer constructions, excitonic pairing of the composite fermions across

N

layers can also generate the family of Blok-Wen states with

U{(k)}_{2}

topological order.