We describe a coupled-chain construction for chiral spin liquids in two-dimensional spin systems. Starting from a one-dimensional zigzag spin chain and imposing SU(2) symmetry in the framework of non-Abelian bosonization, we first show that our approach faithfully describes the low-energy physics of an exactly solvable model with a three-spin interaction. Generalizing the construction to the two-dimensional case, we obtain a theory that incorporates the universal properties of the chiral spin liquid predicted by Kalmeyer and Laughlin: charge-neutral edge states, gapped spin-1/2 bulk excitations, and ground state degeneracy on the torus signalling the topological order of this quantum state. In addition, we show that the chiral spin liquid phase is more easily stabilized in frustrated lattices containing corner-sharing triangles, such as the extended kagome lattice, than in the triangular lattice. Our field theoretical approach invites generalizations to more exotic chiral spin liquids and may be used to assess the existence of the chiral spin liquid as the ground state of specific lattice systems.We describe a coupled-chain construction for chiral spin liquids in two-dimensional spin systems. Starting from a one-dimensional zigzag spin chain and imposing SU(2) symmetry in the framework of non-Abelian bosonization, we first show that our approach faithfully describes the low-energy physics of an exactly solvable model with a three-spin interaction. Generalizing the construction to the two-dimensional case, we obtain a theory that incorporates the universal properties of the chiral spin liquid predicted by Kalmeyer and Laughlin: charge-neutral edge states, gapped spin-1/2 bulk excitations, and ground state degeneracy on the torus signalling the topological order of this quantum state. In addition, we show that the chiral spin liquid phase is more easily stabilized in frustrated lattices containing corner-sharing triangles, such as the extended kagome lattice, than in the triangular lattice. Our field theoretical approach invites generalizations to more exotic chiral spin liquids and may be used to assess the existence of the chiral spin liquid as the ground state of specific lattice systems.

Bulletin of the American Physical SocietyMar 17, 20160 citations

Triangular lattice models of the Kalmeyer-Laughlin spin liquid from coupled wires

Tingyu Gao, Niklas Tausendpfund, Erik Lennart Weerda +3 more

Chiral spin liquids (CSLs) are exotic phases of interacting spins in two dimensions, characterized by long-range entanglement and fractional excitations. We construct a local Hamiltonian on the triangular lattice that stabilizes the Kalmeyer-Laughlin CSL without requiring fine-tuning. Our approach employs coupled-wire constructions and introduces a lattice duality to construct a solvable chiral sliding Luttinger liquid, which is driven toward the CSL phase by generic perturbations. By combining symmetry analysis and bosonization, we make sharp predictions for the ground states on quasi-one-dimensional cylinders and tori, which exhibit a fourfold periodicity in the circumference. Extensive tensor network simulations demonstrating ground-state degeneracies, fractional quasiparticles, nonvanishing long-range order parameters, and entanglement signatures confirm the emergence of the CSL in the lattice Hamiltonian.Chiral spin liquids (CSLs) are exotic phases of interacting spins in two dimensions, characterized by long-range entanglement and fractional excitations. We construct a local Hamiltonian on the triangular lattice that stabilizes the Kalmeyer-Laughlin CSL without requiring fine-tuning. Our approach employs coupled-wire constructions and introduces a lattice duality to construct a solvable chiral sliding Luttinger liquid, which is driven toward the CSL phase by generic perturbations. By combining symmetry analysis and bosonization, we make sharp predictions for the ground states on quasi-one-dimensional cylinders and tori, which exhibit a fourfold periodicity in the circumference. Extensive tensor network simulations demonstrating ground-state degeneracies, fractional quasiparticles, nonvanishing long-range order parameters, and entanglement signatures confirm the emergence of the CSL in the lattice Hamiltonian.

arXiv:2502.13223Physical review. B./Physical review. BNov 8, 20250 citations

Lattice spin models for non-Abelian chiral spin liquids

P. Lecheminant, Alexei M. Tsvelik

We suggest a class of two-dimensional lattice spin Hamiltonians describing non-Abelian SU(2) chiral spin liquids---spin analogs of fractional non-Abelian quantum Hall states---with gapped bulk and gapless chiral edge excitations described by the SU(2) ${}_{n}$ Wess-Zumino-Novikov-Witten conformal field theory. The models are constructed from an array of generalized spin- $n/2$ ladders with multi-spin-exchange interactions which are coupled by isolated spins. Such models allow a controllable analytic treatment starting from the one-dimensional limit and are characterized by a bulk gap and non-Abelian SU(2) ${}_{n}$ gapless edge excitations.We suggest a class of two-dimensional lattice spin Hamiltonians describing non-Abelian SU(2) chiral spin liquids---spin analogs of fractional non-Abelian quantum Hall states---with gapped bulk and gapless chiral edge excitations described by the SU(2)${}_{n}$ Wess-Zumino-Novikov-Witten conformal field theory. The models are constructed from an array of generalized spin-$n/2$ ladders with multi-spin-exchange interactions which are coupled by isolated spins. Such models allow a controllable analytic treatment starting from the one-dimensional limit and are characterized by a bulk gap and non-Abelian SU(2)${}_{n}$ gapless edge excitations.

arXiv:1608.05977Physical review. B./Physical review. BApr 26, 201739 citations

Gapped and gapless spin liquid phases on the Kagome lattice from chiral three-spin interactions

Bela Bauer, Brendan P. Keller, Michele Dolfi +2 more

We argue that a relatively simple model containing only SU(2)-invariant chiral three-spin interactions on a Kagome lattice of S=1/2 spins can give rise to both a gapped and a gapless quantum spin liquid. Our arguments are rooted in a formulation in terms of network models of edge states and are backed up by a careful numerical analysis. For a uniform choice of chirality on the lattice, we realize the Kalmeyer-Laughlin state, i.e. a gapped spin liquid which is identified as the nu=1/2 bosonic Laughlin state. For staggered chiralities, a gapless spin liquid emerges which exhibits gapless spin excitations along lines in momentum space, a feature that we probe by studying quasi-two-dimensional systems of finite width. We thus provide a single, appealingly simple spin model (i) for what is probably the simplest realization of the Kalmeyer-Laughlin state to date, as well as (ii) for a non-Fermi liquid state with lines of gapless SU(2) spin excitations.We argue that a relatively simple model containing only SU(2)-invariant chiral three-spin interactions on a Kagome lattice of S=1/2 spins can give rise to both a gapped and a gapless quantum spin liquid. Our arguments are rooted in a formulation in terms of network models of edge states and are backed up by a careful numerical analysis. For a uniform choice of chirality on the lattice, we realize the Kalmeyer-Laughlin state, i.e. a gapped spin liquid which is identified as the nu=1/2 bosonic Laughlin state. For staggered chiralities, a gapless spin liquid emerges which exhibits gapless spin excitations along lines in momentum space, a feature that we probe by studying quasi-two-dimensional systems of finite width. We thus provide a single, appealingly simple spin model (i) for what is probably the simplest realization of the Kalmeyer-Laughlin state to date, as well as (ii) for a non-Fermi liquid state with lines of gapless SU(2) spin excitations.

arXiv:1303.6963Mar 27, 20136 citations

Network construction of non-Abelian topological phases

Hernán Guillermo Bueno Xavier

In this thesis we use a network of chiral junctions to realize non-Abelian topological phases in two dimensions. Our construction is based on boundary fixed points of low-energy field theories describing Y junctions of critical spin-chain models. To set the stage, we first study a single junction of gapless spin-1 chains belonging to the SU(2)2 universality class, whose spectrum includes fractional excitations such as Majorana fermions. We find that chiral fixed points appear as special points on a transition line that separates two regimes described by open boundary conditions. Remarkably, along this transition the junction behaves as a tunable spin circulator as the spin conductance varies continuously with the coupling constant of a marginal boundary operator. We then construct a family of topological chiral spin liquids departing from a honeycomb network made of critical spin-S chains. The chiral spin liquid phase harbors SU(2)k=2S anyons, which stem from the underlying WessZumino-Witten models that describe the constituent spin chains of the network. The network exhibits quantized spin and thermal Hall conductances. We illustrate our construction by inspecting the topological properties of the SU(2)2 model. We find that this model has emergent Ising anyons, with spinons acting as vortex excitations that bind Majorana zero modes. We also show that the ground state of this network is threefold degenerate on the torus, asserting its non-Abelian character. Our work provides a controllable analytical framework to study non-Abelian topological phases, sheding new light on the stability of such phases in artificial quantum materials.In this thesis we use a network of chiral junctions to realize non-Abelian topological phases in two dimensions. Our construction is based on boundary fixed points of low-energy field theories describing Y junctions of critical spin-chain models. To set the stage, we first study a single junction of gapless spin-1 chains belonging to the SU(2)2 universality class, whose spectrum includes fractional excitations such as Majorana fermions. We find that chiral fixed points appear as special points on a transition line that separates two regimes described by open boundary conditions. Remarkably, along this transition the junction behaves as a tunable spin circulator as the spin conductance varies continuously with the coupling constant of a marginal boundary operator. We then construct a family of topological chiral spin liquids departing from a honeycomb network made of critical spin-S chains. The chiral spin liquid phase harbors SU(2)k=2S anyons, which stem from the underlying WessZumino-Witten models that describe the constituent spin chains of the network. The network exhibits quantized spin and thermal Hall conductances. We illustrate our construction by inspecting the topological properties of the SU(2)2 model. We find that this model has emergent Ising anyons, with spinons acting as vortex excitations that bind Majorana zero modes. We also show that the ground state of this network is threefold degenerate on the torus, asserting its non-Abelian character. Our work provides a controllable analytical framework to study non-Abelian topological phases, sheding new light on the stability of such phases in artificial quantum materials.

May 5, 20230 citations

SU(2)1 chiral edge modes of a critical spin liquid

Didier Poilblanc, Norbert Schuch, Ian K. Affleck

Protected chiral edge modes are a well-known signature of topologically ordered phases, such as the fractional quantum Hall states (FQHS). Using the framework of projected entangled pair states (PEPS) on the square lattice, the authors construct a family of chiral spin- $\frac{1}{2}$ quantum spin liquids with ${\mathbb{Z}}_{2}$ gauge symmetry and analyze in full detail the properties of the edge modes. Surprisingly, the results show that the latter can be well described by a chiral conformal field theory of free bosons (SU(2) ${}_{1}$ ), as is the case for the $\ensuremath{\nu}$= $\frac{1}{2}$ (bosonic) gapped Laughlin state, despite the fact that the numerical data here suggest a critical bulk. The authors thus propose that our family of PEPS physically describes a boundary between a chiral topological phase and a trivial phase and might be closely connected to an (unknown) analogous FQHS.Protected chiral edge modes are a well-known signature of topologically ordered phases, such as the fractional quantum Hall states (FQHS). Using the framework of projected entangled pair states (PEPS) on the square lattice, the authors construct a family of chiral spin-$\frac{1}{2}$ quantum spin liquids with ${\mathbb{Z}}_{2}$ gauge symmetry and analyze in full detail the properties of the edge modes. Surprisingly, the results show that the latter can be well described by a chiral conformal field theory of free bosons (SU(2)${}_{1}$), as is the case for the $\ensuremath{\nu}$=$\frac{1}{2}$ (bosonic) gapped Laughlin state, despite the fact that the numerical data here suggest a critical bulk. The authors thus propose that our family of PEPS physically describes a boundary between a chiral topological phase and a trivial phase and might be closely connected to an (unknown) analogous FQHS.

arXiv:1602.05969Physical review. B./Physical review. BMay 18, 201643 citations

{}_{n}

Wess-Zumino-Novikov-Witten conformal field theory. The models are constructed from an array of generalized spin-

n/2

ladders with multi-spin-exchange interactions which are coupled by isolated spins. Such models allow a controllable analytic treatment starting from the one-dimensional limit and are characterized by a bulk gap and non-Abelian SU(2)

{}_{n}

gapless edge excitations.

\frac{1}{2}

quantum spin liquids with

{\mathbb{Z}}_{2}

gauge symmetry and analyze in full detail the properties of the edge modes. Surprisingly, the results show that the latter can be well described by a chiral conformal field theory of free bosons (SU(2)

{}_{1}

), as is the case for the $\ensuremath{\nu}$=

\frac{1}{2}

(bosonic) gapped Laughlin state, despite the fact that the numerical data here suggest a critical bulk. The authors thus propose that our family of PEPS physically describes a boundary between a chiral topological phase and a trivial phase and might be closely connected to an (unknown) analogous FQHS.