We extend the scope of Kitaev spin liquids to non-Archimedean lattices. For the pentaheptite lattice, which results from the proliferation of Stone-Wales defects on the honeycomb lattice, we find an exactly solvable non-Abelian chiral spin liquid with spontaneous time reversal symmetry breaking due to lattice loops of odd length. Our findings call for potential extensions of exact results for Kitaev models which are based on reflection positivity, which is not fulfilled by the pentaheptite lattice. We further elaborate on potential realizations of our chiral spin liquid proposal in strained $α$-RuCl$_3$.
In chiral magnets a magnetic helix forms where the magnetization winds around a propagation vector $\mathbf{q}$. We show theoretically that a magnetic field $\mathbf{B}_{\perp}(t) \perp \mathbf{q}$, which is spatially homogeneous but oscillating in time, induces a net rotation of the texture around $\mathbf{q}$. This rotation is reminiscent of the motion of an Archimedean screw and is equivalent to a translation with velocity $v_{\text{screw}}$ parallel to $\mathbf{q}$. Due to the coupling to a Goldstone mode, this non-linear effect arises for arbitrarily weak $\mathbf{B}_{\perp}(t) $ with $v_…
We suggest a class of two-dimensional lattice spin Hamiltonians describing non-Abelian SU(2) chiral spin liquids - spin-analogues of fractional non-Abelian quantum Hall states- with gapped bulk and gapless chiral edge excitations described by the SU(2)$_n$ Wess-Zumino-Novikov-Witten conformal field theory. The models are constructed from an array of a generalized spin-$n/2$ ladders with multi-spin exchange interaction 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-Abeli…
We study $S=1$ spin liquid states on the kagome lattice constructed by Gutzwiller-projected $p_x+ip_y$ superconductors. We show that the obtained spin liquids are either non-Abelian or Abelian topological phases, depending on the topology of the fermionic mean-field state. By calculating the modular matrices $S$ and $T$, we confirm that projected topological superconductors are non-Abelian chiral spin liquid (NACSL). The chiral central charge and the spin Hall conductance we obtained agree very well with the $SO(3)_1$ (or, equivalently, $SU(2)_2$) field theory predictions. We propose a local H…
Abelian and non-Abelian topological phases exhibiting protected chiral edge modes are ubiquitous in the realm of the Fractional Quantum Hall (FQH) effect. Here, we investigate a spin-1 Hamiltonian on the square lattice which could, potentially, host the spin liquid analog of the (bosonic) non-Abelian Moore-Read FQH state, as suggested by Exact Diagonalisation of small clusters. Using families of fully SU(2)-spin symmetric and translationally invariant chiral Projected Entangled Pair States (PEPS), variational energy optimization is performed using infinite-PEPS methods, providing good agreemen…
We construct a two-dimensional (2D) lattice model that is argued to realize a gapped chiral spin liquid with (Ising) non-Abelian topological order. The building blocks are spin-1/2 two-leg ladders with $SU(2)$-symmetric spin-spin interactions. The two-leg ladders are then arranged on rows and coupled through SU(2)-symmetric interactions between consecutive ladders. The intra-ladder interactions are tuned so as to realize the c = 1/2 Ising conformal field theory, a fact that we establish numerically via Density Matrix Renormalization Group (DMRG) studies. Time-reversal breaking inter-ladder int…
We study the temperature (T) dependence of the dynamical structure factor of the chiral spin liquid phase of Kitaev's honeycomb model. We find, using a recently developed analytical approach, direct signatures of the Majorana modes associated with the non-Abelian Z_2 vortices (visons). In particular, a characteristic multiplet of discrete peaks at low but nonzero-T and its field dependence reflect the existence of thermally activated visons whose varying separation yields different splittings of the Majorana `zero' modes. The resonant processes involved are specific to the Majorana modes, and …
We consider a chiral spin liquid constructed using the parton theory. This state supports non-Abelian anyons and neutral fermions, which share some similarities with the celebrated Moore-Read state. The edge physics of the parton state and the Moore-Read state is very different. Based on conformal field theory (CFT) analysis, it is proposed that the edge states exhibit an emergent SO(5) symmetry. The counting of edge states in the low-lying SO(5)$_{1}$ CFT towers is computed. The chiral spin liquid is generated using tensor network methods, which allows us to confirm the SO(5)$_{1}$ counting u…
We propose resonating valence bond wave functions for a spin-1 system on the torus that realize a non-Abelian chiral spin liquid. The wave functions take the form of infinite dimensional matrix product states constructed from conformal blocks of the $\mathrm{SO}(3)_{1}$ Wess-Zumino-Witten model. This means that they are lattice analogues of the bosonic Moore-Read state introduced in fractional quantum Hall systems. The topological order of this system is revealed by explicit construction of three-fold degenerate ground states and analytical computation of the modular S and T matrices.
We study the ground-state properties of a spin-1 Heisenberg model on a square lattice with the first- and second-nearest-neighbor antiferromagnetic couplings $J_1$ and $J_2$ and a three-spin scalar chirality term $J_χ$. Using the density matrix renormalization group calculation, we map out a global phase diagram including various magnetic order phases and an emergent quantum spin liquid phase. The nature of the spin liquid is identified as a bosonic non-Abelian Moore-Read state from the fingerprint of the entanglement spectra and identification of a full set of topological sectors. We further …
We provide numerical evidence in favor of spontaneous chiral symmetry breaking and the concomitant appearance of an Abelian chiral spin liquid for three-component fermions on the triangular lattice described by an SU(3) symmetric Hubbard model with hopping amplitude $-t$ ($t>0$) and on-site interaction $U$. This chiral phase is stabilized in the Mott phase with one particle per site in the presence of a uniform $π$-flux per plaquette, and in the Mott phase with two particles per site without any flux. Our approach relies on effective spin models derived in the strong-coupling limit in powers o…
We construct a parent Hamiltonian for the family of non-Abelian chiral spin liquids proposed recently by two of us [PRL 102, 207203 (2009)], which includes the Abelian chiral spin liquid proposed by Kalmeyer and Laughlin, as the special case S=1/2. As we use a circular disk geometry with an open boundary, both the annihilation operators we identify and the Hamiltonians we construct from these are exact only in the thermodynamic limit.
Thermodynamic properties of chiral spin liquids are investigated for a variant of the Kitaev model defined on a decorated honeycomb lattice. Using the quantum Monte Carlo simulation, we find that the model exhibits a finite-temperature phase transition associated with the time reversal symmetry breaking, in both topologically trivial and nontrivial regions. While changing the exchange constants, the phase transition changes from continuous to discontinuous one, apparently correlated with the change in the excitations from Abelian to non-Abelian anyons. We show this coincidence by computing the…
We provide new insights into the Abelian and non-Abelian chiral Kitaev spin liquids on the star lattice using the recently proposed loop gas (LG) and string gas (SG) states [H.-Y. Lee, R. Kaneko, T. Okubo, N. Kawashima, Phys. Rev. Lett. 123, 087203 (2019)]. Those are compactly represented in the language of tensor network. By optimizing only one or two variational parameters, accurate ansatze are found in the whole phase diagram of the Kitaev model on the star lattice. In particular, the variational energy of the LG state becomes exact(within machine precision) at two limits in the model, and …
We study the finite temperature nature of a quantum phase transition between an Abelian and a non-Abelian topological phase in an exactly solvable model of a chiral spin liquid. By virtue of the exact solvability, this model can serve as a testbed for developing better measures for describing topological quantum phase transitions. We characterize this phase transition in terms of the global flux and entanglement entropy, and discuss to what extent the existence of a topologically ordered ground state with non-Abelian excitations is revealed at finite temperature.
We establish the existence of a chiral spin liquid (CSL) as the exact ground state of the Kitaev model on a decorated honeycomb lattice, which is obtained by replacing each site in the familiar honeycomb lattice with a triangle. The CSL state spontaneously breaks time reversal symmetry but preserves other symmetries. There are two topologically distinct CSLs separated by a quantum critical point. Interestingly, vortex excitations in the topologically nontrivial (Chern number $\pm 1$) CSL obey non-Abelian statistics.
Surface chiral Abelian topological order on multilayer cluster Mott insulators
University of Chinese Academy of Sciences（中国科学院大学）；Southeast University（东南大学）；Peking University（北京大学）；Collaborative Innovation Center of Quantum Matter（量子物质协同创新中心）
We present a two-dimensional lattice model for quantum spin-1/2 for which the low-energy limit is governed by four flavors of strongly interacting Majorana fermions. We study this low-energy effective theory using two alternative approaches. The first consists of a mean-field approximation. The second consists of a Random Phase approximation (RPA) for the single-particle Green's functions of the Majorana fermions built from their exact forms in a certain one-dimensional limit. The resulting phase diagram consists of two competing chiral phases, one with Abelian and the other with non-Abelian t…
We construct a family of 1D and 2D long-range SU(2) spin models as parent Hamiltonians associated with infinite dimensional matrix product states that arise from simple current correlation functions in $SU(2)_k$ WZW models. Our results provide a conformal field theory foundation for recent proposals by Greiter and coauthors regarding the realization of non-abelian chiral spin liquids. We explain, in particular, how the symmetrization procedure for the amplitudes of the ground state wave function suggested in Ref. arXiv:0903.4547 originates from the conformal field theory description.
We develop a coupled wire construction of chiral spin liquids. The starting point are individual wires of electrons in the Mott regime that are subject to a Zeeman field and Rashba spin-orbit coupling. Suitable spin-flip couplings between the wires yield an Abelian chiral spin liquid state which supports spinon excitations above a bulk gap, and chiral edge states. The approach generalizes to non-Abelian chiral spin liquids at level k with parafermionic edge states.
We extend the Yao-Kivelson decorated honeycomb lattice Kitaev model [Phys. Rev. Lett. 99,247203 (2007)] of an exactly solvable chiral spin liquid by including disordered exchange couplings. We have determined the phase diagram of this system and found that disorder enlarges the region of the topological non-Abelian phase with finite Chern number. We study the energy level statistics as a function of disorder and other parameters in the Hamiltonian, and show that the phase transition between the non-Abelian and Abelian phases of the model at large disorder can be associated with pair annihilati…
In the physics of the Fractional Quantum Hall (FQH) effect, a zoo of Abelian topological phases can be obtained by varying the magnetic field. Aiming to reach the same phenomenology in spin-like systems, we propose a family of SU($N$)-symmetric models in the fundamental representation, on the square lattice with short-range interactions restricted to triangular units, a natural generalization for arbitrary $N$ of an SU($3$) model studied previously where time-reversal symmetry is broken explicitly. Guided by the recent discovery of SU($2$)$_1$ and SU($3$)$_1$ chiral spin liquids (CSL) on simil…
In the previous paper we studied Dirac fermions in a non-Abelian random vector potential by using lattice supersymmetry. By the lattice regularization, the system of disordered Dirac fermions is defined without any ambiguities. We showed there that at strong-disorder limit correlation function of the fermion local density of states decays algebraically at the band center. In this paper, we shall reexamine the multi-flavor or multi-species case rather in detail and argue that the correlator at the band center decays {\em exponentially} for the case of a {\em large} number of flavors. This means…
We examine the star lattice Kitaev model whose ground state is a a chiral spin liquid. We fermionize the model such that the fermionic vacua are toric code states on an effective Kagome lattice. This implies that the Abelian phase of the system is inherited from the fermionic vacua and that time reversal symmetry is spontaneously broken at the level of the vacuum. In terms of these fermions we derive the Bloch-matrix Hamiltonians for the vortex free sector and its time reversed counterpart and illuminate the relationships between the sectors. The phase diagram for the model is shown to be a sp…
Chiral Graviton Modes in Non-Abelian lattice Fractional Quantum Hall states
SISSA — International School for Advances Studies（国际高等研究院）；ICTP — The Abdus Salam International Centre for Theoretical Physics（阿卜杜斯·萨拉姆理论物理中心）；The University of Hong Kong（香港大学）；State Key Laboratory of Optical Quantum Materials（光学量子材料国家重点实验室）；New Cornerstone Science Lab（新基石科学实验室）；Università di Bologna（博洛尼亚大学）；INFN, Sezione di Bologna（意大利国家核物理研究所博洛尼部分所）
Chiral superconductivity from parent Chern band and its non-Abelian generalization
University of Maryland, College Park（马里兰大学帕克分校）；University of California, Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；Johns Hopkins University（约翰斯·霍普金斯大学）；University of Pittsburgh（匹兹堡大学）
