Session J15: Focus Session: Quantum Spin Liquid Theory

$Z_2$-vortex lattice in the ground state of the triangular Kitaev-Heisenberg model

Investigating the classical Kitaev-Heisenberg Hamiltonian on a triangular lattice, we establish the presence of an incommensurate non-coplanar magnetic phase, which is identified as a lattice of $Z_2$ vortices. The vortices, topological point defects in the SO(3) order parameter of the nearby Heisenberg antiferromagnet, are not thermally excited but due to the spin-orbit coupling and arise at temperature $T\to 0$. This $Z_2$-vortex lattice is stable in a parameter regime relevant to iridates. We show that in the other, strongly anisotropic, limit a robust nematic phase emerges.   

Dislocations in the Kitaev honeycomb model

We study the effects of introducing dislocations into the Kitaev honeycomb model [1]. In the gapped phase, dislocations are $Z_2$ ``twist defects'' associated with the transmutation of electric and magnetic excitations, studied previously in the context of $Z_N$ rotor models [2,3]. We show that each dislocation hosts one unpaired Majorana mode. As a consequence, twist defects have the statistics of Ising anyons. Because dislocations are confined, an additional phase is accumulated due to the change in system's energy during the braiding process. This means that the result of braiding can only be defined up to a phase. Therefore, twists are said to have projective non-Abelian statistics. \\[4pt][1] Alexei Kitaev, Annals of Physics \textbf{321}, 2 (2006) \\[0pt][2] Hector Bombin, Phys. Rev. Lett. \textbf{105}, 030403 (2010) \\[0pt][3] Yi-Zhuang You and Xiao-Gang Wen, Phys. Rev. B \textbf{86}, 161107 (2012)   

Quantum Phase Transition in Heisenberg-Kitaev Model

We explore the nature of the quantum phase transition between a magnetically ordered state with collinear spin pattern and a gapless $Z_2$ spin liquid in the Heisenberg-Kitaev model. We construct a slave particle mean field theory for the Heisenberg-Kitaev model in terms of complex fermionic spinons. It is shown that this theory, formulated in the appropriate basis, is capable of describing the Kitaev spin liquid as well as the transition between the gapless $Z_2$ spin liquid and the so-called stripy antiferromagnet. Within our mean field theory, we find a discontinuous transition from the $Z_2$ spin liquid to the stripy antiferromagnet. We argue that subtle spinon confinement effects, associated with the instability of gapped $U(1)$ spin liquid in two spatial dimensions, play an important role at this transition. The possibility of an exotic continuous transition is briefly addressed.   

Changing topology by knotting in the three-dimensional Toric Code

A novel way to study the ground state degeneracy (GSD) of topological matter is through lattice dislocations: When a second copy of a lattice model is introduced through translation by half a lattice constant ($|\vec{b}|=a/2$), then a lattice dislocation with Burgers vector $\vec{b}$ locally smoothly connects the two model copies. Such dislocations are ``genon'' defects, effectively changing the topology of lattice. In three dimensions (3d), dislocations are closed loops that can be linked and knotted, leading to complex three dimensional manifolds on which the topological theory is defined. We give an analytical construction, supported by exact numerical calculations, for the dependence of GSD on dislocations of such a doubled version of the exactly solvable Kitaev's Toric Code (having $Z_2$ topological order) in both 2d and 3d. Surprisingly, we find that GSD of the 3d model depends only on the total number of dislocation loops, no matter their linking or knotting. The analytical proof is extended to $Z_n$ generalizations of the model. Additionally, we consider the phase in which dislocations become dynamical through proliferation of double dislocations (2$\vec{b}$) in 2d: the resulting gauge theory is non-Abelian, in the special case of $Z_2$ Toric Code it is $D_4$.   

Kibble-Zurek Scaling and String-Net Coarsening in Topologically Ordered Systems

We consider the non-equilibrium dynamics of topologically ordered systems, such as spin liquids, driven across a continuous phase transition into proximate phases with no, or reduced, topological order. This dynamics exhibits scaling in the spirit of Kibble and Zurek but now without the presence of symmetry breaking and a local order parameter. The non-equilibrium dynamics near the critical point is universal in a particular scaling limit. The late stages of the process are seen to exhibit slow, quantum coarsening dynamics for the extended string-nets characterizing the topological phase, a potentially interesting signature of topological order. Certain gapped degrees of freedom that could potentially destroy coarsening are, at worst, dangerously irrelevant in the scaling limit. We also note a time dependent amplification of the energy splitting between topologically degenerate states on closed manifolds. We illustrate these phenomena in the context of particular phase transitions out of the abelian $Z_2$ topologically ordered phase of the toric code, and the non-abelian $SU(2)_k$ ordered phases of the relevant Levin-Wen models.   

A classification of symmetry enriched topological phases with exactly solvable models

Recently a new class of quantum phases of matter: symmetry protected topological phases, such as topological insulators, attracted much attention. In presence of interactions, group cohomology provides their classification. These phases are only short-range entangled, while phases with long-range entangled topological order (having topological ground state degeneracy and/or anyons in the bulk) in presence of global symmetries are much less understood. We present a classification of bosonic gapped quantum phases with or without long-range entanglement, in the presence or absence of on-site global symmetries. In 2+1 dimensions, the quantum phases with global symmetry group $SG$, and with topological order described by finite gauge group $GG$, are classified by the cohomology group $H^3(SG\times GG, U(1))$. We present an exactly solvable local bosonic model for each class. When global symmetry is absent our models are described by Dijkgraaf-Witten discrete gauge theories. When topological order is absent, they become models for symmetry protected topological phases. When both global symmetry and topological order are present, the models describe symmetry enriched topological phases. Our classification includes, but goes beyond the projective symmetry group classification.   

Classifying fractionalization: symmetry classification of gapped Z2 spin liquids in two dimensions

Quantum number fractionalization is a remarkable property of topologically ordered states of matter, such as the fractional quantum Hall liquids, and certain quantum spin liquid states. For a given type of topological order, there are generally many ways to fractionalize the quantum numbers of a given symmetry. Not all distinct fractionalizations will necessarily correspond to distinct phases of matter, however. In this work, we establish a formalism for characterizing fractionalization in gapped, two-dimensional Z2 spin liquids, which leads immediately to a classification of these topologically ordered phases.   

Realization of symmetry classes for gapped $Z_{2}$ spin liquids in simple models

Recently it has been proposed that gapped $Z_{2}$ spin liquids in two dimensions can be partially classified by the distinct types of fractional quantum numbers carried by the $Z_{2}$ charge and flux excitations. On the square lattice with space group and time reversal symmetry, there are about $2^{19}$ symmetry classes. It is an open question which of these classes can be realized in simple models and, more fundamentally, whether all of these classes can actually be realized. We will present results on a class of exactly solvable models addressing these issues.   

Chern-Simons theory for frustrated quantum magnets

We study the problem of frustrated quantum magnets by mapping models with Heisenberg spins, which are hard-core bosons, onto a problem of fermions coupled to a Chern-Simons gauge field [1]. Similar methods have been used successfully in the case of unfrustrated systems like the square lattice [2]. However, in the case of frustrated systems there always exists some arbitrariness in defining the problem. At the mean-field level these issues can be over looked but the effects of fluctuations, which are generally strong in these systems, are expected to alter the mean-field physics [3-4]. We discuss the difficulties involved in setting up this problem on a triangular or kagome lattice and some approaches to tackle these issues. We study the effects of fluctuations in these systems and the possibility of spin-liquid type phases.\\[4pt] [1] E. Fradkin, Phys. Rev. Lett. 63, 322-325 (1989)\\[0pt] [2] A. Lopez, A. G. Rojo, and E. Fradkin, Phys. Rev. B 49, 15139 (1994)\\[0pt] [3] G. Misguich, Th. Jolicoeur, and S. M. Girvin Phys. Rev. Lett. 87, 097203 (2001)\\[0pt] [4] Kun Yang, L. K. Warman and S. M. Girvin, Phys. Rev. Lett. 70, 2641 (1993)   

Identifying Topological Quantum Spin Liquid in Physical Realistic Models

Quantum spin liquids (QSLs) are elusive magnets without magnetism, resisting symmetry breaking even at zero temperature due to strong quantum fluctuations and geometric frustration. The simplest QSLs known theoretically are characterized by topological order, i.e., topological quantum spin liquid, and support fractionalized excitations. However, there is no practical way to directly determine the topological nature of states, such as QSLs. We propose a practical and extremely simple approach, i.e., cylinder construction, to numerically calculate the topological entanglement entropy (TEE), and thereby identify topological order of the state [H. C. Jiang, Z. Wang, and L. Balents, arXiv:1205.4289]. We have successfully applied this approach to a variety of lattice models and S$=$1/2 Kagome Heisenberg model. By extracting an accurate TEE, we identify a quantum spin liquid with topological order for the first time in physically realistic SU(2)-invariant lattice model. We emphasize that the TEE provides positive, ``smoking gun'' evidence for a topological quantum spin liquid, and excludes any topologically trivial states, including the valence bound solid state. Besides the Kagome Heisenberg model, based on large-scale accurate density-matrix renormalization group studies of numerous long cylinders with circumferences up to 14 lattice spacings, our results [H. C. Jiang, H. Yao, and L. Balents, Physical Review B 86, 024424(2012)], through a combination of the absence of magnetic or VBS order, nonzero spin singlet and triplet gaps, as well as a finite TEE extremely close to ln(2), provide compelling evidence that the two-dimensional ground state of the square J$_{1}$-J$_{2}$ Heisenberg model is a topological quantum spin liquid.   

Finite size scaling of entanglement entropy at the Anderson transition with interactions

We study the entanglement entropy(EE) of disordered one-dimensional spinless fermions with attractive interactions. With intensive numerical calculation of the EE using the density matrix renormalization group method, we find clear signatures of the transition between the localized and delocalized phase. In the delocalized phase, the fluctuations of the EE becomes minimum and independent of the system size. Meanwhile the EE's logarithmic scaling behavior is found to recover to that of a clean system. We present a general scheme of finite size scaling of the EE at the critical regime of the Anderson transition, from which we extract the critical parameters of the transition with good accuracy, including the critical exponent, critical point and a power-law divergent localization length.   

Three dimensional symmetry protected topological phase and algebraic spin liquid

It is well-known that one dimensional spin chains are described by O(3) nonlinear sigma models with a topological $\Theta-$term, and $\Theta = 2\pi S$. A pin-1/2 chain (described by $\Theta = \pi$) must be either gapless or degenerate, while a spin-1 chain (described by $\Theta = 2\pi$) is a symmetry protected topological phase, namely its bulk is gapped and nondegenerate, while its boundary is a free spin-1/2 with two fold degeneracy. We prove that these phenomena also occur in arbitrary odd dimensions. For example, in three dimensional space, we construct a series of SU(N) antiferromagnet models, whose low energy field theories are nonlinear sigma models with a 3+1d $\Theta-$term. We will also prove that when $\Theta = \pi$, the disordered phase of this system cannot be gapped and nondegenerate, namely it can be an algebraic liquid phase. When $\Theta = 2\pi$, the system is a three dimensional symmetry protected topological phase, whose 2+1d boundary must be either gapless or degenerate.   

Continuous phase transition between N\'eel and spin liquid states with topological order

It is well known that on square lattice N\'eel and valence bond solid states are connected by a continuous phase transition, and the critical theory consists fractionalized spinons and an emergent U(1) gauge field. Motivated by recent numerical works revealing N\'eel and gapped spin liquid states in $J_1$-$J_2$ model on square lattice, we study other phases that can be obtained after destroying the N\'eel order. We show that by condensing fields that carry both electric charge and magnetic flux of the emergent gauge field, one can obtain spin liquid phases with topological order and no lattice symmetry breaking.