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arXiv:1307.8194

Transitions Between Chiral Spin Liquids and Z2 Spin Liquids

Maissam Barkeshli

Published Jul 31, 2013• 9 citations
arXivHTMLPDF

Abstract

The Kalmeyer-Laughlin chiral spin liquids (CSL) and the Z2 spin liquids are two of the simplest topologically ordered states. Here I develop a theory of a direct quantum phase transition between them. Each CSL is characterized by an integer n and is topologically equivalent to the 1/2n Laughlin fractional quantum Hall (FQH) state. Depending on the parity of n, the transition from the CSL is either to a "twisted" version of the Z2 spin liquid, the "doubled semion" model, or the conventional Z_2 spin liquid ("toric code"). In the presence of SU(2) spin symmetry, the triplet gap remains open through the transition and only singlet operators acquire algebraic correlations. An essential observation is that the CSL/Laughlin FQH states can be understood in terms of bosonic integer quantum Hall (BIQH) states of Schwinger bosons or vortices, respectively. I propose several novel many-body wave functions that can interpolate through the transition.

DOI
10.48550/arxiv.1307.8194

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The Kalmeyer-Laughlin chiral spin liquids (CSL) and the Z2 spin liquids are two of the simplest topologically ordered states. Here I develop a theory of a direct quantum phase transition between them. Each CSL is characterized by an integer n and is topologically equivalent to the 1/2n Laughlin fractional quantum Hall (FQH) state. Depending on the parity of n, the transition from the CSL is either to a "twisted" version of the Z2 spin liquid, the "doubled semion" model, or the conventional Z_2 spin liquid ("toric code"). In the presence of SU(2) spin symmetry, the triplet gap remains open through the transition and only singlet operators acquire algebraic correlations. An essential observation is that the CSL/Laughlin FQH states can be understood in terms of bosonic integer quantum Hall (BIQH) states of Schwinger bosons or vortices, respectively. I propose several novel many-body wave functions that can interpolate through the transition.
We consider the spin rotationally invariant Kalmeyer-Laughlin chiral spin liquid (CSL) in systems with broken time-reversal symmetry and explore symmetry constraints on possible conventional spin crystal states accessible via a direct transition. These constraints provide a framework to identify topological invariants of the magnetically ordered state. We show that the existence of a direct transition from a CSL requires a precise compatibility condition between the topological invariants of the ordered state and the anomaly of the CSL. The lattice symmetries also constrain the functional form of the low-energy theory to describe these transitions. This allows us to construct explicit Chern-Simons-matter field theories for the transition into a class of noncoplanar orders identified as candidates directly accessible from the CSL, including the octahedral spin crystal on the kagom\'e lattice and the tetrahedral order on the triangular and honeycomb lattice. These transitions can either be described using coupled fractionalized CP1{\mathbb{CP}}^{1}CP1 theories or fractionalized matrix principal chiral models. We also discuss extensions to more general magnetic ordering transitions out of the CSL.
The fractional quantum Hall effect (FQHE) realized in two-dimensional electron systems under a magnetic field is one of the most remarkable discoveries in condensed matter physics. Interestingly, it has been proposed that FQHE can also emerge in time-reversal invariant spin systems, known as the chiral spin liquid (CSL) characterized by the topological order and the emerging of the fractionalized quasiparticles. A CSL can naturally lead to the exotic superconductivity originating from the condense of anyonic quasiparticles. Although CSL was highly sought after for more than twenty years, it had never been found in a spin isotropic Heisenberg model or related materials. By developing a density-matrix renormalization group based method for adiabatically inserting flux, we discover a FQHE in a spin-½ isotropic kagome Heisenberg model. We identify this FQHE state as the long-sought CSL with a uniform chiral order spontaneously breaking time reversal symmetry, which is uniquely characterized by the half-integer quantized topological Chern number protected by a robust excitation gap. The CSL is found to be at the neighbor of the previously identified Z2 spin liquid, which may lead to an exotic quantum phase transition between two gapped topological spin liquids.
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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-12\frac{1}{2}21​ quantum spin liquids with Z2{\mathbb{Z}}_{2}Z2​ 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{}_{1}1​), as is the case for the $\ensuremath{\nu}$=12\frac{1}{2}21​ (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.
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.
We study entanglement properties of candidate wave functions for SU(2) symmetric gapped spin liquids and Laughlin states. These wave functions are obtained by the Gutzwiller projection technique. Using topological entanglement entropy $\ensuremath{\gamma}$ as a tool, we establish topological order in chiral spin liquid and Z2{\mathbb{Z}}_{2}Z2​ spin liquid wave functions, as well as a lattice version of the Laughlin state. Our results agree very well with the field theoretic result $\ensuremath{\gamma}=\mathrm{log}D$ where DDD is the total quantum dimension of the phase. All calculations are done using a Monte Carlo technique on a $12\ifmmode\times\else\texttimes\fi{}12$ lattice enabling us to extract $\ensuremath{\gamma}$ with small finite-size effects. For a chiral spin liquid wave function, the calculated value is within 44%4 of the ideal value. We also find good agreement for a lattice version of the Laughlin $\ensuremath{\nu}=1/3$ phase with the expected $\ensuremath{\gamma}=\mathrm{log}\sqrt{3}$.