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arXiv:cond-mat/0512427 · Physical Review B

Theory of the algebraic vortex liquid in an anisotropic spin- 12 triangular antiferromagnet

Jason Alicea, Olexei I. Motrunich, Matthew P. A. Fisher

Published May 24, 2006• 41 citations
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Abstract

We explore spin-$\frac{1}{2}$ triangular antiferromagnets with both easy-plane and lattice exchange anisotropies by employing a dual vortex mapping followed by a fermionization of the vortices. Over a broad range of exchange anisotropy, this approach leads naturally to a ``critical'' spin liquid---the algebraic vortex liquid---which appears to be distinct from other known spin liquids. We present a detailed characterization of this state, which is described in terms of noncompact QED3 with an emergent SU(4) symmetry. Descendant phases of the algebraic vortex liquid are also explored, which include the Kalmeyer-Laughlin spin liquid, a variety of magnetically ordered states such as the well-known coplanar spiral state, and supersolids. In the range of exchange anisotropy where the ``square lattice'' N\'eel ground state arises, we demonstrate that anomalous ``roton'' minima in the excitation spectrum recently reported in series expansions can be accounted for within our approach.

DOI
10.1103/physrevb.73.174430

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We explore spin-12\frac{1}{2}21​ triangular antiferromagnets with both easy-plane and lattice exchange anisotropies by employing a dual vortex mapping followed by a fermionization of the vortices. Over a broad range of exchange anisotropy, this approach leads naturally to a critical'' spin liquid---the algebraic vortex liquid---which appears to be distinct from other known spin liquids. We present a detailed characterization of this state, which is described in terms of noncompact QED3 with an emergent SU(4) symmetry. Descendant phases of the algebraic vortex liquid are also explored, which include the Kalmeyer-Laughlin spin liquid, a variety of magnetically ordered states such as the well-known coplanar spiral state, and supersolids. In the range of exchange anisotropy where the square lattice'' N'eel ground state arises, we demonstrate that anomalous ``roton'' minima in the excitation spectrum recently reported in series expansions can be accounted for within our approach.
Although magnetically ordered at low temperatures, the spin-12\frac{1}{2}21​ triangular antiferromagnet Cs2CuCl4{\mathrm{Cs}}_{2}\mathrm{Cu}{\mathrm{Cl}}_{4}Cs2​CuCl4​ exhibits remarkable spin dynamics that strongly suggest proximity to a spin-liquid phase. Here we ask whether a proximate spin liquid may also occur in an applied magnetic field, leaving a similar imprint on the dynamical spin correlations of this material. Specifically, we explore a spatially anisotropic Heisenberg spin-12\frac{1}{2}21​ triangular antiferromagnet at 13\frac{1}{3}31​ magnetization from a dual vortex perspective, and indeed find a ``critical'' spin-liquid phase described by quantum electrodynamics in (2+1)(2+1)(2+1)-dimensions with an emergent SU(6) symmetry. A number of nontrivial predictions follow for the dynamical spin structure factor in this ``algebraic vortex liquid'' phase, which can be tested via inelastic neutron scattering. We also discuss how well-studied ``up-up-down'' magnetization plateaus can be captured within our approach, and further predict the existence of a stable gapless solid phase in a weakly ordered up-up-down state. Finally, we predict several anomalous ``roton'' minima in the excitation spectrum in the regime of lattice anisotropy where the canted N\'eel state appears.
Motivated by inelastic neutron scattering data on Cs2CuCl4, we explore spin-1/2 triangular lattice antiferromagnets with both spatial and easy-plane exchange anisotropies, the latter due to an observed Dzyaloshinskii-Moriya interaction. Exploiting a duality mapping followed by a fermionization of the dual vortex degrees of freedom, we find a novel critical spin-liquid phase described in terms of Dirac fermions with an emergent global SU(4) symmetry minimally coupled to a noncompact U(1) gauge field. This "algebraic vortex liquid" supports gapless spin excitations and universal power-law correlations in the dynamical spin structure factor which are consistent with those observed in Cs2CuCl4. We suggest future neutron scattering experiments that should help distinguish between the algebraic vortex liquid and other spin liquids and quantum critical points previously proposed in the context of Cs2CuCl4.
There is growing evidence from both experiment and numerical studies that low half-odd integer quantum spins on a kagome lattice with predominant antiferromagnetic near-neighbor interactions do not order magnetically or break lattice symmetries even at temperatures much lower than the exchange interaction strength. Moreover, there appears to be a plethora of low-energy excitations, predominantly singlets but also spin carrying, which suggests that the putative underlying quantum spin liquid is a gapless ``critical spin liquid'' rather than a gapped spin liquid with topological order. Here, we develop an effective field theory approach for the spin-12\frac{1}{2}21​ Heisenberg model with easy-plane anisotropy on the kagome lattice. By employing a vortex duality transformation, followed by a fermionization and flux smearing, we obtain access to a gapless yet stable critical spin liquid phase, which is described by (2+1)(2+1)(2+1)-dimensional quantum electrodynamics (QED3)({\mathrm{QED}}_{3})(QED3​) with an emergent SU(8) flavor symmetry. The specific heat, thermal conductivity, and dynamical structure factor are extracted from the effective field theory, and contrasted with other theoretical approaches to the kagome antiferromagnet.
We reexamine two-dimensional frustrated quantum magnetism with the aim of exploring new critical points and critical phases. We study easy-plane triangular antiferromagnets using a dual vortex approach, fermionizing the vortices with a Chern-Simons field. Herein we develop this technique for integer-spin systems which generically exhibit a simple paramagnetic phase as well as magnetically ordered phases with coplanar and collinear spin order. Within the fermionized-vortex approach, we derive a low-energy effective theory containing Dirac fermions with two flavors minimally coupled to a U(1) and a Chern-Simons gauge field. At criticality we argue that the Chern-Simons gauge field can be subsumed into the U(1) gauge field, and up to irrelevant interactions one arrives at quantum electrodynamics in 2+12+12+1 dimensions (QED3). Moreover, we conjecture that critical QED3 with full SU(2) flavor symmetry describes the O(4) multicritical point of the spin model where the paramagnet and two magnetically ordered phases merge. The remarkable implication is that QED3 with flavor SU(2) symmetry is dual to ordinary critical ${\ensuremath{\Phi}}^{4}$ field theory with O(4) symmetry. This leads to a number of unexpected, verifiable predictions for QED3. A connection of our fermionized-vortex approach with the dipole interpretation of the $\ensuremath{\nu}=1∕2$ fractional quantum Hall state is also demonstrated. The approach introduced in this paper will be applied to spin-1∕21∕21∕2 systems in a forthcoming publication.
The algebraic spin liquid is a long-sought-after phase of matter characterized by the absence of quasiparticle excitations, a low-energy description in terms of emergent Dirac fermions and gauge fields interacting according to (2+1)(2+1)(2+1)-dimensional quantum electrodynamics (QED3)({\mathrm{QED}}_{3})(QED3​), and power-law correlations with universal exponents. The prototypical algebraic spin liquid is the Affleck-Marston $\ensuremath{\pi}$-flux phase, originally proposed as a possible ground state of the spin-12\frac{1}{2}21​ Heisenberg model on the two-dimensional square lattice. While the latter model is now known to order antiferromagnetically at zero temperature, recent sign-problem-free quantum Monte Carlo simulations of spin-12\frac{1}{2}21​ fermions coupled to a compact U(1)\text{U}(1)U(1) gauge field on the square lattice have shown that quantum fluctuations can destroy N\'eel order and drive a direct quantum phase transition to the $\ensuremath{\pi}$-flux phase. We show this transition is in the universality class of the chiral Heisenberg QED3{\mathrm{QED}}_{3}QED3​-- Gross-Neveu-Yukawa model with a single SU(2)\text{SU}(2)SU(2) doublet of four-component Dirac fermions (i.e., Nf=1){N}_{f}=1)Nf​=1), pointing out important differences with the corresponding putative transition on the kagome lattice. Using an $\ensuremath{\epsilon}$ expansion below four space-time dimensions to four-loop order, and a large-Nf{N}_{f}Nf​ expansion up to second order, we show the transition is continuous and compute various thermodynamic and susceptibility critical exponents at this transition, setting the stage for future numerical determinations of these quantities. As a by-product of our analysis, we also obtain charge-density-wave and valence-bond-solid susceptibility exponents at the semimetal-N\'eel transition for interacting fermions on the honeycomb lattice.
We present a systematic numerical study of the spin-1 antiferromagnetic Heisenberg model on the triangular lattice in an out-of-plane magnetic field, using Density Matrix Renormalization Group (DMRG) methods. By mapping out the quantum phase diagram as a function of the single-ion anisotropy DzD_zDz​ and magnetic field, we identify distinct dipolar and quadrupolar spin supersolid states, characterized by spontaneous U(1) symmetry breaking with finite spin superfluid stiffness coexisting with longitudinal translational symmetry breaking. At zero field, the dipolar spin supersolid with a 'Y'-type spin configuration persists down to Dz=0D_z = 0Dz​=0, whereas the quadrupolar spin supersolid prevails at large DzD_zDz​. At intermediate fields, the phase diagram is dominated by an up-up-down phase. At high fields below saturation, a quadrupolar spin superfluid emerges in the large-DzD_zDz​ regime, whereas a dipolar spin supersolid with a 'V'-type spin configuration dominates at small DzD_zDz​. These phases are characterized through their order parameters and spin superfluid stiffness using calculations on various system sizes. Furthermore, the dynamical spin structure factor is obtained across the phase diagram, where characteristic spectral signatures of different phases are observed, including the gapless Goldstone mode and the roton-like minima. These features are directly accessible to inelastic neutron scattering experiments. Our results provide a theoretical understanding of the interplay between frustrations, anisotropy, and Zeeman interactions in driving distinct spin supersolid phases in the spin-1 system, which are relevant to various triangular-lattice antiferromagnets such as Na2_22​BaNi(PO4_44​)2_22​ and K2_22​Ni(SeO3_33​)2_22​.