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arXiv:0804.0786 · Physical Review B

Monopole quantum numbers in the staggered flux spin liquid

Jason Alicea

Published Jul 25, 2008• 33 citations
arXivHTMLPDF

Abstract

Algebraic spin liquids, which are exotic gapless spin states preserving all microscopic symmetries, have been widely studied due to potential realizations in frustrated quantum magnets and the cuprates. At low energies, such putative phases are described by quantum electrodynamics in $2+1$ dimensions. While significant progress has been made in understanding this nontrivial interacting field theory and the associated spin physics, one important issue which has proved elusive is the quantum numbers carried by so-called monopole operators. Here we address this issue in the ``staggered flux'' spin liquid which may be relevant to the pseudogap regime in high ${T}_{c}$. Employing general analytical arguments supported by simple numerics, we argue that proximate phases encoded in the monopole operators include the familiar Neel and valence bond solid orders, as well as other symmetry-breaking orders closely related to those previously explored in the monopole-free sector of the theory. Surprisingly, we also find that one monopole operator carries trivial quantum numbers, and briefly discuss its possible implications.

DOI
10.1103/physrevb.78.035126

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Algebraic spin liquids, which are exotic gapless spin states preserving all microscopic symmetries, have been widely studied due to potential realizations in frustrated quantum magnets and the cuprates. At low energies, such putative phases are described by quantum electrodynamics in 2+12+12+1 dimensions. While significant progress has been made in understanding this nontrivial interacting field theory and the associated spin physics, one important issue which has proved elusive is the quantum numbers carried by so-called monopole operators. Here we address this issue in the ``staggered flux'' spin liquid which may be relevant to the pseudogap regime in high Tc{T}_{c}Tc​. Employing general analytical arguments supported by simple numerics, we argue that proximate phases encoded in the monopole operators include the familiar Neel and valence bond solid orders, as well as other symmetry-breaking orders closely related to those previously explored in the monopole-free sector of the theory. Surprisingly, we also find that one monopole operator carries trivial quantum numbers, and briefly discuss its possible implications.
We study the properties of a class of two-dimensional interacting critical states---dubbed algebraic spin liquids---that can arise in two-dimensional quantum magnets. A particular example that we focus on is the staggered flux spin liquid, which plays a key role in some theories of underdoped cuprate superconductors. We show that the low-energy theory of such states has much higher symmetry than the underlying microscopic spin system. This symmetry has remarkable consequences, leading in particular to the unification of a number of seemingly unrelated competing orders. The correlations of these orders---including, in the staggered flux state, the N\'eel vector, and the order parameter for the columnar and box valence-bond solid states---all exhibit the same slow power-law decay. Implications for experiments in the pseudogap regime of the cuprates and for numerical calculations on model systems are discussed.
The U(1)U(1)U(1) Dirac spin liquid might realize an exotic phase of matter whose low-energy properties are described by quantum electrodynamics in 2+12+12+1 dimensions, where gapless modes exist but spinons and gauge fields are strongly coupled. Its existence was proposed in frustrated Heisenberg models in the presence of frustrating superexchange interactions by the (Abrikosov) fermionic representation of the spin operators [Wen, Phys. Rev. B 65, 165113 (2002)], supplemented by the Gutzwiller projection. Here, we construct charge-QQQ monopole excitations in the Heisenberg model on the triangular lattice with nearest-neighbor (J1{J}_{1}J1​) and next-nearest-neighbor (J2{J}_{2}J2​) couplings. In the highly frustrated regime, singlet and triplet monopoles with Q=1Q=1Q=1 become gapless in the thermodynamic limit; in addition, the energies for generic QQQ agree with field-theoretical predictions, obtained for a large number of gapless fermion modes. Finally, we consider localized gauge excitations, in which magnetic $\ensuremath{\pi}$ fluxes are concentrated in the triangular plaquettes (in analogy with Z2{\mathbb{Z}}_{2}Z2​ visons), showing that these kinds of states do not play a relevant role at low energies. All our findings lend support to a stable U(1)U(1)U(1) Dirac spin liquid in the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the triangular lattice.
We describe spin correlations in the vicinity of a generalized impurity in a wide class of fractionalized spin liquid states. We argue that the primary characterization of the impurity is its electric charge under the gauge field describing singlet excitations in the spin liquid. We focus on two gapless U(1)U(1)U(1) spin liquids described by (2+1)(2+1)(2+1)-dimensional conformal field theories (CFT): the staggered flux (sF) spin liquid, and the deconfined critical point between the N\'eel and valence-bond-solid (VBS) states. In these cases, the electric charge is argued to be an exactly marginal perturbation of the CFT. Consequently, the impurity susceptibility has a 1∕T1∕T1∕T temperature dependence, with an anomalous Curie constant, which is a universal number associated with the CFT. One unexpected feature of the CFT of the sF state is that an applied magnetic field does not induce any staggered spin polarization in the vicinity of the impurity (while such a staggered magnetization is present for the N\'eel-VBS case). These results differ significantly from earlier theories of vacancies in the sF state, and we explicitly demonstrate how our gauge theory corrects these works. We discuss implications of our results for the cuprate superconductors, organic Mott insulators, and graphene.
Symmetry-resolved monopoles and stability criteria for an Nf=12N_f=12Nf​=12 Dirac spin liquid on the maple-leaf lattice
The U(1)\mathrm{U}(1)U(1) Dirac spin liquid provides a useful organizing framework for frustrated magnets: it offers an algebraic parent state from which competing orders, confinement patterns, and low-energy spectral features can be understood. Whether such a state can occur as a stable ground state of a two-dimensional spin Hamiltonian remains an open question, because monopole operators of the compact gauge field can proliferate and confine the spinons. Here we determine the symmetry quantum numbers of the fundamental monopoles of a maple-leaf-lattice Dirac spin liquid whose generic six-node regime realizes QED3_33​ with Nf=12N_f=12Nf​=12 two-component Dirac fermions. The charge-one monopoles span a 924924924-dimensional complex representation. In the spin-singlet, zero-momentum, zero-C6C_6C6​-angular-momentum sector we find five complex charge-one directions, of which three are mapped to their Hermitian conjugates under time reversal. Their real and imaginary Hermitian combinations therefore supply six independent monopole perturbations that are trivial under translations, C6C_6C6​, time reversal, and spin rotation. The Dirac spin liquid phase is thus not protected by symmetry exclusion of charge-one monopoles: its stability depends on whether these allowed perturbations are relevant in the renormalization group sense. Existing large-NfN_fNf​ and numerical estimates of the charge-one monopole scaling dimension lie close to the relevance threshold, making the maple-leaf lattice a concrete platform for testing the stability of compact QED3_33​ in a quantum magnet. Our complete monopole classification also identifies the spin, momentum, and rotation sectors in which monopole excitations should appear, providing symmetry-resolved benchmarks for exact-diagonalization and variational tests of the Nf=12N_f=12Nf​=12 Dirac spin liquid.
The Dirac spin liquid (DSL) is a two-dimensional (2D) fractionalized Mott insulator featuring massless Dirac spinon excitations coupled to a compact U(1) gauge field, which allows for flux-tunneling instanton events described by magnetic monopoles in (2+1)D Euclidean spacetime. The state-operator correspondence of conformal field theory has been used recently to define associated monopole operators and determine their quantum numbers, which encode the microscopic symmetries of conventional ordered phases proximate to the DSL. In this work, we utilize semiclassical instanton methods not relying on conformal invariance to construct monopole operators directly in (2+1)D spacetime as instanton-induced ’t Hooft vertices, i.e., fermion-number-violating effective interactions originating from zero modes of the Euclidean Dirac operator in an instanton background. In the presence of a flavor-adjoint fermion mass, resummation of the instanton gas is shown to select the correct monopole to be proliferated, in accordance with predictions of the state-operator correspondence. We also show that our instanton-based approach is able to determine monopole quantum numbers on bipartite lattices.
We establish that spin liquids described in terms of gapless fermionic (Dirac) spinons and gapless U(1)U(1)U(1) gauge fluctuations can be stable in two dimensions, at least when the physical SU(2)SU(2)SU(2) spin symmetry is generalized to SU(N)SU(N)SU(N). Equivalently, we show that compact QED3{\mathrm{QED}}_{3}QED3​ has a deconfined phase for a large number of fermion fields, in the sense that monopole fluctuations can be irrelevant at low energies. A precise characterization is provided by an emergent global topological U(1)U(1)U(1) symmetry corresponding to the conservation of gauge flux. Beginning with an SU(N)SU(N)SU(N) generalization of the S=1∕2S=1∕2S=1∕2 square-lattice Heisenberg antiferromagnet, we consider the $\ensuremath{\pi}$-flux spin liquid and, via a systematic analysis of all operators, show that there are no relevant perturbations (in the renormalization-group sense) about the large-NNN spin-liquid fixed point, which is thus a stable phase. We provide a further illustration of this conclusion with an approximate renormalization-group calculation that treats the gapless fermions and the monopoles on an equal footing. This approach directly points out some of the flaws in the erroneous ``screening'' argument for the relevance of monopoles in compact QED3{\mathrm{QED}}_{3}QED3​.