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arXiv:cond-mat/0609620

Continuous quantum phase transitions beyond Landau's paradigm in a large-N spin model

Ying Ran, Xiao-Gang Wen

Published Sep 25, 2006• 8 citations
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Abstract

We study a large-N generalization of $J_1$-$J_2$ Heisenberg model on square lattice -- an $Sp(2N)$ spin model. The possible quantum spin liquid phases of the $Sp(2N)$ model are studied using the SU(2) projective construction. We find several spin liquid states at least in large $N$ limit, which include SU(2) $π$-flux state, SU(2) chiral spin state and $Z_2$ spin liquid states. All those spin liquid states have non-trivial quantum orders. We show how projective symmetry group, which characterizes quantum order, protects the stability of even gapless spin liquids. We also study the continuous quantum phase transition from the SU(2) $π$-flux state to the SU(2) chiral spin liquid state, and from the SU(2) $π$-flux state to the $Z_2$ spin liquid state. We show that those phase transitions are beyond the paradigm of Landau symmetry-breaking theory. The first phase transition, although a $Z_2$ symmetry breaking transition, contains critical exponents that are different from those obtained from the Ginzburg-Landau theory of Ising universality class. The second transition does not even involve a change of symmetry and has no symmetry breaking order parameter.

DOI
10.48550/arxiv.cond-mat/0609620

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We study a large-N generalization of J1J_1J1​-J2J_2J2​ Heisenberg model on square lattice -- an Sp(2N)Sp(2N)Sp(2N) spin model. The possible quantum spin liquid phases of the Sp(2N)Sp(2N)Sp(2N) model are studied using the SU(2) projective construction. We find several spin liquid states at least in large NNN limit, which include SU(2) πππ-flux state, SU(2) chiral spin state and Z2Z_2Z2​ spin liquid states. All those spin liquid states have non-trivial quantum orders. We show how projective symmetry group, which characterizes quantum order, protects the stability of even gapless spin liquids. We also study the continuous quantum phase transition from the SU(2) πππ-flux state to the SU(2) chiral spin liquid state, and from the SU(2) πππ-flux state to the Z2Z_2Z2​ spin liquid state. We show that those phase transitions are beyond the paradigm of Landau symmetry-breaking theory. The first phase transition, although a Z2Z_2Z2​ symmetry breaking transition, contains critical exponents that are different from those obtained from the Ginzburg-Landau theory of Ising universality class. The second transition does not even involve a change of symmetry and has no symmetry breaking order parameter.
Recent numerical studies of the J1{J}_{1}J1​-J2{J}_{2}J2​ model on a square lattice suggest a possible continuous phase transition between the N\'eel state and a gapped spin-liquid state with Z2{Z}_{2}Z2​ topological order. We show that such a phase transition can be realized through two steps: First bring the N\'eel state to the U(1) deconfined quantum critical point, which has been studied in the context of N\'eel--valence bond solid (VBS) state phase transition. Then condense the spinon pair--skyrmion/antiskyrmion bound state, which carries both gauge charge and flux of the U(1) gauge field emerging at the deconfined quantum critical point. We also propose a Schwinger boson projective wave function to realize such a Z2{Z}_{2}Z2​ spin liquid state and find that it has a relatively low variational energy ($\ensuremath{-}0.4893{J}_{1}$/site) for the J1{J}_{1}J1​-J2{J}_{2}J2​ model at J2=0.5J1{J}_{2}=0.5{J}_{1}J2​=0.5J1​. The spin liquid state we obtain breaks the fourfold rotational symmetry of the square lattice and therefore is a nematic spin liquid state. This direct continuous phase transition from the N\'eel state to a spin liquid state may be realized in the J1{J}_{1}J1​-J2{J}_{2}J2​ model, or the anisotropic J1x{J}_{1x}J1x​-J1y{J}_{1y}J1y​-J2{J}_{2}J2​ model.
A concept---quantum order---is introduced to describe a new kind of orders that generally appear in quantum states at zero temperature. Quantum orders that characterize the universality classes of quantum states (described by complex ground-state wave functions) are much richer than classical orders that characterize the universality classes of finite-temperature classical states (described by positive probability distribution functions). Landau's theory for orders and phase transitions does not apply to quantum orders since they cannot be described by broken symmetries and the associated order parameters. We introduced a mathematical object---projective symmetry group---to characterize quantum orders. With the help of quantum orders and projective symmetry groups, we construct hundreds of symmetric spin liquids, which have SU(2), U(1), or Z2{Z}_{2}Z2​ gauge structures at low energies. We found that various spin liquids can be divided into four classes: (a) Rigid spin liquid---spinons (and all other excitations) are fully gapped and may have bosonic, fermionic, or fractional statistics. (b) Fermi spin liquid---spinons are gapless and are described by a Fermi liquid theory. (c) Algebraic spin liquid---spinons are gapless, but they are not described by free fermionic-bosonic quasiparticles. (d) Bose spin liquid---low-lying gapless excitations are described by a free-boson theory. The stability of those spin liquids is discussed in detail. We find that stable two-dimensional spin liquids exist in the first three classes (a)--(c). Those stable spin liquids occupy a finite region in phase space and represent quantum phases. Remarkably, some of the stable quantum phases support gapless excitations even without any spontaneous symmetry breaking. In particular, the gapless excitations in algebraic spin liquids interact down to zero energy and the interaction does not open any energy gap. We propose that it is the quantum orders (instead of symmetries) that protect the gapless excitations and make algebraic spin liquids and Fermi spin liquids stable. Since high-Tc{T}_{c}Tc​ superconductors are likely to be described by a gapless spin liquid, the quantum orders and their projective symmetry group descriptions lay the foundation for a spin liquid approach to high-Tc{T}_{c}Tc​ superconductors.
We present the critical theory of a number of zero-temperature phase transitions of quantum antiferromagnets and interacting boson systems in two dimensions. The most important example is the transition of the S=1∕2S=1∕2S=1∕2 square lattice antiferromagnet between the N\'eel state (which breaks spin rotation invariance) and the paramagnetic valence bond solid (which preserves spin rotation invariance but breaks lattice symmetries). We show that these two states are separated by a second-order quantum phase transition. This conflicts with Landau-Ginzburg-Wilson theory, which predicts that such states with distinct broken symmetries are generically separated either by a first-order transition, or by a phase with co-existing orders. The critical theory of the second-order transition is not expressed in terms of the order parameters characterizing either state, but involves fractionalized degrees of freedom and an emergent, topological, global conservation law. A closely related theory describes the superfluid-insulator transition of bosons at half filling on a square lattice, in which the insulator has a bond density wave order. Similar considerations are shown to apply to transitions of antiferromagnets between the valence bond solid and the Z2{Z}_{2}Z2​ spin liquid: the critical theory has deconfined excitations interacting with an emergent U(1)\mathrm{U}(1)U(1) gauge force. We comment on the broader implications of our results for the study of quantum criticality in correlated electron systems.
We present a general review of the projective symmetry group classification of fermionic quantum spin liquids for lattice models of spin S=1/2S=1/2S=1/2. We then introduce a systematic generalization of the approach for symmetric Z2{\mathbb{Z}}_{2}Z2​ quantum spin liquids to the one of chiral phases (i.e., singlet states that break time reversal and lattice reflection, but conserve their product). We apply this framework to classify and discuss possible chiral spin liquids on triangular and kagome lattices. We give a detailed prescription on how to construct quadratic spinon Hamiltonians and microscopic wave functions for each representation class on these lattices. Among the chiral Z2{\mathbb{Z}}_{2}Z2​ states, we study the subset of U(1) phases variationally in the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{d}$ Heisenberg model on the kagome lattice. We discuss static spin structure factors and symmetry constraints on the bulk spectra of these phases.
We study possible spin liquids on square lattice that respect all lattice symmetries and time-reversal symmetry within the framework of Schwinger boson (mean-field) theory. Such spin liquids have spin gap and emergent Z2{Z}_{2}Z2​ gauge field excitations. We classify them by the projective symmetry group method, and find six spin-liquid states that are potentially relevant to the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model. The properties of these states are studied under mean-field approximation. Interestingly we find a spin-liquid state that can go through continuous phase transitions to either the N\'eel magnetic order or magnetic orders of the wave vector at the Brillouin zone edge center. We also discuss the connection between our results and the Abrikosov fermion spin liquids.
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