Gapped
Z2 spin liquid as the simplest spin liquid has been proved to be the most difficult to realize in realistic models. Here we show that the frustration from a third-neighbor exchange
J3 on the spin-
21 J1−J2 model on the square lattice may serve to stabilize such a long-sought state. We argue that a Bosonic RVB description is more appropriate than a Fermonic RVB description for such a gapped spin liquid phase. We show that while the mean field ansatz of the proposed state has a
U(1) gauge symmetry, the gauge fluctuation spectrum on it is actually gapped and the state should be understood rather as a
Z2 spin liquid state with topological order. The state is thus locally stable with respect to gauge fluctuation and can emerge continuously from a collinear Neel ordered phase.
We study the ground-state phase diagram of the spin-
1/2${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the square lattice with an accurate bosonic resonating-valence-bond (RVB) wave function. In contrast to the RVB ansatz based on Schwinger fermions, the representation based on Schwinger bosons, supplemented by a variational Monte Carlo technique enforcing the exact projection onto the physical subspace, is able to describe a fully gapped spin liquid in the strongly frustrated regime. In particular, a fully symmetric
Z2 spin liquid is stable between two antiferromagnetic phases; a continuous transition at
J2=0.4J1, when the Marshall sign rule begins to be essentially violated, and a first-order transition around
J2=0.6J1 are present. Most importantly, the triplet gap is found to have a nonmonotonic behavior, reaching a maximum around
J2=0.51J1, when the lowest spinon excitation moves from the $\ensuremath{\Gamma}$ to the
M point, i.e., $k=(\ensuremath{\pi},0)$.
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 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 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 superconductors.
We examine a spinor Bose gas confined by an elongated trap. Since a spin-independent energy is much higher than a spin-dependent energy in alkali species, the system exhibits different properties by changing a radial confinement. We show that if a spin-dependent coupling is positive, a spin-liquid condensate, which breaks the charge
U(1) symmetry but preserves the spin rotational symmetry, can be realized in an intermediate confinement regime. Properties of the spin-liquid condensate are visible if a temperature is lower than a spin gap to characterize the spin-disorder property. If a temperature is higher than the gap but lower than a spin-dependent coupling energy, on the other hand, a regime in which a spin sector is described by a semiclassical wave emerges. A characterization in each regime by means of correlation functions and topological solitons is also discussed.
Due to strong geometric frustration and quantum fluctuation, the
S=1/2 quantum Heisenberg antiferromagnet on the
kagome lattice has long been considered as an ideal platform to realize a spin liquid (SL), a phase exhibiting fractionalized excitations without any symmetry breaking. A recent numerical study (Yan et al., e-print arXiv:1011.6114) of the Heisenberg
S=1/2,kagome lattice model (HKLM) shows, in contrast to earlier results, that the ground state is a singlet-gapped SL with signatures of
Z2 topological order. Motivated by this numerical discovery, we use the projective symmetry group to classify all 20 possible Schwinger fermion mean-field states of
Z2 SLs on the
kagome lattice. Among them we found only one gapped
Z2 SL (which we call the ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state) in the neighborhood of the U(1) Dirac SL state. Since its parent state, i.e., the U(1) Dirac SL, was found [Ran et al., Phys. Rev. Lett. 98, 117205 (2007)] to be the lowest among many other candidate U(1) SLs, including the uniform resonating-valence-bond states, we propose this ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state to be the numerically discovered SL ground state of the HKLM.
A quantum ground state of matter is realized in a bosonic model on a three-dimensional fcc lattice with emergent low energy excitations. The phase obtained is a stable gapless boson liquid phase, with algebraic boson density correlations. The stability of this phase is protected against the instanton effect and superfluidity by self-duality and large gauge symmetries on both sides of the duality. The gapless collective excitations of this phase closely resemble the graviton, although they have a soft $\ensuremath{\omega}\ensuremath{\sim}{k}^{2}$ dispersion relation. There are three branches of gapless excitations in this phase, one of which is gapless scalar trace mode, the other two have the same polarization and gauge symmetries as the gravitons. The dynamics of this phase is described by a set of Maxwell's equations. The defects carrying gauge charges can drive the system into the superfluid order when the defects are condensed; also the topological defects are coupled to the dual gauge field in the same manner as the charge defects couple to the original gauge field, after the condensation of the topological defects, the system is driven into the Mott insulator phase. In the two-dimensional case, the gapless soft graviton as well as the algebraic liquid phase are destroyed by the vertex operators in the dual theory, and the stripe order is most likely to take place close to the two-dimensional quantum critical point at which the vertex operators are tuned to zero.