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arXiv:1204.6658 · New Journal of Physics

Evidence of a spin liquid with hard-core bosons in a square lattice

Y-H Chan, L-M Duan

Published Nov 28, 2012• 9 citations
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Abstract

We show that laser-assisted hopping of hard-core bosons in a square optical lattice can be described by an antiferromagnetic J 1 – J 2 XY model with a tunable ratio of J 2 / J 1 . We numerically investigated the phase diagram of the J 1 – J 2 XY model using both the tensor network algorithm for infinite systems and the exact diagonalization for small clusters and found strong evidence that in the intermediate region around J 2 / J 1 ∼ 0.5, there is a spin liquid phase with vanishing magnetization and valence bond orders, which interconnects the Néel state on the J 2 ≪ J 1 side and the stripe antiferromagnetic phase on the J 2 ≫ J 1 side. This finding opens up the possibility of studying the exotic spin liquid phase in a realistic experimental system using ultracold atoms in an optical lattice.

DOI
10.1088/1367-2630/14/11/113039

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We show that laser-assisted hopping of hard-core bosons in a square optical lattice can be described by an antiferromagnetic J 1 – J 2 XY model with a tunable ratio of J 2 / J 1 . We numerically investigated the phase diagram of the J 1 – J 2 XY model using both the tensor network algorithm for infinite systems and the exact diagonalization for small clusters and found strong evidence that in the intermediate region around J 2 / J 1 ∼ 0.5, there is a spin liquid phase with vanishing magnetization and valence bond orders, which interconnects the Néel state on the J 2 ≪ J 1 side and the stripe antiferromagnetic phase on the J 2 ≫ J 1 side. This finding opens up the possibility of studying the exotic spin liquid phase in a realistic experimental system using ultracold atoms in an optical lattice.
We investigate the phase diagram of hard-core bosons on a square lattice with competing interactions. The hard-core bosons can also be represented by spin-1/21/21/2 operators and the model can therefore be mapped onto an anisotropic J1{J}_{1}J1​-J2{J}_{2}J2​ Heisenberg model. We find the N\'eel state and a collinear antiferromagnetic state as classical ordered phases to be suppressed by the introduction of ferromagnetic exchange terms in the xxx-yyy plane that result in a ferromagnetic phase for large interactions. For an intermediate regime, the emergence of new quantum states like valence bond crystals or supersolids is predicted for similar models. We do not observe any signal for long-range order in terms of conventional order or dimer correlations in our model and find an exponential decay in the spin correlations. Hence, all evidence is pointing toward a quantum-disordered ground state for a small region in the phase diagram.
We study the spin 1/2 J1-J2 antiferromagnetic Heisenberg model on square lattice by using a recently proposed cluster update method for tensor product states (TPSs). The ground state energies in the thermodynamic limit are in good agreement with the state of art exact diagonalization study, and the energy differences between these two studies are of the order of 10J1 per site. For small bond dimension D (D < 5), we find a paramagnetic ground state with mixed columnar and staggered dimer orders of a magnitude 10−2 in the range of 0.47 < J2/J1 < 0.6, which agrees with previous results. For large bond dimension D (D ≥ 5), we observe all these orders vanish, implying the emergence of a spin liquid phase.
Unique features of the nonmagnetic insulator phase are revealed, and the phase diagram of the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}$ Hubbard model containing the diagonal transfers ${t}^{\ensuremath{'}}$ on a square lattice is presented. Using the path-integral renormalization group method, we find an antiferromagnetic phase for small next-nearest-neighbor transfer ${t}^{\ensuremath{'}}$ and a stripe (or collinear) phase for large ${t}^{\ensuremath{'}}$ in the Mott insulating region of the strong on-site interaction UUU. For intermediate ${t}^{\ensuremath{'}}∕t\ensuremath{\sim}0.7$ at large U∕t>7U∕t>7U∕t>7, we find a longer-period antiferromagnetic-insulator phase with $2\ifmmode\times\else\texttimes\fi{}4$ structure. In the Mott insulating region, we also find a quantum spin liquid (in other words, a nonmagnetic insulator) phase near the Mott transition to paramagnetic metals for the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}$ Hubbard model on the square lattice as well as on the anisotropic triangular lattice. Correlated electrons often crystallize to the Mott insulator usually with some magnetic orders, whereas the ``quantum spin liquid" has been a long-sought issue. We report numerical evidence that a nonmagnetic insulating phase gets stabilized near the Mott transition with remarkable properties: The two-dimensional Mott insulators on geometrically frustrated lattices contain a phase with gapless spin excitations and degeneracy of the ground state in the whole Brillouin zone of the total momentum. The obtained vanishing spin renormalization factor suggests that spin excitations do not propagate coherently in contrast to conventional phases, where there exist either magnons in symmetry-broken phases or particle-hole excitations in paramagnetic metals. It imposes a constraint on the possible pictures of quantum spin liquids and supports an interpretation for the existence of an unconventional quantum liquid. The present concept is useful in analyzing a variety of experimental results in frustrated magnets including organic BEDT-TTF compounds and 3He^{3}\mathrm{He}3He atoms adsorbed on graphite.
We perform an in-depth investigation of the phase diagram of the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the square lattice. We take advantage of density matrix renormalization group and fully augmented matrix product states methods and reach unprecedented accuracy with large bond dimensions. We utilize excited-level crossing analysis to pinpoint the phase transition points. It was believed before that there exists a narrow spin liquid phase sandwiched by the N\'eel antiferromagnetic (AFM) and valence bond solid (VBS) phases. Through careful finite-size scaling of the level crossing points, we find a direct phase transition between the N\'eel AFM and VBS phases at J2/J1=0.535(3){J}_{2}/{J}_{1}=0.535(3)J2​/J1​=0.535(3), suggesting the absence of an intermediate spin liquid phase. We also provide accurate results for ground-state energies for a variety of sizes, from which we find that the transition between the N\'eel AFM and VBS phases is continuous. These results indicate the existence of a deconfined quantum critical point at J2/J1=0.535(3){J}_{2}/{J}_{1}=0.535(3)J2​/J1​=0.535(3) in the model. From the crossing of the first derivative of the energies with J2{J}_{2}J2​ for different sizes, we also determine the precise location of the first-order phase transition between the VBS and stripe AFM phases at J2/J1=0.610(5){J}_{2}/{J}_{1}=0.610(5)J2​/J1​=0.610(5).
Using random phase approximation we build finite-temperature phase diagrams and calculate thermodynamic functions of a square lattice hard-core boson model with nearest (nn) and next-nearest-neighbor (nnn) interactions in terms of equivalent to it an anisotropic spin-1/2 XXZXXZXXZ model in a magnetic field. The system undergoes liquid-solid phase transitions that can be either of the first or second order. Depending on the hopping value and ratio between nn and nnn interactions the system displays two types of critical behavior. The line of the first-order transitions terminates in a bicritical end point inside the solid phase or ends in a tricritical point continuously giving way to the second-order phase transition line. The connection of the hopping and the ratio between the nn and nnn interactions with criticality of the system is investigated. The critical line separating the regions of specific critical behavior is built. Reentrant liquid-solid-liquid phase transitions are found and discussed.
The ground state phase of spin-1/2 J_1-J_2 antiferromagnetic Heisenberg model on square lattice in the maximally frustrated regime (J_2 ~ 0.5J_1) has been debated for decades. Here we study this model by using a recently proposed novel numerical method - the cluster update algorithm for tensor product states (TPSs). The ground state energies at finite sizes and in the thermodynamic limit (with finite size scaling) are in good agreement with the state of art exact diagonalization study, and the energy differences between these two studies are of the order of 10^{-3}J_1 per site. At the largest bond dimension available D (D=9), we find a paramagnetic ground state without any valence bond solid order in the thermodynamic limit in the range of 0.5 < J_2/J_1 < 0.6, which implies the emergence of a spin-liquid phase. Furthermore, we investigate the topologically ordered nature of such a spin-liquid phase by measuring a nonzero topological entanglement entropy.