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arXiv:1201.0804 · Physical Review B

Néel-VBS phase boundary of the extended J1 - J2 model with biquadratic interaction

Yoshihiro Nishiyama

Published Jan 4, 2012• 5 citations
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Abstract

The ${J}_{1}$-${J}_{2}$ model with the biquadratic (plaquette-four-spin) interaction was simulated with the numerical diagonalization method. Some limiting cases of this model have been investigated thoroughly. Taking advantage of the extended parameter space, we survey the phase boundary separating the N\'eel and valence-bond-solid phases. According to the deconfined-criticality scenario, the singularity of this phase boundary is continuous, accompanied with unconventional critical indices. Diagonalizing the finite-size cluster with $N\ensuremath{\leqslant}36$ spins, we observe a signature of continuous phase transition. Our tentative estimate for the correlation-length critical exponent is $\ensuremath{\nu}=1.1(3)$. In order to elucidate a nonlocal character of criticality, we evaluated the Roomany-Wyld $\ensuremath{\beta}$ function around the critical point.

DOI
10.1103/physrevb.85.014403

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The J1{J}_{1}J1​-J2{J}_{2}J2​ model with the biquadratic (plaquette-four-spin) interaction was simulated with the numerical diagonalization method. Some limiting cases of this model have been investigated thoroughly. Taking advantage of the extended parameter space, we survey the phase boundary separating the N'eel and valence-bond-solid phases. According to the deconfined-criticality scenario, the singularity of this phase boundary is continuous, accompanied with unconventional critical indices. Diagonalizing the finite-size cluster with N\ensuremath⩽36N\ensuremath{\leqslant}36N\ensuremath⩽36 spins, we observe a signature of continuous phase transition. Our tentative estimate for the correlation-length critical exponent is \ensuremathν=1.1(3)\ensuremath{\nu}=1.1(3)\ensuremathν=1.1(3). In order to elucidate a nonlocal character of criticality, we evaluated the Roomany-Wyld \ensuremathβ\ensuremath{\beta}\ensuremathβ function around the critical point.
The quantum S=1S=1S=1 spin model on the spatially anisotropic triangular lattice is investigated numerically. The nematic and valence-bond-solid (VBS) phases are realized by adjusting the spatial anisotropy and the biquadratic interaction. The phase transition between the nematic and VBS phases is expected to be a continuous one with unconventional critical indices (deconfined criticality). The geometrical character (spatial anisotropy) is taken into account by imposing the screw-boundary condition (Novotny's method). Diagonalizing the finite-size cluster with $N\ensuremath{\leqslant}20$ spins, we observe a clear indication of continuous phase transition. The correlation-length critical exponent is estimated as $\ensuremath{\nu}=0.92(10)$.
We study the spin-1/2 Heisenberg model on the square lattice with first- and second-neighbor antiferromagnetic interactions J(1) and J(2), which possesses a nonmagnetic region that has been debated for many years and might realize the interesting Z(2) spin liquid. We use the density matrix renormalization group approach with explicit implementation of SU(2) spin rotation symmetry and study the model accurately on open cylinders with different boundary conditions. With increasing J(2), we find a Néel phase and a plaquette valence-bond (PVB) phase with a finite spin gap. From the finite-size scaling of the magnetic order parameter, we estimate that the Néel order vanishes at J(2)/J(1)≃0.44. For 0.5<J(2)/J(1)<0.61, we find dimer correlations and PVB textures whose decay lengths grow strongly with increasing system width, consistent with a long-range PVB order in the two-dimensional limit. The dimer-dimer correlations reveal the s-wave character of the PVB order. For 0.44<J(2)/J(1)<0.5, spin order, dimer order, and spin gap are small on finite-size systems, which is consistent with a near-critical behavior. The critical exponents obtained from the finite-size spin and dimer correlations could be compatible with the deconfined criticality in this small region. We compare and contrast our results with earlier numerical studies.
The spatially anisotropic triangular antiferromagnet is investigated with the numerical diagonalization method. As the anisotropy varies, the model changes into a variety of systems such as the one-dimensional, triangular, and square-lattice antiferromagnets. Taking into account such a geometrical character, we impose the screw-boundary condition, which interpolates smoothly the one- and two-dimensional lattice structures. Diagonalizing the finite clusters with N=16,20,…,32N=16,20,\dots{},32N=16,20,…,32 spins, we observe an intermediate phase between the valence-bond solid (VBS) and N\'eel phases. Suppressing the intermediate phase by applying the ring exchange, we realize a direct VBS-N\'eel transition. The simulation data indicate that the transition is a continuous one with the correlation-length critical exponent $\ensuremath{\nu}=0.80(15)$. These features are in agreement with the deconfinement-criticality scenario advocated by Senthil and co-workers in the context of the high-temperature superconductivity.
The ground state phase of a spin-$\frac{1}{2} {J}_{1}\text{\ensuremath{-}}{J}_{2}$ antiferromagnetic Heisenberg model on a square lattice around the maximally frustrated regime $({J}_{2}\ensuremath{\sim}0.5{J}_{1})$ has been debated for decades. Here we study this model using 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 exact diagonalization study. Through finite size scaling of the spin correlation function, we find the critical point J2c1=0.572(5)J1{J}_{2}^{{c}_{1}}=0.572(5){J}_{1}J2c1​​=0.572(5)J1​ and critical exponents $\ensuremath{\nu}=0.50(8),\phantom{\rule{0.16em}{0ex}}{\ensuremath{\eta}}_{s}=0.28(6)$. In the range of $0.572<{J}_{2}/{J}_{1}\ensuremath{\le}0.6$ we find a paramagnetic ground state with an exponentially decaying spin-spin correlation. Up to a $24\ifmmode\times\else\texttimes\fi{}24$ system size, we observe power law decaying dimer-dimer and plaquette-plaquette correlations with an anomalous plaquette scaling exponent ${\ensuremath{\eta}}_{p}=0.24(1)$ and an anomalous columnar scaling exponent ${\ensuremath{\eta}}_{c}=0.28(1)$ at J2/J1=0.6{J}_{2}/{J}_{1}=0.6J2​/J1​=0.6. These results are consistent with a potential gapless U(1)U(1)U(1) spin-liquid phase. However, since the U(1)U(1)U(1) spin liquid is unstable due to the instanton effect, a valence bond solid order with very small amplitude might develop in the thermodynamic limit. Thus, our numerical results strongly indicate a deconfined quantum critical point at J2c1{J}_{2}^{{c}_{1}}J2c1​​. Remarkably, all the observed critical exponents are consistent with the $J\text{\ensuremath{-}}Q$ model.
XY-VBS phase boundary for the square-lattice J1J_1J1​-J2J_2J2​ XXZ model with the ring exchange
Abstract The square-lattice J1J_1J1​ J 1 - J2J_2J2​ J 2 XXZ model with the ring-exchange interaction K was investigated numerically. As for the hard-core-boson model with the nearest-neighbor hopping J1/2J_1/2J1​/2 J 1 / 2 , namely, the J1J_1J1​ J 1 -KXY model, it has been reported that the ring exchange leads to a variety of exotic phases such as the valence-bond-solid (VBS) phase. In this paper, we extend the parameter space to investigate the phase boundary between the XY (superfluid) and VBS phases. A notable feature is that the phase boundary terminates at the fully frustrated point, J2/J1→0.5−J_2/J_1 \rightarrow 0.5^-J2​/J1​→0.5− J 2 / J 1 → 0 . 5 - . As a scaling parameter for the multi-criticality, the distance from the multi-critical point δ(≥0)\delta (\ge 0)δ(≥0) δ ( ≥ 0 ) is introduced. To detect the phase transition, we employed the high-order fidelity susceptibility χF(3)\chi ^{(3)}_FχF(3)​ χ F ( 3 ) , which is readily evaluated via the exact-diagonalization scheme. As a demonstration, for a fixed value of δ\delta δ δ , the XY-VBS criticality was analyzed by the probe χF(3)\chi ^{(3)}_FχF(3)​ χ F ( 3 ) . Thereby, with properly scaling δ\delta δ δ , the χF(3)\chi ^{(3)}_FχF(3)​ χ F ( 3 ) data were cast into the crossover-scaling formula to determine the multi-criticality. Graphic abstract
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).