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arXiv:1101.1999 · Physical Review B

Deconfined criticality for the S=1 spin model on the spatially anisotropic triangular lattice

Yoshihiro Nishiyama

Published Feb 15, 2011• 6 citations
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Abstract

The quantum $S=1$ 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 $\ensuremath{\nu}=0.92(10)$.

DOI
10.1103/physrevb.83.054417

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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⩽20N\ensuremath{\leqslant}20N\ensuremath⩽20 spins, we observe a clear indication of continuous phase transition. The correlation-length critical exponent is estimated as \ensuremathν=0.92(10)\ensuremath{\nu}=0.92(10)\ensuremathν=0.92(10).
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 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 $N\ensuremath{\leqslant}36$ 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.
We study a spin-1/2 SU(2) model on the honeycomb lattice with nearest-neighbor antiferromagnetic exchange J that favors Néel order and competing six-spin interactions Q that favor a valence-bond-solid (VBS) state in which the bond energies order at the "columnar" wave vector K=(2π/3,-2π/3). We present quantum Monte Carlo evidence for a direct continuous quantum phase transition between Néel and VBS states, with exponents and logarithmic violations of scaling consistent with those at analogous deconfined critical points on the square lattice. Although this strongly suggests a description in terms of deconfined criticality, the measured threefold anisotropy of the phase of the VBS order parameter shows unusual near-marginal behavior at the critical point.
We continue recent efforts to discover examples of deconfined quantum criticality in one-dimensional models. In this work we investigate the transition between a Z3{\mathbb{Z}}_{3}Z3​ ferromagnet and a phase with valence bond solid (VBS) order in a spin chain with ${\mathbb{Z}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{3}$ global symmetry. We study a model with alternating projective representations on the sites of the two sublattices, allowing the Hamiltonian to connect to an exactly solvable point having VBS order with the character of SU(3)SU(3)SU(3)-invariant singlets. Such a model does not admit a Lieb-Schultz-Mattis theorem typical of systems realizing deconfined critical points. Nevertheless, we find evidence for a direct transition from the VBS phase to a Z3{\mathbb{Z}}_{3}Z3​ ferromagnet. Finite-entanglement scaling data are consistent with a second-order or weakly first-order transition. We find in our parameter space an integrable lattice model apparently describing the phase transition, with a very long, finite, correlation length of 190878 lattice spacings. Based on exact results for this model, we propose that the transition is extremely weakly first order and is part of a family of deconfined quantum critical points described by walking of renormalization group flows.
We consider spin-half quantum antiferromagnets in two spatial dimensions in the quantum limit, where the spins are in a valence bond solid (VBS) phase. The transition between two such VBS phases is studied. In some cases, an interesting second-order transition controlled by a fixed line with varying critical exponents is found. A specific example is provided by an antiferromagnetically coupled bilayer system on the honeycomb lattice where a continuous quantum phase transition can generically exist between two VBS phases. Furthermore, these critical points are deconfined, in the sense that gapped spin-1∕21∕21∕2 spinon excitations emerge right at the transition. The low-energy physics of this critical point (up to marginally irrelevant interactions) contains just a free quadratically dispersing ``photon.'' The phase structure on one side of this continuous transition is very intricate, consisting of a series of infinitely closely spaced further transitions in a ``devil's staircase'' form. Analogies with previous examples of deconfined quantum criticality are emphasized. Closely related transitions in single layer systems are explored. These are second order only at some multicritical points. The solvable Rokshar-Kivelson point of quantum dimer models of single layer systems is found to correspond to a nongeneric multicritical point.
We perform a numerical study of a spin-1/2 model with ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry in one dimension which demonstrates an interesting similarity to the physics of two-dimensional deconfined quantum critical points (DQCP). Specifically, we investigate the quantum phase transition between Ising ferromagnetic and valence bond solid (VBS) symmetry-breaking phases. Working directly in the thermodynamic limit using uniform matrix product states, we find evidence for a direct continuous phase transition that lies outside of the Landau-Ginzburg-Wilson paradigm. In our model, the continuous transition is found everywhere on the phase boundary. We find that the magnetic and VBS correlations show very close power-law exponents, which is expected from the self-duality of the parton description of this DQCP. Critical exponents vary continuously along the phase boundary in a manner consistent with the predictions of the field theory for this transition. We also find a regime where the phase boundary splits, as suggested by the theory, introducing an intermediate phase of coexisting ferromagnetic and VBS order parameters. Interestingly, we discover a transition involving this coexistence phase which is similar to the DQCP, being also disallowed by the Landau-Ginzburg-Wilson symmetry-breaking theory.