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arXiv:0902.2060 · Physical Review B

Deconfinement criticality in the spatially anisotropic triangular antiferromagnet with ring exchange

Yoshihiro Nishiyama

Published Feb 20, 2009• 11 citations
arXivHTMLPDF

Abstract

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 $N=16,20,\dots{},32$ 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.

DOI
10.1103/physrevb.79.054425

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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 \ensuremathν=0.80(15)\ensuremath{\nu}=0.80(15)\ensuremathν=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 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 use quantum Monte Carlo methods to study the ground-state phase diagram of a S=1/2S=1/2S=1/2 honeycomb lattice magnet in which a nearest-neighbor antiferromagnetic exchange JJJ (favoring N\'eel order) competes with two different multispin interaction terms: a six-spin interaction Q3{Q}_{3}Q3​ that favors columnar valence-bond solid (VBS) order, and a four-spin interaction Q2{Q}_{2}Q2​ that favors staggered VBS order. For ${Q}_{3}\ensuremath{\sim}{Q}_{2}\ensuremath{\gg}J$, we establish that the competition between the two different VBS orders stabilizes N\'eel order in a large swath of the phase diagram even when JJJ is the smallest energy scale in the Hamiltonian. When ${Q}_{3}\ensuremath{\gg}({Q}_{2},J)$ $\mathrm{[}{Q}_{2}\ensuremath{\gg}({Q}_{3},J)\text{]}$, this model exhibits at zero temperature phase transition from the N\'eel state to a columnar (staggered) VBS state. We establish that the N\'eel-columnar VBS transition is continuous for all values of Q2{Q}_{2}Q2​, and that critical properties along the entire phase boundary are well characterized by critical exponents and amplitudes of the noncompact CP1{\mathrm{CP}}^{1}CP1 (NCCP1)({\mathrm{NCCP}}^{1})(NCCP1) theory of deconfined criticality, similar to what is observed on a square lattice. However, a surprising threefold anisotropy of the phase of the VBS order parameter at criticality, whose presence was recently noted at the Q2=0{Q}_{2}=0Q2​=0 deconfined critical point, is seen to persist all along this phase boundary. We use a classical analogy to explore this by studying the critical point of a three-dimensional XYXYXY model with a fourfold anisotropy field which is known to be weakly irrelevant at the three-dimensional XYXYXY critical point. In this case, we again find that the critical anisotropy appears to saturate to a nonzero value over the range of sizes accessible to our simulations.
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 the N\'eel to fourfold columnar valence bond solid (cVBS) quantum phase transition in a sign-free S=1S=1S=1 square-lattice model. This is the same kind of transition that for S=1/2S=1/2S=1/2 has been argued to realize the prototypical deconfined critical point. Extensive numerical simulations of the square-lattice S=1/2S=1/2S=1/2 N\'eel-VBS transition have found consistency with the deconfined critical point scenario with no direct evidence for first-order behavior. In contrast to the S=1/2S=1/2S=1/2 case, in our quantum Monte Carlo simulations for the S=1S=1S=1 model, we present unambiguous evidence for a direct conventional first-order quantum phase transition. Classic signs of a first-order transition demonstrating coexistence including double-peaked histograms and switching behavior are observed. The sharp contrast from the S=1/2S=1/2S=1/2 case is remarkable; we hypothesize that this is a striking demonstration of the role of the size of the quantum spin in the phase diagram of two-dimensional lattice models.
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 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.