Deconfined criticality for the spin model on the spatially anisotropic triangular lattice
The quantum 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 spins, we observe a clear indication of continuous phase transition. The correlation-length critical exponent is estimated as .
pacs:75.10.Jm 05.30.-d 75.40.Mg 74.25.Ha
According to the deconfined-criticality scenario,Senthil04 (); Senthil04b (); Levin04 () in two dimensions, the phase transition separating the valence-bond-solid (VBS) and antiferromagnetic phases is continuous; naively,Senthil04b () such a transition should be discontinuous, because the adjacent phases possess distinctive order parameters such as the VBS coverage pattern and the sublattice magnetization, respectively. A good deal of theoretical investigationsMotruich04 (); Tanaka06 (); Liu05 (); Dellenschneider06 (); Ghaemi06 (); Thomas07 (); Kim97 (); Singh10 () has been made to support this scenario. (On the contrary, in Refs. Kotov09, ; Isaev10, ; Kotov10, ; Isaev10b, ; Kuklov04, ; Kuklov08, ; Jian08, ; Kruger06, , it was claimed that the transition would be a weak first-order one.)
The magnetic frustration is a clue to the realization of the VBS phase. Actually, the square-lattice antiferromagnet with the next-nearest-neighbor interaction (- model) exhibits the VBS phase around the fully frustrated () regime.Oitmaa96 () The quantum Monte Carlo method is not applicable to this problem because of the negative-sign problem. So far, the - model has been studied with the series-expansionOitmaa96 (); Sirker06 () and numerical-diagonalizationPoilblanc06 () methods.
Alternatively, one is able to realize the VBS phase through incorporating the biquadratic interactionRead89 (); Kawashima07 (); correspondingly, one has to enlarge the magnitude of spin to . The biquadratic interaction (unlike the magnetic frustration) is tractable with the quantum Monte Carlo method. The deconfined criticality is realized by tuning the spatial anisotropy.Harada07 () That is, as the interchain interaction increases, a transition from the VBS phase to either nematic or antiferromagnetic phase occurs.Harada07 () Meanwhile, it turned out that the ring-exchange (plaquette-four-spin) interaction also induces the VBS phase even for . Extensive Monte Carlo simulationsSandvik07 (); Melko08 () support the deconfined-criticality scenario; the results are compared with ours in Sec. IV. The transition occurs for a considerably large ring exchange; the antiferromagnetic order would be far more robust than that of VBS. As mentioned above, the character of the singularity is controversial Kotov09 (); Isaev10 (); Kotov10 (); Isaev10b (); Kuklov04 (); Kuklov08 (); Jian08 (); Kruger06 (); possibly, the correctionsSandvik10 () affect the scaling analysis. Even for the spatially anisotropic triangular antiferromagnet,Trumper99 () The ring exchange gives rise to the VBS phase.Misguich99 (); Nishiyama09 () This model is intractable with the quantum Monte Carlo method, and analytical considerations provide valuable information as to the deconfined criticality.Xu09 (); Qi09 (); Nakane09 ()
In this paper, we investigate the spatially-anisotropic-triangular-lattice model with the biquadratic interaction by means of the numerical diagonalization method. To be specific, the Hamiltonian is given by
Here, the quantum spins are placed at each triangular-lattice point ; see Fig. 1 (a). The summation () runs over all possible nearest-neighbor (skew-diagonal) pairs. The parameter () denotes the corresponding coupling constant. Hereafter, we consider as the unit of energy (). Along the bond, both quadratic and biquadratic interactions exist, and the parameter controls a strength of the former component. The -bond interaction is purely biquadratic. The interaction interpolates the one-dimensional () and square-lattice () structures. Correspondingly, the VBS and spin-nematic phases appear, as the interaction varies; see a schematic phase diagram, Fig. 2. In order to take into account such a geometrical character, we implement the screw-boundary condition [Fig. 1 (b)] through resorting to Novotny’s method (Sec. II).
The rest of this paper is organized as follows. In Sec. II, we explain the simulation scheme. We also make an overview on the biquadratic-interaction spin models relevant to ours. In Sec. III, we demonstrate that the present model exhibits a clear indication of deconfined criticality at a moderate value of . We also analyze the criticality with the finite-size-scaling theory. In Sec. IV, we present the summary and discussions.
Ii Screw-boundary condition: Novotny’s method
Before commencing a explanation of technical details, we present a basic idea of Novotny’s method. We implement the screw-boundary condition for a finite cluster with spins; see Fig. 1 (b). Basically, the spins, (), constitute a one-dimensional () structure, and the dimensionality is lifted to by the bridges over the long-range pairs. The present system (1) has a spatial anisotropy governed by . We take into account such a geometrical character through imposing the screw-boundary condition. According to Novotny, the long-range interactions are introduced systematically by the use of the translation operator ; see Eq. (3), for instance. The operator satisfies the formula
Here, the base diagonalizes each of ; namely, the relation holds.
Novotny’s method was adapted to the quantum model in dimensions.Nishiyama08 () Our simulation scheme is based on this formalism. In the following, we present the modifications explicitly for the sake of selfconsistency. The interaction , Eq. (4) of Ref. Nishiyama08, , has to be replaced with the Heisenberg interaction
Additionally, we introduce the biquadratic interaction
The definition of and an algebra are presented in the Appendix. Based on these expressions, we replace Eq. (3) of Ref. Nishiyama08, with
We diagonalize this matrix for spins in Sec. III. The above formulae complete the formal basis of our simulation scheme. However, in order to evaluate the above Hamiltonian-matrix elements efficiently, one may refer to a number of techniques addressed in Refs. Novotny92, ; Nishiyama08, .
Last, we overview the biquadratic-interaction spin models. As mentioned in the Introduction, the present model reduces to the one-dimensional and square-lattice models in the limiting cases and , respectively. Each of these limiting cases has been studied extensively. Here, we devote ourselves to the nearest-neighbor interaction of the generic form, , parameterized by . The regime is relevant to the present research. As for (Ref. Fath95, ), the dimer phase appears in . Namely, around , the stability of the dimer (VBS) phase would be maximal. For (Ref. Harada02, ), the ferromagnetic, nematic, and antiferromagnetic phases appear in , , and , respectively. The phase diagram in Fig. 2 is based on these preceding studies.
Iii Numerical results
In this section, we present the simulation results. We employed the simulation scheme developed in Sec. II. We devote ourselves to the analysis of the transition between the nematic and VBS phases (deconfined criticality); see Fig. 2. We treat a variety of system sizes ( is the number of spins within a cluster). The linear dimension of the cluster is given by
iii.1 Critical point
In this section, we investigate a location of the phase boundary separating the VBS and nematic phases.
In Fig. 3, we plot the scaled energy gap for various and . The quadratic-interaction strength is fixed to . The symbol denotes the first excitation gap. According to the finite-size scaling, the scaled energy gap should be scale-invariant at the critical point. In fact, we observe an intersection point at , which indicates an onset of the -driven phase transition. As mentioned in the Introduction, the nature of this singularity is under current interest. The present result indicates that the singularity is a continuous one; the critical phenomena (below the upper critical dimension) should be described by the finite-size scaling.
In Fig. 4, we plot the approximate transition point for with (). The parameters are the same as those of Fig. 3. Here, the approximate transition point denotes a scale-invariant point with respect to a pair of system sizes . Namely, the following relation holds;
The least-squares fit to the data of Fig. 4 yields an estimate in the thermodynamic limit . In order to appreciate possible extrapolation errors, we made an alternative extrapolation with the -abscissa scale. Thereby, we obtain . The extrapolation errors appear to be negligible. (As a matter of fact, we surveyed a wide range of , and found that the parameter yields an optimal finite-size behavior.) Hence, we obtain an estimate .
Making simular analyses for various values of , we arrived at a phase diagram, Fig. 2. The phase boundary around and is ambiguous because of finite-size errors. Possibly, around , the magnetic structure (antiferromagnetic order) conflicts with the screw-boundary condition, resulting in an enhancement of finite-size errors. On the one hand, in , the character of the transition changes to a discontinuous one, and the finite-size-scaling method becomes invalid. Hence, the finite-size behavior improves around the midst () of .
Last, we address a number of remarks. First, in the scaling analysis, Fig. 3, we assumed the dynamical critical exponent , following the conclusion of the Monte Carlo simulations.Sandvik07 (); Melko08 () Second, we argue a possible systematic error for the data in Fig. 4. As a matter of fact, for large system sizes, the data exhibit an enhancement, suggesting that the extrapolated value of should be larger than . However, in the subsequent analyses, the extrapolated value of is no longer used, and systematic deviations are less influential. Nevertheless, as to the singularity of , it has to be mentioned that the present data cannot exclude a possibility of weak-first-order transition, for which the scaling approach becomes invalidated.
iii.2 Correlation-length critical exponent
In this section, we analyze the criticality found in Sec. III.1.
In Fig. 5, we plot the approximate critical exponent
for with . The parameters are the same as those of Fig. 3. The least-squares fit to these data yields in the thermodynamic limit. This estimate may be affected by systematic errors. (The statistical error would be an underestimate.) In order to appreciate an error margin, we made an alternative extrapolation with the -abscissa scale. Thereby, we obtain . The discrepancy indicates an amount of systematic errors. As a result, we arrive at an estimate
which covers the above results obtained via independent extrapolations.
This is a good position to address a remark on the scaling behaviors of Figs. 3-5. As mentioned in the Introduction, notorious corrections were observed for the square-lattice antiferromagnet with the ring exchange. Our data, on the contrary, appear to exhibit moderate corrections to scaling, particularly, in Figs. 3 and 4. For the former antiferromagnet, the VBS phase emerges for a considerably large ring exchange, indicating that the antiferromagnetic phase is robust. On the one hand, our model exhibits a stable VBS state owing to the spatial anisotropy (one-dimensionality in the limit), and the nematic phase turns into the VBS phase at a moderate coupling strength, . We suspect that these peculiarities of the present (rather artificially designed) model bring about improved scaling behaviors.
Last, we make a consideration of the abscissa scale of Fig. 5 (4). The critical exponent (point) has a leading correction of O [O]; here, the symbol denotes the index for corrections to scaling. At present, the index for the deconfined criticality is unclear. As a reference, one may refer to that of the Heisenberg universality class, (Ref. Hasenbusch01, ). Making use of this value, we set the abscissa scale to that depicted in Fig. 5 (4).
Iv Summary and discussions
The spin model on the spatially anisotropic triangular lattice, Eq. (1), was investigated numerically. As the spatial anisotropy changes, the VBS and nematic phases appear (Fig. 2). Hence, this rather artificial model provides a candidate for the analysis of the deconfined criticality, which is arousing much attention recently. We employed Novotny’s method (screw boundary condition) in order to take into account such a geometrical character (spatial anisotropy).
As a result, we observe a clear indication of the -driven criticality through the finite-size-scaling analysis (Fig. 3). This result supports the deconfined-criticality scenario. Thereby, we estimate the correlation-length critical exponent as .
As a reference, we overview related studies. For the square-lattice antiferromagnet with the ring exchange, the estimates, (Ref. Sandvik07, ) and (Ref. Melko08, ), were reported. As for the spatially-anisotropic-triangular antiferromagnet with the ring exchange, the exponent was obtained.Nishiyama09 () These results are to be compared with the index, (Ref. Campostrini02, ), for the Heisenberg universality class. Our result indicates a tendency toward an enhancement for the correlation-length critical exponent, as compared to that for the Heisenberg universality class. Taking the advantage of the numerical diagonalization method, we are able to extend the interactions so as to eliminate finite-size errors. A frustrated interaction along the -bond direction may stabilize (extend the regime of) the VBS phase substantially. This problem will be addressed in the future study.
Appendix A A reduction formula for the biquadratic interaction
The biquadratic interaction reduces to a seemingly quadratic form
Here, the operators are given by the relations, , , , , and . This reduction formula is a key ingredient in Sec. II.
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