Deconfined-critical behavior of the VBS- and nematic-order parameters for the spatially anisotropic -spin model
The phase transition between the valence-bond-solid (VBS) and nematic phases, the so-called deconfined criticality, was investigated for the quantum -spin model on the spatially anisotropic triangular lattice with the biquadratic interaction by means of the numerical diagonalization method. We calculated both VBS- and nematic-order parameters, aiming to clarify the nature of this transition from complementary viewpoints. Simulating the clusters with spins, we estimate the correlation-length critical exponent as . We also calculated Fisher’s exponent (anomalous dimension) for each order parameter.
keywords:75.10.Jm Quantized spin models, 05.30.-d Quantum statistical mechanics (for quantum fluids aspects, see 67.10.Fj), 75.40.Mg Numerical simulation studies, 74.25.Ha Magnetic properties
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.) Extensive computer simulations have been made to support this claim Sandvik07 (); Melko08 (). However, still, one cannot exclude a possibility of a weak-first-order transition accompanied with an appreciable latent-heat release Kotov09 (); Isaev10 (); Kotov10 (); Isaev10b (); Kuklov04 (); Kuklov08 (); Jian08 (); Kruger06 (); Sirker06 ().
The magnetic frustration is a good clue to the realization of the VBS phase Sirker06 (); Poilblanc06 (); Nishiyama12 (). Alternatively, one is able to stabilize the VBS state with the spatial anisotropy and the biquadratic interaction Harada07 (); Nishiyama09 (). In our preceding paper Nishiyama11 (), we investigated the quantum -spin model on the spatially anisotropic triangular lattice with the biquadratic interaction, Eq. (1), and analyzed the singularity of the VBS-nematic phase transition; a schematic phase diagram is presented in Fig. 1, where the parameters and control the spatial anisotropy and the biquadratic interaction, respectively. Scrutinizing the scaling behavior of the excitation gap, we obtained the correlation-length critical exponent Nishiyama11 (). In this paper, we calculate both VBS- and nematic-order parameters, and survey the criticality from complementary viewpoints.
To be specific, we present the Hamiltonian for the -spin model on the spatially anisotropic triangular lattice with the biquadratic interaction
Here, the quantum spins are placed at each triangular-lattice point ; see Fig. 2 (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, there appear the VBS Fath95 () and spin-nematic Harada02 () phases (Fig. 1). In order to take into account such a geometrical character, we implement the screw-boundary condition [Fig. 2 (b)] through resorting to Novotny’s method Novotny92 (); Nishiyama08 () (Section 2.1).
The critical indices for the deconfined criticality have been investigated thoroughly. We overview a number of related studies. First, the -spin model on the spatially anisotropic square lattice with the biquadratic interaction Harada07 () was simulated by means of the quantum Monte Carlo method; here, the spatial-anisotropy axis is set to be parallel to the primitive vector of the unit cell. At the VBS-nematic phase boundary, the authors estimated the (reciprocal) correlation-length critical exponent as and for the VBS- and nematic-order parameters, respectively. These indices coincide, as anticipated; namely, a relation holds. Additionally, they obtained Fisher’s exponent for the nematic order ; postulating the dynamical critical exponent (see below), one arrives at . (Its VBS-order counterpart was not given.) In our model, Eq. (1), we impose the spatial anisotropy along the skew-diagonal direction [Fig. 2 (a)], aiming to make an alternative approach to this problem. Second, the internal symmetry of each spin is extended from an ordinary SU to SU with an arbitrary integral number Read89 (); Kawashima07 (); Lou09 (); Kaul11 () or even a continuously variable parameter Beach09 (); note that the -spin model with the (finely-tuned) biquadratic interaction can possess the SU symmetry, and this problem is quite relevant to ours. The latter model Beach09 () demonstrates a clear evidence of a deconfined criticality with the critical indices, , , and at ; here, the index denotes the ordinary Fisher’s exponent for the constituent SU moment. It would be noteworthy that Fisher’s exponent acquires a considerable enhancement. Third, pioneering considerations Sandvik07 (); Melko08 () on the deconfined criticality were set forward for the square-lattice model with the biquadratic (plaquette-four-spin) interaction. Sets of critical indices, , , and Sandvik07 (); , , and Melko08 (), have been reported. A closely related model, the so-called model Lou09b (), yields almost similar results, , , and ; here, the indices, and , are the magnetization critical exponents for the constituent moment and the VBS-order parameter, respectively. Last, we recollect a number of field-theoretical considerations. A formalism for the -spin model is provided in Ref. Grover11 (), where a microscopic origin of an enhancement of Fisher’s exponent is argued. An extensive simulation on the hedgehog-suppressed O model Motruich04 () revealed a clear evidence for the novel critical indices, and . As a reference, we quote the critical exponents for the three-dimensional Heisenberg universality class Compostrini02 (), and .
2 Numerical results
In this section, we present the simulation results. Before commencing detailed analyses of criticality, we explain the simulation algorithm in a self-contained manner. Throughout this section, we fix the parameter to an intermediate value, , where the finite-size-scaling behavior improves Nishiyama11 (); see the phase diagram, Fig. 1. The number of spins (system size) ranges in . The linear dimension of the cluster is given by
because spins form a rectangular cluster.
The Hamiltonian (1) possesses a number of symmetries, with which one is able to reduce (block-diagonalize) the size of the Hamiltonian matrix. Here, aiming to eliminate the Hilbert-space dimensionality, we look into the subspace with the total longitudinal-spin moment , even parity, and the wave number with respect to the internal-spin-rotation-, spin-inversion- (), and lattice-translation-symmetry groups, respectively. The ground state belongs to this subspace. The size of the reduced (block-diagonal) Hamiltonian is for .
2.1 Simulation method
As mentioned in the Introduction, we impose the screw-boundary condition on a finite cluster with spins; see Fig. 2 (b). Basically, the spins, (), constitute a one-dimensional () structure, and the dimensionality is lifted to by the bridges over the long-range pairs. According to Novotny Novotny92 (), the long-range interactions are introduced systematically by the use of the translation operator ; see Eq. (4), 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 self-consistency. The interaction , Eq. (4) of Ref. Nishiyama08 (), has to be replaced with the Heisenberg interaction
Additionally, we introduce the biquadratic interaction
Here, we utilized an equality
with the quadrapole moments, , , , , and . Based on these expressions, we replace Eq. (3) of Ref. Nishiyama08 () with
We diagonalize this matrix for spins. 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 ().
2.2 Critical behavior of the VBS-order parameter
In this section, we investigate the critical behavior of the VBS-order parameter; in the VBS phase, along the -bond direction, the staggered-dimer order develops Fath95 ().
In Fig. 3, we present the Binder parameter
for various and . Here, the quadrature of the VBS-order parameter is given by
and the symbol denotes the ground-state average. The interaction parameter is set to an intermediate value , as mentioned above. As the system size increases, the Binder parameter increases (decreases) in the long- (short-) range-order phase. Hence, in Fig. 3, we see that the VBS order develops in the small- regime; low-dimensionality promotes the formation of the VBS state. The intersection point of the curves indicates a location of the critical point.
In order to extrapolate the critical (intersection) point to the thermodynamic limit, in Fig. 4, we plot the approximate critical point (pluses) 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 relation
with holds. The least-squares fit to the data of Fig. 4 yields an estimate in the thermodynamic limit, . A remark is in order. The wavy character in Fig. 4 is an artifact due to the screw-boundary condition. Actually, there appears a bump (drop) at (, ); in other words, an undulation comes out, depending on the condition whether the linear dimension of the cluster, , is close to an even number () or an odd one (, ). Possible systematic error of is argued in Section 2.5.
We turn to the analysis of the correlation-length critical exponent . In Fig. 5, we plot the approximate critical exponent (pluses)
for with and (). The parameters are the same as those of Fig. 3. Again, there emerges a wavy character intrinsic to the screw-boundary condition; a notable bump at would be an artifact, preventing us to access to the thermodynamic limit systematically. The least-squares fit to these data yields in the thermodynamic limit. Uncertainty of this result is argued in Section 2.5.
2.3 Critical behavior of the nematic- (quadrupolar-) order parameter
In this section, we investigate the critical behavior of the nematic- (quadrupolar-) order parameter.
In Fig. 6, we present the Binder parameter
for various and . Here, the nematic-order parameter is given by
The parameters are the same as those of Fig. 3. The nematic order appears to develop in the large- regime, offering a sharp contrast to the VBS order (Fig. 3). We amended the definition of , Eq. (13), so as to attain an intersection point of the curves in Fig. 6; otherwise, the intersection point disappears. Namely, we discarded the short-distance contributions within the correlator in order to get rid of corrections to scaling. As a byproduct, the location of intersection point drifts significantly with respect to ; a subtlety of -based result is reconsidered in Section 2.5.
In Fig. 4, we plot the approximate transition point (crosses), Eq. (10), for with . The parameters are the same as those of Fig. 3. The finite-size errors seem to be larger than those of . The least-squares fit to the data of Fig. 4 yields an estimate in the thermodynamic limit; the error margin is considered in Section 2.5
We turn to the analysis of the correlation-length critical exponent , Eq. (11). In Fig. 5, we plot the approximate critical exponent (crosses) for with . The parameters are the same as those of Fig. 3. The least-squares fit to these data yields in the thermodynamic limit; a possible systematic error is appreciated in Section 2.5.
2.4 Fisher’s critical exponent
At the critical point , the quadratic moment obeys the power law, , with Fisher’s exponent (anomalous dimension) and system size .
First, we consider the case of . In Fig. 7, we plot the approximate critical exponent
for , (plusses), and (crosses) with . The parameters are the same as those of Fig. 3. The least-squares fit to these data yields and in the thermodynamic limit.
2.5 Extrapolation errors of , and
First, we consider the critical point . In Fig. 4, we made independent extrapolations, which yield and . The discrepancy (systematic error) between these results, , is larger than the insystematic (statistical) errors, O(). Hence, we consider the former as the main source of the extrapolation error. Taking a mean value of and , we estimate . We address a number of remarks. First, this estimate is consistent with the preceding one Nishiyama11 () within the error margin; note that the preceding estimate Nishiyama11 () was calculated through the scaling of the excitation gap. Second, the result would be more reliable than ; actually, the slope of in Fig. 4 appears to be smaller than that of , suggesting that the former extrapolation would be trustworthy. Last, the extrapolated critical point does not affect the subsequent analyses; rather, the approximate critical point was fed into the formulas for the critical indices, Eqs. (11), (14), and (15).
Second, we turn to the correlation-length critical exponent . In Fig. 5, two independent extrapolations yield and . The discrepancy, , is comparable with the statistical error, . Taking a mean value, we obtain an estimate
Here, the error margin comes from (propagation of uncertainty) with systematic () and insystematic () errors. The estimate, Eq. (16), agrees with the preceding one Nishiyama11 (). Our result may support the deconfined-criticality scenario, suggesting that the exponent acquires an enhancement, as compared with that of the three-dimensional Heisenberg universality class, Compostrini02 ().
Last, we consider Fisher’s exponent . In Fig. 7, the results, and , are obtained. The discrepancy between them, , dominates the statistical error, . Considering the former as a error margin, we obtain . On the one hand, in Fig. 8, the discrepancy between and is negligible. Considering the statistical error as a main source of uncertainty, we arrive at .
This is a good position to address a remark. As mentioned above, the VBS-order-based results, and , are more reliable than the nematic-order-based ones, and . In general, the discrete symmetry, , is more robust than the continuous one, , allowing us to make a systematic scaling analysis even for the system size tractable with the numerical diagonalization method. In fact, as claimed in Section III F 2 of Ref. Manousakis91 (), the numerical diagonalization method up to is incapable of providing a conclusive evidence for the spontaneous magnetization of the two-dimensional Heisenberg antiferromagnet. Hence, tentatively, we discard the nematic-order-based results, and refer to the VBS-order-based ones to obtain crude estimates, and . These conclusions are comparable with the large-scale-Monte-Carlo results Lou09b (), and , for the model; here, we made use of the scaling relation, , to evaluate and from and Lou09b (). Possibly Lou09b (), the exponent suffers from “drift” (scaling corrections), and may restore through taking into account of yet unidentified scaling corrections. As mentioned in the Introduction, in the preceeding study of the -spin model Harada07 (), an estimate was reported. It is suggested that the anomalous dimension for the nematic order would be almost vanishing.
3 Summary and discussions
The phase transition separating the VBS and nematic phases (Fig. 1), the so-called deconfined criticality, was investigated for the -spin model on the spatially anisotropic triangular lattice with the biquadratic interaction, Eq. (1). So far, the criticality has been analyzed via the scaling of the first excitation gap Nishiyama11 (). In this paper, evaluating both VBS- and nematic-order parameters, we made complementary approaches to this criticality. As a result, we estimate the correlation-length critical exponent ; this estimate agrees with Nishiyama11 (), supporting the deconfined-criticality scenario.
Encouraged by this finding, we put forward the analysis of Fisher’s exponent for each order parameter. As overviewed in the Introduction, so far, the exponent Harada07 () has been reported as to the -spin model; corrections to scaling did not admit to the estimation of . We obtained a set of indices ; again, the exponent suffers from scaling corrections. Alternatively, setting tentatively (Sec. 2.5), we arrive at crude results . These results might be reminiscent of the large-scale-simulation results Lou09b (), , , and , for the model. We suspect that the exponent would almost vanish through taking into account of corrections to scaling properly. In Ref. Banerjee10 (), Fisher’s exponent was estimated rather accurately through scrutinizing the local-moment distribution around a magnetic impurity; this idea has a potential applicability to a wide class of systems. This problem will be addressed in future study.
- (1) T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303 (2004) 1490.
- (2) T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher, Phys. Rev. B 70 (2004) 144407.
- (3) M. Levin and T. Senthil, Phys. Rev. B 70 (2004) 220403.
- (4) A. W. Sandvik, Phys. Rev. Lett. 98 (2007) 227202.
- (5) R. G. Melko and R. K. Kaul, Phys. Rev. Lett. 100 (2008) 017203.
- (6) V. N. Kotov, D.-X. Yao, A. H. Castro Neto, and D. K. Campbell, Phys. Rev. B 80 (2009) 174403.
- (7) L. Isaev, G. Ortiz, and J. Dukelsky. Phys. Rev. B 82 (2010) 136401.
- (8) V. N. Kotov, D.-X. Yao, A. H. Castro Neto, and D. K. Campbell, Phys. Rev. B 82 (2010) 136402.
- (9) L. Isaev, G. Ortiz, and J. Dukelsky, J. Phys.: Condens. Matter 22 (2010) 016006.
- (10) A. Kuklov, N. Prokof’ev, and B. Svistunov, Phys. Rev. Lett. 93 (2004) 230402.
- (11) A.B. Kuklov, M. Matsumoto, N.V. Prokof’ev, B.V. Svistunov, and M. Troyer, Phys. Rev. Lett. 101 (2008) 050405.
- (12) F.-J. Jiang, M. Nyfeler, S. Chandrasekharan, and U.-J. Wiese, J. Stat. Mech. (2008) P02009.
- (13) K. Krüger and S. Scheidl, Europhys. Lett. 74 (2006) 896.
- (14) J. Sirker, Z. Weihong, O.P. Sushkov, and J. Oitmaa, Phys. Rev. B 73 (2006) 184420.
- (15) D. Poilblanc, A. Läuchli, M. Mambrini, and F. Mila, Phys. Rev. B 73 (2006) 100403(R).
- (16) Y. Nishiyama, Phys. Rev. B 85 (2012) 014403.
- (17) K. Harada, N. Kawashima, and M. Troyer, J. Phys. Soc. Japan 76 (2007) 013703.
- (18) Y. Nishiyama, Phys. Rev. B 79 (2009) 054425.
- (19) Y. Nishiyama, Phys. Rev. B 83 (2011) 054417.
- (20) G. Fáth and J. Sólyom, Phys. Rev. B 51 (1995) 3620.
- (21) K. Harada and N. Kawashima, Phys. Rev. B 65 (2002) 052403.
- (22) M. A. Novotny, J. Appl. Phys. 67 (1992) 5448.
- (23) Y. Nishiyama, Phys. Rev. E 78 (2008) 021135.
- (24) N. Read and S. Sachdev, Phys. Rev. Lett. 62 (1989) 1694.
- (25) N. Kawashima and Y. Tanabe, Phys. Rev. Lett. 98 (2007) 057202.
- (26) J. Lou, A. W. Sandvik, and N. Kawashima, Phys. Rev. B 80 (2009) 180414(R).
- (27) R. K. Kaul, Phys. Rev. B 84 (2011) 054407.
- (28) K. S. D. Beach, F. Alet, M. Mambrini, and S. Capponi, Phys. Rev. B 80 (2009) 184401.
- (29) J. Lou and A. W. Sandvik, Phys. Rev. B 80 (2009) 212406.
- (30) T. Grover and T. Senthil, Phys. Rev. Lett. 107 (2011) 077203.
- (31) O. I. Motrunich and A. Vishwanath, Phys. Rev. B 70 (2004) 075104.
- (32) M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, and E. Vicari, Phys. Rev. B 65 (2002) 144520.
- (33) E. Manousakis, Rev. Mod. Phys. 63 (1991) 1.
- (34) A. Banerjee, K. Damle, and F. Alet, Phys. Rev. B 82 (2010) 155139.