Two-dimensional easy-plane SU(3) magnet with the transverse field: Anisotropy-driven multicriticality
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Two-dimensional easy-plane SU (3) magnet with thetransverse field: Anisotropy-driven multicriticality
Yoshihiro Nishiyama
Department of Physics, Faculty of Science, Okayama University, Okayama 700-8530,Japan
Abstract.
The two-dimensional easy-plane SU(3) magnet subjected to thetransverse field was investigated with the exact-diagonalization method. So far, asto the XY model (namely, the easy-plane SU(2) magnet), the transverse-field-drivenorder-disorder phase boundary has been investigated with the exact-diagonalizationmethod, and it was claimed that the end-point singularity (multicriticality) at the XX -symmetric point does not accord with large- N -theory’s prediction. Aiming to reconcilethe discrepancy, we extend the internal symmetry to the easy-plane SU(3) with theanisotropy parameter η , which interpolates the isotropic ( η = 0) and fully anisotropic( η = 1) cases smoothly. As a preliminary survey, setting η = 1, we analyze the order-disorder phase transition through resorting to the fidelity susceptibility χ F , whichexhibits a pronounced signature for the criticality. Thereby, with η scaled carefully,the χ F data are cast into the crossover-scaling formula so as to determine the crossoverexponent φ , which seems to reflect the extension of the internal symmetry group toSU(3).
1. Introduction
The one-dimensional XY model subjected to the transverse field H and anisotropy η with the Hamiltonian H XY = − P i [(1 + η ) σ xi σ xi +1 + (1 − η ) σ yi σ yi +1 + Hσ zi ] ( σ i : Paulimatrices at site i ) is attracting much attention [1, 2, 3, 4] in the context of the quantuminformation theory [5, 6]. A key ingredient is that the model covers both XX - ( η = 0)and Ising-symmetric ( η = 1) cases, and there appear rich characters as to the transverse-field-driven order-disorder phase transition. Recently, the multicriticality at η = 0, i.e. ,the end-point singularity of the phase boundary toward the XX -symmetric point, wasexplored in depth from the quantum-information-theoretical viewpoint [7].Meanwhile, its extention to the two-dimensional counterpart has been made [8, 9].According to the large- N theory [9], namely, for sufficiently large internal symmetrygroup, the phase boundary should exhibit a reentrant (non-monotonic) behavior. Thereentrant behavior leads to a counterintuitive picture such that the disorder phase isinduced by the lower internal symmetry group. On the contrary, as to the XY (namely,easy-plane SU(2)) model, the exact-diagonalization study [8, 9, 10] claimed that thephase boundary rises up linearly (monotonically) in proximity to the multicritical point; wo-dimensional easy-plane SU (3) magnet with the transverse field: Anisotropy-driven multicriticality z = 2( = 1) [12] at the multicritical point.The aim of this paper is to reconcile the discrepancy between the large- N - [9]and easy-plane-SU(2)-based [8, 9, 10] results; namely, so far, only the extremum caseshave been considered. For that purpose, we extend the internal symmetry group tothe easy-plane SU(3) [13], and explore how the extention of the internal symmetrygroup affects the multicriticality. Additionally, as a probe to detect the phase transition[14, 15, 16, 17, 18, 19, 20, 21], we resort to the fidelity [22, 23, 24, 25] F ( H, H + ∆ H ) = |h H | H + ∆ H i| , (1)where the vectors, | H i and | H + ∆ H i , denote the ground states with the proximateinteraction parameters, H and H + ∆ H , respectively. The fidelity (1) is readilyaccessible via the exact-diagonalization method, which yields the ground-state vector | H i explicitly. According to the elaborated exact-diagonalization study of the two-dimensional XXZ and Ising models [17], the fidelity-mediated analysis admits a reliableestimate for the criticality, although the available system size N ≤
20 is rather restricted.To be specific, we present the Hamiltonian for the two-dimensional easy-plane SU(3)magnet subjected to the transverse field H = − J X h ij i [(1+ η )( S xi S xj + S yi S yj )+(1 − η )( Q zxi Q zxj + Q yzi Q yzj )]+ H N X i =1 Q z i , (2)with the S = 1-spin operator S i placed at each square-lattice point i = 1 , , . . . , N .Likewise, the quadrupolar moments at site i , Q x − y i = ( S xi ) − ( S yi ) , Q z i = √ S zi ) − √ , Q xyi = S xi S yi + S yi S xi , Q yzi = S yi S zi + S zi S yi , and Q zxi = S zi S xi + S xi S zi , are incorporated.These eight operators constitute the SU(3) algebra [26] just like the Gell-Mann matrices.The summation P h ij i runs over all possible nearest-neighbor pairs h ij i , and the couplingconstant J sets the unit of energy, J = 1, throughout this study. The parameters, η and H , denote the anisotropy of the internal symmetry and the transverse field, respectively.The anisotropy η = 0 gives rise to the asymmetry between the S x,yi and Q yz,zxi sectors.Irrespective of the anisotropy η , the Hamiltonian commutes with the z -axis-rotationgenerator P Ni =1 S zi , and hence, the Hamiltonian (2) describes the easy-plane sector ofthe SU(3) magnet.In Fig. 1, we present a schematic phase diagram of the easy-plane SU(3) magnet(2) for the anisotropy η and the transverse field H ; the overall character would resemblethat of the XY model [8]. The phase boundary H c ( η ) separates the order ( H < H c ( η ))and disorder ( H > H c ( η )) phases. In the order phase, there appears the spontaneousmagnetization of the in-plane moment ( S xi , S yi ) [( Q zxi , Q yzi )] for the η > ( < )0 regime.Therefore, the phase boundary H = H c ( η ) should belong to the three-dimensional (3D) XY universality class except at the multicritical point η = 0. At η = 0, the Hamiltonian wo-dimensional easy-plane SU (3) magnet with the transverse field: Anisotropy-driven multicriticality P Ni =1 Q z i . Hence, as the transverse field H increases, the successive level crossings take place as to the ground-state energy up to H < H c (0)(= 8 / √
3) [7], and above the threshold
H > H c (0), the transverse moment P Ni =1 Q z i saturates eventually; see Appendix. This transition mechanism is the sameas that of the magnetization plateau (saturation of magnetization), and the power-lawsingularities have been investigated in depth [12]. Around the threshold H = H c (0), theefficiency of the quantum Monte Carlo sampling suffers from the slowing-down problem[27], and the aforementioned exact-diagonalization study [8] circumvented this difficulty.Then, there arises a problem how the phase boundary H c ( η ) terminates at themulticritical point η = 0; see Fig. 2. The exact-diagonalization analysis of the XY model indicates that the phase boundary H c ( η ) rises up linearly (monotonically) [8, 9]around the multicritical point η = 0. Actually, the power-law singularity of the phaseboundary [28, 29] H c ( η ) − H c (0) ∼ | η | /φ , (3)is characterized by the crossover exponent φ ≈ XY magnet [8, 9, 10]. Onthe one hand, the large- N theory [9] suggests that the phase boundary H c ( η ) showsa reentrant behavior. That is, the phase boundary H c ( η ) exhibits a non-monotonicdependence on the anisotropy η . So far, only the limiting cases such as the XY [8, 9, 10]and spherical [9] models have been considered, and no information has been provided asto the multicriticality in between. As the internal symmetry group is enlarged graduallyfrom SU(2), the phase boundary may become curved convexly, accompanying with thesuppressed crossover exponent φ <
1. In this paper, considering the SU(3) version ofthe easy-plane magnet, we investigate the crossover exponent φ quantitatively by theagency of the fidelity F ( H, H + ∆ H ) (1).The rest of this paper is organized as follows. In Sec. 2, we present the numericalresults. Prior to the detailed analysis of the multicriticality, we investigate the caseof the fully anisotropic limit η = 1, aiming to demonstrate the performance of oursimulation scheme. In the last section, we address the summary and discussions.
2. Numerical results
In this section, we present the numerical results for the easy-plane SU(3) magnetsubjected to the transverse field (2). We employed the exact-diagonalization methodfor the finite-size cluster with N ≤ × L = √ N , which sets the fundamental length scale in the subsequent finite-size-scaling analyses. As a probe to detect the phase transition, we utilized the fidelitysucceptibility [14, 15, 16, 17, 18, 19, 20, 21] χ F ( H ) = − L ∂ H F ( H, H + ∆ H ) | ∆ H =0 , (4)with the fidelity F ( H, H + ∆ H ) (1). As was demonstrated in Ref. [17] for the two-dimensional XXZ and Ising models under the transverse field with N ≤
20 spins, the wo-dimensional easy-plane SU (3) magnet with the transverse field: Anisotropy-driven multicriticality χ F (4) admits a reliable estimate of the criticality, even thoughthe available system size N ≤
20 is rather restricted.In fairness, it has to be mentioned that the similar scheme was applied to the one-dimensional XY magnet under the transverse field [7]. In this pioneering study [7]. theauthors took a direct route toward the multicritical point, ( η, H ) → (0 , H c (0)). The χ F data exhibit the intermittent peaks, reflecting the successive level crossings along theordinate axis η = 0, as shown in Fig. 1. In this paper, to avoid such a finite-size artifact,we took a different route to the multicritical point, keeping the anisotropy η to a finitevalue. That is, based on the the crossover-scaling theory [28, 29], the anisotropy η isscaled properly, as the system size L changes. As a byproduct, we are able to estimatethe crossover exponent φ quantitatively, which characterizes the power-law singularity ofthe phase boundary; see Fig. 2. Before commencing detailed crossover-scaling analysesof χ F , we devote ourselves to the fully anisotropic case η = 1 so as to examine theperformance of our simulation scheme. H c (1) at η = 1 : Fidelity-susceptibility analysis In this section, as a preliminary survey, setting the anisotropy parameter to the fullyanisotropic limit η = 1, we investigate the order-disorder phase transition of the easy-plane SU(3) magnet under the transverse field (2) via the fidelity susceptibility χ F (4). At this point η = 1, the model (2) reduces to the spin- S = 1 XY modelwith the single-ion anisotropy D , for which a variety of preceding results are available[30, 31, 32, 33, 34, 35].In Fig. 3, we present the fidelity susceptibility χ F (4) for various values of thetransverse field H , and the system sizes, (+) L = 3, ( × ) 4, and ( ∗ ) 5, with the fixed η = 1. We see that the fidelity susceptibility exhibits a pronounced signature for theorder-disorder phase transition around H ≈ H ∗ c ( L ) for 1 /L /ν with η = 1fixed. Here, the approximate critical point H ∗ c ( L ) denotes the location of the fidelity-susceptivity peak ∂ H χ F ( L ) | H = H ∗ c ( L ) = 0 , (5)for each system size L . The power of the abscissa scale 1 /ν comes from the scalingdimension of the parameter H [20], and the correlation-length critical exponent ν is setto the value of the 3D- XY universality class, ν = 0 . H c ( η = 1) = 6 . , (6)in the thermodynamic limit L → ∞ .This is a good position to address an overview of the related studies. In Table1, we recollect a number of preceding results for the critical point H c (1) at the fullyanisotropic case η = 1. As mentioned above, at η = 1, our model (2) reduces to the wo-dimensional easy-plane SU (3) magnet with the transverse field: Anisotropy-driven multicriticality S = 1 XY model with the single-ion anisotropy D , and the D -based results areconverted via the relation H = 2 D/ √ H c (1) was estimated as H c = 6 .
300 [30], 6 .
628 [31, 32], and 6 . H c . Our exact-diagonalization(ED) result H c = 6 . χ F appearsto be accordant with these preceding studies. Particularly, our result H c = 6 . . χ F -mediated simulation scheme. Encouraged by this finding, we further explore thecritical behavior of χ F in the next section. η = 1In this section, we investigate the criticality of the fidelity susceptibility χ F (4) at thefully anisotropic limit, η = 1. To begin with, we set up the finite-size-scaling formulafor the fidelity susceptibility [20] χ F = L α F /ν f (cid:16) ( H − H c ) L /ν (cid:17) , (7)with a scaling function f , and the fidelity-susceptibility critical exponent α F ; namely,the index α F describes the singularity χ F ∼ | H − H c | − α F at the critical point H c .According to Ref. [20], the critical exponent α F satisfies the relation α F /ν = α/ν + 1 , (8)with the specific-heat critical index α ; namely, the index α describes the singularity ofthe specific heat as C ∼ | H − H c | − α . We postulate that the the criticality belongs tothe 3D- XY universality class [36, 37] for the anisotropic regime η = 0. Putting theexisting values [36], α = 0 . ν = 0 . XY universality class intothe scaling relation (8), we arrive at α F /ν = 0 . . (9)Notably, this index α F /ν = 0 . α/ν = − . α F /ν and ν ,appearing in the expression (7) are all fixed, and we are able to carry out the scalinganalysis of χ F unambiguously.In Fig. 5 we present the scaling plot, ( H − H c ) L /ν - χ F L − α F /ν , for various H andsystem sizes, (+) L = 3, ( × ) 4, and ( ∗ ) 5, with the fixed η = 1. Here, the scalingparameters are set to H c = 6 .
53 [Eq. (6)], ν = 0 . α F /ν = 0 . wo-dimensional easy-plane SU (3) magnet with the transverse field: Anisotropy-driven multicriticality ad hoc parameter adjustment is undertaken in the present scaling analysis.We address a number of remarks. First, the scaling plot, Fig. 5, indicates that thephase transition belongs to the 3D- XY universality class. So far, as for the Heisenbergantiferromagnet with the single-ion anisotropy D , it has been claimed that the D -drivensingularity belongs to the 3D- XY universality class [33, 35]. Our simulation result showsthat the quantum XY ferromagnet is also under the reign of the 3D- XY universalityclass. Second, from the scaling plot, Fig. 5, we see that corrections to finite-size scalingare rather suppressed. Actually, as first noted by Ref. [17], the fidelity susceptibilitydetects the underlying singularity clearly out of the subdominant contributions. Last,a key ingredient is that χ F ’s singularity α F /ν = 0 . α/ν = − . XY universality. In this sense, the former is appropriate as aquantifier for the criticality. η = 0 : Analysis of thecrossover exponent φ We then turn to the analysis of the multicriticality at η = 0. For that purpose, weextend the above scaling formalism (7) to the crossover-scaling formula [28, 29] χ F = L ˙ α F / ˙ ν g (cid:16) ( H − H c ( η )) L / ˙ ν , ηL φ/ ˙ ν (cid:17) , (10)with yet another controllable parameter η , the accompanying crossover exponent φ , anda scaling function g . Here, the indices, ˙ α F and ˙ ν , denote the fidelity-susceptibility andcorrelation-length critical exponents, respectively, right at the multicritical point η = 0.The meaning of the second argument of the crossover-scaling formula (10), ηL φ/ ˙ ν , isas follows. The correlation-length critical exponent ˙ ν describes the singularity of thelength scale L as L ∼ | H − H c (0) | − ˙ ν . This relation leads to the physically convincingexpression ηL φ/ ˙ ν ∼ η/ | H − H c (0) | φ . Now, it is apparent that the crossover exponent φ describes the mutual relationship between H and the new entity η .Before carrying out the crossover-scaling analyses, we fix the values of thecritical indices appearing in the formula (10). As mentioned in Introduction, thephase transition along the ordinate axis η = 0 is essentially the same as that ofthe magnetization plateau, and the power-law singularities have been investigatedin considerable detail [12] as follows. The corelation-length critical exponent wasdetermined as ˙ ν = 1 / α F / ˙ ν = 3 via thescaling relation ˙ α F / ˙ ν = ˙ α/ ˙ ν + z [20]. Here, the dynamical critical exponent takes thevalue z = 2 [12], and the specific-heat critical exponent ˙ α = 1 / ∼ √ H − H c [12], is regarded as the internal energy.(Note that the first derivative of the ground-state energy corresponds to the “internalenergy” in the classical statistical mechanical context.) The above consideration nowcompletes the prerequisite for the crossover-scaling analysis. The remaining index φ isto be adjusted so as to achieve an alignment of the crossover-scaled χ F data. wo-dimensional easy-plane SU (3) magnet with the transverse field: Anisotropy-driven multicriticality H − H c ( η )) L / ˙ ν - χ F L − ˙ α F / ˙ ν , forvarious system sizes, (+) L = 3, ( × ) 4, and ( ∗ ) 5, with ˙ ν = 1 / α F / ˙ ν = 3determined above. Here, we fixed the second argument of the scaling formula (10) to aconstant value ηL φ/ ˙ ν = 5 with an optimal crossover exponent φ = 0 .
8, and the parameter H c ( η ) was determined via the same scheme as that of Sec. 2.1. From Fig. 6, we seethat the crossover-scaled data fall into the scaling curve satisfactorily. Particularly, the L = 4 ( × ) and 5 ( ∗ ) data are about to overlap each other. Such a feature suggests thatthe choice φ = 0 . φ = 0 .
7, in Fig. 7, we present thecrossover-scaling plot, ( H − H c ( η )) L - χ F L − , for various system sizes L = 3 , ,
5; thesymbols are the same as those of Fig. 6. Here, the second argument of the crossover-scaling formula (10) is fixed to ηL φ = 3 .
62 with a proposition φ = 0 .
7. For such asmall value of φ = 0 .
7, the crossover-scaled data get scattered; particularly, the left-sideslope becomes dispersed, as compared to that of Fig. 6. Similarly, under the setting φ = 0 .
9, in Fig. 8, we present the crossover-scaling plot, ( H − H c ( η )) L - χ F L − , forvarious system sizes L = 3 , ,
5; the symbols are the same as those of Fig. 6. Here, thesecond argument of the crossover-scaling formula is set to a constant value ηL φ = 6 . φ = 0 .
9. For such a large value of φ = 0 .
9, the crossover-scaled data becomescattered; actually, the hill-top data split up. Considering that the above cases, Figs. 7and 8, yield the lower and upper bounds, respectively, for the estimate of φ , we concludethat the crossover exponent lies within φ = 0 . . (11)Our result excludes the possibility that the phase boundary H c ( η ) rises up linearly φ ≈
1, as observed for the XY (namely, easy-plane SU(2)) magnet [8, 9, 10]. Rather,as for the extended internal symmetry group, the phase boundary is curved convexly,accompanying with a slightly suppressed crossover exponent (11).A number of remarks are in order. First, the underlying physics behind thecrossover-scaling plot, Fig. 6, differs from that of the fixed- η scaling, Fig. 5. Actually,the former scaling dimension ˙ α F / ˙ ν = 3 is much larger than that of the latter, α F /ν = 0 . φ has to be adjusted carefully, andthe collapse of the crossover-scaled data points, Fig. 6, is by no means coincidental.In this sense, the present crossover-scaling analysis captures the characteristics of themulticritical behavior. Second, because the fidelity-susceptibility approach does not relyon any presumptions as to the order parameter involved, it detects both singularities,˙ α/ ˙ ν = 3 and α/ν = 0 . N [9] andeasy-plane-SU(2) [8, 9, 10] symmetry groups, have been considered, and the multicriticalbehavior in between remains unclear. Our result φ = 0 . <
1) [Eq. (11)] indicatesthat the phase boundary H c ( η ) is curved convexly, as the internal symmetry group wo-dimensional easy-plane SU (3) magnet with the transverse field: Anisotropy-driven multicriticality < d ≤ . . . . dimensions even for the large- N case [9], it is reasonable that the slightly enlarged SU(3) group does not lead to the reentrant behavior immediately in d = 2 dimensions.
3. Summary and discussions
The two-dimensional easy-plane SU(3) magnet under the transverse field (2) wasinvestigated with the exact-diagonalization method. Because the method allows directaccess to the ground state, we do not have to care about the anisotropy between thereal-space and imaginary-time directions rendered by the dynamical critical exponent z = 2( = 1) [12] at the multicritical point η = 0. Our main concern is to reconcile thediscrepancy between the large- N theory [9] and the exact-diagonalization analysis of the XY magnet [8, 9, 10] as to the multicriticality (Fig. 2). For that purpose, we considerthe SU(3) version of the easy-plane magnet (2), and performed the exact-diagonalizationsimulation by the agency of the fidelity susceptibility χ F (4). The fidelity susceptibilityhas an advantage in that it detects the phase transition sensitively, as compared to thatof the specific heat [17, 20]. As a preliminary survey, setting the anisotropy parameterto η = 1, we estimate the critical point via χ F as H c (1) = 6 . .
300 [30], 6 .
628 [31, 32], and 6 . H c (1) = 6 . . χ F -mediated simulation scheme. Thereby, we cast the χ F data intothe crossover-scaling formula (10) with the anisotropy η scaled properly. Adjusting thecrossover exponent φ carefully, we attain an alignment of the crossover-scaled data pointsfor φ = 0 . φ < H c ( η ) getscurved convexly around the multicritical point η = 0, as the internal symmetry groupis extended to SU(3). Actually, only in 1 < d ≤ . . . . dimensions, the reentrantbehavior is realized even for the large- N case [9]. Hence, it is reasonable that the slightlyenlarged internal symmetry does not lead to the reentrant behavior immediately.We conjecture that the phase boundary may exhibit a quadratic curvature, φ = 0 . < d <
2) system realized effectively by the power-law-decayinginteractions [40]. In such a fractional-dimensional system, the reentrant behavior maycome out even for the XY model. This problem is left for the future study. Acknowledgments
This work was supported by a Grant-in-Aid for Scientific Research (C) from JapanSociety for the Promotion of Science (Grant No. 20K03767). wo-dimensional easy-plane SU (3) magnet with the transverse field: Anisotropy-driven multicriticality Table 1.
A number of preceding results for the critical point H c (1) at η = 1are recollected. At η = 1, the model (2) reduces to the spin- S = 1 XY modelwith the single-ion anisotropy D . These D -based results are converted so as tomatch our notation via the relation H = 2 D/ √
3. So far, a variety of technieus,such as the bosonic mean-field approximation (BMFA) [30], self-consistent harmonicapproximation (SCHA) [31, 32], and quantum Monte Carlo (QMC) [33] methods, havebeen employed, and these results are comparable to the present exact diagonalization(ED) result. (The BMFA estimate is read off from Fig. 1 of Ref. [30].) In the respectivestudies, as a probe to detect the phase transition, various quantifiers, such as thespontaneous magnetization, energy gap, correlation length and fidelity susceptibility,were utilized. Because “the anisotropy does not have large effects” [30], the large-scaleQMC result [33] for the Heisenberg model [34, 35] is shown as well.
Method Quantifier Model H c ( η = 1)BMFA [30] spontaneous magnetization XY . XY . . XY . Appendix A. Transition point H c = 8 / √ at η = 0At the isotropic point η = 0, the location of the multicritical point H c (0) = 8 / √ H , the ground state is givenby the direct product ⊗ Ni | i i of the local base | i i , which satisfies S zi | m i i = m | m i i ( m = − , ,
1) at each site i . Due to the strict selection rule at η = 0, magnons’pair creation is prohibited. Thus, the single magnon at site j , | j i = | i j ⊗ ( ⊗ Ni = j | i i ),propagates coherently through the transfer amplitude − J over the nearest neighbors,obeying the dispersion relation − J (cos k x + cos k y ) + √ H ( k : wave number) above theground state. Therefore, at H c ( η = 0) = 8 / √
3, the band gap closes in a way reminiscentof the metal-insulator transition. This transition mechanism is precisely the same asthat of the magnetization plateau [12] (saturation of the magnetization). A notablepoint is that the dynamical critical exponent takes z = 2, because the quadratic bandbottom touches the ground-state energy level. In other words, the symmetry betweenthe real-space and imaginary-time directions is violated by z = 1. It is a benefit of theexact-diagonalization method that the method allows direct access to the ground state,for which the imaginary-time system size is infinite. References [1] J. Maziero, H. C. Guzman, L. C. C´eleri, M. S. Sarandy, and R. M. Serra, Phys. Rev. A (2010)012106.[2] Z.-Y. Sun, Y.-Y. Wu, J. Xu, H.-L. Huang, B.-F. Zhan, B. Wang, and C.-B. Duanpra, Phys. Rev.A (2014) 022101.[3] G. Karpat, B. C¸ akmak, and F. F. Fanchini, Phys. Rev. B (2014) 104431. wo-dimensional easy-plane SU (3) magnet with the transverse field: Anisotropy-driven multicriticality level crossingsQ moment saturation H η
3D XY universality order phasedisorder phase
Q -symmetric multicriticality(end-point singularity) H c ( η ) zz H c (0) Figure 1.
A schematic phase diagram of the easy-plane SU(3) magnet (2) forthe anisotropy η and transverse field H is presented. The phase boundary H c ( η )terminates at the multicritical point H = H c ( η = 0) = 8 / √ η = 0, and this end-point singularity is our concern. For small H < H c ( η ), the in-plane order, ( S xi , S yi )[( Q yzi , Q zxi )] develops in the η > ( < )0 side, whereas for large H > H c ( η ), the disorderphase extends. At the isotropic point η = 0 in between, the transverse moment P Ni =1 Q z i commutes with the Hamiltonian, and the ground-state level crossing occurssuccessively [7] up to H < H c (0). Above this threshold H > H c (0), the magnetizationplateau [12] (saturation of the moment P Ni =1 Q z i ) sets in.[4] Q. Luo, J. Zhao, and X. Wang, Phys. Rev. E (2018) 022106.[5] A. Steane, Rep. Prog. Phys. (1998) 117.[6] C.H. Bennett and D.P. DiVincenzo, Nature (2000) 247.[7] V. Mukherjee, A. Polkovnikov, and A. Dutta, Phys. Rev. B (2011) 075118.[8] M. Henkel, J. Phys. A: Mathematical and Theoretical (1984) L795.[9] S. Wald and M. Henkel, J. Stat. Mech.: Theory and Experiment (2015) P07006.[10] Y. Nishiyama, Eur. Phys. J. B (2019) 167.[11] S. Jalal, R. Khare, and S. Lal, arXiv:1610.09845.[12] V. Zapf, M. Jaime, and C. D. Batista, Rev. Mod. Phys. (2014) 563.[13] J. D’Emidio and R. K. Kaul, Phys. Rev. B (2016) 054406.[14] H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P. Sun, Phys. Rev. Lett. (2006) 140604.[15] P. Zanardi and N. Paunkovi´c, Phys. Rev. E (2006) 031123.[16] H.-Q. Zhou, and J. P. Barjaktarevi˜c, J. Phys. A: Math. Theor. (2008) 412001. wo-dimensional easy-plane SU (3) magnet with the transverse field: Anisotropy-driven multicriticality η H c ( η ) H c (0) φ =1 φ <1 reentrant Figure 2.
The multicriticality (end-point singularity) of the phase boundary H c ( η )at η = 0 is characterized by the crossover exponent φ [28, 29] such as H c ( η ) − H c (0) ∼| η | /φ (3). As for the XY model (namely, the easy-plane SU(2) magnet), the exact-diagonalization simulation suggests that the phase boundary rises up linearly with φ ≈ N analysis admits the reentrant(non-monotonic) behavior [9]. As indicated, the reentrant behavior leads to acounterintuitive picture such that the disorder phase is induced by the lower internalsymmetry group η = 0 around the multicritical point. It is anticipated that for theSU(3) case, the phase boundary is curved convexly with a slightly suppressed crossoverexponent φ < (2009) 021108.[18] W.-L. You and Y.-L. Dong, Phys. Rev. B (2011) 174426.[19] D. Rossini and E. Vicari, Phys. Rev. E (2018) 062137.[20] A. F. Albuquerque, F. Alet, C. Sire, and S. Capponi, Phys. Rev. B (2010) 064418.[21] D. Schwandt, F. Alet, and S. Capponi, Phys. Rev. Lett. (2009) 170501.[22] A. Uhlmann, Rep. Math. Phys. (1976) 273.[23] R. Jozsa, J. Mod. Opt. (1994) 2315.[24] A. Peres, Phys. Rev. A (1984) 1610.[25] T. Gorin, T. Prosen, T. H. Seligman, and M. ˇZnidariˇc, Phys. Rep. (2006) 33. wo-dimensional easy-plane SU (3) magnet with the transverse field: Anisotropy-driven multicriticality χ F H Figure 3.
The fidelity susceptibility χ F (4) is plotted for various values of thetransverse field H and the system sizes, (+) L = 3, ( × ) 4, and ( ∗ ) 5, with the fixedanisotropy parameter η = 1. The fidelity susceptibility indicates a notable peak aroundthe critical point H ≈ H c * ( L ) ν Figure 4.
The approximate critical point H ∗ c ( L ) (5) is plotted for 1 /L /ν with thefixed anisotropy parameter η = 1. Here, the correlation-length critical exponent is setto the 3D- XY -universality value ν = 0 . H c = 6 . L → ∞ . wo-dimensional easy-plane SU (3) magnet with the transverse field: Anisotropy-driven multicriticality χ F L - α F / ν (H-H c )L ν Figure 5.
The scaling plot, ( H − H c ) L /ν - χ F L − α F /ν , is presented for various systemsizes, (+) L = 3, ( × ) 4, and ( ∗ ) 5, with the fixed η = 1; see the scaling formula (7).Here, the scaling parameters are set to H c = 6 .
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The crossover-scaling plot, ( H − H c ( η )) L / ˙ ν - χ F L − ˙ α F / ˙ ν , is presented forvarious system sizes, (+) L = 3, ( × ) 4, and ( ∗ ) 5, with ˙ ν = 1 / α F / ˙ ν = 3;see text for details. Here, the second argument of the crossover-scaling formula (10)is fixed to ηL φ/ ˙ ν = 5 with the crossover exponent φ = 0 .
8. The crossover-scaled datafall into the scaling curve satisfactorily; particularly, the L = 4 ( × ) and 5 ( ∗ ) data areabout to overlap each other under the setting, φ = 0 . wo-dimensional easy-plane SU (3) magnet with the transverse field: Anisotropy-driven multicriticality χ F L - (H-H c )L Figure 7.
The crossover-scaling plot, ( H − H c ( η )) L / ˙ ν - χ F L − ˙ α F / ˙ ν , is presented forvarious system sizes, (+) L = 3, ( × ) 4, and ( ∗ ) 5, with ˙ ν = 1 / α F / ˙ ν = 3. Here,the second argument of the crossover-scaling formula (10) is fixed to ηL φ/ ˙ ν = 3 .
62 withthe crossover exponent φ = 0 .
7. The left-side slope gets scattered under such a smallvalue of φ = 0 . wo-dimensional easy-plane SU (3) magnet with the transverse field: Anisotropy-driven multicriticality χ F L - (H-H c )L Figure 8.
The crossover-scaling plot, ( H − H c ( η )) L / ˙ ν - χ F L − ˙ α F / ˙ ν , is presented forvarious system sizes, (+) L = 3, ( × ) 4, and ( ∗ ) 5, with ˙ ν = 1 / α F / ˙ ν = 3. Here,the second argument of the crossover-scaling formula (10) is fixed to ηL φ/ ˙ ν = 6 . φ = 0 .
9. The hill-top data split up for such a large value of φ = 0 ..