Quantum Monte Carlo study of the transverse-field Ising model on a frustrated checkerboard lattice
aa r X i v : . [ c ond - m a t . s t r- e l ] J u l Quantum Monte Carlo study of the transverse-fieldIsing model on a frustrated checkerboard lattice
H. Ishizuka, Y. Motome, N. Furukawa
A,B , and S. Suzuki A Department of Applied Physics, University of Tokyo, Japan A Department of Physics and Mathematics, Aoyama Gakuin University, Japan B Multiferroics Project, ERATO, Japan Science and Technology Agency (JST),c/o Department of Applied Physics, University of Tokyo, JapanE-mail: [email protected]
Abstract.
We present the numerical results for low temperature behavior of the transverse-field Ising model on a frustrated checkerboard lattice, with focus on the effect of both quantumand thermal fluctuations. Applying the recently-developed continuous-time quantum MonteCarlo algorithm, we compute the magnetization and susceptibility down to extremely lowtemperatures while changing the magnitude of both transverse and longitudinal magnetic fields.Several characteristic behaviors are observed, which were not inferred from the previously-studied quantum order from disorder at zero temperature, such as a horizontal-type stripeordering at a substantial longitudinal field and a persistent critical behavior down to lowtemperature in a weak longitudinal field region.
1. Introduction
Frustration due to the geometrical structure of systems has been one of the major topicsin condensed matter physics. The interest covers a wide range of fields, such as frustratedmagnetism [1], proton ordering in hydrogen-bonded systems [2, 3], and anomalous transportand multiferroics in transition metal oxides [4, 5, 6]. The geometrical frustration often preventsthe system from selecting a unique ground state, instead gives rise to classically degenerateground states; i.e., a macroscopic number of different configurations of the system variables leadto the same ground state energy at the classical level. As a result, the ground state remainsto be disordered and bears finite residual entropy. Such macroscopic degeneracy provides afertile ground for various peculiar phenomena as it makes the system to be extremely sensitiveto perturbations, such as remnant interactions, external fields, and fluctuations.Among such degeneracy-lifting mechanisms, quantum and thermal fluctuations have attractedparticular interest. These fluctuations sometimes lift the ground-state degeneracy and inducesome particular ordering, as known by order from disorder [7]. There, the ordering is selectedfrom the manifold to maximize the associated entropy (zero-point entropy in the case of quantumfluctuation). This entropic effect plays a decisive role at low temperatures ( T ) and causes manyfascinating phenomena in the frustrated systems.One of the minimal models for studying the order-from-disorder phenomenon is thetransverse-field Ising model (TIM). In the absence of geometrical frustration, in general, themodel develops a long-range order at low T , while the ordering is suppressed and a quantumparamagnetic state is induced by the transverse field. When the classical ground state isacroscopically degenerate under strong frustration, quantum fluctuations by the transversefield and/or thermal fluctuations by temperature can induce a particular ordering via the order-from-disorder mechanism. Effect of thermal fluctuations has been studied mainly in the absenceof the transverse field, i.e., for pure Ising models without the transverse field [1, 8]. Meanwhile,effect of quantum fluctuations has also been studied. For example, it was shown that TIM on avariety of frustrated lattices exhibit several nontrivial behaviors at T = 0, such as a bond orderingand Kosterlitz-Thouless transition [9, 10]. In general, the thermal and quantum fluctuations donot necessarily lead to the same effect, and the relation between them is of particular interestto explore yet another order-from-disorder phenomenon.In this contribution, we present our numerical results for the order-from-disorder phenomenain TIM in a wide range of temperature and the transverse/longitudinal magnetic fields. Weconsider the TIM on a two-dimensional checkerboard lattice and study its low- T physics bya sophisticated quantum Monte Carlo (QMC) technique. As a result, in the intermediatelongitudinal-field regime, we find instability toward a Neel order, in accordance with the previousreport [10]. On the other hand, for both the weaker and stronger longitudinal fields, our resultsindicate different behaviors from the previous report. Under the weak field, featureless magneticsusceptibility is observed down to extremely low T , which implies a very weak proximity effectto the predicted Neel ground state or a possibility of another state in the T = 0 limit. On thecontrary, in the strong field, the system indicates instability toward an unexpected horizontal-type stripe ordering.
2. Model and method
To investigate the effect of quantum and thermal fluctuations on the classically degenerateground state in frustrated systems, here we focus on the frustrated checkerboard Ising modelwith transverse and longitudinal magnetic fields. The Hamiltonian is given as: H = J X h i,j i s zi s zj + Γ X i s xi + h X i s zi , (1)where s αi is the Pauli spin operator at site i , and J is the Ising interaction between nearest-neighbor sites h i, j i on the checkerboard lattice (see Fig. 1); Γ and h are the transverse andlongitudinal fields, respectively. In the following, we consider the antiferromagnetic case, J > J = 1.When Γ = h = 0, the model remains to be disordered down to zero T [8]. The lowest energyis achieved by enforcing a simple local constraint — a two-up two-down local spin configurationfor all the plaquettes with crisscrossing interaction, similar to the so-called ice rule in water ice[2]. Consequently, a macroscopic number of different spin configurations give the same lowestenergy: The ground state is macroscopically degenerate and the residual entropy is estimated tobe ∼ N log ( N is the number of spins). The spin correlation, however, is critical in the sensethat it decays algebraically as a function of distance. The situation is unaltered as h increasesup to h = 2 J at Γ = 0. (For larger h , the ground state consists of “three-up one-down”configurations, which is also macroscopically degenerate.) When the quantum fluctuation setsin by switching on Γ, it was predicted, from perturbative considerations in Γ /h , that a Neelorder is induced in the 0 < h < J region at T = 0 [10] [see Fig. 1(a)]. Hereafter, we focus onthe 0 < h < J region.To investigate the low- T behaviors of this model with taking account of both quantumand thermal fluctuations, we conducted numerical calculations by QMC method based on thepath-integral approach [11, 12]. Here, we employed the recently-developed continuous-timealgorithm [13] to approach extremely low T . At low T , the MC sampling suffers from the slowrelaxation due to the severe frustration. To accelerate the relaxation process, we applied the a) JJ J (b)
Figure 1.
Schematic pictures ofthe transverse-field Ising modelon the checkerboard lattice. (a)Neel type order and (b) horizon-tal stripe order.loop-flip algorithm developed for frustrated Ising-type spin systems [14, 15]. In our calculations,the loops are formed in the real space at a particular imaginary-time slice, and flipped togetherwith all the spin variables along the imaginary-time direction, i.e., on the torus defined by theloop in space and time. In addition, the replica exchange method was used to further suppressthe slowing down [16]. For the present system, the replicas are exchanged along a constant-Γ /T line because the Boltzmann weight depends on the number of domain walls in the imaginary-timedirection which is proportional to Γ /T .Calculations were done for N = 4 × L site systems with L = 12 under the periodic boundaryconditions. Observables were computed typically for 120000 samplings after 30000 steps ofthermalization. The results were divided into eight bins to evaluate the statistical errors.
3. Results and discussions
Figure 2 shows the QMC results for the susceptibility at a relatively weak longitudinal field h = 0 . χ Neel = NT ( h m i − h| m Neel |i ) ; m Neel = 1 N X i s zi ( − i x + i y , (2)while Fig. 2(b) is that for the horizontal-type stripe order parameter [Fig. 1(b)], which weconsider as another candidate of order from disorder, χ stripe = NT ( h m i − h m stripe i ) ; m = (cid:16) N X i s zi ( − i x (cid:17) + (cid:16) N X i s zi ( − i y (cid:17) . (3)Note that this stripe state belongs to the two-up two-down manifold and is different from thediagonal one discussed in the three-up one-down manifold for 2 J < h < J [10]. Figure 2(c)shows the local correlation parameter given by ρ = N P p f ( p ), where the sum runs over allplaquettes. f ( p ) is a function defined for each plaquette p as f ( p ) = 1 for P i ∈ p s zi = 0 andotherwise f ( p ) = − / ρ → ρ → T → ∞ and/orΓ → ∞ . As shown in Figs. 2(a) and 2(b), for weak Γ . .
5, both the susceptibilities increasewith decreasing T in Curie-law like behavior, i.e., χ ∝ T − . A shoulder-like feature observedat T ∼ . χ stripe corresponds to a formation of the ice-rule type local configurations, asindicated by the saturation of ρ in Fig. 2(c). These behaviors suggest that the system is inthe correlated regime with satisfying the local constraint, but remains critical, without choosingeither Neel or stripe ordering, presumably because of the strong frustration. We also calculatethe momentum dependence of the susceptibility, and confirm that it remains featureless withoutshowing any peak in the momentum space. It is surprising that the critical behavior is robustlyobserved down to very low T ∼ .
01. This implies that the order-from-disorder mechanism isextremely weak or ineffective in this region. On the other hand, the results for Γ & . (a) Γ=0.11Γ=0.21Γ=0.30Γ=0.41Γ=0.51Γ=0.61 (b)
Γ=0.11Γ=0.21Γ=0.30Γ=0.41Γ=0.51Γ=0.61 χ N ee l χ s t r i p e ρ (c) -2 -1 -2 -1 -2 -1 Γ=0.11Γ=0.21Γ=0.30Γ=0.41Γ=0.51Γ=0.61
T T
Figure 2. T dependence of (a) χ Neel , (b) χ stripe , and (c) local correlation parameter ρ at h = 0 . .
11 to 0 . T . In this region, Γ is strong enough to disturb the ice-rule typeconfigurations [Fig. 2(c)], and the ground state turns into a quantum paramagnetic state.With increasing the longitudinal field, a contrasting behavior between the two susceptibilitiesshows up in the vicinity of phase transition to the quantum paramagnetic state. Figure 3 showsthe results at h = 0 .
9. In the region Γ . .
3, both χ Neel and χ stripe diverge toward T = 0, ina similar manner to the results for Γ . . & . χ Neel and χ stripe saturate, a qualitativelydifferent behavior is observed: At Γ ∼ . χ stripe begins to deviate from the Curie-law likebehavior and tends to saturate, while χ Neel remains to diverge (indicated by the arrows in Fig. 3).This implies that the stripe-type fluctuation is suppressed and the Neel-type ordering is likelyfavored in the low- T limit. Similar behavior is observed in a finite region on the verge of thequantum paramagnetic phase for 0 . < h < . T (a) (b) Γ=0.11Γ=0.21Γ=0.30Γ=0.41Γ=0.51Γ=0.61 Γ=0.11Γ=0.21Γ=0.30Γ=0.41Γ=0.51Γ=0.61 χ N ee l χ s t r i p e -2 -1 -2 -1 (c) -2 -1 ρ Γ=0.11Γ=0.21Γ=0.30Γ=0.41Γ=0.51Γ=0.61
T T
Figure 3. T dependence of (a) χ Neel , (b) χ stripe , and (c) local correlation parameter ρ at h = 0 . .
11 to 0 .
61. The arrows in (a) and (b)indicate a contrasting behavior between χ Neel and χ stripe in the intermediate Γ region.For larger longitudinal fields, yet another behavior appears. For h & .
0, the MC relaxationbecomes very slow at low T even with the use of the loop-flip algorithm and the replica-exchangetechnique. The situation is demonstrated in Fig. 4. The results show the relaxation of the orderparameters m Neel and m stripe at h = 1 . /T = tan(85 ◦ ) axis; the data aremeasured for 50000 MC steps after particular thermalization steps of N therm starting from thenitial configuration with corresponding perfect order: Figures 4(a) and 4(b) show the resultsstarting from the perfectly-Neel-ordered state and the perfectly-horizontal-stripe-ordered state,respectively. As shown in Fig. 4, m Neel ∼ N therm = 10 thermalization independent of theinitial state. On the other hand, the m stripe shows strong dependence on the initial state evenafter N therm ∼ steps. A possible origin of the freezing behavior is strong first order transitionto the horizontal-stripe-ordered state. Although the extremely slow relaxation prevents us fromobtaining numerically-converged results, the results suggest that the system has an instabilitytoward the horizontal stripe ordering at low T in this high- h region. T (a) (b) m m Neel, N =100000 therm
Neel, N =150000 therm
Neel, N =200000 therm
Stripe, N =100000 therm
Stripe, N =150000 therm
Stripe, N =200000 therm
Γ ΓT
Neel, N =100000 therm
Neel, N =150000 therm
Neel, N =200000 therm
Stripe, N =100000 therm
Stripe, N =150000 therm
Stripe, N =200000 therm
Figure 4. T dependence of m Neel and m stripe at h = 1 . /T = tan(85 ◦ ) forvarying thermalization steps N therm . (a) is the results starting from the perfectly-Neel-orderedconfiguration and (b) is from the perfectly-horizontal-stripe-ordered one.Summarizing the results above, we deduce a ‘phase diagram’ in Fig. 5, which indicates the low- T instability anticipated from the MC data. In the strong longitudinal field region, 1 . . h . . . .
2, our MC results indicate an instability toward the horizontal-stripe-type orderingbelow some particular temperature, as exemplified in Fig. 4(b). Although we could not directlyconfirm the phase transition because of the extremely slow MC relaxation, we expect this phaseto appear at a finite temperature. In the intermediate longitudinal field region, 0 . . h . . . . Γ . .
5, the results appear to favor the Neel-type instability. The Neel temperatureappears to be extremely low, less than T = 0 .
01. In the remaining low- h region for Γ . . T > .
01; the system is strongly correlated but remains to be critical under the frustration.Let us compare the result to the T = 0 argument by Moessner and Sondhi [10]. In the previousstudy, by a perturbation in Γ /h , it was predicted that a Neel-type order occurs in the entireregion of our consideration. Our result in Fig. 5 shows a similar tendency in the intermediate h , where the Neel-type order appears to be favored. However, we could not detect any sign ofa particular ground state in the lower- h region, although the Neel-type order was expected toextend down to h → T = 0 phase diagram in the previous study [10]. Thisapparent disagreement can be due to very weak degeneracy lifting by the order from disorderfor the expected Neel ground state. Another possibility is that, for large Γ /h , the perturbativeargument in Γ /h no longer holds and a different phase emerges in the region. In order to identifythe nature of the system in this region, further analysis down to lower T in larger system sizes isnecessary. On the other hand, under relatively strong field 1 . . h . .
8, we detected instabilitytoward another phase from the slow relaxation behavior, namely, the horizontal-stripe phase.This might be related with the competition in the ground states between the two-up two-downNeel order and three-up one-down diagonal-stripe order at h ≃
2. Our preliminary calculationshow a finite-size effect in this regime; the horizontal-stripe type instability appears to extend toslightly lower- h region in larger size systems. More detailed results will be discussed elsewhere. h Γ Figure 5. ‘Phase diagram’ for the model inEq. (1) as functions of transverse field Γ andlongitudinal field h . The results indicate the low- T instabilities which are deduced from the QMCresults down to T ≃ .
01. The blue squares(orange circles) show the region in which theNeel(horizontal stripe)-type instability shows up.The green triangles show the region where both χ Neel and χ stripe remain to show the Curie-lawlike divergence down to the lowest T . The areawithout symbols for Γ >
4. Summary
To summarize, we have investigated the effect of quantum and thermal fluctuations on thethermodynamics of the frustrated checkerboard Ising model with transverse and longitudinalmagnetic fields. We have examined the magnetic instabilities induced by order-from-disordermechanism by employing a continuous-time quantum Monte Carlo method with the loop-flipupdate and replica exchange algorithm. We identified characteristic behaviors, which were notinferred from the previous study for the quantum order from disorder in the ground state. Oneis the instability toward a horizontal-stripe-type ordering in a relatively high longitudinal-fieldregime. This is distinguished from the diagonal stripe predicted for a higher field. The other is noclear indication of dominant ordering or fluctuation down to T ≃ . J in a relatively low-fieldregime; the susceptibility shows a persistent Curie-law-like divergence at all the momenta. Thiscould be due to the emergence of a novel state or simply by surprisingly weak order-from-disordereffect in the Neel ground state expected in the previous theory.H.I. and Y.M. thank T. Misawa, Y. Motoyama, H. Shinaoka, and M. Udagawa for helpfulcomments. This research was supported by KAKENHI (No. 19052008 and No. 22540372), andGlobal COE Program“the Physical Sciences Frontier”. References [1] For a recent review, Diep H T 2004
Frustrated Spin Systems (World Scientific Publishing)[2] Pauling L 1935
J. Am. Chem. Soc. J. Chem. Phys. Phys. Rev.
Science
Nature
J. Phys. (Paris) Statistical Mechanics of Periodic Frustrated Ising Systems (Springer-Verlag, Berlin-Heidelberg-New York-Tokyo)[9] Moessner R, Sondhi S L and Chandra P 2000
Phys. Rev. Lett. Phys. Rev. B Prog. Theor. Phys. Phys. Lett. A Phys. Rev. Lett.
J. Chem. Phys. Phys. Rev. Lett. J. Phys. Soc. Jpn.65