Dynamics of first-order quantum phase transitions in extended Bose-Hubbard model: From density wave to superfluid and vice-versa
Keita Shimizu, Takahiro Hirano, Jonghoon Park, Yoshihito Kuno, Ikuo Ichinose
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] A ug Dynamics of first-order quantum phase transitionsin extended Bose-Hubbard model:From density wave to superfluid and vice-versa
Keita Shimizu , Takahiro Hirano , Jonghoon Park , YoshihitoKuno and Ikuo Ichinose Department of Applied Physics, Nagoya Institute of Technology, Nagoya 466-8555,Japan Department of Physics, Graduate School of Science, Kyoto University, Kyoto606-8502, JapanE-mail: [email protected]
Abstract.
In this paper, we study the nonequilibrium dynamics of the Bose-Hubbardmodel with the nearest-neighbor repulsion by using time-dependent Gutzwiller (GW)methods. In particular, we vary the hopping parameters in the Hamiltonian as afunction of time, and investigate the dynamics of the system from the density wave(DW) to the superfluid (SF) crossing a first-order phase transition and vice-versa.From the DW to SF, we find scaling laws for the correlation length and vortex densitywith respect to the quench time. This is a reminiscence of the Kibble-Zurek scalingfor continuous phase transitions and contradicts the common expectation. We givea possible explanation for this observation. On the other hand from the SF to DW,the system evolution depends on the initial SF state. When the initial state is theground-state obtained by the static GW methods, a coexisting state of the SF andDW domains forms after passing through the critical point. Coherence of the SF orderparameter is lost as the system evolves. This is a phenomenon similar to the glasstransition in classical systems. When the state starts from the SF with small localphase fluctuations, the system obtains a large-size DW-domain structure with thindomain walls.PACS numbers: 67.85.Hj, 03.75.Kk, 05.30.Rt
Keywords : Ultra-cold atomic gases, Bose-Hubburd model, Quantum dynamical phasetransition, First-order phase transition, Superfluid, Density wave. uench dynamics of the extended Bose-Hubbard model
1. Introduction
In recent years, dynamics of quantum-many body systems is one of the most activelystudied subjects in physics. Process in which a system approach to an equilibrium is offundamental interests, and also evolution of system under a quench has attracted manyphysicists. Nowadays, ultra-cold atomic gas systems play a very important role for thestudy on these subjects because of their versatility, controllability and observability [1].Theoretical ideas proposed to understand transient phenomena are to be tested byexperiments on ultra-cold atomic systems. This is one of examples of so-called quantumsimulations [2, 3, 4, 5].For the second-order thermal phase transition, time-evolution of systems undera change in temperature has been studied extensively so far. From the view pointof cosmology, Kibble [6, 7] claimed that the phase transitions lead to disparate localchoices of the broken symmetry state and as a result, topological defects called cosmicstrings are generated. Later, Zurek [8, 9, 10] pointed out that a similar phenomenon isrealized in laboratory experiments on the condensed matter systems like the superfluid(SF) of He. After the above seminal works, many theoretical and experimental studieson the Kibble-Zurek (KZ) mechanism have appeared [11]. Concerning to experimentson Bose-condensed ultra-cold atomic gases, the correlation length of the SF and therate of topological defect formation were measured and the KZ scaling hypothesis wasexamined [12, 13].To study dynamics of quantum many-body systems, the parameters in theHamiltonian are varied through a quantum phase transition (QPT), i.e., the quantumquench [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], and the system evolutionis observed. Experiments on this problem have been already done using the variousultra-cold atomic gases [27, 28, 29, 30, 31, 32]. Among them, works in Refs. [27, 28]questioned the applicability of the KZ scaling theory to the QPT, whereas Refs. [29, 30]concluded that the observed results were in good agreement with the KZ scaling law.In this paper, we focus on the two-dimensional (2D) Bose-Hubbard model(BHM) [33, 34], which is a canonical model of the bosonic ultra-cold atomic gas systemsin an optical lattice. In particular, we add nearest-neighbor (NN) repulsions betweenatoms. Then, the resultant system is described by an extended Bose-Hubbard model(EBHM). As a result, a parameter region corresponding to the density wave (DW)appears in the ground-state phase diagram, in addition to the Mott insulator and SF.Near the half-filling, there exists a first-order phase transition between the SF and DW[35]. We shall study the quench dynamics of the EBHM on passing across the SFand DW phase boundary. There are only a few works for the dynamical propertiesof quantum systems at first-order phase transitions under a quench [36, 38, 39], andtherefore detailed study on that problem is desired.This paper is organized as follows. In Sec. 2, we introduce the EBHM and explainthe Gutzwiller (GW) methods, which are used in the present work. In Sec. 3, quenchdynamics of the first-order phase transition from the DW to SF is studied. Behavior uench dynamics of the extended Bose-Hubbard model τ Q . Contrary to the common expectation, we find that scalinglaws hold for the correlation length and vortex density. In Sec. 4, we give a possibleexplanation of the observed results from viewpoint of the SF bubble-nucleation process.We employ a time-dependent Ginzburg-Landau theory and show that scaling laws withsmall deviations from the KZ scaling hold in the vicinity of a triple point in the phasediagram. Applicability of the GW methods is also discussed there. In Sec. 5, we studythe time evolution of the system from the SF to DW crossing the first-order phasetransition. We find that even for very slow quench, a genuine DW does not form ifwe start the time evolution with the ground-state obtained by the static GW methods.Numerical result shows that a coexisting state of the SF and DW appears instead. Onthe other hand, if SF states with small coherent phase fluctuations are employed as aninitial state, the system acquires a DW domain structure of large size with thin domainwalls. Section 6 is devoted for conclusion. In appendix, we show the results obtained forthe hard-core Bose-Hubbard model, in which the first-order phase transition betweenthe DW and SF exists as in the soft-core system of the present work. We discuss thebehavior of the correlation length and vortex density compared to the soft-core case.
2. Extended Bose-Hubbard model and slow quench
We consider the EBHM whose Hamiltonian is given by [42], H BH = − J X h i,j i ( a † i a j + H.c.) + U X i n i ( n i − V X h i,j i n i n j − µ X i n i , (1)where h i, j i denotes NN sites of a square lattice, a † i ( a i ) is the creation (annihilation)operator of boson at site i , n i = a † i a i , and µ is the chemical potential. J ( >
0) and U ( >
0) are the hopping amplitude and the on-site repulsion, respectively. We also addthe NN repulsion with the coefficient V , which plays an important role in the presentwork.In this study, we are interested in cases near the half filling, i.e., ρ ≡ N s P i h n i i ≈ /
2, where N s is the total number of the lattice sites, and we take N s = 64 ×
64 or100 ×
100 for the practical calculation. We set U = 1 as the energy unit, and time t ismeasured in the unit ~ /U . We investigated the system in Eq.(1) by using the static GWapproximation and show obtained ground-state phase diagram in Fig. 1 for V /U = 0 . V /U = 0 .
05. We also show the system energy,particle density and amplitude of the SF order parameter, | Ψ | ≡ N s P i | Ψ i | , whereΨ i ≡ h a i i , in Fig. 2 for µ/U = 0 .
1. From the results in Fig. 2, it is obvious that the uench dynamics of the extended Bose-Hubbard model Figure 1.
Ground-state phase diagram of the extended Bose-Hubbard model for V = 0 .
05 obtained by the static GW methods. There exist three phases, the densitywave (DW), superfluid (SF) and supersolid (SS). Mean particle density ρ ≈ / Figure 2.
Physical quantities in the DW and SF critical region in various systemsizes; the hopping J -term energy, amplitude of SF order ( | Ψ | ), and mean density ( ρ ).The obtained results show that the phase transition is of first order as dictated byLandau-Ginzburg-Wilson paradigm. Critical point is estimated as J c /U ≈ . system exists near the half filling ρ ≈ /
2, and a first-order phase transition between theDW and SF takes place at J c /U ≃ .
022 as a finite jump in | Ψ | indicates. The existenceof the first-order phase transition is quite plausible as the DW and SF have both theown long-range order. In recent paper [40], we studied the EBHM for V /U = 0 . ρ ≈
1. There exists a substantially finite region of the SS inaddition to the DW and SF. These three phases are separated by two second-order phasetransitions. This result is in agreement with the quantum Monte-Carlo study [41].In the following, we shall study dynamics of the system under “slow quenchs”. uench dynamics of the extended Bose-Hubbard model H i , which is derived by introducing the expectation value Ψ i = h a i i , H GW = X i H i ,H i = − J X j ∈ i NN ( a † i Ψ j + H.c.) + U n i ( n i − V X j ∈ i NN n i h n j i − µn i , (2)where i NN denotes the NN sites of site i , and Hartree-Fock type approximation hasbeen used for the hopping and NN repulsion. To solve the quantum system H GW inEq.(2), we introduce GW wave function, | Φ GW i = N s Y i (cid:16) n c X n =0 f in ( t ) | n i i (cid:17) , ˆ n i | n i i = n | n i i , (3)where n c is the maximum number of particle at each site, and we mostly take n c = 6in the present work. Some quantities are calculated with n c = 10 to verify that n c = 6is large enough for the study of the half filling case. See Fig. 3 and Fig. 7. In terms of { f in ( t ) } , the order parameter of the SF is given as,Ψ i = h a i i = n c X n =1 √ nf i ∗ n − f in , (4)and { f in ( t ) } are determined by solving the following Schr¨odinger equation for variousinitial states, i ~ ∂ t | Φ GW i = H GW ( t ) | Φ GW i . (5)The time dependence of H GW ( t ) in Eq.(5) comes from the quench J → J ( t ) with fixed U and V as explained in the following section. Practically, the time evolution above iscalculated by the fourth-order Runge-Kutta method.
3. Dynamics of phase transition from density wave to superfluid
We first study the dynamics from the DW to SF. In this section, the hopping amplitudeis varied as J ( t ) − J c J c ≡ ǫ ( t ) = tτ Q , (6)where τ Q is the quench time, which is a controllable parameter in experiments. Weemployed 10 samples as the initial state at t = − τ Q (i.e., J ( − τ Q ) = 0), which have theDW order with small local density fluctuations from the perfect DW. Then, we solveEq.(5) to obtain | Φ GW i . Physical quantities for which scaling lows are examined areobtained by averaging over samples. The linear quench in Eq.(6) is terminated at t = t f uench dynamics of the extended Bose-Hubbard model DW (1,0) SFSS
Figure 3. (Upper panel) Phase of the SF order parameter Ψ i for τ Q = 300 as afunction of time. (Middle panel) Amplitude of the SF order parameter Ψ i for τ Q = 300as a function of time. Relevant times ˆ t and t eq are ˆ t ≈
70 and t eq ≈ t ex ≈ | Ψ | terminates. From t eq to t ex , coarsening process of the phase of Ψ i takes place in large scales [26]. (Lowerpanel) Calculation of | Ψ | in the n c = 10 case is also shown. It is in good agreementwith that of n c = 6. with J ( t f ) = 0 . > J c ) in the numerical study. Subsequent behavior of the system isalso observed to see how the system approaches to an equilibrium.We show the typical behavior of | Ψ | as a function of t in Fig. 3 for τ Q = 300. At t = 0, the system crosses the critical point at J c /U ≃ . | Ψ | remains vanishingly small for some period, and then it develops very rapidly.After the rapid increase, | Ψ | starts to fluctuate and coarsening of the phase of the SForder parameter takes place there [26]. ˆ t in Fig. 3 is defined as | Ψ(ˆ t ) | = 2 | Ψ(0) | , and t eq is the time at which the oscillation of | Ψ | starts. Similarly, t ex is the time at which thatoscillation terminates.Similar behavior to the above was observed in the Mott to SF quench dynamics and uench dynamics of the extended Bose-Hubbard model -0.4-0.2 0 0.2 0.4 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.05 0.1 0.150.015 0.02 0.025 0.03 0.035 0.04 Figure 4. ∆ DW , ∆ cDW and ∆ SF as a function of time for τ Q = 300. After passingthe equilibrium critical point J c /U ≃ . examined carefully [26]. Compared with the Mott to SF dynamics, the SF amplitude | Ψ | is smaller, e.g., for t > t eq , | Ψ | ∼ (0 . − .
9) in the Mott to SF transition, whereas | Ψ | ∼ . ρ ∼ DW ≡ N s P i ( − ) i h n i i , ∆ CDW ≡ N s P h i,j i |h ( n i − n j ) i| ,and the even-odd deference of the SF order parameter defined as ∆ SF ≡ N s P h i,j i || Ψ i |−| Ψ j || are shown in Fig. 4. These quantities exhibit fluctuations as a function of timeuntil J ≈ . τ Q exhibits a similar behavior,although the reaction of the system starts at larger value of J/U for smaller value ofthe quench time τ Q .It is interesting to study the correlation length ξ of the SF order parameter and thevortex density N v as a function of the quench time τ Q . These quantities are defined asfollows; h Ψ ∗ i Ψ j i ∝ exp( −| i − j | /ξ ) ,N v = X i | Ω i | , Ω i = 14 h sin( θ i +ˆ x − θ i ) + sin( θ i +ˆ x +ˆ y − θ i +ˆ x ) − sin( θ i +ˆ x +ˆ y − θ i +ˆ y ) − sin( θ i +ˆ y − θ i ) i , (7) uench dynamics of the extended Bose-Hubbard model Figure 5.
Scaling laws observed for the correlation length ξ , vortex number N v at t = ˆ t , and ˆ t with respect to τ Q . where θ i is the phase of Ψ i (Ψ i = | Ψ i | e iθ i ) and ˆ x (ˆ y ) is the unit vector in the x ( y )direction. For continuous second-order phase transitions, the KZ hypothesis predictsa scaling law such as ξ ∝ τ b Q and N v ∝ τ − d Q . Recently, applicability of the above KZscaling law for second-order quantum phase transition has been discussed for severalquantum systems. On the other hand for first-order phase transitions, it is commonlyexpected that such a scaling law does not hold as the relaxation time cannot be definedproperly. For a classical statistical model, another type of scaling law was proposed forfirst-order phase transitions [36]. It should be also noted that off-equilibrium dynamicsof a quantum Ising ring was investigated recently and finite-size scaling laws for first-order phase transitions were proposed [37]. There, off-equilibrium scaling variables weregiven in terms of an energy gap and quench time, and physical quantities were obtainedas a function of time.To see if scaling law exists or not, we measured ξ and N v at t = ˆ t and t = t eq . Inthe original KZ hypothesis for continuous phase transitions [11], ˆ t is the time at whichthe system re-enters an equilibrium after the freezing (or impulse) period. On the otherhand, t eq is the time at which a coarsening process of the SF phase coherence starts [26].We show the obtained results in Figs. 5 and 6. The results show that at t = ˆ t , both ξ and N v satisfy the scaling law with exponents b = 0 .
25 and d = 0 .
26, respectively, uench dynamics of the extended Bose-Hubbard model Figure 6.
Scaling lows observed for the correlation length, vortex number at t = t eq ,and t eq with respect to τ Q . and also ˆ t ∝ τ . . On the other hand at t = t eq , data at each τ Q exhibits slightlylarge fluctuations but scaling laws for the correlation length, N v and t eq seem to existfor τ Q >
20. The above results indicate that besides the KZ mechanism, there existsanother mechanism to generate the scaling laws. Possible explanation is given in Sec. 4.It should be noted that after passing the critical point, ∆ DW and ∆ SF have even-oddsite fluctuations, and therefore, the system is not homogeneous. We think that becauseof this inhomogeneity, the critical exponents of ξ and N v at t = ˆ t do not satisfy theexpected relation such as b = d/
2. On the other hand at t = t eq , the system is ratherhomogeneous, and therefore b ∼ d/ τ Q in Fig. A.2. There, ξ (ˆ t ) and N v (ˆ t )fluctuate rather strongly. This behavior comes from the fact that fluctuations of theparticle number at each site is smaller compared with the soft-core case, and as a result,the stability of the phase degrees of freedom of the SF order parameter is weakened.We terminate the linear quench at t f = τ Q = 300. After t f , the system approachesto an equilibrium as the results in Figs. 3 and 4 indicate. It is interesting to see howthe correlation length of the SF develops. As the results in Fig. 7 show, the correlation uench dynamics of the extended Bose-Hubbard model Figure 7. (Upper left panel) For a typical initial state at t = − τ Q , the correlationlength is calculated as a function of time. After passing t = t eq , increase of thecorrelation length becomes weak. (Upper right panel) We also show the results for the n c = 10 case. (Lower panels) The correlation functions G ( r ) = N s P i h a † i a i + r i exhibitvery close behavior in the n c = 6 and n c = 10 cases. length increases after passing the critical point as it is expected. However, its increasegets weak at t ∼ t eq , and it saturates at t ∼
500 and keeps a finite value. To study theresultant phase, we measured N v and found that there exist no vortices at t > finite-temperature ( T ) SF phase for sufficientlylarge t with an effective T , T eff . The finite- T SF in 2D has a quasi-long range orderand the correlation length diverges, i.e., the Kosterlitz-Thouless (KT) phase. The aboveresult seems to indicate that some other state is realized in the final stage of the presentprocess. However, the system behavior may strongly depend on the average particledensity ρ . Further study is needed to clarify this interesting problem. In fact, we studiedthis problem in the case of the mean particle density ρ ≈ V /U = 0 .
375 [40]. Inthe quench process such as the DW → SS → SF, the correlation length continues toincrease even for large t . This result seems to indicate that a KT phase of the SF isrealized there. uench dynamics of the extended Bose-Hubbard model
4. Consideration by the Ginzburg-Landau theory
In the previous section, we showed that the results obtained by the GW methods indicatethe scaling laws of ˆ t , t eq and the correlation length with respect to the quench time τ Q .It is interesting and also important to study the origin of these observations from moreuniversal and intuitive point of view. To this end, the Ginzburg-Landau (GL) theory isquite useful. In fact very recently, it was pointed out that the GL theory can drive thescaling laws for the second-order phase transition by analytical transformation of theassociated equations of motion [50]. In this section, we first review the above derivationof the scaling laws for the ordinary second-order phase transition, and then give anintuitive picture of the scaling laws by using a classical solution representing decay ofthe false vacuum. Then, we extend the methods to the present case involving the SFand DW order parameters. This consideration also gives an insight about the physicalmeaning and limitation of the GW methods. Let us start with the stochastic GL equation for a complex order parameter (condensate) φ ( ~r, t ), ∂φ∂t = ∇ r φ − ǫ ( t )2 φ − | φ | φ + Θ( ~r, t ) , (8)where Θ( ~r, t ) represents the uncorrelated white-noise variables with h Θ( ~r, t )Θ( ~r ′ , t ′ ) i = T δ ( ~r − ~r ′ ) δ ( t − t ′ ) and T is the temperature of particles ensemble not participating theBose-Einstein condensate. As in Ref. [50], we consider the critical parameter ǫ ( t ) suchas ǫ ( t ) = − (cid:12)(cid:12)(cid:12) tτ Q (cid:12)(cid:12)(cid:12) λ sgn( t ) , (9)where λ is a parameter for the quench protocol. Then, let us change variables as follows, η = αt, ~ℓ = ( α ) / ~r, ˜ φ = φ/ ( α ) / , (10)where α = τ − λ/ ( λ +1)Q . In terms of the new variables, the equation of motion (8) leads to ∂ ˜ φ∂η = ∇ ℓ ˜ φ − | η | λ sgn( η ) ˜ φ − | ˜ φ | ˜ φ + 1 α Θ( ~ℓ, η ) . (11)In Eq.(11), the τ Q -dependence in Eq.(9) disappears except the last white-noise term.From the above fact, it is concluded in Ref. [50] that the τ Q -dependence of ˆ t and ξ (ˆ t )are expected to follow the transformation in Eq.(10), and they are given as follows forsufficiently low T ,ˆ t ∝ α − = τ λ/ ( λ +1)Q , ξ (ˆ t ) ∝ α − / = τ λ/ λ +1)Q . (12)For the linear quench λ = 1, ˆ t ∝ τ / and ξ (ˆ t ) ∝ τ / . The above estimations agree withthose of the KZ scaling with the mean-field exponents such as ν = 1 / z = 2.As we show, the above scaling transformation gives an intuitive picture that derivesthe KZ scaling law. To this end, we put Θ( ~r, t ) = 0 in Eq.(8) and consider a static uench dynamics of the extended Bose-Hubbard model ǫ ( t ) = − ǫ <
0. In this case, the static ground state is given as φ = √ ǫ . To study the sudden quench dynamics , we consider the decay of the falsevacuum φ = 0 to the true ground state φ = √ ǫ . In 1D case, a classical solutionrepresenting the decay is obtained as follows [39], φ ( t, x ) = √ ǫ h (cid:16) √ ǫ x − v t ) (cid:17)i − , (13)where v = √ ǫ , and φ ( t, −∞ ) = √ ǫ and φ ( t, ∞ ) = 0. The solution Eq.(13) obviouslyrepresents the situation in which the true vacuum φ = √ ǫ born in the false vacuumexpands with the speed v .Let us consider the “slow” quench dynamics and study bubble nucleation-evolutionprocess in the SF formation. We expect that this process corresponds to the numericalstudies in the previous sections. We have to find the solution to Eq.(8) that describesa single SF-bubble evolution in the false vacuum φ = 0, but we cannot find an exactsolution. However, the above solution in Eq.(13) suggests that a spherically-symmetricsolution in higher dimensions and also for the time-dependent ǫ ( t ) has the followingform for ǫ ( t ) < ‡ φ s ( ~r, t ) = p | ǫ ( t ) | F (cid:16)p | ǫ ( t ) | ( r − v t t ) (cid:17) , r > , (14)where v t = C p | ǫ ( t ) | with a certain constant C , and F ( x ) is a decreasing functionsuch as F ( −∞ ) = 1 and F ( ∞ ) = 0. In fact, we can show that the function φ s ( ~r, t )in Eq.(14) satisfies the scaling transformation in Eq.(10) for the time-dependent ǫ ( t ) inEq.(9), i.e., ˜ φ s ( η, ~ℓ ) = φ s / ( α ) / = p η λ F (cid:16)p η λ ( ℓ − v ( η ) η ) (cid:17) , v ( η ) = C p η λ , (15)does not depend on τ Q . As far as the above picture holds in the time evolution of thesystem, Eq.(15) implies that typical events and phenomena are observed similarly insystems with various τ Q ’s, and corresponding times have τ Q -dependence such as τ λ/ ( λ +1)Q .For example, we numerically obtained ˆ t and t eq for various τ Q ’s in Sec. 3 by startingwith qualitatively the same initial states. These values are related to τ Q -independentˆ η and η eq that are obtained by the rescaled picture from Eq.(15), i.e., ˆ t and t eq in the τ Q -system are given by ˆ t = τ λ/ ( λ +1)Q ˆ η and t eq = τ λ/ ( λ +1)Q η eq . § Furthermore, a typicallinear size of the bubble at t , i.e., the correlation length at t , ξ ( t ), is given as ξ ( t ) = Z t v t dt ∝ τ λ/ t λ/ , (16) ‡ Solution in Eq.(14) might be regarded a solution in the slow quench limit, in which the time-derivativeof ǫ ( t ) is small. However, it also satisfies the scaling transformation with ǫ ( t ) in Eq.(9). See thediscussion below. § Rough estimation of ˆ η and η eq are the followings. As ˆ t is determined by the condition such as | Ψ(ˆ t ) | = 2 | Ψ(0) | , p ˆ η λ ( v (ˆ η )ˆ η ) =constant for the 2D case. On the other hand, as t eq is the time atwhich the overlap of SF bubbles starts [26], v ( η eq ) η eq =constant. Simulation for various λ ’s is a futurework. uench dynamics of the extended Bose-Hubbard model ξ (ˆ t ) ∝ ( τ Q ) λ/ λ +1) ˆ η ( λ +2) / and ξ ( t eq ) ∝ ( τ Q ) λ/ λ +1) η ( λ +2) / . After t eq , themerging and coarsening process of SF bubbles takes place [26], and therefore the abovepicture and also the resultant scaling laws do not hold anymore.4.2. GL theory, GW methods and quantum Monte-Carlo simulation Here, it is suitable to comment on the GW approximation. The GL theory and alsothe Gross-Pitaevskii (GP) equation consider only the mean field and totally ignorefluctuations around it. On the other hand in the GW approximation, we focus ona wave function of site factorization, and wave function at each site is obtained bysolving the site-factorized Hamiltonian in which the NN operators are replaced withtheir expectation values [26]. The uncertainty relation between the particle number andphase at each site is faithfully taken into account although an equation of motion similarto the GL (GP) equation is derived by the GW methods. This is an advantage of theGW approximation over the GL and GP theories.As more reliable methods, let us consider the quantum Monte-Carlo (MC)simulations of the coherent-state path integral in the imaginary-time formalism. Inthis MC simulations, quantum operators are reduced into classical variables and thequantum superpositions are treated by the fluctuations in the imaginary-time direction.Large number of configurations are generated by the MC updates and physical quantitiesare calculated by averaging them over generated configurations. In the Metropolis MCalgorithm, the local updates are applied to variables at each site by calculation a localenergy around that site. In the vicinity of a phase transition point, a large numberof configurations contribute equally, and calculations by large CPU times are requiredin order to take into account all relevant configurations. On the other hand awayfrom the critical point, the number of important configurations is not so large. Fromthe viewpoint of the MC simulation with the local update, we can get an interestinginsight into the GW approximation. That is, let us imagine that we perform a GWcalculation for a system with size 10 × . When we calculate expectation values,we divide the 10 × system into 10 number of 10 × subsystems. We obtainthe expectation values by averaging values calculated in each subsystem. Comparedwith the path-integral MC simulation, this method is more reliable as the uncertaintyrelation is faithfully respected. [In the path-integral MC simulation, this relates tothe problem how accurately effects of the Berry phase are taken into account. See forexample, Ref. [51].] However in the vicinity of the phase transition, 10 configurationsare not sufficient to obtain physical quantities closely related to the singularities ofthe phase transition. The above consideration suggests that the GW methods are afairly good approximation for calculating physical quantities that are finite even forthe critical regime, e.g., finite order parameters. In other words, the estimation of thecritical exponents by the GW methods is not reliable even for using very large systems.The above consideration may over estimate the reliability and applicability of theGW methods, but it explains why the GW methods often succeed in obtaining correct uench dynamics of the extended Bose-Hubbard model As the phase diagram in Fig. 1 shows, the present first-order phase transition is locatedin the vicinity of the triple point of the DW, SF and SS. The GL theory for the quenchdynamics in Sec. 4.1 can be applied to this case with some modification. Besidesthe SF order parameter, we introduce a coarse-grained real DW order parameter, D ( ~r, t )[ ∼ ( − ) i n i ]. GL equations are given as ∂φ∂t = ∇ r φ − ǫ ( t ) φ − g | φ | φ − g D φ, (17) ∂D∂t = ∇ r D + m ( t ) D − g D − g D | φ | , (18)where the positive parameters g , g and g are phenomenological ones, which are to bedetermined by the parameters U and V . The positivity of g comes from the fact thatthe SF and DW are competing orders in the original EBHM. On the other hand, ǫ ( t )and m ( t ) are parameters that are determined by J ( t ), U and V . In the quench fromthe DW to SF, both ǫ ( t ) and m ( t ) are decreasing functions of t .Let us consider a slow quench, and denote the phase transition time from theDW to SF by t c . At t = t c − δ ( δ → +0), the system is in the DW and then, ǫ ( t c ) + g D ( t c ) = ǫ ( t c ) + g g m ( t c ) > φ = 0 and D = m ( t c ) g . On the other hand at t = t c + δ ( δ → +0), the system is in the SF, and m ( t c ) − g | φ | = m ( t c ) + 2 g g ǫ ( t c ) < | φ | = − ǫ ( t c ) g and D = 0. From the above equations, we obtain the constraintfor the occurrence of the direct DW to SF transition such as 2 g > g g , and ǫ ( t c ) < , m ( t c ) >
0. The critical time, t c , is determined by the condition that thepotential energy V = ǫ ( t ) | φ | + g | φ | + g D | φ | − m ( t ) D + g D has the same value inthe DW and SF states at t = t c . This condition gives ǫ ( t c ) = g g m ( t c ). On the otherhand, the triple point is realized by ǫ ( t c ) = m ( t c ) = 0 or 2 g = g g .Let us focus on the SF for t ≥ t c . In this case, D = 0 and we only consider the GLequation in Eq.(17) with D = 0. We assume the same protocol with Eq.(9) and then,the transformation in Eq.(12) can be applied as in the case of the second-order phasetransition. Correlation length at time t is estimated as ξ ( t ) = Z tt c v t dt = 1 τ λ/ ( t λ/ − t λ/ c ) . (19)The second term on the RHS in Eq.(19) comes from the finite jump of φ at thecritical point and indicates the deviation from the genuine second-order phase transition.However for sufficiently small t c such as t c ≪ ˆ t, t eq , the correlation length satisfies almostthe same scaling law with the KZ one. uench dynamics of the extended Bose-Hubbard model
5. Dynamics of phase transition from superfluid to density wave
This section considers the temporal evolution of the system under a quench from theSF to DW. We found that behaviors of the system strongly depend on the initial state.We shall show the results in the following two subsections.
SS SFDW(1,0)
Figure 8.
Transition from SF to DW with J ( t f ) = 0, Case A. The system passesthrough the critical point J c at t = 0. Even for t >
0, both the SF amplitude and DWorder parameter do not exhibit the typical behaviors of the DW.
SF density Density
Phase
Figure 9.
Snapshots of SF local density (amplitude), particle density, SF phasedegrees of freedom, and vortex density at t = τ Q ( J/U = 0). Global coherence of Ψ i does not exist, and finite-size domains of the DW partially form as indicated by thered circles. uench dynamics of the extended Bose-Hubbard model J c − J ( t ) J c ≡ − ǫ ( t ) = tτ Q . (20)In order to clarify the quench dynamics, we shall consider three cases in this subsection.In the first case, Case A, we start with configurations at J ( t = − τ Q ) = 2 J c = 0 .
044 andterminate the quench at t = τ Q with J ( τ Q ) = 0. We employ the tGW methods to studythe system. In Case A, as well as Cases B and C in the later study in this subsection, the initial state is the lowest-energy state obtained by the static GW methods .The obtained results of | Ψ | , ∆ DW and ∆ SF are shown in Fig. 8 for τ Q = 300. | Ψ | exhibits fluctuations in the SF for t <
0, whereas it becomes stable in the region
J < J c (i.e., t > i has a phase coherence inthe SF, which induces amplitude fluctuations, as the amplitude and phase of the SForder parameter are quantum conjugate variables with each other. On the other handin the would-be DW region for t >
0, the phase coherence is lost, and then the SFamplitude is stable. The DW order parameter ∆ DW does not have a stable finite valueeven after passing through the critical point at t = 0. These results indicate that somekind of domain structure forms there, i.e., small DW domains may coexist with localSF regions. Calculations of the amplitude of Ψ i and the particle density at t = τ Q areshown in Fig. 9. As expected above, DW domains and regions with finite SF amplitudecoexist without overlapping with each other.In Case A, the quench stops with J ( τ Q ) = 0, and therefore no movement of particlesoccurs after the quench, and the particle-density snapshot in Fig. 9 continues to describethe states for t > τ Q . Similarly, we expect that the coherence of the phase of Ψ i isdestroyed at t = τ Q because J ( τ Q ) = 0 and also τ Q = 300 is a slow quench. See Fig. 9.In order to verify the expected behavior of Ψ i , we measured the vortex density as afunction of time. At t = τ Q , N v ∼
300 is sufficiently large. In summary, in Case A with τ Q = 300, an inhomogeneous state with local DW and SF domains forms after quench.SF order parameter gradually loses its phase coherence during the slow quench.On the other hand for cases of smaller τ Q = 100 and 50, the SF order parameterΨ i is finite even at t = τ Q , and it varies after t = τ Q . The phase of Ψ i gradually losesits long-range coherence by the existence of the repulsive interactions for t > τ Q .As Case B, we consider a quench such as J ( − τ Q ) = 0 .
044 and J (0) = J c = 0 .
022 asbefore but it terminates at t = t f with J ( t f ) = 0 .
01, i.e., t f = 0 . τ Q (see Fig. 10). Wealso study how the system evolves after t f . Observed quantities are shown in Fig. 10for τ Q = 50. The DW order parameter ∆ DW develops but its value fluctuates in ratherlong period after passing J c as in Case A. The total energy slightly decreases until t f ,and the kinetic and on-site energies exhibit fluctuating behavior for t < t f although theNN interaction energy is rather stable. This behavior mostly originates from the localdensity fluctuations, and the stability of the NN interaction comes from the cancellationmechanism between NN sites j ∈ i NN. After passing the critical point at t = 0, theΨ i keeps a coherent SF order for some period as the calculation of the vortex number uench dynamics of the extended Bose-Hubbard model -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 50 150 250 350 450 0 0.01 0.02 0.03 0.04 E n e r gy total energykinetic energyon-site energyNN interaction energy Figure 10.
Transition from SF to DW with J ( t f ) = 0 .
01, Case B. Genuine global DWorder does not form. After passing J c at t = 0, N v keeps a small value for a while, andthe SF order survives there. After passing t f = 0 . τ Q = 27 .
5, the total energy of thesystem keeps a constant value as the system is and isolated one.
Figure 11.
Transition from SF to DW with J ( t f ) = 0 .
02, Case C. Increase of N v isslow compared to the cases J ( t f ) = 0 and J ( t f ) = 0 .
01. SF amplitude | Ψ | also keepsa finite value even for t → large. However, N v increases smoothly, and therefore, thesupercooled state formed in the quench is not a meta-stable state. N v indicates. At t ≈ N v indicates. The state at t ∼ t f is a supercooled state , and a coexistingphase of local domains of the DW and SF is realized there. The observed phenomenonafter t > t f , therefore, has very similar nature to the glass transition , in which the phasecoherence and superfluidity are getting lost as the supercooled state evolves after thequench. We call it quantum glass transition (QGT) as the hopping amplitude J , insteadof temperature, is the controlled physical quantity and the relevant transition is quantummechanical one instead of thermal one. We have verified that similar phenomenon isobserved for other values of τ Q , e.g., τ Q = 20 and 200.In both Case A and Case B, the above mentioned QGT is observed dynamicallyas a nonequilibrium phenomenon , i.e., the QGT point is passed through as the systemevolves. Therefore as the next problem, it is interesting to see whether there exits agenuine glass transition point, J g ( < J c ). Below J g , the supercooled state is meta-stable uench dynamics of the extended Bose-Hubbard model Figure 12. (Upper-left) Vortex number as a function of time. Each point denotesthe following time; (a) t = −
50, (b) t = 0, (c) t = 150, and (d) t = 450. (Upper-right) Particle density snapshot in Case C. At t=0, a typical DW domain appears asindicated in the red circle. (Lower-left) SF density snapshot in Case C. (Lower-right)Snapshot of phase degrees of freedom of SF order parameter in Case C. or at least has a long life time, and the SF survives without losing its phase coherence.For Cases A and B, J < J g . Then as Case C, we studied the quench whose finial point is J ( t f ) = 0 .
02, i.e., very close to the equilibrium critical point. Obtained order parameter | Ψ | and vortex number N v are shown in Fig. 11 for τ Q = 50, and time evolution of theparticle density, amplitude and phase of Ψ i are shown in Fig. 12. After passing thecritical point J = J c at t = 0, the domain formation of the DW starts as shown by theparticle-density snapshot in Fig. 12, whereas the long-range coherence of the SF orderparameter Ψ i exists there. Compared with the cases of J ( t f ) = 0 and J ( t f ) = 0 . t > J g cannot beobserved. Similar results are obtained for the case of τ Q = 20 and τ Q = 200. uench dynamics of the extended Bose-Hubbard model (a) t = -300 (b) t = 115 (c) t = 300 D e n s it y (a) t = -300 (b) t = 115 (c) t = 300 P h a s e (a) t = -300 (c) t = 300(b) t = 115 SF d e n s it y Figure 13. (First) SF order parameter as a function of time. Each point denotes thefollowing time; (a) t = − t = 115, and (c) t = 300. (Second) Particle densitysnapshot in Case D. (Third) SF density snapshot in Case D. (Lowest) Snapshot of phaseof SF order parameter in Case D. At t = 300, a large scale DW domain structure withthin domain walls forms. Coherence of SF phase is lost there. In Sec. 5.1, we studied dynamical evolution of the system from the SF to DW. Inthat study, the initial state is set to the ground-state obtained by the equilibrium GWmethods. It is interesting to see how the dynamical phenomena depend on the initialstate as we are considering the first-order phase transition. In order to study thisproblem, we consider a SF state that is uniform and has almost perfect phase coherence uench dynamics of the extended Bose-Hubbard model j = √ ρe iδθ j with random numbers { δθ j } from a uniform distribution [ − . , . × π . The other condition is the samewith the Case A, (please refer to the left panel in Fig. 8). We call the present studyCase D.We investigated the time evolution of the system by the tGW methods, andobtained results are shown in Fig. 13. Interestingly enough, the system behavior afterpassing across the critical point J c is substantially different from that in Cases A. TheSF order parameter | Ψ | decreases a finite amount at t ∼ t ∼ t ∼
115 and it develops to thewhole system at t ∼ t < t ∼
6. Conclusion
In this work, we studied dynamical behavior of the EBHM in 2D by using the tGWmethods. In the ground-state phase diagram, there are three phases, the SF, DW, andSS. In particular, we are interested in the first-order phase transition between the SFand DW under a slow quench of the hopping amplitude.First, we investigated the dynamics of the EBHM in the transition from the DW toSF. In the practical calculation, we fix the strength of the one-site and NN repulsions,and vary the hopping parameter J . After passing through the equilibrium critical point J c , the amplitude of the SF order parameter, | Ψ | , remains vanishingly small until t = ˆ t .After ˆ t , it develops quite rapidly. Therefore, ˆ t has the meaning of the reentry time to theadiabatic region passing from the frozen regime although the present phase transitionis of first order. At t eq ( > ˆ t ), | Ψ | stars to oscillate until t = t ex . This behavior is quitesimilar to that in the second-order phase transition from the Mott insulator to SF, whichwe observed in the previous work [26]. Then we are interested in whether some kind uench dynamics of the extended Bose-Hubbard model τ Q exist. Our numerical study shows that the scaling laws such as ξ ∝ τ b Q and N v ∝ τ − d Q infact hold. This result is against to the simple expectation that such scaling laws do notexist in the first-order phase transitions because the simple relaxation-time picture andthe concept of the (dynamical) critical exponents are not applicable. From this result,we think that there exists another mechanism, besides the KZ mechanism, to generatethe scaling laws. As a possible explanation, we studied the present system by using theGL-type theory suggested by Ref. [50]. This consideration indicates that the observedscaling laws come from the fact that the present phase transition point is located in thevicinity of the triple point.In the second half, we studied the dynamics of the EBHM in the quench of theopposite direction, i.e., from the SF to DW. We focused on how the final value of thehopping amplitude of the quench, J ( t f ), influences the dynamics of the system duringand after the quench.Our numerical study showed very interesting phenomena. First, in the case for theGW ground-state as the initial state, the genuine DW state does not form even for veryslow quench τ Q = 300. Instead, the coexisting state composed of DW and SF domainsappears and spatially inhomogeneous structure of that state is stable after the quench.In cases with J ( t f ) >
0, the SF order parameter has a phase coherence at t = t f , andafter the quench, the SF order is getting weak by the generation of vortices. Obviously,the quench produces a supercooled state in which the domain structure of the DW andSF local (i.e., short-range) coherent state forms. These two domains have an off-setstructure with each other. Then, after termination of the quench, the SF is destroyed.This phenomenon is a reminiscent of the glass transition in classical polymers etc, andwe call the observed phenomenon quantum glass transition.On the other hand, if we start with the uniform SF state with tiny fluctuations inthe phase of the SF order, the system evolves into the DW with thin domain walls.In the phase diagram of the EBHM near the half-filling shown in Fig. 1, there isthe SS phase, and the SS has two phase boundaries with the DW and SF. In the case ofthe mean particle density ρ = 1 and strong NN repulsion, the region of the SS is largeand two second-order phase transitions are observed clearly from the SS to the DW andSF, respectively. It is interesting to study the dynamics in that region, that is, howthe system develops crossing through two second-order phase boundaries. Some relatedproblem was recently studied in classical systems, and a modified KZ scaling law wasproposed [52]. We studied the above problem in the EBHM by using tGW methods,and results are published in Ref. [40]. Appendix A. Hard-core Bose-Hubbard model
In this work, we study the EBHM of the soft-core boson. Hard-core extended Bose-Hubbard model (HCEBHM) is also an interesting model and its relationship to the s = 1 / uench dynamics of the extended Bose-Hubbard model Figure A.1.
Equilibrium physical quantities obtained by the GW methods. ρ = 1 / V = 1. The results show the existence of a first-order phase transition as thequantum simulations in Refs. [53, 54] proved. Figure A.2.
Quench dynamics of the HCEBHM with ρ = 1 / τ Q = 300. Thecorrelation length ξ and vortex density N v fluctuate rather strongly compared to thesoft-core cases. This result comes from the fact that particle-number fluctuation ateach site is restricted by the hard-core constraint, and as a result, fluctuation in thephase of the SF order parameter Ψ i is getting large. square lattice is given as H HC = − J X h i,j i ( a † i a j + H.c.) + V X h i,j i n i n j − µ X i n i , (A.1) uench dynamics of the extended Bose-Hubbard model ρ = 1 / τ Q -dependenceof ˆ t , etc in the quench dynamics in Fig. A.2. The results in Fig. A.1 obviously showthat there is a first-order phase transition from the DW to SF for increasing J/V as thequantum MC simulations in Ref. [53, 54] proved. On the other hand, the correlationlength ξ and vortex density N v fluctuate rather strongly compared to the soft-corecases. This result comes from the fact that the HCEBHM has a small fluctuations inthe particle number at each site, and as a result, the phase of the SF order parameterfluctuates rather randomly. Acknowledgments
Y. K. acknowledges the support of a Grant-in-Aid for JSPS Fellows (No.17J00486).
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