Complex semiclassical analysis of the Loschmidt amplitude and dynamical quantum phase transitions
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y Complex semiclassical analysis of the Loschmidt amplitude anddynamical quantum phase transitions
Tomoyuki Obuchi , ∗ Sei Suzuki , and Kazutaka Takahashi Department of Mathematical and Computing Science,Tokyo Institute of Technology, Yokohama 226-8502, Japan Department of Liberal Arts, Saitama Medical University, Moroyama, Saitama 350-0495, Japan Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan (Dated: September 3, 2018)We propose a new computational method of the Loschmidt amplitude in a generic spin systemon the basis of the complex semiclassical analysis on the spin-coherent state path integral. Wedemonstrate how the dynamical transitions emerge in the time evolution of the Loschmidt amplitudefor the infinite-range transverse Ising model with a longitudinal field, exposed by a quantum quenchof the transverse field Γ from ∞ or 0. For both initial conditions, we obtain the dynamical phasediagrams that show the presence or absence of the dynamical transition in the plane of transversefield after a quantum quench and the longitudinal field. The results of semiclassical analysis areverified by numerical experiments. Experimental observation of our findings on the dynamicaltransition is also discussed. I. INTRODUCTION
Triggered by experiments using ultracold atomic sys-tems, dynamics of a closed quantum many-body sys-tem has been one of the fascinating topics in condensedmatter physics . In particular, the time evolution af-ter a sudden change of the Hamiltonian has attracteda lot of attention as a basic setting of a problem onthe out-of-equilibrium quantum state. One of the inter-esting phenomenon associated with this so-called quan-tum quench is the dynamical quantum phase transition(DQPT). While the equilibrium quantum phase transi-tion is usually associated with a singularity of the ground-state energy in the axis of a parameter contained in theHamiltonian, the DQPT involves a singularity in time.The present paper focuses on such a dynamical singu-larity appearing in the return probability to the initialstate, which is directly related to the Loschmidt ampli-tude defined below .The phenomena of the DQPT are observed not only inthe Loschmidt amplitude but also in the time average oflocal physical quantities such as order parameters. Al-though a certain correspondence is pointed out , thesetwo kinds of quantities are generally different. The localphysical quantities represent the properties of the steadystate in the long time limit after a quantum quench. Theybring a clear physical consequence and are easy to accessby experiments. The DQPT of them corresponds to aphase transition with the parameter in the Hamiltonianafter a quantum quench. The Loschmidt amplitude, onthe other hand, can be seen as an extension of the par-tition function on the imaginary axis corresponding totime. The DQPT here is defined as a singular behaviorwith time in the rate function of it, as an analogy withthe thermodynamic phase transition accompanied by thesingularity of the free energy as a function of the temper-ature. However, the Loschmidt amplitude involves deli-cate points in several aspects: physical meaning of thesingularities, experimental implementations, and even technicalities for theoretical computations. Several re-cent works have been devoted to resolve the first del-icate point based on statistical mechanical concepts suchas renormalization group, symmetry breaking, universal-ity, and scaling . They have provided solid advances.For instance, the singularity has been tied with a behav-ior of the order parameter and entanglement productionin systems with symmetry-broken phases . However, ageneral comprehension including the relation of the sin-gularities to other local quantities with a generic initialstate is still lacking. One of the origins of the difficultyin obtaining a general description lies, in our opinion, inthe limitation on theoretical techniques to compute theLoschmidt amplitude. Most of theoretical works so fardepend on the result of specific models being analyticallytractable, and generic properties of the Loschmidt ampli-tude’s singularity have been speculated from the result.Hence, a more versatile computational method will be agreat help to understand the Loschmidt amplitude.Under this circumstance, here we propose a new the-oretical framework for computing the Loschmidt ampli-tude for a generic spin system on the basis of a semiclas-sical computation. This can be regarded as a mean-fieldmethod and is expected to be exact in the infinite dimen-sion, though it is still applicable as an approximation toa generic spin system in any dimension with an arbitrarystate. The static approximation is often used with themean-field method and is known to give a correct re-sult for quantities in the equilibrium in the system withan infinite-range interaction . However, the static ap-proximation does not work for the computation of theLoschmidt amplitude. In this sense, our method goesbeyond the static approximation and can be useful forcomputation of out-of-equilibrium quantities.Our semiclassical method is essentially the same asthe one used in Refs. , but their analysis has beenonly on local physical quantities. This is presumablydue to the lack of general prescriptions to compute theLoschmidt amplitude so far. The present work comple- ΓΓ h Γ c PMFM0
FIG. 1. Phase diagram of the ground state in the infinite-range transverse Ising model (1) in the plane of the transversefield Γ and the symmetry-breaking longitudinal field h . Theferromagnetic (FM) phase lies on the axis of Γ from 0 to Γ c = J with h = 0, while other parameter area is a paramagnetic(PM) phase. ments this point. The key difference of our method fromthe preceding studies lies in the determination ofthe semiclassical path that follows the initial and finalconditions properly. In our method, the range of dynam-ical variables is extended from real to complex numbers,and matching the semiclassical path with the boundaryconditions is achieved in the complex space. This ideahas been proposed in Refs. for single-spin systemsand we extensionally apply it to many-spin systems. Ac-cordingly, there emerge multiple solutions in the bound-ary value problem, and the solution that gives the largestreturn probability is selected. We find that it is this se-lection that gives the singularity in the Loschmidt ampli-tude. In this sense, the singularity of Loschmidt ampli-tude is very similar to an equilibrium phase transition.Using the complex semiclassical method, in the presentpaper, we study the infinite-range transverse Ising modelHamiltonian with uniform coupling J and longitudinalfield h :ˆ H = − N J N N X i =1 σ zi ! − Γ N X i =1 σ xi − h N X i =1 σ zi , (1)where σ αi ( i = 1 , , . . . , N ; α = x, z ) is the Pauli matrixand N is the number of spins. For this model, we considera quantum quench of the transverse field Γ from Γ i to Γ f at t = 0. As shown in Fig. 1, this system shows two dif-ferent phases in equilibrium and both inter- and intra-phase protocols of quench are examined. The Loschmidtamplitude is defined by L ( t | ψ ) = h ψ | e − it ˆ H | ψ i , (2)where the state | ψ i is chosen as the ground state of theHamiltonian with Γ = Γ i . The Loschmidt amplitude isexpected to exhibit a large deviation nature, and henceits rate function at N → ∞ is the primary object of ouranalysis. The rate function is defined as f ( t | ψ ) = − N log L ( t | ψ ) . (3) Note that its real part, f r = ℜ f , accounts for the returnprobability P ( t | ψ ) = |L ( t | ψ ) | as 2 f r = − N log P ( t | ψ ),while the imaginary part has no direct physical conse-quence.The rest of the paper is organized as follows. In Sec. II,we describe the formulation and procedures needed tomake the problem computationally tractable. In Sec. III,the analytical solutions computed from the inventedmethod are shown and are compared to numerical ex-periments on finite size systems. Exact derivation of therate function, available only on some specific parameters,is also given to justify the result. Section IV is devotedto discussion and summary. The relevance of the presentwork to experiments, quantum engineering, and compu-tation is discussed there. II. FORMULATIONA. Spin coherent states and path integrals
We start from reviewing the path integral formulationfor spin systems. An arbitrary state of a single spin isrepresented by a spin-coherent state as | θ, ϕ i = e ib (cid:18) e − i ϕ cos θ |↑i + e i ϕ sin θ |↓i (cid:19) , (4)where |↑i and |↓i are the eigenstates of σ z with eigen-values +1 and −
1, respectively. Hereafter the gauge b isfixed to be 0 and is disregarded, since it does not affectany physical consequences. As is well known, the aver-age of spin variables σ = ( σ x , σ y , σ z ) over a spin-coherentstate corresponds to three-dimensional polar representa-tion as h θ, ϕ | σ | θ, ϕ i = (sin θ cos ϕ, sin θ sin ϕ, cos θ ) . (5)The spin-coherent state constitutes an overcomplete ba-sis: Z − d cos θ Z π dϕ π | θ, ϕ i h θ, ϕ | = |↑i h↑| + |↓i h↓| = I, (6)where I denotes the 2 × θ, ϕ ) are not orthogonal in general.We apply this spin-coherent state formulation to N -spin systems and write the variables as ( θ , ϕ ) = { ( θ i , ϕ i ) } Ni =1 . Using the spin-coherent states, we writeany propagator with arbitrary time-dependent Hamilto-nian ˆ H ( t ) as G ( t | Ω ′ , Ω ′′ ) ≡ h Ω ′′ | T e − i R t ds ˆ H ( s ) | Ω ′ i , where T is the time-ordering operator, and Ω ′ = ( θ ′ , ϕ ′ ) andΩ ′′ = ( θ ′′ , ϕ ′′ ) are initial and final states respectively.This propagator is rewritten in a path integral form as G ( t | Ω ′ , Ω ′′ ) = Z Ω ′′ Ω ′ N Y i =1 D cos θ i D ϕ i e S [ θ , ϕ ] . (7)This is an integral over all possible paths of the variables( θ ( s ) , ϕ ( s )). The action functional S [ θ , ϕ ] is given by S [ θ , ϕ ] = i Z t ds ( X i ˙ ϕ i ( s ) cos θ i ( s ) − H ( θ , ϕ , s ) ) , (8)where the dot symbol denotes the time derivative and H ( θ , ϕ , s ) = h θ ( s ) , ϕ ( s ) | ˆ H| θ ( s ) , ϕ ( s ) i . B. Complex semiclassical analysis
The path integral formalism gives the exact result ifwe can perform the integration over all paths literally.However, this is difficult in general, and the semiclassicalapproximation is here employed.
1. Boundary value problem
The basic idea of the semiclassical method is to takeinto account only the dominant stationary paths amongall the paths. The stationary condition in the action S leads to the following equations of motion (EOMs):12 ˙ θ i sin θ i = ∂ H ∂ϕ i ,
12 ˙ ϕ i sin θ i = − ∂ H ∂θ i . (9)We naively expect that the solution of these EOMs, sat-isfying the boundary conditions ( θ (0) , ϕ (0)) = Ω ′ =( θ ′ , ϕ ′ ) and ( θ ( t ) , ϕ ( t )) = Ω ′′ = ( θ ′′ , ϕ ′′ ), is the de-sired semiclassical path. If there are multiple semiclas-sical paths, we give indices to them as { ( ¯ θ ( ν ) , ¯ ϕ ( ν ) ) } ν where the symbol ¯ · represents a generic semiclassical pathhereinafter. Semiclassical actions corresponding to thosepaths are defined as S cl [ ¯ θ ( ν ) , ¯ ϕ ( ν ) ] = S [ ¯ θ ( ν ) , ¯ ϕ ( ν ) ]. Theygive an approximation of the propagator as G ( t | Ω ′ , Ω ′′ ) ∼ X ν A ν e S cl [ ¯ θ ( ν ) , ¯ ϕ ( ν ) ] , (10)where A ν denotes a possible amplitude factor.Unfortunately, this procedure does not work in thepresent problem. The solution of Eq. (9) cannot satisfy,in general, both the boundary conditions ( θ (0) , ϕ (0)) =Ω ′ and ( θ ( t ) , ϕ ( t )) = Ω ′′ . For a given initial condition( θ (0) , ϕ (0)) = Ω ′ , the time evolution of the system isuniquely determined by the EOMs, and the final values( θ ( t ) , ϕ ( t )) do not necessarily coincide with the boundaryone Ω ′′ . This is the reason why the Loschmidt amplitudehas been difficult to be evaluated by the semiclassicalor similar methods, though some exceptions are foundwhen the semiclassical path is constant in time .To overcome this problem, following the prescription inRefs. , we below introduce the so-called Wiener regu-larization term making the path integral well defined inthe action, and deal with the unregularized action as thevanishing limit of the regularization term. This yields adifferent boundary condition.
2. Wiener regularization and modified boundary condition
By using the prescription by Klauder , Alscher andGrabert demonstrated that the exact propagator canbe computed in single-spin systems with arbitrary time-dependent magnetic fields . Here we apply this to many-spin systems.The Wiener regularization is defined as W [ θ , ϕ ] = − m Z t ds X i (cid:16) ˙ θ i + ˙ ϕ i sin θ i (cid:17) , (11)where m represents a constant. Adding this term to theaction, S [ θ , ϕ ] → S [ θ , ϕ ] + W [ θ , ϕ ], and taking the sta-tionary condition, we obtain the modified semiclassicalEOMs as12 ˙ θ j sin θ j = ∂ H ∂ϕ j + i m (cid:16) ¨ ϕ j sin θ j + 2 ˙ θ j ˙ ϕ j sin θ j cos θ j (cid:17) , (12a)12 ˙ ϕ j sin θ j = − ∂ H ∂θ j − i m (cid:16) ¨ θ j sin θ j − ˙ ϕ j sin θ j cos θ j (cid:17) . (12b)Due to the regularization term, the higher-order deriva-tives appear in the EOMs and its general solution hasmore arbitrary constants, which naturally enables us tohave a solution connecting to both the boundary val-ues Ω ′ and Ω ′′ . Meanwhile, the terms coming from theregularization introduce the imaginary number into theEOMs. Hence the corresponding semiclassical path be-comes complex in general and loses a clear physical inter-pretation. Bloch sphere representation is not applicableto visualize the semiclassical path. From a formal corre-spondence, the Wiener regularization can be regarded asa kinetic energy of spins with a pure imaginary mass.To recover the original action, we take the zero masslimit m →
0. For small m , the time span s ∈ [0 , t ] isdivided into three characteristic regions : T = [0 , m ], T cl = [ m, t − m ], and T = [ t − m, t ]. In T cl , the mass termsproportional to m become irrelevant and the time evo-lution is essentially driven by the original unregularizedEOMs. In T and T , the trajectory is strongly hingedby the mass terms to match the boundary conditions.As a result, in the m → s = 0 and s = t from the boundary values to the edges ofthe semiclassical path in T cl → [0 , t ]. These jumps givea condition for the values at the boundary ( ¯ θ (0) , ¯ ϕ (0))and ( ¯ θ ( t ) , ¯ ϕ ( t )), which has a simple explicit form:tan (cid:18) ¯ θ i (0)2 (cid:19) e i ¯ ϕ i (0) = tan (cid:18) θ ′ i (cid:19) e iϕ ′ i , (13a)tan (cid:18) ¯ θ i ( t )2 (cid:19) e − i ¯ ϕ i ( t ) = tan (cid:18) θ ′′ i (cid:19) e − iϕ ′′ i . (13b)This condition implies that there can be multiple semi-classical paths to satisfy Eq. (13) and that they can becomplex even in the m → that the solu-tion of the unregularized EOMs (9) under the condition(13) gives the exact propagator.
3. Solving the boundary value problem
The boundary value problem becomes well-defined nowand we can find solutions matching both the boundaryvalues Ω ′ and Ω ′′ in a generic situation. A practical wayfor solving the problem is to employ the following variabletransformation : ζ j ( s ) = tan (cid:18) θ j ( s )2 (cid:19) e iϕ j ( s ) , (14a) η j ( s ) = tan (cid:18) θ j ( s )2 (cid:19) e − iϕ j ( s ) . (14b)These variables are, if ( θ j ( s ) , ϕ j ( s )) are real, a stereo-graphic representation of a point on the unit sphere pro-jected from the south pole onto the equatorial plane.Hence we call them stereographic variables. The bound-ary condition is now written as ζ j (0) = ζ ′ j ≡ tan (cid:18) θ ′ j (cid:19) e iϕ ′ j , (15a) η j ( t ) = η ′′ j ≡ tan (cid:18) θ ′′ j (cid:19) e − iϕ ′′ j , (15b)and the remaining boundary values, ζ i ( t ) and η i (0), arenot specified. The spin variables in the Hamiltonian areconverted to the stereographic variables through the re-lation h θ j , ϕ j | σ j | θ j , ϕ j i = 11 + ζ j η j ζ j + η j − i ( ζ j − η j )1 − ζ j η j , (16)and the semiclassical EOMs (9) are˙ ζ j = − i (1 + ζ j η j ) ∂ H ∂η j , (17a)˙ η j = − i (1 + ζ j η j ) ∂ H ∂ζ j . (17b)Using the solution of the EOMs, (¯ ζ j , ¯ η j ), we can writethe semiclassical action as e S cl [¯ ζ , ¯ η ] = N Y j =1 (s (1 + ¯ ζ j (0)¯ η j (0))(1 + ¯ ζ j ( t )¯ η j ( t ))(1 + ζ ′ j η ′ j )(1 + ζ ′′ j η ′′ j ) × (cid:18) ζ ′ j η ′ j ζ ′′ j η ′′ j ¯ ζ j (0)¯ η j (0)¯ ζ j ( t )¯ η j ( t ) (cid:19) ) × exp Z t ds ( N X j =1 (1 − ¯ ζ j ¯ η j )( ˙¯ ζ j ¯ η j − ¯ ζ j ˙¯ η j )¯ ζ j ¯ η j (1 + ¯ ζ j ¯ η j ) − i H ( ¯ ζ , ¯ η , s ) ) . (18)
4. Spatially uniform solutions
A problem arises when we compute the semiclassicalpaths satisfying Eq. (15). We need to fix both the ini-tial conditions on ζ i and the final ones on η i . The initialconditions on η i must be selected so as to meet the finalconditions. This requires us to solve the EOMs manytimes, and results in a bottleneck of the present methodto compute the propagator. This is because the computa-tional cost for searching such an initial condition growsexponentially with the number of spins. Therefore, inpractice, we need an assumption that reduces the degreeof freedom, namely, the computational cost of searchingthe initial value of η i .In the present paper, we assume the spatial unifor-mity. Our Hamiltonian (1) has infinite-range interac-tions and the mean-field ansatz gives the exact resultfor static systems. Although it is not evident whetherthe spatial uniformity holds for dynamical systems, weexamine this ansatz in the following. The boundaryvalues of ( ζ i ( s ) , η i ( s )) are identical for all i ’s, so that( ζ ′ i , η ′ i ) = ( ζ ′ , η ′ ) , and ( ζ ′′ i , η ′′ i ) = ( ζ ′′ , η ′′ ). Then, onlytwo functions, ζ ( s ) and η ( s ), are sufficient to describethe dynamics, and the exhaustive search of η (0) is now areasonable task. Moreover, as far as the Loschmidt am-plitude is concerned, the initial and final boundary valuesare common: ζ ′ = ζ ′′ = ζ b and η ′ = η ′′ = η b . Summa-rizing these particular conditions, we obtain the explicitformulas of the EOMs as˙ ζ = i Γ (cid:0) − ζ (cid:1) − iζ (cid:18) h + J − ζη ζη (cid:19) , (19a)˙ η = − i Γ (cid:0) − η (cid:1) + 2 iη (cid:18) h + J − ζη ζη (cid:19) . (19b)For a given t , these EOMs are solved under the conditions ζ (0) = ζ b and η ( t ) = η b . The other boundary values ζ ( t )and η (0) are not specified and are determined uniquelyfrom the above conditions. We also note that the relation ζ ( s ) = η ∗ ( s ) does not necessarily hold in general.The solutions (¯ ζ ( ν ) ( s ) , ¯ η ( ν ) ( s )) are not unique and wecan represent the Loschmidt amplitude as L ( t | Ω b ) = h Ω b | e − i ˆ H t | Ω b i ∼ X ν A ν e − Nf [¯ ζ ( ν ) , ¯ η ( ν ) ] , (20)where f [¯ ζ, ¯ η ] = −
12 log (1 + ζ b ¯ η (0))(1 + ¯ ζ ( t ) η b )(1 + ζ b η b ) − i Z t ds (cid:18) Γ2 (¯ ζ + ¯ η ) + h + J ζ ¯ η − ζ ¯ η (1 + ¯ ζ ¯ η ) (cid:19) . (21)The time derivative terms are eliminated by performingthe integration by parts or using the EOMs. We alsonote that the amplitude A ν is not important to calculatethe rate function in Eq. (3) at N → ∞ .
5. Dominant semiclassical paths and a heuristic searchprocedure
Equation (13) has a countably infinite number of so-lutions, and the EOMs do as well. Among those manysemiclassical solutions, the one that makes the real partof f [¯ ζ ( ν ) , ¯ η ( ν ) ] the smallest gives the rate function inEq. (3). How can we find such a dominant solution? Theexhaustive search of ¯ η (0) in the whole complex space isnot plausible even under the spatial uniformity. To over-come the situation, we here give a heuristic procedure toobtain such a dominant path. Since the correct initialcondition ¯ η (0) depends on the end time t , we hereafteruse a notation C ( t ) = ¯ η (0; ¯ η ( t ) = η b ). The basic idea ofthe heuristic is starting from a trivial solution at a spe-cific time t ∗ and extending it with changing the time t from t ∗ gradually.The first trivial solution is obtained at t ∗ = 0, where C (0) = η b . Then, for a small time step ∆ t , C (∆ t ) isobtained as follows. We examine several values as theinitial condition for ¯ η ( s ) around η b and solve the EOMs.We select the best one for C (∆ t ) that makes the finalvalue ¯ η ( s = ∆ t ) closest to η b . For the next time step t = 2∆ t , we examine the values around C (∆ t ) and re-peat the same procedures, giving C (2∆ t ). We repeatthis procedure until we reach a desired end time t , yield-ing the sequence of the initial condition. We write thissequence as C ( t ).To obtain the second trivial solution, an important ob-servation is that the dynamics is periodic at most of pa-rameters . There exists a specific period τ and the orderparameters at t n = t + nτ are identical for ∀ n ∈ N . Thisimplies that at t ∗ = τ the final condition ¯ η ( s = t ∗ ) = η b isrealized by having the initial condition ¯ η (0) = η b , yield-ing C ( t ∗ = τ ) = η b . Extending C ( t ) back from t = τ to t = 0 based on the same procedure for C ( t ), we getanother sequence of the initial condition, and write it as C ( t ). In Fig. 2, a schematic picture of this heuristic isgiven.The question is whether these two sequences of ini-tial conditions, C ( t ) and C ( t ), are identical or not. Ifthey are different, they give two different semiclassicalpaths. In such a situation, there should be a switch be-tween two paths at a certain critical time t c in the period[0 , τ ], that yields a singularity of the Loschmidt ampli-tude. Meanwhile, if they are identical, only one dominantsemiclassical path exists and is analytic with respect to t . For longer time t > τ , we repeat the above procedure.For the next period [ τ, τ ], C ( t ) is obtained by extending C ( t ) from t = τ to 2 τ with the trivial value C ( τ ) = η b ,and C ( s ) is given by an extension from t = 2 τ to τ with C (2 τ ) = η b . We note that by construction C ( s ) and C ( s ) are continuously connected. The solutions for thewhole time axis are obtained along this way.We adopt the above scenario to search the solution.This may give a wrong result in general, but, as far aswe have investigated, the result shows a good agreement s η η b η Δ t ( s ) C (2 Δ t )C (3 Δ t )C ( Δ t ) η Δ t ( s ) η Δ t ( s ) Δ t Δ t Δ t (0) s η η b C ( τ -2 Δ t )C ( τ - Δ t ) η τ ( s ) τ -2 Δ t τ - Δ t ( τ ) η τ - Δ t ( s ) η τ -2 Δ t ( s ) τ - - - -- - FIG. 2. Schematic pictures of the heuristic to obtain appro-priate initial conditions C ( t ) (left panel) and C ( t ) (rightpanel) of ¯ η t . The complex plane of η is schematically mappedto the horizontal axis. Here, ¯ η t ( s ) denotes the semiclassi-cal path satisfying the final condition ¯ η ( t ) = η b for given t .The initial condition, ¯ η (0) for given t is accordingly searched,starting from t = 0 ( C ) or t = τ ( C ). with numerical experiments as we see in the following.Our heuristic procedure is constructed under the assump-tion that the system shows a periodic behavior and onlyone transition at most in one cycle. As long as this as-sumption is true, our heuristic can find the correct dom-inant path. For more general cases, e.g. spin glasseswithout periodicity , other heuristics should be tai-lored. Investigation of such cases is beyond the scope ofthis paper and will be an interesting future work. III. RESULT
We present the results of our semiclassical compu-tation. We study two cases: quenches from Γ i = ∞ (Sec. III A) and quenches from Γ i = 0 (Sec. III B). Thefirst case is the quench from Γ i = ∞ to a finite valueΓ f < ∞ , where the boundary condition is the groundstate at Γ i = ∞ , namely, | Ω b i = ⊗ i |→i i with |→i i be-ing the eigenstate of σ xi for eigenvalue +1. The othercase is the opposite quench, from Γ i = 0 to Γ f >
0. Weset h ≥
0+ and thus | Ω b i = ⊗ i |↑i i . Since exact calcula-tion is possible for a quench from Γ f = ∞ to Γ i = 0, weshow its result in Sec. III A as well. We also show the re-sults of numerical studies in Sec. III C to confirm that thecomplex semiclassical analysis gives a reasonable result. A. Quench from Γ i = ∞ In this case, the boundary condition is given by( θ ′ , ϕ ′ ) = ( θ ′′ , ϕ ′′ ) = ( π/ , ζ b , η b ) = (1 , h = 0, the state doesnot evolve and the semiclassical path is written as ¯ ζ ( s ) =¯ η ( s ) = 1 for ∀ s . Hence, we consider the case h > < τ < ∞ ispresent. In fact, we see several patterns of the rate func-tion and DQPT as well. We obtain the correspondingphase diagram.
1. A solvable case: Γ f = 0 We first investigate the quench to Γ f = 0. In this case,the state is evolved under the classical Ising Hamiltonianand an analytical solution of Eq. (19) is available. Wesolve the equation under the conditions ζ (0) = 1 and η ( t ) = 1. Putting the initial condition as ( ζ (0) , η (0)) =(1 , C ), we get the explicit solution of the dynamics as¯ ζ ( s ) = exp (cid:18) − is (1 + C ) h + (1 − C ) J C (cid:19) , (22a)¯ η ( s ) = C exp (cid:18) is (1 + C ) h + (1 − C ) J C (cid:19) . (22b)Then the condition η ( t ) = 1 gives us C exp (cid:18) it (1 + C ) h + (1 − C ) J C (cid:19) = 1 , (23)which yields C ( t ).This example clearly shows the presence of multiplepaths satisfying the boundary condition. As declaredin Sec. II B 5, we investigate two paths associated withthe initial conditions C ( t ) and C ( t ), each of which iscontinuously extended from C (0) = η b = 1 and from C ( τ ) = η b = 1 respectively, where τ is the period ofthe dynamics. The period τ can be obtained by putting C = 1 and t = τ in the solution (22) as τ = πh . (24)The solutions of Eq. (23) connecting to C (0) = 1 and C ( τ ) = 1, C ( t ) and C ( t ) respectively, are shown inFig. 3 for h/J = 0 .
1. As a reference, the solutions in thenext period [ τ, τ ], C and C , are also displayed. Given atime t , the rate function f k ( t ) corresponding to the initialcondition C k ( t ) is evaluated by inserting Eq. (22) with C = C k ( t ) into Eq. (21) and performing the integrationwith respect to s . The result is shown in the right panelof Fig. 3. This exhibits the DQPT at t = τ / f to f occurs. Similarly, the switch from f to f occurs at t = τ , showing another DQPT.Apart from the analytical solution of the EOMs, therate function itself can be computed exactly for the caseΓ f = 0. Dashed lines in the right panel of Fig. 3 rep-resent the result. In terms of the total spin operatorˆ S z = P i σ zi , our Hamiltonian after the quench is writ-ten as ˆ H = − (cid:18) JN ˆ S z + h ˆ S z (cid:19) . (25)The eigenvalue of this Hamiltonian is characterized bythat of ˆ S z denoted by M taking the value N − k with k = 0 , , . . . , N . For a given M , there are (cid:0) Nk (cid:1) degeneratestates. Let us define a normalized vector (cid:12)(cid:12) N − k (cid:11) in this subspace, which is the equal-weight sum of the (cid:0) Nk (cid:1) basisvectors. Using this basis, we can write the initial stateas | Ω b i = N X k =0 (cid:18) (cid:19) N s(cid:18) Nk (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) N − k (cid:29) . (26)Applying the time-evolution operator e − i ˆ H t only gives aphase factor for each term. The Loschmidt amplitude iswritten as L ( t ) = N X k =0 (cid:18) (cid:19) N (cid:18) Nk (cid:19) × exp ( i " JN (cid:18) N − k (cid:19) + h (cid:18) N − k (cid:19) t ) . (27)Using an approximation valid for N ≫ (cid:18) (cid:19) N (cid:18) Nk (cid:19) ∼ r πN exp " − N (cid:18) k − N (cid:19) , (28)we write the amplitude as L ( t ) ∼ r πN X k = − N , − N +1 ,..., N exp (cid:20) − N (1 − iJt ) k − ihtk (cid:21) . (29)For N ≫ k = −∞ to ∞ to yield L ( t ) ∼ r πN ∞ X n = −∞ Z ∞−∞ dφ × exp (cid:20) − N (1 − iJt ) φ − ihtφ + 2 iπnφ (cid:21) = r − iJt ∞ X n = −∞ exp (cid:20) − N ( ht − πn ) − iJt ) (cid:21) , (30)where the Poisson summation formula is used in thefirst line . We then define n that minimizes f r ( t ; n ) as n ∗ ( t ) = arg min n ∈ Z f r ( t ; n ), where f r ( t ; n ) = ( ht − πn ) J t ) . (31)The contribution from n = n ∗ dominates the sum inEq. (30) and we obtain the real part of the rate functionas f r ( t ) = f r ( t, n ∗ ( t )). f r ( t ) exhibits singularities because n ∗ ( t ) changes discretely as t grows. Thus the transitiontime t c is obtained by equating two neighboring values f r ( t ; n ) = f r ( t ; n + 1) as t c ( n ) = πh (cid:18) n + 12 (cid:19) . (32)Hence the period is given by τ = π/h . The branches of f r ( t ; n ) with n = 0 , , Jt R e ( C ( t )) -1-0.500.51 C C C C Jt I m ( C ( t )) -0.1-0.0500.050.10.150.2 C C C C Jt R e ( f ( t )) × -3 f f f f n=0n=1 n=2 FIG. 3. (Left, Center) The initial condition C ( t ) = η (0; η ( t ) = η b ) satisfying Eq. (13) is plotted against t at Γ i = ∞ , h/J = 0 . f = 0. The left and center panels are for the real and imaginary parts, respectively. Two periods [0 , τ ] and [ τ, τ ] arefocused on, and two branches in each period are displayed and the vertical straight line denotes the period τ = π/h . Theperiod τ is given by Jτ = 10 π in the present case. (Right) The real part of the rate function f r ( t ) corresponding to those fourbranches are shown. The analytical solution (31) is shown by the dashed lines. The actual rate function follows the smallestbranch at each t as in the upper left panel of Fig. 10. Fig. 3, which exhibits the perfect agreement with f k ( t )evaluated by the integration in Eq. (21) with the solutionof EOMs (22) and C k ( t ).Two noteworthy consequences are provided by this an-alytical solution. One is that the DQPT always exists forany h >
0, while it does not for h = 0. Some earlier workshave pointed out that a DQPT appears when quenchcrosses an equilibrium quantum phase transition . Thepresent results reveal the existence of the opposite situa-tion. The other is that the real part of the rate function f r ( t ) = f r ( t ; n ∗ ( t )) shrinks by the speed of O ( t − ) as t grows and finally vanishes in the limit t → ∞ . The van-ishing rate function may be thought to imply |L ( t ) | → / √ − iJt in Eq. (30). The modulus of this factordecreases as t grows, so that |L ( t ) | goes to zero. This im-plies that there exists a crossover time t × determined bycomparing the O (1) factor and the exponentially scal-ing one e − Nf . For t > t × the O (1) factor dominatesthe Loschmidt amplitude. However, the crossover time t × is expected to a unbounded increasing function of N .Hence in the large size limit our computation of the ratefunction is meaningful in the whole time region.
2. General Γ f > Let us proceed to general final values Γ f >
0. The ana-lytical solution of the EOMs is not available in this case.Hence we numerically search the initial conditions C ( t )and C ( t ), and evaluate the corresponding rate functions f ( t ) and f ( t ).Our heuristic procedure starts from evaluating the pe-riod τ of the dynamics. For this purpose, we run thenumerical simulation of the EOMs (19) using the naiveinitial condition, ζ (0) = ζ b = 1 and η (0) = η b = 1.We employ the Runge-Kutta method of the fourth or-der. As an example, we show the result for the case with h/J = 0 . f /J = 0 . Jτ ≈ . η (0) = η b = 1 does not satisfy the boundary condi-tion (13) for a generic end time t . Given an end time t , we need to estimate the appropriate initial condition η (0) = C ( t ), and then compute the path (¯ ζ ( s ) , ¯ η ( s )). Asa result, the semiclassical paths satisfying Eq. (13) arevery different from the naive ones. Putting the end timeas Jt = Jτ ≈ .
8, we plot the real parts of such pathsin the center panel of the same figure. As explained inSec. II B 5, we have two different sequences of the ini-tial conditions, yielding two different paths (¯ ζ ( s ) , ¯ η ( s ))and (¯ ζ ( s ) , ¯ η ( s )). Both paths satisfy ¯ ζ (0) = ζ b = 1 and¯ η ( t ) = η b = 1 as they should. The real parts of thecorresponding two rate functions are plotted in the rightpanel. The smaller branch at each time corresponds tothe true rate function, leading to the DQPT at Jt c ≈ . f to f . Note that this panel is plot-ted against the end time t while the center one is plottedagainst the dummy time s , given the end time t = τ .The DQPT observed here has the nature of the first or-der transition, in a sense that the first order time deriva-tive of the rate function jumps at the transition time.By examining the several parameters, we have realizedthat this first order nature tends to be stronger as Γ f in-creases, but suddenly vanishes at a certain critical valueΓ fc ( h ). For Γ > Γ fc ( h ), the curve of the rate function hasa smooth peak without singularity. In Fig. 5, we plot C , C , f , and f for h/J = 0 . f , Γ f /J = 1 . .
6. They clearly showthat the critical value Γ fc ( h ) is present between these twovalues of Γ f . In the same way, computing the rate func-tion in a range of h and Γ f , we draw a phase diagram inthe case of quench from Γ i = ∞ in Fig. 6. The phaseboundary approaches to the equilibrium transition pointΓ c = J in the limit h →
0. This is reasonable because theperiod of the dynamics τ diverges as h → f < Γ c ,and DQPTs do not exist according to the present sce-nario. Jt ¯ ζ ( t ) , ¯ η (t) -0.200.20.40.60.81 Re( ζ )Im( ζ )Im( η ) Js R e ( ¯ ζ ( s )) , R e ( ¯ η ( s )) ¯ ζ ¯ η ¯ ζ ¯ η Jt R e ( f ( t )) f f FIG. 4. Semiclassical paths and the rate functions at Γ i = ∞ , h/J = 0 .
1, and Γ f /J = 0 .
6. (Left) The paths with theinitial condition ζ (0) = ζ b and η (0) = η b . The period Jτ ≈ . η is omitted because ℜ{ η ( s ) } = ℜ{ ζ ( s ) } . (Center) Given the end time t = τ , the real parts of semiclassical paths with the modified initial conditions η (0) = C ( t ) to satisfy Eq. (13) are plotted against the dummy time s . Two different paths corresponding to different initialconditions, C and C , are shown. The real parts of ζ and η are identical and are overlapping. As a guide to the eye, twohorizontal straight lines are drawn at unity and zero. (Right) Two branches of the rate function f ( t ) and f ( t ). A DQPToccurs around Jt c ≈ . Jt C ( t ) -1-0.500.511.5 Re(C )Im(C )Re(C )Im(C ) Jt R e ( f ( t )) f f Jt C ( t ) -0.200.20.40.60.81 Re(C )Im(C )Re(C )Im(C ) Jt R e ( f ( t )) FIG. 5. Plot of the initial conditions C and C (Left) and thecorresponding rate functions f and f (Right) at Γ i = ∞ and h/J = 0 . f /J = 1 . f /J = 1 . f /J = 1 .
5, two different branches exist and a DQPToccurs at Jt c ≈ .
76, while they are merged and only oneanalytic curve is present for Γ f /J = 1 . B. Quench from Γ i = 0 We next study the opposite quench from Γ i = 0. Theboundary condition is now given by ζ b = η b = 0.As in the previous case, the numerical search of C ( t )and C ( t ) brings the behavior of the rate function andthe DQPT in this case. However the results are ratherdifferent. In the previous case, the DQPT was the firstorder like and there was a prominent cusp in a period[0 , τ ]. When going across the DQPT boundary, the cuspturned into a smooth peak and the bifurcation or mergeof the two initial conditions C ( t ) and C ( t ) occurs in the h/J Γ f / J Γ i = ∞ No DQPTDQPT
FIG. 6. The phase diagram for the quench from Γ i = ∞ . Thephase boundary line is the interpolation of the data points middle of the period [0 , τ ]. For the quench from Γ i = 0,however, this is not the case and the DQPT emerges ina more delicate form.Figure 7 is the plots of C , C , f , and f for h/J = 0 . f , Γ f /J = 0 . .
7. This figure demonstrates that the bifurcation ofthe two initial conditions C ( t ) and C ( t ) occurs around t ≈ τ in a rather continuous manner. As a result, dis-criminating the two branches of the solution is harderthan the quench from Γ i = ∞ . This tendency holdsfor the range of h and Γ f we have searched, which re-quires us to conduct a more precise numerics to obtainthe phase diagram. Moreover, as we see from the bot-tom panels (Γ f /J = 0 . C ( t ) tends to show a rathersingular behavior: a smooth curve suddenly changes intoa plateau as t decreases and finally it vanishes for small t . Although we cannot completely reject a possibilitythat these behaviors are caused by certain numerical er-rors, we have carefully checked and confirmed that the Jt C ( t ) -0.200.20.40.6 Re(C )Im(C )Re(C )Im(C ) Jt R e ( f ( t )) Jt C ( t ) -0.200.20.4 Re(C )Im(C )Re(C )Im(C ) Jt R e ( f ( t )) Re(f (t))Re(f (t)) FIG. 7. Plot of the initial conditions C and C (Left) and thecorresponding rate functions f and f (Right) at Γ i = 0 and h/J = 0 . f /J = 0 . f /J = 0 . τ of the dynamics is Jτ ≈ . Jτ ≈ . f /J = 0 . f /J = 0 .
7, respectively. ComparingΓ f /J = 0 . .
7, we can see a new branch emerges around t ≈ τ , which gives a DQPT at a very close time to τ . Forthe bottom right panel, f r coming from the plateau region of C ( t ) is out of the range in the shown scale, meaning that itis irrelevant for the DQPT. h/J Γ f / J Γ i =0 No DQPTDQPT
FIG. 8. The phase diagram for the quench from Γ i = 0. shown C ( t ) satisfies the boundary condition in a goodprecision for the intermediate and large t region, and nobranches exists continuously connected to the plateau forsmall t . Hence, these singular behaviors are expected tobe true. Fortunately, they are irrelevant for locating theDQPT point since the DQPT occurs at larger t where nopathological behavior appears. The resultant phase dia-gram for Γ i = 0 is given in Fig. 8. The phase boundaryΓ fc ( h ) approaches to Γ d /J = 1 / h → d is the dynamical transitionpoint of an order parameter, m z = h Ψ( t ) | σ z | Ψ( t ) i where | Ψ( t ) i = e − it ˆ H | Ω b i , with this particular choice of Γ i22 .Upon approaching to Γ d , the period τ of the dynam-ics diverges and DQPTs should vanish. Note that this dynamical value Γ d does not have any meaning for theequilibrium transition. This is in contrast to the quenchfrom Γ i = ∞ where the equilibrium transition point Γ c works as the DQPT transition point at h = 0.Unlike in the Γ i = ∞ case, the dynamics does not stopeven at h = 0. This enables us to see an interestingbehavior of the Loschmidt amplitude at h = 0 and Γ f =Γ d . This point is on the separatrix in the phase spaceand the order parameter monotonically decreases as t grows. No periodicity exists (or τ = ∞ ). Hence, we onlyexamine the first sequence of the initial condition for η ( s )and C ( t ), and compute the corresponding rate function.The result is shown in Fig. 9. This figure shows that therate function asymptotically vanishes as t → ∞ , but thisdoes not necessarily imply |L ( t ) | → C. Comparison with numerical experiments
To validate our semiclassical computations, we hereshow the results of numerical experiments and comparethem with the semiclassical results for several parame-ters. Our Hamiltonian (1) commutes with the squaredtotal spin operator ˆ S = ˆ S x + ˆ S y + ˆ S z . For both thequenches from Γ i = 0 and ∞ , the initial state is inthe subspace of the total spin S = N/
2. Hence thestate of our system preserves this total spin and wemay consider the time-dependent state inside this sub-space. In the basis of the eigenvalues of ˆ S z , our Hamil-tonian is represented in a tridiagonal matrix form andwe can easily evaluate the time evolution of the state | Ψ( t ) i = e − it ˆ H | Ω b i by the LU decomposition. The di-mension of the subspace is N + 1 and we can treat fairlylarge size systems. However, the computation requiresus to take a lot of sums of complex numbers and thenumerical precision tends to be degraded as N becomeslarge. This computational difficulty sensitively dependson the parameters and below the simulated system sizesare adaptively changed for this reason.Figure 10 is the plots of the rate functions for thequench from Γ i = ∞ . The results of numerical simu-lation show a good agreement with the theoretical curvedenoted by the solid black line, both below and abovethe transition point Γ fc ( h/J = 0 . /J ≈ .
53. Thisjustifies our semiclassical computation. The upper leftpanel in Fig. 10 for Γ f = 0 and h/J = 0 . Jτ = πJh ≈ .
4. We see the consistent agreementbetween the numerical and semiclassical computations.The deviation for the whole time and the oscillating be-havior at large t are considered to be due to the finitesize effect.Figure 11 represents the result of a quench from Γ i = 0at h = 0+. Again, the numerical results show a fairlygood agreement with the semiclassical curve. At the sep-aratrix, Γ f /J = 1 /
2, the monotonic decay of the ratefunction after a single peak is well reproduced by the nu-0 Jt C -0.100.10.20.30.4 Re(C )Im(C ) Jt R e ( f ( t )) Jt m z ( t ) FIG. 9. Plot of the initial conditions C (Left) and the corresponding rate function f (Center) at Γ i = 0 and h = 0+ withΓ f /J = Γ d /J = 1 /
2. The dynamics on the separatrix is not periodic, which is demonstrated by the right panel plotting themagnetization m z ( t ) = h Ψ( t ) | σ z | Ψ( t ) i computed by the semiclassical method in Ref. . Jt R e ( f ( t )) × -3 h=0.1, Γ i = ∞ , Γ f =0 N=800N=1000N=2000N=4000 Jt R e ( f ( t )) h=0.1, Γ i = ∞ , Γ f =0.5 N=600N=800N=1000N=2000 Jt R e ( f ( t )) h=0.1, Γ i = ∞ , Γ f =1 N=200N=400N=600N=700 Jt R e ( f ( t )) × -3 h=0.1, Γ i = ∞ , Γ f =2 N=600N=800N=1000N=2000
FIG. 10. The real part of the rate function for Γ f /J = 0 (Up-per left), 0 . . . h/J = 0 . i = ∞ . The threepanels except for the lower right one show the DQPT, whichis in agreement with the semiclassical computation given bythe black solid line. merics, validating our semiclassical computation even ata special point of the dynamics. IV. DISCUSSION AND SUMMARY
In this paper, we have invented a computationalmethod for the Loschmidt amplitude based on the com-plex semiclassical approach, and applied it to the trans-verse field Ising model with a symmetry breaking fieldin the infinite dimension. Two quantum quenches, fromzero and infinite transverse fields, have been examined.From the behavior of the rate function, the presence orabsence of the DQPTs have been captured. The phasediagrams have been mapped out in the plane of finaltransverse field and symmetry breaking field. These re-sults have been examined by numerical simulations inde- Jt R e ( f ( t )) h=0, Γ i =0, Γ f =0.25 N=100N=200N=300N=400 Jt R e ( f ( t )) h=0, Γ i =0, Γ f =0.5 N=200N=400N=500N=600 Jt R e ( f ( t )) h=0, Γ i =0, Γ f =0.75 N=100N=200N=300N=400 Jt R e ( f ( t )) h=0, Γ i =0, Γ f =1.25 N=100N=200N=300N=400
FIG. 11. The real part of the rate function for Γ f /J = 0 . . .
75 (Lower left), and 1 . h = 0+ for the quench from Γ i = 0. Theupper two panels do not show any DQPT while the lowerones do. The upper right panel is for the separatrix and thecorresponding rate function shows a monotonic decay after asmooth peak. pendently that solve the Schr¨odinger equation literally,which fully supports our semiclassical computations.Although our computational method has succeeded inunveiling several properties of the Loschmidt amplitude,its physical implications are still unclear. ˇZunkoviˇc etal. have pointed a connection between the Loschmidtamplitude and the order parameter in the steady statelong after the quench . However we have not found sucha connection as far as the quench from Γ i = ∞ is con-cerned. Therefore the presented result might add a fur-ther mystery on the DQPT. Disentangling DQPTs of theLoschmidt amplitude and an order parameter may opena new comprehension on quantum dynamics.An experimental observation of a DQPT is a fascinat-ing topic. A very recent work has actually observedDQPTs using a certain topological nature of the singu-1larity . Unfortunately, this is possible only in non-interacting systems and its generalization to interactingsystems is unclear. Although there are some other ex-periments observing the Loschmidt amplitude, theirmethods rely on the smallness of the system or certainlocality of the phenomena. The application of their meth-ods to global phenomena in many-spin systems is againnontrivial. Our model, the Ising model with long rangeinteractions, itself can be realized in a trapped ion sys-tem . Another recent experiment on this system hasobserved nontrivial cusps in the probability to returnto the ground-state manifold, giving a clear evidence ofthe DQPT . Their setup corresponds to Γ i = 0 and h = 0+ in the present paper, and we expect that fur-ther nontrivial results can be obtained in other setupsaccording to our findings. Such additional experimentsare encouraged.A more direct application of our method might befound in quantum engineering or computing. In those disciplines, it is an important problem to estimate theprobability achieving a desired state in certain quantumprocesses. For example in quantum annealing , theprobability to find the ground state is an important ob-ject to be calculated. Using techniques from the spinglass theory combined with the present method, itstypical value might be evaluated. This will provide atheoretical challenge for both quantum mechanics andrandom spin systems. ACKNOWLEDGEMENTS
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