Characterizing the excited-state quantum phase transition via the dynamical and statistical properties of the diagonal entropy
aa r X i v : . [ qu a n t - ph ] A ug Characterizing the excited-state quantum phase transition via the dynamical andstatistical properties of the diagonal entropy
Qian Wang
Department of Physics, Zhejiang Normal University, Jinhua 321004, China andCAMTP-Center for Applied Mathematics and Theoretical Physics,University of Maribor, Mladinska 3, SI-2000 Maribor, Slovenia
Francisco P´erez-Bernal
Departamento de Ciencias Integradas y Centro de Estudios Avanzados en F´ısica,Matem´aticas y Computaci´on, Universidad de Huelva, Huelva 21071, Spain andInstituto Carlos I de F´ısica Te´orica y Computacional, Universidad de Granada, Granada 18071, Spain (Dated: August 21, 2020)Using the diagonal entropy we analyze the dynamical signatures of the excited-state quantumphase transitions (ESQPT) in the Lipkin-Meshkov-Glick (LMG) model. We first show that thetime evolution of the diagonal entropy behaves as an efficient indicator of the presence of ESQPT.We further consider the diagonal entropy as a random variable over a certain time interval and focuson the statistical properties of the diagonal entropy. We find that the probability distribution of thediagonal entropy provides a clear distinction between different phases of ESQPT. In particular, weobserve that at the critical point the probability distribution of the diagonal entropy has a universalform, which can be well described by the so-called beta distribution. We finally demonstrate thecentral moments of the diagonal entropy can reliably detect the critical point of ESQPT.
I. INTRODUCTION
The notion of excited-state quantum phase transition(ESQPT) [1, 2] was first introduced to describe the non-analytical properties in excited states of quantum sys-tems and was soon identified, both theoretically [3–9] andexperimentally [10–13], in various many-body systems.As the generalization of the ground-state quantum phasetransitions (QPTs) [14, 15], ESQPTs are manifested bythe singularity in the density of states at the criticalenergy, for fixed Hamiltonian parameters. It has beenfound that ESQPT plays an important role in severalcontexts including quantum decoherence process [16–18],quantum chaos [19–22], and quantum thermodynamics[23, 24]. Many efforts have been devoted to understandthe intriguing properties, both statical [25–31] and dy-namical [32–39], of this new kind phase transition.With the progress of the experimental technologies, thestudy of nonequilibrium dynamics of isolated quantumsystems has attracted upsurge of interest in the past fewyears [40, 41]. Along this direction, it is natural and im-portant to explore the effects produced in the nonequi-librium dynamics of an isolated system by the ESQPT.Several remarkable dynamical effects of ESQPT, such asenhance the decay of the survival probability [36–39, 42],the exponential growth of the out-of-time-order correla-tors [43], and the singularities in the time evolution ofobservables [33], are revealed. Moreover, the investiga-tion of how to dynamically probe ESQPTs is also underan active development [18, 32, 34, 39] and implies thepossible experimental exploration of the dynamical in-fluences of ESQPT in quantum isolated many-body sys-tems. In spite of these numerous works, many aspectsof the dynamical signatures of ESQPT are still remainedopen. More works are required in order to get a deeper understanding of the properties of ESQPTs.In this work, we consider the Lipkin-Meshkov-Glick(LMG) model [44] and study the dynamical features ofan ESQPT by means of the diagonal entropy. As theShannon information entropy of the probability distribu-tion corresponding to the energy eigenstates, the diag-onal entropy is defined as S d = − P n ρ nn ln ρ nn , where ρ nn are the diagonal elements of the density matrix ρ in the energy eigenstates [45]. The diagonal entropy ex-hibits most of the properties of a thermodynamic entropy,including additivity, conserved in the adiabatic processand increases when an equilibrium system is taken out ofequilibrium. Hence, it is an appropriate entropy for thestudies of the nonequilibrium dynamics in an isolatedquantum systems and has been employed in diverse ar-eas of physics [46–50]. Moreover, S d is consistent withthe well-known von Neumann’s entropy for systems inequilibrium. It is also worth mentioning that since thediagonal entropy only involves the diagonal part of thedensity matrix, it can be experimentally measured in anefficient way [50].We first focus on the time evolution of the diagonalentropy following a cyclic quench and show that the ES-QPT can be well detected by the distinct behaviors in thetime evolution of diagonal entropy. Then by consideringthe diagonal entropy as a random variable over a cer-tain time interval, we further investigate the statisticalproperties of the diagonal entropy. We will demonstratehow the underlying ESQPT is imprinted on the featuresof the probability distribution of the diagonal entropy.In particular, at the critical point of ESQPT, we showthat the probability distribution of the diagonal entropyfollows a universal form, independent of the system sizeand the Hamiltonian parameters. Additionally, we illus-trate this universal distribution is in good agreement withthe so-called beta distribution and show that the centralmoments of the diagonal entropy are able to detect thepresence of ESQPT.The article is structured as follows. In Sec. II, we de-scribe the protocol used in this work and introduce themodel, briefly review some of its properties. In Sec. III,we present our main results and discuss how the signa-tures of ESQPT can be identified in the dynamics of thediagonal entropy as well as its statistical properties. Fi-nally, we summarize the main conclusions of this work inSec. IV. II. PROTOCOL AND MODELA. Protocol and diagonal entropy
Assuming the system is described by a Hamiltonian H ( g ) with g being the control parameter of the sys-tem. We consider a cycle protocol by suddenly changingthe control parameter at different times. As depicted inFig. 1(a), the protocol consists of the following processes.(i) Initially, the system is in the state ρ i = | ψ in ih ψ in | with the initial control parameter g i and Hamiltonian H i . Here, | ψ in i is the n th eigenstate of H i with eigen-value E in . (ii) At time t = 0, the control parameter g issuddenly changed (quenched) from the initial value g i toa final value g f . Then the Hamiltonian of the system isvaried to a new one H f with eigenstates | ψ fn i and eigen-values E fn . The dynamics of the system is, therefore,governed by the Hamiltonian H f . (iii) At time t = τ ,the system undergoes another quench, which changes thecontrol parameter from g f to its initial value g i , complet-ing the cycle protocol, and let the system evolves under H i for t ≥ τ .The state of the system at t = τ is given by ρ τ = e − iH f τ ρ i e iH f τ . Then the diagonal entropy in the eigen-states of the final Hamiltonian H i can be written as S d ( τ ) = − X k C k ( τ ) ln C k ( τ ) , (1)where C k ( τ ) = |h ψ ik | e − iH f τ | ψ in i| , with | ψ ik i is the k theigenstates of H i . The diagonal entropy has been ar-gued to fulfill the second law of the thermodynamics,namely, it grows when an equilibrium system is takenout of equilibrium and it reaches a constant value at theequilibration time scale. Note that C k ( τ ) reduces to thewell-known survival probability when we take k = n and P k C k ( τ ) ≡ τ and H f .It is known that the non-linear dependencies of S d ( τ )on the density matrix results in the difference betweenthe long-time averaged diagonal entropy, denoted by S d ( τ ), and the diagonal entropy of the long-time averagedstate, which we denote as hS d i [47, 51]. For an initial purestate, it has been conjectured that the deviation between hS d i and S d ( τ ) satisfies ∆ = hS d i − S d ( τ ) ≤ − γ , where γ = 0 . . . . is the Euler’s constant [51]. As ∆ has the FIG. 1. (a) The quench protocol studied in this work, as de-scribed in the text. (b) Schematic representation of the LMGmodel. Spins are fully connected through an infinite rangecoupling and in an external magnetic field with strength α along the z direction. (c) Rescaled even-parity energy spec-trum of the Lipkin model as a function of α with N = 50 (leftpanel) and the rescaled density of states of the LMG modelfor α = 0 . N = 5000 (right panel). The green dotsdenote the numerical results, while the solid line is obtainedvia Eq. (3). All quantities are dimensionless.FIG. 2. Time evolution of S d ( τ ) for different values of λ with α = 0 . N = 1000. The inset in each panel showsthe long time behavior of S d ( τ ) with τ changes from 10 to1 . × for the corresponding λ . The axes in all figures aredimensionless. minimum fluctuations when the system in equilibrium, ithas been employed to study the connection between therelaxation process and the transition from integrabilityand chaos in various quantum systems [47, 48]. Here toexplore the signatures of ESQPT in nonequilibrium dy-namics of the quantum isolated system, we investigatethe dynamical and statistical properties of diagonal en-tropy of the LMG model, in which the above mentionedcycle protocol is implemented. B. Lipkin-Meshkov-Glick (LMG) model
As the transverse Ising model with infinite-range in-teractions, the LMG model describes N fully connectingparticles of 1 / α , see Fig. 1(b) for a schematic repre-sentation of the LMG model. It was originally introducedas a toy model in nuclear physics [44], but nowadays itwas studied in many areas of physics [23, 43, 52–56] andwas realized in different experimental platforms with highprecision [57–60]. In particular, it has been used as aparadigmatic model in the studies of excited-state quan-tum phase transitions [2, 16–18, 24, 25, 34, 38].Employing the collective spin operators J β = P l σ lβ / β = { x, y, z } and σ β are the Pauli spin matrices, theHamiltonian of the LMG model can be written as H = − − α ) N J x + α (cid:18) J z + N (cid:19) , (2)where N is the total number of spins, α ∈ [0 ,
1] is thestrength of the magnetic field along the z direction. TheHamiltonian in Eq. (2) conserves the total spin J = J x + J y + J z , whose eigenvalues are j ( j +1) with 0 ≤ j ≤ N/ j = N/
2, with dimension N + 1.Moreover, as the parity operator Π = e iπ ( J z + j ) is alsocommutated with H , the Hamiltonian matrix in j = N/ N/ N/
2. We further restrict to even parity block,which includes the ground state.The elements of the Hamiltonian matrix in the basis | j, j z i , which are the eigenstates of J z and with − N/ ≤ j z ≤ N/
2, are given by h j, j z | H | j, j z i = q α (cid:18) N j z (cid:19) + α − , h j, j z + 2 | H | j, j z i = − − αN s(cid:18) N − j z − (cid:19) × s(cid:18) N − j z (cid:19) (cid:18) N j z + 1 (cid:19) (cid:18) N j z + 2 (cid:19) , where q α = [2(1 − α ) j z /N ] + 2 α −
1. For simplicity, weconsider ~ = 1 throughout this work and set the quanti-ties studied in this article as dimensionless.The LMG model in Eq. (2) undergoes a second-orderquantum phase transition at the critical point α c = 0 . α <α c , while it belongs to the symmetric phase as α > α c .Another remarkable feature that exhibited by the LMGmodel is the ESQPT for α < α c [2, 16, 17]. The excited-state quantum phase transitions are characterized by theconcentrating of the eigenvalues at the critical energy E c .For the LMG model, this is illustrated in the left panel inFig. 1(c), where we see that the energy levels are pilingin the neighborhood of E = 0 and thus we have E c = 0. The cluster of the eigenvalues at E c = 0 leads to acusp singularity in the density of states ν ( E ) defined as ν ( E ) = P n δ ( E − E n ). In the semiclassical limit N → ∞ , ν ( E ) can be analytically calculated as [17, 18] ν ( E ) = N π Z δ [ E − H cl ( x, p )] dxdp, (3)where H cl is the classical counterpart of H in Eq. (2).The right panel of Fig. 1(c) plots the density of statesfor the case of α = 0 . N = 5000. We observe that ν ( E ) obtained by means of Eq. (3) matches very well thenumerical data and it shows a clearly cusp divergence at E c = 0, as expected. This further verifies the existenceof the ESQPT at E c = 0 in LMG model when α < α c .In the following, we focus on the identification of thesignatures of this ESQPT in the dynamical and statisticalproperties of the diagonal entropy. III. DYNAMICS AND STATISTICS OF THEDIAGONAL ENTROPY
Below we first focus on the dynamics of the diago-nal entropy S d ( τ ), and then consider S d ( τ ) as a uni-formly distributed random variable with τ ≥
0. We aremainly interested in how the signatures of ESQPT man-ifest themselves in the time evolution of S d ( t ) and theprobability distribution of S d ( t ), as well as the momentsof this distribution.In our study, the above described cycle protocol isachieved as follows. Initially, the system is at the groundstate | ψ i of the Hamiltonian (2) with H i = H , g i = 0and ρ i = | ψ ih ψ | . At time t = 0, we turn on an externalmagnetic field along the z direction with strength λ . Wethus have g f = λ and H f = H + λ ( J z + N/ t = τ toback to the starting point, completing the closed cycle.The diagonal entropy at time t = τ , S d ( τ ), is given byEq. (1) with C k ( τ ) = |h ψ k | e − iH f τ | ψ i| = (cid:12)(cid:12)(cid:12)(cid:12)Z dE Ω k ( E ) e iEτ (cid:12)(cid:12)(cid:12)(cid:12) . (4)Here, | ψ k i is the k th eigenstate of H in Eq. (2) andΩ k ( E ) = X m h ψ k | ψ fm ih ψ fm | ψ i δ ( E − E fm ) , (5)with | ψ fm i denotes the m th eigenstate of H f correspond-ing to the eigenvalue E fm .The system can drive through the critical energy ofESQPT by varying the strength of the external magneticfield. When the system initially in the ground state, thecritical strength, denoted by λ αc , which makes the systemreach the critical energy E c = 0, of the LMG model canbe obtained using the semiclassical approach with resultreads [16, 17] λ αc = 12 (4 − α ) . (6) FIG. 3. Panels (a)-(c): Ω k ( E ) as a function of the rescaled energy ε fm = ( E fm − E f ) / ( E fmax − E f ) for several values of λ with k increases from 0 (ground state) to 20 in steps of 2, marked by different colors. Here, E f is the ground state energyof H f , while E fmax denotes the maximum energy of H f . To offer the three-dimensional-like visualization, Ω k ( E ) are shiftedin the y -direction by 2 − (1+ k/ k (a), 2 − (1+ k/ k (b), and 2 − k/ k (c). Also, Ω k ( E ) in each panel have been multiplied by2 (a), 45 (b) and 45 (c). The green dashed line in panel (b) indicates the rescaled critical energy of ESQPT. Panels (d)-(f):Heat mapping depicting ln[ C k ( τ )] as a function of k and τ for the same values of λ as in panels (a)-(c). White color indicates C k ( τ ) = 0. The other parameters are: α = 0 . N = 1000. All quantities are dimensionless. Here, α satisfies 0 < α < /
5. We would like to point outthat the critical strength, λ αc , of the ESQPT differs fromthe critical strength, λ αc , for the ground state quantumphase transition [17]. A. Dynamical behaviors of S d ( τ ) As the start point, we first investigate the signaturesof ESQPT in the dynamics of the diagonal entropy. InFig. 2, we have plotted the time evolution of S d ( τ ) fordifferent values of λ with α = 0 . N = 1000. In thiscase, according to Eq. (6), we have λ αc = 1. We first notethat the behavior of S d ( τ ) as a function of τ dependsstrongly on the value of λ . Specifically, for λ < λ αc , S d ( τ )periodically oscillates around a small value, as shown inFig. 2(a). Increasing λ leads to increase in S d ( τ ). Inaddition, the regular oscillations in S d ( τ ) are graduallychanging to an irregular pattern as λ increases. At λ = λ αc = 1, as can be seen from Fig. 2(b), S d ( τ ) shows a fastgrowth which then rapidly saturates to a greater valuewith very small fluctuations. Above the critical pointwith λ = 2 [see Fig. 2(c)], we observe that S d ( τ ) increasesslowly with time and then irregularly oscillates around asteady value with larger amplitude.These observed features in the dynamics of the diago- nal entropy indicate that the underlying ESQPT of thesystem has strong impact on the equilibration process ofthe system after a quench. On the other hand, these fea-tures would enable us to detect the occurring of ESQPTin the system through the singular behavior of S d ( τ ) at λ = λ αc . Moreover, different phases of an ESQPT canalso be identified by the distinct behaviors of the diago-nal entropy at λ < λ αc and λ > λ αc , respectively.To understand the features exhibited by S d ( τ ), as in-dicated in Eq. (4), we note that C k ( τ ) is the squaremodulus of the Fourier transform of Ω k ( E ) defined inEq. (5). Therefore, the remarkable distinct behaviors in S d ( τ ) originate from the fact that the properties of Ω k ( E )have a dramatic change as the system passes through theESQPT. The particular behavior of S d ( τ ) at the criticalpoint of ESQPT is rooted in the singular structure inΩ k ( E ). To make this statement lucid, we plot Ω k ( E )and the corresponding C k ( τ ) for different λ with α = 0 . N = 1000 in Fig. 3.For the case of λ = 0 . < λ αc , as shown in Fig. 3(a),Ω k ( E ) are rather localized at the low eigenenergies of H f and the main contribution comes from the the states with k ≤
3. This simply structure in Ω k ( E ) gives rise to theregular oscillations in time evolution of C k ( τ ) for small k and C k ( τ ) = 0 for greater values of k , as is evidentfrom Fig. 3(d). This means that the time dependence of FIG. 4. Panels (a)-(c): Probability distributions of the diagonal entropy for λ = 0 . λ = λ αc = 1 . λ = 2 (c)with α = 0 .
2. Panels (d)-(f): Probability distribution P ( S d ) for λ = 0 . λ = λ αc = 1 (b), and λ = 2 with α = 0 .
4. In allcases N = 1000. Red solid line in each panel denotes the corresponding fitted beta distribution in Eq. (9) with fitting param-eters ( a, b, S , S m ) are given by (a) (1 . , . , , . . , . , . , . . , . , . , . . , . , , . . , . , . , . . , . , . , . R as a function of λ for different system sizes N with α = 0 . α = 0 . R as afunction of system size N with α = 0 . λ αc = 1 .
5, whilethe inset in (b) plots R as a function of N with α = 0 . λ αc = 1 .
0. The axes in all figures are dimensionless. S d ( τ ) behaves as a periodical function, see Fig. 2(a). As λ increases, the non-zero contribution to Ω k ( E ) involvesmore and more states with greater value of k , indicatingthe increases in S d ( τ ). At the critical point λ = λ αc =1, we can see from Fig. 3(b) that the greater the valueof k is, the complexity of Ω k ( E ) is. In this case, onremarkable feature of Ω k ( E ) is the cusp-like shape nearthe critical energy (marked by the green dashed line) of FIG. 6. Probability distribution of the shifted and rescaleddiagonal entropy, i. e., ( S d − S d ) / √ Σ, at different λ αc with sev-eral system sizes N . Here, S d denotes the averaged S d and Σ isthe variance of S d . The cyan solid line denotes the fitted betadistribution [cf. Eq. (9)] with fitting parameters ( a, b, S , S m )given by (22 . , . , − , R be-tween P [( S d − S d ) / √ Σ] and the fitted beta distribution as afunction of λ αc for different system sizes N . The axes in allfigures are dimensionless. ESQPT, regardless of the value of k . It is known that thesame structure in Ω ( E ) leads to the survival probability, C ( τ ), shows a fast decay which then followed by therandom oscillations with tiny amplitude [36]. Here, we FIG. 7. Second, third, and fourth central moments (as la-belled) of P ( S d ) as a function of λ for 0 . ≤ α ≤ . N = 500 (left column) and N = 1000 (rigth col-umn). The vertical dashed lines in each panel denote thecritical values λ αc for each corresponding α . The axes in allfigures are dimensionless. find that the cusps in Ω k ( E ) with k > C k ( τ ), asillustrated in Fig. 3(e). Therefore, the behavior of S d ( τ )at λ = λ αc can be traced back to the cusp-like structuresin Ω k ( E ) at the critical energy of ESQPT. When λ =2 > λ αc , the structures of Ω k ( E ) at small values of k arevery regular, whereas increasing k leads the structures ofΩ k ( E ) become more and more complexity, as observed inFig. 3(d). As a consequence, the initial regular behaviorin C k ( τ ) is followed by small irregular oscillations with C k ( τ ) ≈ S d ( t ) is slow at the initial time for the case of λ = 2[see Fig. 2(c)].These results allow us to draw a conclusion that theunderlying ESQPT in the system has significant impacton the quench induced equilibration process and the timedependent behavior of the diagonal entropy can reliablydistinguish the different phases of ESQPT. In addition,at the critical point, the particular dynamical behaviorof the diagonal entropy acts as a good indicator of thepresence of ESQPT. B. Statistical properties of S d ( τ ) To gain further insights into the signatures of ES-QPT in the quench induced nonequilibrium dynamics ofa quantum many-body system, in this subsection we ex-plore the statistical properties of the diagonal entropy.We will consider S d ( τ ) as a random variable which obeysthe uniform distribution. Then, for the time window[ τ , τ + ∆ τ ], the probability distribution function of thediagonal entropy is defined as P ( S d ) = lim ∆ τ →∞ τ Z τ +∆ ττ δ [ S d ( τ ) − S d ] dτ, (7)where the value of τ is much larger than the initial timescale. To calculate this distribution, all the the intrica-cies of S d ( τ ) should be captured. This means we need to evolve the system in a long period of time. In our simula-tion, we take τ = ∆ τ = 10 . We have carefully checkedthat the results obtained for larger τ and ∆ τ values arenot change. The cumulative distribution function of S d is given by F ( S d ) = Z S d S P ( x ) dx, (8)where S is the minimal value of the distribution rangeand P ( x ) is the probability distribution function inEq. (7).In Fig. 4, we plot the probability distribution of thediagonal entropy and the corresponding cumulative dis-tribution for α = 0 . α = 0 . N = 1000 at the value of λ is below, at,and above the critical value λ αc . We see that P ( S d ) is adouble-peaked distribution at smaller λ due to the peri-odic oscillations in S d ( τ ). Meanwhile, the small ampli-tude in the oscillations of S d ( τ ) in this case also givesrise to the lower values of S d . The growing in S d ( τ ) withincreasing λ shifts P ( S d ) to higher values of S d . We fur-ther observe that the increase in λ transforms P ( S d ) froma double-peak form to a bell shape with an asymmetricstructure. This comes from the fact that the evolution of S d ( τ ) at long-time has a strong random oscillations forgreater λ .Further understanding of the properties of P ( S d ) canbe obtained by fitting P ( S d ) to the so-called beta distri-bution, defined as [61–63] ϕ B ( x ) = ( x − S ) a − ( S m − x ) b − ( S m − S ) a + b − B ( a, b ) , (9)where S m denotes the maximal value of the distributionrange, a, b are the shape parameters of the distribution,and B ( a, b ) = R u a − (1 − u ) b − du is the beta function.The cumulative distribution function of the beta distri-bution is given byΦ B ( x ) = Z xS ϕ B ( y ) dy, (10)where x is such that S ≤ x ≤ S m .In Fig. 4, the fitted beta distribution in Eq. (9) and itscumulative distribution are, respectively, denoted by thered solid line in the main panel and the red dashed linein the inset for each correspondent case. One can im-mediately identify the obvious deviations between P ( S d )and the beta distribution when the value of λ is far awayfrom the critical value λ αc , see the first and last columnsof Fig. 4. However, at the critical point with λ = λ αc , thebeta distribution agrees extremely well with the numeri-cal results, as illustrated in panels (b) and (e) of Fig. 4.To quantitatively examine the differences between P ( S d )and the beta distribution, we employ the root-square er-ror (RMSE), which measures the deviations between thepredicted values and the observed values [64]. For ourpurpose, we consider the RMSE, denoted by R , betweenthe cumulative distribution function of the diagonal en-tropy and beta distribution, defined as R = s(cid:18) S m − S (cid:19) Z S m S [ F ( z ) − Φ B ( z )] dz, (11)where F ( z ) and Φ B ( z ) are given by Eqs. (8) and (10),respectively.In Fig. 5, we plot the variation of R with increasing λ for different system sizes with several α . As we cansee, R shows an obvious dip at the critical value λ αc , itsminimum value decreases with an increase in N . Thismeans P ( S d ) is in good agreement with the beta distri-bution at the critical point of ESQPT, consistence withthe behavoior of P ( S d ) as a function of λ (see Fig. 4).Moreover, the degree of agreement can be improved byincreasing the system size N . At the critical point, wefurther find that the decrease in R with the system size N is replaced by the tiny fluctuation around a vanish-ingly small value when N > λ αc , as shown in the insets of Fig. 5.A further natural and important question is whetherthe probability distribution of the diagonal entropy, P ( S d ), has an universal form at the critical point of ES-QPT. In what follow, we show that this is indeed in ourcase. To this end, we consider the shifted and rescaleddiagonal entropy, denoted by S d , defined as S d = S d − S d √ Σ , (12)where S d = R dS d P ( S d ) S d is the averaged S d and Σ = R dS d P ( S d )( S d − S d ) is the variance of S d . Then, weinvestigate the probability distribution of S d , P ( S d ), atdifferent critical values of λ αc for several system sizes N .Our numerical results are shown in Fig. 6. We observethat the numerical data for different λ αc and N show anexcellent collapse, indicating that P ( S d ) is the univer-sal distribution for the ESQPT. Moreover, the distribu-tion P ( S d ) can also be well fitted by the beta distribu-tion in Eq. (9), with fitting parameters ( a, b, S , S m ) =(22 . , . , − , P ( S d )and the fitted beta distribution are vanishingly small atdifferent λ αc and almost independent of the system size N , as depicted in the inset of Fig. 6. This further con-firms the universality of P ( S d ) at the critical point ofESQPT. Central moments of P ( S d ) Having studied the probability distribution of the di-agonal entropy we now turn to identify the signatures ofESQPT in the statistical properties of P ( S d ), by inves-tigating the central moments of P ( S d ). The n th centralmoment of P ( S d ) is defined as µ Sn = E [( S d − S d ) n ] = Z + ∞−∞ dS d P ( S d )( S d − S d ) n . (13) FIG. 8. Critical value λ αc , extracted from different centralmoments (as labelled), as a function of α for different systemsizes N . For each central moment, the location of its extremevalue has been identified as the critical value λ αc . The solidline denotes the analytical result, which gives by Eq. (6). Theaxes in the figure are dimensionless. The first central moment µ S is always zero, thus wemainly focus on the moments with n = 2 , ,
4, whichare the variance, skewness, and kurtosis of a distribu-tion, respectively. These central moments provide theinformation about the shape of the distribution.In Fig. 7, we plot µ S , µ S and µ S as a function of λ fordifferent α and system sizes N . We can see that all ofthe considered central moments have the non-analyticalpoints, which arise as the cusps, in the neighborhoodof the critical values λ αc , regardless of the system size.Specifically, the cusps in µ S and µ S corresponds to theirminimum value, which approaches zero as N increases.As the second and fourth central moments measure thefluctuations and the heaviness of the tail of a probabilitydistribution. Hence, the minimal values in the behaviorsof µ S and µ S indicate that P ( S d ) has small fluctuationsand becomes a light-tailed distribution near the criticalpoint of ESQPT, in according with the results observedin panels (b) and (e) of Fig. 4. However, for the thirdcentral moment µ S , we first note that it always less thanzero, independent of the values of α and the system size N . It is known that the third central moment quantifiesthe asymmetry of a distribution. Therefore, the nega-tive values of µ S imply that the area underneath the theleft tail of P ( S d ) is larger than the one under the righttail, as shown in Fig. 4. Around λ = λ αc , the third cen-tral moment µ S undergoes a cusp toward zero, which issharper for larger values of N . This quantitatively meansthat the distribution P ( S d ) for the system at the criticalpoint of ESQPT is more symmetric than if the system faraway form the critical point, as is evident from Figs. 4(b)and 4(e).These observed features in the behaviors of the cen-tral moments suggesting that the critical value of λ canbe numerically extracted from the extreme values of thecentral moments. By identifying the critical point as thelocation of the extreme values in the central moments,we have plotted the numerically estimated λ αc as a func-tion of α in Fig. 8. We also depict the analytical resultof λ αc in Eq. (6). As can be seen from the figure, thenumerical results show a good agreement with the ana-lytical ones, in particular for the results provided by µ S and µ S , respectively. Moreover, the agreement can beenhanced by increasing the system size N . Therefore, weconfirm that the ESQPT has strong effects on the sta-tistical properties of the probability distribution, P ( S d ),of the diagonal entropy. On the other hand, the centralmoments of P ( S d ) can reliably detect the critical pointof ESQPT. IV. CONCLUSION
We have studied in detail the effects of the ESQPTon the dynamics and statistics of the diagonal entropy ina quantum many-body system, which undergoes an ES-QPT at a certain critical energy. We have shown that thediagonal entropy exhibits a distinct change in its dynam-ical behaviors as the system passes through the criticalpoint of ESQPT. The particular behavior in the dynam-ics of the diagonal entropy has been observed at the crit-ical point of ESQPT. Therefore, the existence of ESQPTcan be detected through the calculation of the dynamicsof the diagonal entropy, which also allows us to efficientlydistinguish between the different phases of ESQPT. Tounderstand the different dynamical behaviors of the diag-onal entropy, we have explored the connections betweenthe structures in Ω k ( E ) [cf. Eq. (5)] and the dynamicsof the diagonal entropy. The results indicated that thedifferences in the time evolution of the diagonal entropyresulting from the structure changes in Ω k ( E ) and itsnontrivial cusp structures at the critical point lead tothe particular dynamics of the diagonal entropy.The features observed in the dynamics of the diagonalentropy imply that the ESQPT has nontrivial impactson the probability distribution of the diagonal entropy.We have demonstrated that the distribution of the diag-onal entropy transforms from the double-peak form to anasymmetrical bell shape when the system crosses the ES-QPT. In particular, we have found that the distributionof the diagonal entropy can be well described by the so- called beta distribution at the critical point of ESQPT.Hence, the distribution of the diagonal entropy acts as auseful tool to discover the ESQPT. One intriguing andremarkable result in our study is the universal behav-ior exhibited by the distribution of the diagonal entropyat the critical point of ESQPT. We confirmed that thisuniversal distribution of the diagonal entropy indepen-dent of the system parameter and size, and also in goodagreement with the beta distribution. Additionally, toexamine more thoroughly the effects of the ESQPT onthe statistical properties of the diagonal entropy, we haveanalyzed the central moments of the distribution of thediagonal entropy. Our analyses have suggested that thenonanalytical behaviors in the central moments can beemployed as reliable tools to identify the critical point ofESQPT.At the critical point of ESQPT, the universality distri-bution of the diagonal entropy stems from its universaldynamical behavior, which further originates from thecusps in the structure of Ω k ( E ). Since the same cuspshave been found in various ESQPTs [36, 42], we wouldexpect that our results still hold in other quantum many-body systems, such as Dicke model [3], kicked-top model[5], and Rabi model [6]. A very interesting topic forfurther work is to systematically studying the statisti-cal properties of the diagonal entropy in different many-body systems. We hope that our results may help toget a deeper understanding of the properties of ESQPTand shed lights on the effects of ESQPT on the nonequi-librium dynamics in quantum systems. Finally, sincethe measurement of diagonal entropy is quite efficientin quantum simulators [50], we believe that the obtainedresults can be experimentally verified. ACKNOWLEDGMENTS
Q. W. acknowledges support from the National ScienceFoundation of China under Grant No. 11805165, Zhe-jiang Provincial Nature Science Foundation under GrantNo. LY20A050001, and the Slovenian Research Agency(ARRS) under the Grants No. J1-9112 and No. P1-0306. [1] P. Cejnar, M. Macek, S. Heinze, J. Jolie, and J. Dobeˇs,J. Phys. A: Math. Gen. , L515 (2006).[2] M. Caprio, P. Cejnar, and F. Iachello,Ann. Phys. (N. Y.) , 1106 (2008).[3] T. Brandes, Phys. Rev. E , 032133 (2013).[4] M. A. Bastarrachea-Magnani, S. Lerma-Hern´andez, andJ. G. Hirsch, Phys. Rev. A , 032101 (2014).[5] V. M. Bastidas, P. P´erez-Fern´andez, M. Vogl, andT. Brandes, Phys. Rev. Lett. , 140408 (2014).[6] R. Puebla, M.-J. Hwang, and M. B. Plenio,Phys. Rev. A , 023835 (2016).[7] A. Rela˜no, C. Esebbag, and J. Dukelsky,Phys. Rev. E , 052110 (2016). [8] J. E. Garc´ıa-Ramos, P. P´erez-Fern´andez, and J. M.Arias, Phys. Rev. C , 054326 (2017).[9] J. Khalouf-Rivera, F. Prez-Bernal, and M. Car-vajal, “Excited state quantum phase transitionsin the bending spectra of molecules,” (2020),arXiv:2006.13058 [physics.chem-ph].[10] D. Larese, F. Prez-Bernal, and F. Iachello,J. Mol. Struct. , 310 (2013).[11] B. Dietz, F. Iachello, M. Miski-Oglu, N. Pietralla,A. Richter, L. von Smekal, and J. Wambach,Phys. Rev. B , 104101 (2013).[12] J. Khalouf-Rivera, M. Carvajal, L. F. Santos, andF. P´erez-Bernal, J. Phys. Chem. A , 9544 (2019). [13] T. Tian, H.-X. Yang, L.-Y. Qiu, H.-Y.Liang, Y.-B. Yang, Y. Xu, and L.-M. Duan,Phys. Rev. Lett. , 043001 (2020).[14] S. Sachdev, “Quantum phase transitions,” in Handbook of Magnetism and Advanced Magnetic Materials (American Cancer Society, 2007).[15] L. Carr,
Understanding quantum phase transitions (CRCpress, 2010).[16] A. Rela˜no, J. M. Arias, J. Dukelsky, J. E.Garc´ıa-Ramos, and P. P´erez-Fern´andez,Phys. Rev. A , 060102 (2008).[17] P. P´erez-Fern´andez, A. Rela˜no, J. M. Arias,J. Dukelsky, and J. E. Garc´ıa-Ramos,Phys. Rev. A , 032111 (2009).[18] Q. Wang and F. P´erez-Bernal,Phys. Rev. A , 022118 (2019).[19] P. P´erez-Fern´andez, A. Rela˜no, J. M. Arias, P. Ce-jnar, J. Dukelsky, and J. E. Garc´ıa-Ramos,Phys. Rev. E , 046208 (2011).[20] M. A. Bastarrachea-Magnani, S. Lerma-Hern´andez, andJ. G. Hirsch, Phys. Rev. A , 032102 (2014).[21] M. A. Bastarrachea-Magnani, B. L. del Car-pio, S. Lerma-Hern´andez, and J. G. Hirsch,Phys. Scr. , 068015 (2015).[22] C. M. L´obez and A. Rela˜no,Phys. Rev. E , 012140 (2016).[23] R. Puebla and A. Rela˜no,Phys. Rev. E , 012101 (2015).[24] Q. Wang and H. T. Quan,Phys. Rev. E , 032142 (2017).[25] Z.-G. Yuan, P. Zhang, S.-S. Li, J. Jing, and L.-B. Kong,Phys. Rev. A , 044102 (2012).[26] R. Puebla and A. Rela˜no,EPL (Europhysics Letters) , 50007 (2013).[27] P. Strnsk, M. Macek, and P. Cejnar,Ann. Phys. , 73 (2014).[28] P. Strnsk, M. Macek, A. Leviatan, and P. Cejnar,Ann. Phys. , 57 (2015).[29] P. Cejnar and P. Strnsk, Phys. Lett. A , 984 (2017).[30] P. P´erez-Fern´andez and A. Rela˜no,Phys. Rev. E , 012121 (2017).[31] M. ˇSindelka, L. F. Santos, and N. Moiseyev,Phys. Rev. A , 010103 (2017).[32] R. Puebla, A. Rela˜no, and J. Retamosa,Phys. Rev. A , 023819 (2013).[33] G. Engelhardt, V. M. Bastidas, W. Kopylov, andT. Brandes, Phys. Rev. A , 013631 (2015).[34] Q. Wang and F. P´erez-Bernal,Phys. Rev. A , 062113 (2019).[35] W. Kopylov, G. Schaller, and T. Brandes,Phys. Rev. E , 012153 (2017).[36] P. P´erez-Fern´andez, P. Cejnar, J. M. Arias,J. Dukelsky, J. E. Garc´ıa-Ramos, and A. Rela˜no,Phys. Rev. A , 033802 (2011).[37] L. F. Santos and F. P´erez-Bernal,Phys. Rev. A , 050101 (2015).[38] L. F. Santos, M. T´avora, and F. P´erez-Bernal,Phys. Rev. A , 012113 (2016). [39] F. Prez-Bernal and L. F. Santos,Fortschr. Phys. , 1600035 (2017).[40] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalat-tore, Rev. Mod. Phys. , 863 (2011).[41] T. Langen, R. Geiger, and J. Schmiedmayer,Annu. Rev. Condens. Matter Phys. , 201 (2015).[42] M. Kloc, P. Str´ansk´y, and P. Cejnar,Phys. Rev. A , 013836 (2018).[43] S. Pilatowsky-Cameo, J. Ch´avez-Carlos, M. A.Bastarrachea-Magnani, P. Str´ansk´y, S. Lerma-Hern´andez, L. F. Santos, and J. G. Hirsch,Phys. Rev. E , 010202 (2020).[44] H. Lipkin, N. Meshkov, and A. Glick,Nucl. Phys. , 188 (1965).[45] A. Polkovnikov, Ann. Phys. , 486 (2011).[46] L. F. Santos, A. Polkovnikov, and M. Rigol,Phys. Rev. Lett. , 040601 (2011).[47] I. Garc´ıa-Mata, A. J. Roncaglia, and D. A. Wisniacki,Phys. Rev. E , 010902 (2015).[48] O. Giraud and I. Garc´ıa-Mata,Phys. Rev. E , 012122 (2016).[49] E. J. Torres-Herrera and L. F. Santos,Ann. Phys. , 1600284 (2017).[50] Z.-H. Sun, J. Cui, and H. Fan,Phys. Rev. Research , 013163 (2020).[51] T. N. Ikeda, N. Sakumichi, A. Polkovnikov, andM. Ueda, Ann. Phys. , 338 (2015).[52] P. Ribeiro, J. Vidal, and R. Mosseri,Phys. Rev. Lett. , 050402 (2007).[53] F. de los Santos, E. Romera, and O. Casta˜nos,Phys. Rev. A , 043409 (2015).[54] S. Campbell, G. De Chiara, M. Pater-nostro, G. M. Palma, and R. Fazio,Phys. Rev. Lett. , 177206 (2015).[55] A. Russomanno, F. Iemini, M. Dalmonte, and R. Fazio,Phys. Rev. B , 214307 (2017).[56] T. Xu, T. Scaffidi, and X. Cao,Phys. Rev. Lett. , 140602 (2020).[57] M. Albiez, R. Gati, J. F¨olling, S. Hun-smann, M. Cristiani, and M. K. Oberthaler,Phys. Rev. Lett. , 010402 (2005).[58] T. Zibold, E. Nicklas, C. Gross, and M. K. Oberthaler,Phys. Rev. Lett. , 204101 (2010).[59] I. D. Leroux, M. H. Schleier-Smith, and V. Vuleti´c,Phys. Rev. Lett. , 073602 (2010).[60] V. Makhalov, T. Satoor, A. Evrard,T. Chalopin, R. Lopes, and S. Nascimbene,Phys. Rev. Lett. , 120601 (2019).[61] W. Feller, An Introduction to Probability theory and itsapplication Vol II (John Wiley and Sons, 1971).[62] N. L. Johnson, S. Kotz, and N. Balakrishnan,
Contin-uous univariate distributions (John Wiley & Sons, Ltd,1995).[63] A. K. Gupta and S. Nadarajah,
Handbook of beta distri-bution and its applications (CRC press, 2004).[64] M. J. Schervish and M. DeGroot,