Dynamical Quantum Phase Transitions in the Transverse Field Ising Model Stefan Kehrein (Universität Göttingen) Markus Heyl (Technische Universität Dresden) Anatoli Polkovnikov (Boston University) arXiv:1206.2505 week ending PRL 102, 130603 (2009) PHYSICAL REVIEW LETTERS 3 APRIL 2009 anisotropy, while the relaxation time increases with standard techniques (see [4] and references therein), we decreasing Á. Logarithmic divergence of the relaxation express this two-spin correlator as a Pfaffian, with coeffi- time in the limit Á 0 is suggested by the fit shown in cients calculated in a similar manner as for the Ising model Fig. 4(a). The picture! is less clear closer to the isotropic in a transverse field [12]. Exploiting the light-cone effect point. For the range 0:5 Á < 1, there appears to be an [17,20], we are able to evaluate numerically the order additional time scale after which the oscillations start to parameter dynamics up to times of the order of Jt 100. decay even faster than exponentially; simultaneously, the The results are displayed in Fig. 1(b). An analytic expres- period of the oscillations is reduced. Therefore, the relaxa- sion can be derived for Á 0, which is given by m t week ending PRLs 102, 130603 (2009) PHYSICAL REVIEW LETTERS 3 APRIL 2009 tion times plotted in Fig. 4(a) are valid only within an 0:5cos2 Jt . For Á < Á ,¼ exponentially decaying oscilla-ð Þ¼ ð Þ c intermediate time window, whose width shrinks upon ap- tions relaxation time of the order parameter is expected to quires exact convergence to the asymptotic value, as de- proaching the critical point. diverge as the system approaches the critical point [14]. fined in Ref. [15]. From this point of view, the oscillations m t e t= cos2 !t const (6) For intermediate easy-axis anisotropies 1 Á 3, the sð Þ / À ½ ð ÞÀ We find an opposite trend in the dynamics of the prepared in the XX limit are characterized by an infinite relaxation magnetization does not reach a stable regime within the reproduce the numerical results at large times veryNe´el well. state. In the long-time limit, our results suggest that time. numerically accessible time window [Fig. 3(a)]. The com- For Á Á , the staggered magnetization decays exponen-local magnetic order vanishes for all values Á < . XXZ model.—In the general case of Á Þ 0, the problem c The solution of the quench dynamics in the XXZ1 chain is no longer analytically treatable, and we have to resort to plicated behavior of m t in this parameter range can be tially with no oscillations at large times [Eq. (4)]. In con- sð Þ involves, in principle, all energy scales of the Hamiltonian, numerical techniques. We use the infinite-size time- ascribed to the interplay of processes at all energy scales. trast to the XXZ model, the oscillation period inand the approximativeXZ methods become essentially inaccurate evolving block decimation (iTEBD) algorithm [3], which Nevertheless, the numerical data suggest that the relaxation model diverges at the isotropic point Á Á , and the latter ¼ c in many cases. The mean field approximation, for example, implements the ideas of the density matrix renormalization is fastest close to the isotropic point, in the range between exactly marks the crossover between oscillatory andleads non- to contradictions with our results—an algebraic de- group (DMRG) method [16] for an infinite system. The Á 1 and Á 1:6. A simple generic type of behavior is oscillatory behavior of m t . We have extractedcay the for re-Á 1 and a nonvanishing asymptotic value of the algorithm uses an optimal matrix-product representation of sð Þ recovered¼ for¼ large anisotropies Á * 3. The numerical laxation times from exponential fits to the numericalstaggered data, moment for Á > 1 [13]. Renormalization group the infinitely extended chain, keeping only the dominant data in Fig. 3(b) indicate exponential relaxation of the showing a clearly pronounced minimum right atbased the iso- approaches [5] are restricted to low-lying modes, eigenstates of the density matrix of a semi-infinite subsys- which is not sufficient in the present case. The exact tem, in combination with a Suzuki-Trotter decomposition staggered magnetization tropic point [see Fig. 4(b)]. The relaxation time scales as 1 2 numerical results presented in this study go further than of the evolution operator. This method is very efficient for t= ÁÀ for Á Ác and as Á for Á Ác. ms t eÀ : (4) / / the predictions of low-energy theories [5]. small t; however, the increasing entanglement under time ð Þ / Apart from the numerical evaluation of the Pfaffian,Before we delving into a more detailed study, it is instruc- evolution [17] requires one to retain an exponentially The relaxation time scales roughly quadratically with Á can prove rigorously that in the infinite-time limittive to the consider the so-called XX limit (Á 0) of the growing number of eigenstates. We find that the error of [Fig. 4(a)]. Oscillations do persist on top of the exponential staggered magnetization vanishes for all anisotropiesHeisenberg in chain (1), which can be mapped¼ onto the our calculations behaves in a similar way to that of the decay, but they fade out quickly. the range Ác < Á < . Indeed, since the Pfaffianproblem reduces of free fermions. In this case, one easily obtains finite-size DMRG algorithm, and the methodology devel- XZ model.—We now turn to the study of the XZ to a Toeplitz determinant1 at t [12], we can usean Szego analytic¨’s expression for the time evolution of the stag- oped in Ref. [18] can be applied in order to control the !1 gered magnetization: m t J 2Jt =2 (Fig. 1). Here J accuracy [19]. By carefully estimating the runaway time Hamiltonian lemma to calculate the large-distance asymptotics of the sð Þ¼ 0ð Þ 0 z z denotes the zeroth-order Bessel function of the first kind. via comparing results with different control parameters x x two-spin correlator in the above-mentioned regime, obtain- 1 HXZ J 2Sj Sj 1 ÁSjSj 1 : (5) Thus, after a short transient time t JÀ , the staggered [18], the absolute error in the plotted data is kept below ¼ j f þ þ þ g 1 2 n 6 X ing, for n 1, limt C n; t 4 1 1 4=Ámagnetization=2 , displays algebraically decaying oscillations 10À . Using 2000 states and a second-order Suzuki-Trotter !1 ð Þ ½ð þ À Þ 3 1 In this model, a quantum phase transition separates two which immediately implies that m t 0. originating from the finite bandwidth of the free-fermionic decomposition with a time step 10À JÀ for large Á s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 gapped phases at Á 2, with antiferromagnetic order in Discussion.—We have analyzedð the!1 dynamicsÞ¼ model: of the and up to 7000 states with 10À JÀ for small Á, an c intermediate time regime Jt &16 can be reached, which, the z direction for Á ¼> Á and in the x direction for Á < staggered magnetization in the XXZ and XZ models fol- 1 c m t cos 2Jt : (2) in general, far exceeds the short transient time. Á . Unlike the XXZ model, it can be easily diagonalized lowing a quantum quench. Our main result is that in both s c ð Þp4t À 4 An overview of the results is presented in Fig. 1(a). For analytically. In order to study the staggered magnetization models there is a dynamically generated relaxation rate ffiffiffiffiffiffiffiffi small anisotropies we find oscillations of the order parame- which is fastest close to the critical point. This pointIn general, also we are interested in generic behavior of the of the XZ model, we have to calculate the two-spin corre- relaxation dynamics on large time scales. We adopt a ter similar to those in the XX limit but with a decay time n z z decreasing upon approaching the isotropic point Á 1. In lation function C n; t 1 c 0 S0 t Sn t c 0 in the marks a crossover between oscillatory and nonoscillatorydefinition of relaxation which does not rely on time- ð Þ ¼ ðÀ 2Þ h j ð Þ ð Þj i the easy-axis regime Á > 1 of the XXZ model, the¼ relaxa- infinite-range limit, since ms t limn C n; t . Using dynamics of the order parameter. The dynamicsaveraged of the equilibration of the observable but instead re- ð Þ¼ !1 ð Þ magnetic order parameter turns out to be a good observable tion slows down again for increasing Á, and we observe Barmettler et al., PRL ’09 nonoscillatory behavior for Á 1. 0 246 8 10 12 14 16 18 forSpin the chains: quantitative extraction of nontrivial time scales. In 0 246 8 10 12 14 16 18 (a) =0 Figure 2 focuses on easy-plane anisotropy 0 < Á < 1. 0.4 ∆ 0 ∆=1.0 (a) XXZ model ∆=0.5 ∆=1.2 ∆=1 The results for 0 < Á 0:4 are well described, for acces-
s Dynamical Phase Transitionss ∆=2 ∆=1.4 (a) XXZ model (b) XZ model 0.2 sible time scales, by exponentially decaying oscillations m m ∆=1.6 100 ∆=4 ∆=1.8 relaxation time τ period T t= week ending -0.005 ∆=2.0 0 PHYSICAL REVIEW LETTERSms t eÀ cos !t : (3) ] ∆ Initial Hamiltonian H :PRL defines103, 056403state | (2009)Ψ > ð Þ / ð þ 31Þ JULY 2009
critical point i i
-1 log −1 0.5 Quantum ∝ ∝∆ 2 t for =0 The oscillation frequency is almost independent of the (b) 10 ∝∆ 0.25 -0.2 runaway ∆ a 0.2 critical point U=0.5 b ponentially long time scales [18]. It remains to show that ∆=6.0 2 s Quench: H 0.5 | (t)> = exp(-iH t) | > 0.25 Final∝∆ Hamiltonian f: defines time evolution Ψ f Ψi
U=8 d (i) the envelope function R tV of the oscillating term time [J m U=1(b) XZ model ∆ 0.21 0.4 ð Þ =0.2 =3.0 ∆=4.0 dmed ∆ | decays to zero for t 1=V, and (ii) the quasistationary ? s 0.1 d ∆=0.4 0.125 1 0.3 U=1.5 th
|m value d differs from the thermal value d .∆=0.6 (i) Insert- 0.2 4 6 8 U stat 0.1 th
d(t) 0.17 0 2 4 6 8 10 12 14 16 18 0.1 1 10 0.1 1 10 U=6 | Dynamical phase transition = “Sudden” change inU=2 the dynamical behavior ing an eigenbasiss K m k m yields R tV Jt ∆ ∆ 0.1 0 m
|m j i ¼ j i ð Þ¼ U=5 itV km kn 0 U=2.5 m;n n m 0e ð À Þ n K m . In this expression all of observables as function of 0.13quench 0parameter 2 4 6 8 10and/or 12 14 16real 18 time hj ih ji h j þj i Jt U=4 oscillating terms0.01 dephase in the long-time average FIG. 3 (color online). (a) Focus on the XXZ chain close to the FIG. 4 (color online). Relaxation time and oscillation period U=3 U=3.3 P critical point Á 1. (b) Comparison of the XXZ chain (sym- T 2 as a function of anisotropy in the XXZ and XZ models. [13,15], so that only energy-diagonal terms contribute to ! FIG. 1 1 (color online). Dynamics of the staggered magnetiza- ¼ ¼ the sum. But from K0;D 0 it follows that D is a good bols) and the XZ chain (dashed lines) for strong anisotropies; Logarithmic or algebraic laws are emphasized by solidtion lines.ms Int in (a) the XXZU=0.5chain and (b) the XZ chain.d Symbols 0.001 ½ ¼ 0.8ð Þ quantum number0 of 2n 4so that 6 8n K 10n 12 140, and 16 thus 18 solid lines correspond to an exponential fit. The dynamics of the the region close to the criticalEckstein point ofet al., the PRLXXZ ’09model (indicatedcorrespond to numericalU=1 results, and lines represent analytical j i h Jtj þj i ¼ Hubbard model: U=1.5 U=8 R tV vanishes in the long time limit (if it exists and if staggered magnetization of the XXZ and XZ chains converge by the question mark), it becomes impossible to extractresults 0.6 or a fits by corresponding laws (see text). For Á 0 the ð Þ Schiro and Fabrizio, PRL ’10 U=6 ¼ accidental degeneracies between sectors of different D are towards each other in the large-Á limit. relaxation time from the numerical results. typicaln(t) behaviorU=2 of the error is illustrated by comparing the FIG. 2 (color online). Absolute value of the staggered magne- ∆ 0.4 U=5 irrelevant). From Eq. (4) we therefore conclude that d t numerical iTEBD result with 2400 retained states to the exact tization in the XXZ model. Symbols represent2 numericalð results,Þ 130603-3 curve: The absolute deviation from the exact curve is less than equals dstat for times 1=V t U=V , up to corrections 0.2 U=2.5 U=4 solid curves2 correspond2 to fits by the exponential law (3), and 10 6 for t Hamiltonian: • Quantum phase transition at gc = 1 • Exactly solvable via Jordan-Wigner mapping and Bogoliubov transformation S. Sachdev, Quantum Phase Transitions Interaction quench (instantaneous): Magnetic field g0 → g1 • Quenches within same phase: g0, g1 < 1 or g0, g1 > 1 • Quenches across quantum critical point: g0 < 1, g1 > 1 or g0 > 1 , g1 < 1 Initial state: Ground state of H(g0) Eigenoperators Ground state of H(g1) determined by quench (g0, g1) → Mode occupation numbers are time invariant First: Interpretation of quench dynamics in fermionic model Lee-Yang theory and Fisher zeroes Observation: Non-analyticities in real time evolution in thermodynamic limit Rotating phases: How can this become non-analytic? Analogous to Lee-Yang theory of thermodynamic phase transitions: Grand canonical partition function canonical partition function Fugacity of m particles ⇒ Polynomial of degree M in fugacity: Zeroes zi(T) away from positive real axis ⇒ Pressure is analytic Unless one takes the thermodynamic limit V → ∞ : Zeroes can converge to a branch cut on positive real axis indicating a phase transition M. E. Fisher, The nature of critical points Likewise for zeroes in the complex temperature plane: Fisher zeroes Thermodynamic phase transition = free energy/particle is nonanalytic in thermodynamic limit W. van Saarloos, D. A. Kurtze; J. Phys. A (1984) Boundary partition function: • for real z: partition function of equilibrium system with boundaries separated by z [ Le Clair et al., Nucl. Phys. B (1995) ] • for z = it and Amplitude of quench dynamics Thermodynamic limit: Lines of Fisher zeroes in complex plane For quench across quantum critical point: Fisher zeroes on time axis at with new non-equilibrium energy scale t tc Breakdown of short-time exp. in thermodynamic limit: Dynamical Quantum Phase Transition at tc β βc Our definition Breakdown of high-temperature expansion in thermodynamic limit: equilibrium phase transition at βc Work distribution functions Work performed during quench: Talkner et al., Phys. Rev. E (2007) Two energy measurements eigenstates of H(g1) • P(W) becomes δ-function in thermodynamic limit [ Silva, PRL (2008) ] • Fluctuations? Nr(w) • Large deviation form P (W )=e with rate function r(w) ≥ 0 and work density w=W/N [ Expectation value wex = Work density w Work density w - Rate functions asymmetric: Gaussian only around w = wex (central limit theorem) - For w →g 00: =universal 0, g1 =behavior 0.5 ∝ w ln w g0 = 0, g1 = 1.5 Rate function r(w;t) for work distribution of double quench 2 is non-analytic at Fisher zeroes: r(w=0,t) = | f(it) | g0 = 0.5 g1 = 2.0 Influence of non-analytic behavior extends to w>0 (dynamical quantum phase transition) 1 2i✏(k⇤)t 1+2n(k⇤) e 0= uk⇤ (t = 0) uk⇤ (t) = (1) h | i 1+2n(k⇤) “Critical” mode k* 1 • f(it)= ln uk(t = 0) uk(t) 1 (2) Non-analytic behavior of N h | i k Y è related to mode k* with (see also Pollmann et al., PRE (2010)) 2i✏(k⇤)t 1+2n(k⇤) e 0= u (t = 0) u (t) = (1) h k⇤ | k⇤ i 1+2n(k ) ⇤ è n(k*) = 1/2 (“infinite temperature”) 1 f(it)= ln uk(t = 0) uk(t) (2) • Existence of k* guaranteed Nfor quenchh across |QCP sincei k Y n(k=0) = 1 , n(UV) = 0 also for: - non-instantaneous ramping - other dispersion relations with same IR, UV-behavior Equilibrium vs. Dynamics Quench within same phase f(r) equilibrium Wick rotation r f(it) dynamics Quench across QCP “Naive” Wick rotation not possible r Non-equilibrium time evolution is not described by equilibrium properties Subtleties (?) Ground state of fermionic model for g0 < 1 is not thermodynamic pure ground state of the spin model. Thermodynamic pure ground state (obeys cluster decomposition) 1 (g0) >= ( GS(g0) >NS GS(g0) >R ) |± p2 | ±| Neveu-Schwarz sector (even) Ramond sector (odd) Well-known (see e.g. Calabrese et al., arXiv:1204.3911): N N - Even operators Oe x x Oe = Oe 0 j 1 0 j 1 j=1 j=1 Y Y can be evaluated only in@ NS-sectorA @ A è fermionic model from before - Odd operators Oo involve R-sector è difficult Example: Calculate longitudinal magnetization m = < σz > via z z 2 lim i i+r = m r !1h i even operator in fermionic model BUT: (not fully appreciated in previous literature on quench dynamics) Hamiltonian H = He + Ho è Return amplitude (Loschmidt echo) in general iH(g1)t iHf (g1)t +(g0) e +(g0) GS(g0) e GS(g0) = h | iH(g )t | i h | | i 6 (g ) e 1 (g ) ⇢ h 0 | | 0 i fermionic model spin model Careful reinterpretation Quench from paramagnetic phase: Unique ground state in spin model = |GS(g0)>NS i(H +H )t iH t GS(g ) e e o GS(g ) = GS(g ) e e GS(g ) NSh 0 | | 0 iNS NSh 0 | | 0 iNS = previous result from fermionic model è everything unchanged (incl. Fisher zeroes, nonanalytic behavior in time, etc.) Quench from ferromagnetic phase: Return amplitude/Loschmidt matrix 1 iH(g1)t f++(it) f+ (it) lim ln (g0) e (g0) = N N h± | |± i f +(it) f (it) !1 ✓ ◆ [ f++(z)=f (z) ,f+⇤ (z)=f +(z⇤)] Based on numerical evaluation for finite systems (up to N=200): • f++(z) and f+-(z) are analytic • t < first Fisher time: |f++(it)| < |f+-(it)| first Fisher time < t < second Fisher time: |f++(it)| > |f+-(it)| second Fisher time < t < third Fisher time: |f++(it)| < |f+-(it)| Larger overlap between same or other thermodynamic pure ground states alternates at Fisher times. Work distribution function for double quench: w=0 corresponds to return |+> -> |+> and |+> -> |-> è same result as in fermionic model, r(w=0,it) non-analytic at Fisher times |f++(it)| < |f+-(it)| |f++(it)| > |f+-(it)| |f++(it)| < |f+-(it)| Entries in Loschmidt matrix are analytic, but work distribution function shows non-analytic behavior. Equilibrium order parameter: longitudinal magnetization = z Longitudinal Magnetization z t t Quenches within the ferromagnetic phase: exponential relaxation (t) e Quenches within ferromagnetic phase: Exponential decay z [ Calabrese et al., PRL (2011)Calabrese ] et al., PRL ’11 Rigidity of ferromagnetic order: spins are pinned in the yz-plane Quenches across the QCP: oscillatory decay with frequency 100 Longitudinal magnetization has 8 | ) 1 ∗ t 10− = ( x 4 t/t ρ - analytical2 ttime evolution | 2 0 10− -1 1 - shows oscillatory decay with 0 0 1 -1 t sz(t) sy(t) zeroes with period 3 10− = Fisher time 4 t 2 (see also Calabrese et al., 10− | 1 Spin precession arXiv:1204.3911) in yz-plane − 1 f g 5 | 10− Longitudinal magnetization 0 1 - consistent with behavior of -1 0 0 -1 1 sy(t) Note: very similar to XZ model 6 sz(t) rate functions in Loschmidt matrix 10− Barmettler et al., PRL ’09 012345678 time t/t∗ DQPT Sharp changes of dynamical behavior as a function of control parameter Summary I. Existence of dynamical quantum phase transitions in close analogy to equilibrium phase transitions II. Breakdown of short-time expansion in thermodynamic limit ⇒ rate functions non-analytic in real time III. Existence of new non-equilibrium energy scale ε(k*) IV. Dynamical quantum phase transition is protected V. Signatures in other observables (longitudinal magnetization) VI. Main question: Universality away from quadratic/integrable Hamiltonian