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Dynamical Quantum Phase Transitions KITP ! 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].
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