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PHYSICAL REVIEW LETTERS

PRL 112, 070601 (2014)

Quantum Quenches and Work Distributions in Ultralow-Density Systems 1

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Yulia E. Shchadilova,1,2 Pedro Ribeiro,1 and Masudul Haque1

Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany A. M. Prokhorov General Physics Institute, Russian Academy of Sciences, Vavilova 38, 119991 Moscow, Russia (Received 22 March 2013; revised manuscript received 26 December 2013; published 19 February 2014) We present results on quantum quenches in lattice systems with a fixed number of particles in a much larger number of sites. Both local and global quenches in this limit generically have power-law work distributions (“edge singularities”). We show that this regime allows for large edge singularity exponents beyond that allowed by the constraints of the usual thermodynamic limit. This large-exponent singularity has observable consequences in the time evolution, leading to a distinct intermediate power-law regime in time. We demonstrate these results first using local quantum quenches in a low-density Kondo-like system, and additionally through global and local quenches in Bose-Hubbard, Aubry-Andre, and hard-core boson systems at low densities. DOI: 10.1103/PhysRevLett.112.070601

PACS numbers: 05.70.Ln, 03.65.Yz, 05.30.-d, 67.85.-d

Introduction.—Motivated by experimental progress in exploring nonequilibrium physics with cold-atom systems [1–3], there has been increasing interest in the dynamics of thermally isolated systems [4]. Despite the rapidly growing body of research in this topic, many aspects are still poorly understood. For example, what type of equilibration can be expected for various types of local and global quenches? Another question involves the overlap distribution, closely related to the work distribution [5–7] for a quantum quench. What is the typical form of the distribution of overlaps of the initial state with the final eigenstates? What are the effects of these distributions on time-evolving quantities? In experimental settings suitable for exploring nonequilibrium physics, a common situation is to have a fixed number of particles in a large spatial region. This contrasts sharply with the notion of the thermodynamic limit, where large regions are filled with a constant density. As examples, we note first that Ref. [3] involves a fixed number of bosons, initially localized in two groups, undergoing oscillatory dynamics in a larger space. Second, many cold atom experiments involve detection after the sudden release of initially trapped particles, so that a finite number of particles explore a large space through a quench. Ref. [8] has made explicit experimental studies of expansion in the presence of a lattice after turning off a trap. Finally, in a lattice spin system, spins oriented opposite to the background may be regarded as hard-core bosons. A highly polarized magnetic system is thus a natural realization of the regime under discussion. Dynamics of such a system has been explored through a cold-atom realization [9], and might eventually also be studied in solids that are insulating magnets. The study of nonequilibrium issues in this ultralowdensity limit—fixed number of particles, arbitrary large sizes—is clearly of topical importance but has been nearabsent in the theory literature. In this work, we present a 0031-9007=14=112(7)=070601(5)

study of quenches in this limit. We present several dynamical aspects which, through calculations in different low-density systems, we show to be generic features of quantum quenches in this regime where the usual thermodynamic limit is not applicable. One striking result involves the overlap distribution, ðfÞ ðiÞ jhΨð0Þjϕm ij, where jΨð0Þi ¼ jϕ0 i is the initial state (ground state of initial Hamiltonian), and m indexes the eigenstates of the final Hamiltonian. We show that this quantity is dominated by a power-law decay, ∼m−α , generically for quenches involving low-density systems. The associated “edge singularity” in the work distribution has large power-law exponents which would not be compatible with the usual thermodynamic limit. This in turn has a remarkable real-time consequence: in the evolution of observables away from their initial value, there appears an intermediate power-law regime between the initial perturbative time period and the large-time steady-state behavior. Kondo-like model.—The main system we use for demonstrating these general results involves a few (N c ) mobile fermions (“conduction electrons”) in a tight-binding closed chain (Fig. 1). One site of the lattice is Kondo-coupled to a single spin-12 “impurity.” The Hamiltonian is H¼−

X † ðci;s ciþ1;s þ H:c:Þ þ J S⃗ imp · S⃗0 ;

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P where S⃗0 ¼ 12 s;s0 c†0;s ⃗σ ss0 c0;s0 is the spin on site i ¼ 0 (s, s0 are spin indices), and i ∈ ½0; L − 1 is the site index. We study quenches of J, i.e., local quenches, starting from the ground state at J ¼ Ji and studying the dynamics after changing J instantaneously to its new value J f . The ground state is a spin singlet, and quenches of J preserve the spin, so that all dynamics is confined to the spin singlet sector.

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Figure 1 summarizes the equilibrium physics for N c ¼ 1. In an infinite chain, in the ground state, the fermion is localized around the impurity-coupled site (i ¼ 0) with localization length ξðJÞ. At large J (regime C), the itinerant fermion is almost completely localized at site 0 (ξ ≲ 1). At smaller J, the itinerant fermion is spread over multiple sites ξ > 1 (regime B). For any finite size L, there is a boundarysensitive small-J regime (regime A) where the fermion cloud extends over the whole system (ξ ≳ L). For 1 < N c ≪ L, there are additional features, but a similar general picture persists. Observables.—We will present time dependences of the occupancy n0 ðtÞ of site i ¼ 0 for the Kondo-like system, and of the Loschmidt echo LðtÞ ¼ jhΨð0ÞjΨðtÞij2 . The observable n0 ðtÞ is of obvious importance for the model (1), while LðtÞ is well-defined for any model and is closely related to the work distribution [5–7]. Despite the nonlocal nature of LðtÞ, there exist proposals for experimentally measuring this quantity, and related quantities have been measured [10]. Lack of equilibration to new ground state.—The final ˆ saturates is hOi ˆ DE ¼ value at which an observable O P ðfÞ 2 ðfÞ ˆ f jhΨð0Þjϕ ij hϕ j Ojϕ i, the so-called “diagonal m m m m ensemble” (DE) value [11]. In Fig. 1(c) we show n0 ðtÞ after a quench within the C region: it reaches the DE value hnˆ 0 iDE relatively rapidly, and then shows “revivals” at roughly periodic intervals of t ∼ L=2. The DE value where n0 saturates is markedly different from the ground state

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FIG. 1 (color online). Kondo-like model with single mobile fermion. (a) Energy gap as a function of J, L ¼ 100 sites, showing distinct behaviors in three regimes. Left inset shows system geometry. Right inset shows spectrum in C regime. (b) Density profiles nj in three regimes; dashed lines showing exponential localization. (c) Time evolution of occupancy n0 at impurity-coupled site; C → C quench, J i ¼ 102 , J f ¼ 10; L ¼ 100 sites. The long-time average hnˆ 0 iDE (solid black line), around which n0 ðtÞ oscillates, is significantly different from the equilibrium J ¼ J f value. Gray dashed and red dash-dotted lines show equilibrium n0 values for J ¼ J i and J ¼ J f . (d) Size dependence of hnˆ 0 iDE values for J i ¼ 102. Upper blue: J f ¼ 10; lower green: J f ¼ 0.9. Red dots are corresponding J ¼ J f equilibrium values.

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value of n0 for J ¼ Jf. This seemingly contradicts the intuition that a local quench in a large system should lead to relaxation to the final ground state value, because the energy pumped into the system by a local quench is a OðL−1 Þ effect. The reason this does not happen in the C → C quenches is that the itinerant electron occupies only some sites near the impurity. Thus, most lattice sites do not participate in the dynamics, and cannot serve as a bath to absorb the disturbance at site 0. This effect is not restricted to N c ¼ 1, but is true for finite number N c > 1 of fermions for L ≫ N c [12]. Figure 1(d) demonstrates that the lack of equilibration in quenches to C or B regions is not a finite-L effect. This effect represents a loss of the distinction between local and global quenches, which is a generic feature of the L → ∞ limit with finite particle number. Overlap distributions.—Figures 2(a)–2(d) summarize overlap distribution behaviors in quantum quenches between different regimes of the system (1) for N c ¼ 1. These behaviors can be derived from detailed consideration of the eigenfunctions [12]. In C → C quenches, the ground ðiÞ ðfÞ state overlap jhϕ0 jϕm¼0 ij dominates, and the small m ≠ 0 overlaps have the form ∝ sinðc1 mÞ [12]. The most remarkðiÞ ðfÞ able feature is the power-law behavior, jhϕ0 jϕm ij ∼ m−α , in quenches starting from or ending in the A region. The exponent α is 2 for A → A quenches and 1 for A → C quenches. These power-law overlap distributions are a generic phenomenon; we have found such behavior in several other low-density systems, both for local and global quenches. (The behavior is particularly clean for the

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PRL 112, 070601 (2014)

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FIG. 2 (color online). Overlap distribution jhϕ0 jϕm ij, powerlaw exponents α, and work distribution pðωÞ. (a)–(d) Kondo-like model, N c ¼ 1. (a),(b) Quenches from J i ¼ 10−3 to J f ¼ 103 (A → C), J i ¼ 10 to J f ¼ 103 (C → C), J i ¼ 103 to J f ¼ 10−3 (C → A), and J i ¼ 10−3 to J f ¼ 10−2 (A → A). (c),(d) Exponents of power-law fits m−α for quenches ending at and starting from the A regime. (e),(f) Bose-Hubbard chain with N b ¼ 3 bosons in L ¼ 20 sites; interaction U globally quenched from 0.3 to 0.5. (f) Approximations to the work distribution are obtained using Gaussians of width σ to replace the delta function of Eq. (2). (g) Kondo-like model, N c ¼3; quench from J i ¼10−3 to J f ¼ 10−2 .

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N c ¼ 1 system because of its simplicity.) Figure 2(g) shows the overlap distribution for the same model with N c ¼ 3 fermions. There are now additional structures, but the dominant overlaps follow a clear power law. Figure 2(e) shows the overlap distribution for a Bose-Hubbard chain at low density [12]. Again, there are interesting additional structures, but the dominant overlaps follow a clear power law (∼m−α ). The exponent α is approximately 0.5 in Figures 2(e) and 2(g). The generic dependence of α on particle number, density and quench type is currently an open question. Work distribution.—The overlap distribution is related to X ðiÞ ðfÞ 2 pðωÞ ¼ δðω − ϵm Þjhϕ0 jϕm ij ; (2) m ðfÞ

ðfÞ

where ϵm ¼ Em − E0 are the final eigenenergies measured from the final ground state energy. This is the socalled work distribution [5–7], except for a shift between ω and the usual work variable. (The energy prior to the quench plays no role in the temporal dynamics and so is not relevant for this work.) The work distribution is related to R the Loschmidt echo: LðtÞ ¼ j dωpðωÞeiωt j2 . Since Lð0Þ ¼ 1 by definition, pðωÞ must be normalizable. At large sizes (but constant particle number), pðωÞ can be treated as a continuous function starting from ω ¼ Δ, the finite-size gap, which vanishes at large L. We have found that, for quantum quenches in low-density systems, the work distribution generically has behavior pðωÞ ≈ p0 ω−b for ω > Δ, with large exponents b > 1. These power-law divergences are analogs of what would be called “X-ray edge singularities” in systems with a regular thermodynamic limit. In finite-number systems, pðωÞ remains normalizable despite the singularity as Δ → 0 because the magnitude of pðωÞ also vanishes (p0 → 0) in the large-size limit, due to the vanishing density. This contrasts sharply to systems with the usual thermodynamic limit where density remains constant as L → ∞, and pðωÞ itself is a well-defined nonvanishing quantity in the limit. This constrains the singularity pðωÞ ∼ ω−b to generally have smaller exponent, b < 1 (e.g., [6]). The lowdensity systems of interest here have no such constraint; a central result of the present work is that super-linear singularities (b > 1) are signatures of low-density systems. For finite-density systems, the only exceptions known to us involve the special cases of quenching across phase transitions, where b > 1 edge singularities may also occur [7]. For the model (1) with N c ¼ 1, pðωÞ ∼ ω−5=2 (A → A) and pðωÞ ∼ ω−3=2 (A → C) (b > 1 in both cases), Fig. 2(f) shows the work distribution for a global interaction quench in the Bose-Hubbard chain, with the delta function regularized as Gaussian. There is a power law with superlinear (b > 1) singularity. This is another example of the loss of distinction between global and local quenches in the lowdensity limit, as “edge singularities” are often associated primarily with local quenches for finite-density systems

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[6]. We have also found super-linear singularity exponents in other low-density systems [12], e.g., quenches of the strength or position of a weak trapping potential for a BoseHubbard system, quenches of on-site potentials and hopping strengths for hard-core bosons in a ladder geometry, and quenches of quasi-disorder potential strengths in an Aubry-André [13] system. Role of the density of states.—For the model (1) with N c ¼ 1, the behavior ∼m−α implies energy-dependence ∼ω−α=2 for the overlap distribution. Together with a factor of ω−1=2 from the one-dimensional single-particle density of states, this leads to pðωÞ ∼ ω−α−1=2 , i.e., b ¼ 5=2ð3=2Þ for A → AðCÞ quenches. This argument can be generalized: if the overlap distribution follows m−α and the density of states in the relevant lower-energy part of the spectrum behaves as ρðωÞ ∼ ωγ , the work distribution pðωÞ ∼ ω−b will have exponent b ¼ 2γα − γ þ 2α [12]. For single-particle systems, we have γ ¼ −1=2 in one dimension, as in the above example. For a generic system, however, the many-body density of states does not necessarily behave as a power law. We have found cases (Bose-Hubbard chain with trap) where an approximate power-law region with exponent γ~ in ρðωÞ leads to an approximate power law in pðωÞ with exponent b~ ¼ 2~γ α − γ~ þ 2α [12]. Also, if α ¼ 1=2, any power-law form of ρðωÞ implies a linear edge singularity pðωÞ ∼ ω−1. In this case, a super-linear edge singularity can only happen with some non-power-law form of ρðωÞ. This occurs in the Bose-Hubbard chain case of Figs. 2(e),2(f) [12]. The intermediate-time ∼tβ region.—The appearance of larger powers in the edge singularity has novel consequences for real-time dynamics. We have identified an intermediate-time power-law region, that appears as a direct consequence of the large-power edge singularity. At initial times after a quench, the Loschmidt echo LðtÞ and other observables evolve away from their initial value quadratically with time, ∼t2 , as can be explained from generic perturbative arguments. We have found that, when pðωÞ has a large-exponent singularity, there is a region of time (after the initial perturbative times and before the large-time steady-state oscillations), where Lð0Þ − LðtÞ ¼ 1 − LðtÞ follows a new power-law behavior. If pðωÞ ∼ ω−b with b ∈ ð1; 3Þ in the energy range [Δ, Λ] and the contributions outside this energy window can be neglected, then in the time window between t ∼ Λ−1 and t ∼ Δ−1 one sees the behavior 1 − LðtÞ ∼ tb−1 [12]. The same phenomenon is also found in some observables: jOðtÞ − Oð0Þj can have an extended ∼tβOˆ region ˆ (βOˆ < 2) after the initial perturbative ∼t2 region. When O ˆ has the form of a rank-1 projector, R O ¼ jχihχj, we can write the time evolution as OðtÞ ¼ j dωpOˆ ðωÞeiωt j2 , where

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pOˆ ðωÞ ¼

X ðiÞ ðfÞ ðfÞ δðω − ϵm Þhϕ0 jϕm ihϕm jχi m

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differs from the work distribution (2) in that one factor of ðfÞ the overlap is replaced by hϕm jχi. If pOˆ ðωÞ has a power−bOˆ law singularity structure ω with exponent bOˆ ∈ ð1; 3Þ, the time evolution of OðtÞ away from Oð0Þ will show the intermediate-time region ∼tβOˆ (βOˆ ¼ bOˆ − 1). When the ˆ does not have the form O ˆ ¼ jχihχj, it is not operator O simple to formulate an analogous expression. A generic operator for a many-body system will not have this form, but the site occupancies for single-particle systems (e.g., nj for our N c ¼ 1 model) are rank-1 projectors, as is the Loschmidt echo for any system. Currently, little is known about pOˆ ðωÞ behaviors for projector-type observables in different quantum quenches, or about the conditions necessary for having an intermediate-time regime in generic observables. The intermediate-time regime for low-density systems is illustrated in Fig. 3. For the model (1) with N c ¼ 1 particle, this regime is present in the Loschmidt echo, for both A → A and A → C quenches. In the occupancy n0 ðtÞ, the intermediate-exponent regime can be seen for A → A quenches (with form ∼t1=2 ), but not for the A → C quenches, for which case the eigenstate dependence of ðfÞ hϕm jχi does not favor a large enough exponent in pnˆ 0 ðωÞ [12]. Figure 3(g) displays the intermediate-time region for a Bose-Hubbard chain with interaction quenches, and Fig. 3(h) shows the same for the Kondo-like model with N c ¼ 3 itinerant fermions. With hard-core bosons on a ladder-shaped lattice, considering time evolution after various local and global quenches, we find extended intermediate-time regions ∼tβ in 1 − LðtÞ, with exponents matching β ¼ b − 1 where b is the singularity exponent in pðωÞ, calculated with Gaussian regularization [12]. With quenches of a trapping potential, we find quench parameter combinations where

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FIG. 3 (color online). (a)–(f) Time evolution of n0 ðtÞ and LðtÞ in N c ¼ 1 model. (a)–(d) A → A quenches: J i ¼ 10−3 , J f ¼ 10−2 . (e),(f) A → C: J i ¼ 10−3 , J f ¼ 10. Extended intermediate region between ∼t2 region and long-time oscillatory region is seen in n0 ðtÞ for A → A quenches and in LðtÞ for both A → A and A → C quenches, but not in n0 ðtÞ for A → C quenches. (g) Bose-Hubbard, global interaction quench, three bosons in chains of length L ¼ 10 and L ¼ 30. Quench from U i ¼ 0.02L to U f ¼ 0.01L. (h) N c ¼ 3 fermions; J i ¼ 10−3 , J f ¼ 10−2 .

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pðωÞ shows super-linear edge singularities (b > 1) but no intermediate-time regime shows up in the LðtÞ dynamics because the singularity exponents are too large, b > 3 [12]. We have also found an example (Aubry-André system) where there are well-defined pðωÞ ∼ ω−b regions but the contributions from outside the power-law region are so large that the dynamical intermediate-time signature is washed out [12]. Extent of the intermediate-time region.—If the powerlaw window for pðωÞ is ω ∈ ½Δ; Λ, the ∼tβ region with β ∈ ð0; 2Þ extends from t ∼ Λ−1 to t ∼ Δ−1 . The scale Λ is generally of the order of the bandwidth, and so is set by the hopping strength. Since the finite-size gap Δ vanishes with increasing system size L, the intermediate-time region gets more and more extended in time for larger L. This is shown in Fig. 3(g) through a comparison of two different L values. Density regimes.—The large size limit L → ∞ can be taken with the density N=L fixed (usual thermodynamic limit), or with the number N fixed. We call the latter the “ultralow-density limit.” Our explicit resuts involve finite L ≫ 1 and 1 ≲ N ≪ L. The physics in such cases is expected to be better described by the constraints of the ultralow-density limit rather than the usual thermodynamic limit. Since the demarcation of regimes is not strictly defined for finite L, we use the terms “low” and “ultralow” loosely; we expect our findings to be applicable for many N ≪ L situations. Discussion.—For systems that are not well described by the traditional thermodynamic limit but instead have a fixed number of particles in a large size, as is common in setups relevant for large classes of nonequilibrium experiments, we have presented universal features of quantum quenches. These include edge singularities with large exponents not usually occurring in “regular-limit” systems, a loss of the usual distinctions between local and global quenches, and a novel intermediate-time region in the dynamics. Universal behaviors in quantum quenches are generally sought and discussed in asymptotic times. A new universality at intermediate times, visible in widely different systems, is of obvious distinction and interest. Our results open up various research avenues. One issue is to bridge the gap between the regime considered here and the regular thermodynamic limit. A systematic study of overlap or work distributions and associated quench dynamics with varying density and system sizes is currently lacking. A related issue is that of experimental accessibility. Since measurements will likely be in finite-size systems, it is important to demarcate which number or size combinations show features of our fixed-number large-size limit, from those corresponding to the regular thermodynamic limit. It may also be interesting to supplement our results on the Loschmidt echo with time-evolution studies of traditionally measurable observables such as densities and correlation functions. Finally, Eq. (3) highlights the general lack of knowledge about observables in general eigenstates

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of many-body Hamiltonians. Current research is addressing some eigenstate expectation values [11,14], but clearly further investigations are warranted. M. H. and Y. S. thank M. Vojta for collaboration on equilibrium properties of the Hamiltonian (1) [15].

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Quantum quenches and work distributions in ultralow-density systems.

We present results on quantum quenches in lattice systems with a fixed number of particles in a much larger number of sites. Both local and global que...
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