Universal single-frequency oscillations in a quantum impurity system after a local quench

Universal single-frequency oscillations in a quantum impurity system after a local quench

Abolfazl Bayat Department of Physics and Astronomy, University College London, Gower St., London WC1E 6BT, United Kingdom    Sougato Bose Department of Physics and Astronomy, University College London, Gower St., London WC1E 6BT, United Kingdom    Henrik Johannesson Department of Physics, University of Gothenburg, SE 412 96 Gothenburg, Sweden    Pasquale Sodano International Institute of Physics, Universidade Federal do Rio Grande do Norte, 59078-400 Natal-RN, Brazil, and
Departamento de F�isica Teórica e Experimental, � Universidade Federal do Rio Grande do Norte, 59072-970 Natal-RN, Brazil
INFN, Sezione di Perugia, Via A. Pascoli, 06123, Perugia, Italy

Long-lived single-frequency oscillations in the local non-equilibrium dynamics of a quantum many-body system is an exceptional phenomenon. In fact, till now, it has never been observed, nor predicted, for the physically relevant case where a system is prepared to be quenched from one quantum phase to another. Here we show how the quench dynamics of the entanglement spectrum may reveal the emergence of such oscillations in a correlated quantum system with Kondo impurities. The oscillations we find are characterized by a single frequency. This frequency is independent of the amount of energy released by the local quench, and scales with the inverse system size. Importantly, the quench-independent frequency manifests itself also in local observables, such as the spin-spin correlation function of the impurities.

71.10.Hf, 75.10.Pq, 75.20.Hr, 75.30.Hx

I Introduction

The dynamics of an isolated quantum many-body system following an instantaneous change of a Hamiltonian parameter (quantum quench) is a topic of growing interest Eisert-Ther-2014 (). The problem touches on a multitude of subjects, from the foundation of quantum statistical physics to the engineering of quantum states and devices Polkovnikov-NED-2011 (); Eisert2015 (). A quench injects energy which disperses among the interacting degrees of freedom, and as time evolves local observables relax to their equilibrium values Proukakis-Ther-2013 (). When the quench is local, with a sudden change of a local parameter in a Hamiltonian, the energy injected to the system is nonextensive. In such a case one may expect, and indeed finds Eisler-Ent-2007 (), intermediate times at which wave propagation and reflection from boundaries can create slowly decaying oscillatory behaviour. This raises the question whether one could find a physically relevant model where, by tuning a pertinent parameter, the equilibration after a local quench of the Hamiltonian is strongly suppressed, or even totally eliminated.

Here, in point of fact, we show that, for the spin emulator Bayat-TIKM-2012 () of the two-impurity Kondo model Jayprakash (); ALJ (), a local quench into the Kondo-screened phase AffleckReview () induces the onset of long-lived oscillations. By analyzing the quench dynamics of the lowest eigenvalues of the entanglement spectrum Haldane-ES-2008 (); CL (); Sanpera-SG-2012 (), we find that the frequency of these oscillations is sharply determined if one tunes the impurity-spin interaction so that all spins become entangled with an impurity. Remarkably, is independent of the local quench energy and scales as remaining constant when the system size increases. Moreover, we find that the frequency leaves its fingerprints on local observables, including the spin-spin impurity correlation functions. As such, we believe that our finding is potentially relevant to future designs of spin-based quantum devices as the impurity spins do not seem to equilibrate despite being strongly connected to reservoirs.

Ii Model

We consider a spin chain emulator of the two-impurity Kondo model Bayat-TIKM-2012 () with two localized spin-1/2, each coupled to a frustrated spin-1/2 Heisenberg chain, and to each other via a Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. The Hamiltonian can be written as

Figure 1: Schematic picture of the spin emulator of the two-impurity Kondo model. The impurity coupling , together with the RKKY coupling , determines the extension of the entanglement length . The strength of the RKKY coupling can be externally controlled; for instance, in a quantum dot, by tuning gate voltages as indicated in the figure. For any , in the Kondo regime, we define as the impurity coupling for which the entanglement length extends over the entire chain, i.e. . By quenching the RKKY coupling from in the RKKY regime to in the Kondo phase one induces the non-equilibrium dynamics being studied here. For the sake of simplicity the next-nearest coupling is not shown in the figure since we fixed .



Here labels the left and right chains, with the vector of Pauli matrices at site on chain . The couplings and are nearest and next nearest neighbour couplings, respectively. The dimensionless parameter plays the role of an antiferromagnetic Kondo coupling between the impurities and their corresponding bulks and the dimensionless coupling measures the RKKY impurity interaction. The mapping to the Kondo model is valid for (with ) Sorenson-QIE-2007 (); Affleck-KondoSpinChain-2008 () since in this interval the bulk excitations are massless (as in the full electronic version of the Kondo problem). As we will see, our analysis of the quench dynamics applies to the entire Kondo regime, although, for a marginal coupling (in the sense of the renormalization group) produces logarithmic corrections which pollutes the numerical data Affleck-KondoSpinChain-2008 (); Deschner-2011 (). In order to avoid this, we tune for which both and faithfully represent the spin sector of a single-impurity Kondo model. Note that the terms in Eq. (1) each define a spin emulator of the single channel Kondo model, with labels Sorenson-QIE-2007 (). A schematic picture of the model is given in Fig. 1.

The ground state shows a quantum phase transition at a critical value of the RKKY coupling. For a small coupling the Kondo interaction dominates and each impurity spin gets screened by its bulk (Kondo phase), while in the opposite limit the impurities form a local singlet (RKKY phase) and decouple from the rest of the system ALJ ().

Iii Results

iii.1 Entanglement spectrum

Having partitioned the system into two parts, and (see Fig. 1), one can write an arbitrary pure state in the orthogonal Schmidt basis Nielsen-Chuang-2000 () as


where the ordered set of real numbers form the entanglement spectrum. The Schmidt bases and diagonalize the reduced density matrices and of the left and right parts respectively. It has been shown that the two largest Schmidt numbers dominate the entanglement spectrum, with their difference, , behaving as an order parameter at the quantum phase transition Sanpera-SG-2012 (); Bayat-SG-2014 (). Notably, all levels of the entanglement spectrum contribute essentially to the von Neumann entropy


In the Kondo regime , the system supports a length scale which diverges at the critical point in the thermodynamic limit. It may be determined numerically by exploiting its interpretation as the length scale over which the two impurities are entangled with two identical blocks of spins on both sides Bayat-TIKM-2012 (). One thus finds, for a large but finite system with Bayat-SG-2014 (),


Here is a constant, with a critical exponent taking the value in the neighborhood of the critical point Bayat-TIKM-2012 (). According to Eq. (7), for any given in the Kondo regime, it is always possible to find an optimal impurity coupling such that , making each impurity entangled with all spins in its bulk. For the length exceeds the length of the chain and Kondo screening does not take place AffleckReview ().

The time evolution of entanglement spectra following a quantum quench have been the subject of several recent investigations Cardy-Quench-2014 (); Zamora-Splitting-2014 (); DeChiara-EntSpec-2014 (); Vodola-EntSpect-2013 (); Calabrese-LQ-2007 (). As we shall see next, the entanglement dynamics of the present problem exhibits some striking features.

Figure 2: Quench dynamics of the first two largest Schmidt numbers and versus time for in a chain of length with , and (blue solid line); (red dashed line) using exact diagonalization: plots (a) and (b). The Fourier transform for in a chain of length with and (blue line with small solid circles representing data points) and (red line with stars representing data points): plot (c). Quench dynamics of the first two largest Schmidt numbers and versus time for in a chain of length when and using tDMRG: plots (d) and (e). The Fourier transform for in a chain of length with and : plot (f). Quench dynamics of the first two largest Schmidt numbers and versus time for a non-optimal case, (here ), in a chain of length when and : plots (g) and (h). The Fourier transform for in a chain of length with and : plot (i).

iii.2 Entanglement dynamics

We initially prepare the system in the ground state of the Hamiltonian in Eq. (1), choosing the coupling in the RKKY regime. At time the coupling is instantaneously changed to . As a result, the system evolves as


where , and where are eigenvalues of the quench Hamiltonian, defined by Eq. (1) with . The corresponding eigenstates are global singlets, as implied by spin-rotational symmetry. By tracing part (see Fig. 1), one can compute the reduced density matrix of part , from which the entanglement spectrum is obtained.

Figs. 2(a) and (b) show the results for two quantum quenches with different values of using exact diagonalization, with the two largest Schmidt numbers and plotted as functions of time. In both cases , so that . While the amplitudes of the oscillations are different for the two quenches, very surprisingly, the dynamics of is governed by a single frequency , independent of the quench. Moreover, as seen in Figs. 2(a) and (b), the oscillations do not damp.

It is instructive to study the power spectrum


In Fig. 2(c), is plotted for the two given quenches. As revealed by the plot, a sharp peak, independent of the quench, emerges at the frequency . This shows that there exists a a unique frequency excited by quenching. In Figs. 2(d)-(f) the same quantities are plotted for a larger chain, using a time-dependent density-matrix renormalization group (tDMRG) algorithm white-DMRG-RungeKutta (). Again, single-frequency oscillations are uncovered on the numerically accessible time scale. To a high precision, can be fit by a sinusoidal function as


where is the single frequency involved in the dynamics, with the eigenvalues of the states which dominate the expansion in Eq. (8). and are the quench-dependent amplitude and phase.

Of course, the single-frequency oscillations disappear for non-optimal Kondo coupling as shown in Figs. 2(g) and (h). Indeed, these plots clearly display dispersive multi-frequency oscillations, also evidenced by the power spectrum in Fig. 2(i). It is, however, very important to observe that there always exists a dominant frequency in the power spectrum of which exhibit a pertinent scaling with . It follows from Fig. 3(a) that scales with as the Kondo temperature,


using that Jayprakash (). For , i.e. when , this frequency becomes the only one accessible to the system, i.e. . For we see deviations from Eq. (11) due to finite-size effects, as now exceeds the length .

It is important to realize that the single-frequency oscillations unveiled by our simulations are very different from the quench dynamics studied in Refs. Eisler-Ent-2007 (); StephanDubail (); Saleur-Kondo-2014 (); Vasseur-KondoUniversal-2014 (); Divakaran (); Igloi (), which reflects a finite-size effect when instantaneously joining two quantum critical systems. Likewise, the oscillating quench dynamics numerically observed in certain 1D lattice models Barmettler (); Gritsev (); Faribault () are also different from our results, coming from a global quench of an integrable interaction, and being either damped Barmettler () or exhibiting a multi-frequency power spectrum Gritsev (); Faribault (). In all these cases one observes dispersive wave propagation while our scenario is nonperturbative, long-lived and dispersionless.

Figure 3: (a) The dominant frequency as a function of in a semi-logarithmic plot for a chain of fixed length when the impurity coupling varies. The deviation from linearity for is due to the finite size effect as the entanglement length exceeds the system size. (b) The quench-independent frequency as a function of on a semi-logarithmic plot. Each point corresponds to a different length for which the optimal coupling is found and then the quench-independent frequency is determined through time evolution. (c) Data collapse for the dynamics of as a function of for three different lengths.

iii.3 Quench independence

The frequency is quench-independent as it depends neither on nor . Independence from is evident from Eq. (10). Independence from comes about by tuning the Kondo coupling to its “optimal” value , thus entangling the impurities with all spins in their respective bulks, i.e. . For instance, in a chain of length , for and one finds that ; the very same value of is obtained for and .

Figure 4: (a) The von Neumann entropy versus time for in a chain of length when and (blue solid line) and (red dashed line). (b) The Fourier transform of the von Neumann entropy versus for the case of . The case for (not shown in the figure) shows a similar peak at with highly suppressed higher harmonics. (c) The singlet fraction versus time for in a chain of length when and (blue solid line) and (red dashed line). (d) The Fourier transform of the singlet fraction versus for the case of .

The dependence of on the optimal Kondo coupling is plotted in Fig. 3(b), showing that


as expected from Eq. (11). By combining Eq. (12) with Eqs. (7) and (11), it follows that for a chain of arbitrary size ,


To check Eq. (13), we have computed the product for various lengths , keeping fixed; see TABLE 1 which shows that . The scaling in Eq. (13) suggests that we get data collapse onto a universal curve for different lengths if plotting vs. . This is confirmed in Fig. 3(c).

iii.4 Effective model for -

To expound on the resonance mechanism giving rise to the quench-independent frequency , it is crucial to note that, for an optimal quench , only the two singlet eigenstates with eigenvalues are essentially involved in the dynamics, with in Eq. (8) when . Namely, one numerically verifies that there is a small residual overlap for any and when . As one moves away from , other eigenstates rapidly come into play and significantly contribute to the time evolution in Eq. (8).

The dominance of two singlet eigenstates at the optimal quench suggests that the dynamics may be captured by an effective four-spin model. Consider and for the impurity spins in the L/R parts, and and for the spins in the L/R bulks (see Fig. 1). We represent , where . Similarly, if we now make the ansatz that , with and , then the initial state will be . Thus, , which periodically brings back . It is truly remarkable that the quench dynamics of a complex quantum-many body system, when properly tuned, can qualitatively be mimicked by four spins! Intuitively, the entanglement makes the bulk spins collectively behave as two effective spins, forming a dynamically coordinated composite with the impurities. Putting this intuition on firm ground would be extremely interesting, and could open a new vista on quantum engineered quench dynamics.

8 12 16 20 24 28 32 36 40
2.250 2.184 2.192 2.180 2.136 2.100 2.080 2.01 2.000
Table 1: The product as a function of the length : the table shows that independently of the choice of the initial coupling when .

iii.5 von Neumann entropy

Since effectively one frequency governs the dynamics when , one may expect that the matrix elements of oscillate with a few harmonics of . To verify this, we study the von Neumann entropy which depends on all levels of the entanglement spectrum. In Fig. 4(a), is given for two different quantum quenches. When increasing the difference , and thus, the energy released to the system through the quantum quench, deviations from a single frequency sinusoidal function become apparent, with the appearance of higher harmonics in the power spectrum . This is shown in Fig. 4(b). A similar dependence on higher harmonics of the fundamental frequency is observed for the lower multiplets of the entanglement spectrum, i.e. for with .

Singlet fraction and spin correlations.- The emergence of a single frequency (and of its harmonics) is not a feature only of the dynamics of global quantities such as the entanglement spectrum or the von Neumann entropy. To probe for local quantities we trace out the bulks and compute the reduced density matrix of the two impurities . This has the form of a Werner state due to the spin-rotational symmetry of the model,


where is the singlet fraction, is the singlet and are the triplets. The singlet fraction, which is nowadays experimentally accessible Petta-QD-2005 (); Bloch-singlet-triplet-2008 (), determines all local properties of the two-impurity composite, such as the two-point correlation and the two-impurity concurrence concurrence () . In Fig. 4(c) the singlet fraction is plotted versus time for two different quantum quenches. As for the von Neumann entropy, the dynamics can be perfectly matched to the quench-independent frequency for small quenches, and its higher harmonics (essentially the third one) for larger quenches. The Fourier transform is plotted in Fig. 4(d), showing the peaks for and .

Iv Conclusions

In this article we have shown that a local quantum quench across the quantum critical point in a pertinently tuned two-impurity Kondo spin chain may lead to the emergence of long-lived single-frequency oscillations in the dynamics of the entanglement spectrum. The frequency thus revealed is independent from the energy released by the quench, and also shows scaling behavior with the system size, implying data collapse for the time evolution of levels in the entanglement spectrum. Important for possible experiments in the future, the quench-independent frequency leaves distinct fingerprints in local observables, and it can be observed as the dominant frequency even when the system is not tuned to make the entanglement length extend over the full system.

The fact that the single-frequency dynamics is found to be tied to the emergence of an optimal entanglement length may hint at new physics. By identifying the entanglement length with the dynamically generated screening length characteristic of Kondo systems Bayat-TIKM-2012 (), our result points towards novel schemes for measuring it in the laboratory. Recent experimental progress in realizing two-impurity Kondo physics with tunable interactions bork2011 (); chorley2012 (); Spinelli () makes this a tangible challenge.

We conjecture that a single frequency can be associated also to other massless systems. For this to happen there has to be a dynamically generated entanglement length which extends over the full system. In this work we used a spin-chain emulation of the two-impurity Kondo model as a paradigmatic example where this phenomenon occurs. Since the spin-chain emulator is a faithful realization of the spin sector of the two-impurity Kondo model, with a dynamics that effectively decouples from charge at low energies glazman1999 (), we expect that our results will be relevant also for the full two-impurity Kondo model with itinerant electrons. Thus, a spinful double-quantum-dot system, with the confined spins interacting with the conduction electrons in their independent leads, could serve as an experimental setup where the emergence of a long-lived dynamics may be observed.

Acknowledgement: We would like to thank I. Affleck, F. Buccheri, D. K. Campbell, R. Egger, A. Ferraz, L. Glazman, V. Korepin, K. Le Hur, M. Rigol, and A. Tavanfar for useful discussions. A.B. is supported by the EPSRC grant EP/K004077/1. A.B. also acknowledges the hospitality from the International Institute of Physics (Natal) where this work was initiated. S.B. acknowledges support of the ERC grant PACOMANEDIA. The visits of H.J. and P.S. to London were supported by ERC grant PACOMANEDIA and EPSRC grant EP/J007137/1 respectively. H.J. acknowledges support from the Swedish Research Council and STINT. P.S. thanks the Ministry of Science, Technology and Innovation of Brazil, MCTI and UFRN/MEC for financial support and CNPq for granting a ”Bolsa de Produtividade em Pesquisa”. All authors acknowledge the support received from the NORDITA program on ”Quantum Engineering of States and Devices” where part of this work was carried out.


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