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arXiv:1203.1815 · Physical Review Letters

Universal Energy Distribution of Quasiparticles Emitted in a Local Time-Dependent Quench

Pietro Smacchia, Alessandro Silva

Published Jul 17, 2012• 33 citations
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Abstract

We study the emission of quasiparticles in the scaling limit of the 1D quantum Ising chain at the critical point perturbed by a time-dependent local transverse field. We compute exactly and for a generic time dependence the average value of the transverse magnetization, its correlation functions, as well as the statistics of both the inclusive and exclusive work. We show that, except for a cyclic perturbation, the probability distribution of the work at low energies is a power law whose exponent is universal, i.e., does not depend on the specific time-dependent protocol, but only on the final value attained by the perturbation.

DOI
10.1103/physrevlett.109.037202

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We study the emission of quasiparticles in the scaling limit of the 1D quantum Ising chain at the critical point perturbed by a time-dependent local transverse field. We compute exactly and for a generic time dependence the average value of the transverse magnetization, its correlation functions, as well as the statistics of both the inclusive and exclusive work. We show that, except for a cyclic perturbation, the probability distribution of the work at low energies is a power law whose exponent is universal, i.e., does not depend on the specific time-dependent protocol, but only on the final value attained by the perturbation.
We study the statistics of the work done by globally changing in time with a generic protocol the mass in a free bosonic field theory with relativistic dispersion and the transverse field in the one-dimensional Ising chain both globally and locally. In the latter case we make the system start from the critical point and we describe it in the scaling limit. We provide exact formulas in all these cases for the full statistics of the work and we show that the low-energy part of the distribution of the work displays an edge singularity whose exponent does not depend on the specifics of the protocol that is chosen and may only depend on the position of the initial and final values with respect to the critical point of the system. We also show that the condensation transition found in the bosonic system for sudden quenches [A. Gambassi and A. Silva, Phys. Rev. Lett. 109, 250602 (2012)] is robust with respect to the choice of the protocol.
Dynamical correlations of the spin and the energy density are investigated in the critical region of the random transverse-field Ising chain by numerically exact calculations in large finite systems (L ≤ 128). The spin-spin autocorrelation function is found to decay proportional to (log t) −2xm and (log t) −2xs m in the bulk and on the surface, respectively, with xm and x s m the bulk and surface magnetization exponents, respectively. On the other hand the critical energy-energy autocorrelation functions have a power law decay, which are characterized by novel critical exponents ηe ≈ 2.2 in the bulk and η s e ≈ 2.5 at the surface, respectively. The numerical results are compared with the predictions of a scaling theory. 05.50.+q, 64.60.Ak, 68.35.Rh Typeset using REVTEX 1 The asymptotic behavior of the time-dependent correlation functions for interacting many-body systems turned out a very difficult subject of theoretical research. Exact results in this field are scarce, one can mention the one-dimensional spin 1/2 XY-model [1] and the Ising chain in a transverse-field [2]. Both models can be mapped onto a system of non-interacting fermions, where the equal-position correlation functions are calculated by the Pfaffian method utilizing the theory of Töplitz determinants. In this Letter we consider- at the first time- the critical dynamical correlations of an interacting quantum system in the presence of quenched (i.e. time-dependent) disorder. It has recently become clear that quenched disorder has rather different effects on phase transitions in quantum systems [3] than on those thermally driven phase transitions. For example, in the Griffiths phase, which is situated at the disordered side of the critical point, the susceptibility has an essential singularity in classical systems, whereas in a quantum system the corresponding singularity is stronger, it is in a power law form. Here we consider the prototype of random quantum systems the one-dimensional random transverse-field Ising model defined by the Hamiltonian:
We discuss different quench protocols for Ising and XY spin chains in a transverse magnetic field. With a sudden local magnetic field quench as a starting point, we generalize our approach to a large class of local non-sudden quenches. Using finite time path field theory (FTPFT) perturbative methods, we show that the difference between the sudden quench and a class of quenches with non-sudden switching on the perturbation vanishes exponentially with time, apart from non-substantial modifications that are systematically accounted for. As the consequence of causality and analytic properties of functions describing the discussed class of quenches, this is true at any order of perturbation expansion and thus for the resummed perturbation series. The only requirements on functions describing the perturbation strength switched on at a finite time t=0 are as follows: (1) their Fourier transform f(p) is a function that is analytic everywhere in the lower complex semiplane, except at the simple pole at p=0 and possibly others with ℑ(p)<0; and (2) f(p)/p converges to zero at infinity in the lower complex semiplane. A prototypical function of this class is tanh(ηt), to which the perturbation strength is proportional after the switching at time t=0. In the limit of large η, such a perturbation approaches the case of a sudden quench. It is shown that, because of this new type of universal behavior of Loschmidt echo (LE) that emerges in an exponentially short time scale, our previous results for the sudden local magnetic field quench of Ising and XY chains, obtained by the resummation of the perturbative expansion, extend in the long-time limit to all non-sudden quench protocols in this class, with non-substantial modifications systematically taken into account. We also show that analogous universal behavior exists in disorder quenches, and ultimately global ones. LE is directly connected to the work probability distribution, and the described universal behavior is therefore appropriate in potential concepts of quantum technology related to spin chains.
We study the dynamical properties of the random transverse-field Ising chain at criticality using a mapping to free fermions with which we can obtain numerically exact results for system sizes L as large as 256. The probability distribution of the local imaginary time correlation function $S(\ensuremath{\tau})$ is investigated and found to be simply a function of $\ensuremath{\alpha}\ensuremath{\equiv}\ensuremath{-}\mathrm{ln}S(\ensuremath{\tau})/\mathrm{ln}\ensuremath{\tau}.$ This scaling behavior implies that the typical correlation function decays algebraically, ${S}_{\mathrm{typ}}(\ensuremath{\tau})\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}{\ensuremath{\alpha}}_{\mathrm{typ}}},$ where the exponent ${\ensuremath{\alpha}}_{\mathrm{typ}}$ is determined from $P(\ensuremath{\alpha}),$ the distribution of $\ensuremath{\alpha}.$ The precise value for ${\ensuremath{\alpha}}_{\mathrm{typ}}$ depends on exactly how the ``typical'' correlation function is defined. The form of $P(\ensuremath{\alpha})$ for small $\ensuremath{\alpha}$ gives a contribution to the average correlation function ${S}_{\mathrm{av}}(\ensuremath{\tau})$ namely, ${S}_{\mathrm{av}}(\ensuremath{\tau})\ensuremath{\sim}(\mathrm{ln}\ensuremath{\tau}{)}^{\ensuremath{-}{2x}_{m}},$ where xm{x}_{m}xm​ is the bulk magnetization exponent, which was reported recently in Europhys. Lett. 39, 135 (1997). These results represent a type of ``multiscaling'' different from the well-known ``multifractal'' behavior.
By means of free fermionic techniques combined with multiple precision arithmetic we study the time evolution of the average magnetization, , of the random transverse-field Ising chain after global quenches. We observe different relaxation behaviors for quenches starting from different initial states to the critical point. Starting from a fully ordered initial state, the relaxation is logarithmically slow described by , and in a finite sample of length L the average magnetization saturates at a size-dependent plateau here the two exponents satisfy the relation . Starting from a fully disordered initial state, the magnetization stays at zero for a period of time until with and then starts to increase until it saturates to an asymptotic value , with . For both quenching protocols, finite-size scaling is satisfied in terms of the scaled variable . Furthermore, the distribution of long-time limiting values of the magnetization shows that the typical and the average values scale differently and the average is governed by rare events. The non-equilibrium dynamical behavior of the magnetization is explained through semi-classical theory.
We study the time evolution in the transverse-field Ising chain subject to quantum quenches of finite duration, ie, a continuous change in the transverse magnetic field over a finite time. Specifically, we consider the dynamics of the total energy, one- and two-point correlation functions and Loschmidt echo during and after the quench as well as their stationary behaviour at late times. We investigate how different quench protocols affect the dynamics and identify universal properties of the relaxation.