Large deviations and universality in quantum quenches
We study the large deviations statistics of the intensive work done by changing globally a control parameter in a thermally isolated quantum many-body system. We show that, upon approaching a critical point, large deviations well below the mean work display universal features related to the critical Casimir effect in the corresponding classical system. Large deviations well above the mean are, instead, of quantum nature and not captured by the quantum-to-classical correspondence. For a bosonic system we show that in this latter regime a transition from exponential to power-law statistics, analogous to the equilibrium Bose-Einstein condensation, may occur depending on the parameters of the quench and on the spatial dimensionality.
Introduction – Recent experimental progresses in the physics of trapped ultracold atomic gases have stimulated a growing interest in the non-equilibrium behavior of thermally isolated quantum many-body systems PSSV-11 (). A number of aspects are presently being investigated experimentally, ranging from the propagation of correlations after quenches Cheneau2012 () to relaxation and pre-thermalization inferred from the statistical fluctuations of the interference contrast of split condensates Gring2012 (). On the theoretical side, a compelling issue under investigation is that of the role played by universality in the non-equilibrium dynamics PSSV-11 (), since predictions independent of microscopic details make the comparison with experiments a particularly stringent test. Universal behavior can be investigated by studying either the time dependence of correlation functions CC-06 (); CG-11 (), in particular close to criticality, or their statistical fluctuations Kitagawa2011 (). In this context, a number of studies have focused on macroscopic, thermodynamic variables such as work S-08 (); GS-11 (); Heyl2012 (); Bunin2012 (); Smacchia2012 (); HPK-12 () and entropy Polkovnikov2010 (), exploring the emergence of universality in their statistical fluctuations.
Statistical fluctuations are known to provide insight into the physics of classical equilibrium and non-equilibrium systems T-09 (). The statistics of macroscopic extensive variables exhibits a first, obvious form of universality associated to typical, "small" fluctuations, which is however rather insensitive to the underlying properties of the system T-09 (). Indeed, as the mean of a generic extensive quantity (e.g., the magnetization in a spin system) grows proportionally to the number of degrees of freedom, the one of the associated intensive variable (i.e., the magnetization per unit volume) approaches a finite value . The central limit theorem (when applicable) suggests that the typical fluctuations of are suppressed and have a Gaussian distribution around . On the other hand, large fluctuations, though rare, are capable of probing the specific details of the physical system T-09 () and they might provide valuable information on its universal behavior. In order for a large fluctuation to occur, an extensively large number of microscopic fluctuating variables (i.e., the spin, in our example) has to deviate significantly from their corresponding means footnote1 () and this happens with a probability which is exponentially small in the size . Accordingly, for large , one expects to be distributed according to a probability density , where the so-called rate function is non-negative, vanishes for footnote2 (), and characterizes the statistics of both large deviations and Gaussian fluctuations.
Here we show that the statistics of large deviations of the intensive work done during a global quench of a thermally isolated quantum many-body system provides insight into its universal properties. For a global quench one heuristically expects to feature a prominent Gaussian peak centered at a finite mean . By focusing on the tails of this distribution, we demonstrate that there is a clear distinction between large deviations well below () and well above () the mean. The former are determined by the excess free energy of the dimensional classical correspondent in a film GS-11 (); CG-11 () and acquire universal features close to a possible critical point. The latter, instead, are genuinely quantum features, beyond the quantum-to-classical correspondence, which may, however, maintain some tracts of universality. Our analysis encompasses as examples the cases of quenches in the quantum Ising chain S-08 () and in a free bosonic system GSS-12 (). In addition, we show that, depending on the space dimensionality, the large deviation statistics of bosonic systems displays a so-called condensation transition (see, e.g., Ref. MK-10 ()), analogous to the Bose-Einstein condensation.
Statistics of the work – Consider a quantum system with interacting degrees of freedom and Hamiltonian . The extensive work performed on the system during the quench is determined by the initial state, typically the ground state of , and by the eigenvalues and eigenvectors of the post-quench Hamiltonian . In particular, is a stochastic variable with probability density Kurchan2000 (); Talkner2007 ()
where indicates generically the extensive ground-state energy of . As vanishes identically for below , we refer to this threshold so that .
The probability can be conveniently studied via its moment generating function
which, for , exists in the complex half-plane containing (with possible zeros, see, e.g., Ref. HPK-12 ()). For later purposes, we distinguish here a class A of systems in which for large but finite cannot exceed a certain extensive threshold from the class B whithin which can assume arbitrarily large values. Generically, in the former class, is defined for all with for , whereas in the latter is defined only for with a generic singularity in its derivative at . The quantum Ising chain in a transverse field and the free bosonic field belong to classes A and B, respectively.
where is in fact such a partition function of the classical -dimensional system with transfer matrix corresponding to the quantum Hamiltonian , in a film geometry with transverse "surface" area , "thickness" and equal boundary conditions set by . On the basis of one naturally defines the free energy per , where is the temperature of the corresponding classical system, which depends on the parameters of . In terms of the classical system, the variable in Eq. (2) is the distance between the two confining surfaces which we assume to have a large transverse area . Upon increasing , the free energy density per unit area decomposes in decreasing powers of as FSS ()
where is the bulk free energy density and is the surface free energy density, i.e., the energy cost for introducing separately each single boundary into the otherwise bulk system. The corrections "corr." in Eq. (4) vanish for . In order to separate the effects of confinement from the bulk behavior, one usually introduces the so-called excess free energy density per unit area , which plays a fundamental role in what follows and becomes independent of in the limit of large considered hereafter. In terms of the quantum system, one finds from Eqs. (3) and (4), that , GS-11 () and therefore
For , is defined in terms of by this equations and it lacks its thermodynamic interpretation.
Large deviations and universality – Equation (5) is crucial for understanding the emergence of universality in the large deviations statistics of the intensive work . In fact, its distribution for can be determined by a saddle-point approximation of the inverse Laplace transform of , which actually provides a heuristic derivation of the Gärtner-Ellis theorem T-09 (). In particular, Eq. (5) implies that has the form , where the rate function is the Legendre-Fenchel transform of (and viceversa, under certain assumptions T-09 ())
in which the infimum is taken within the domain of definition of and .
The generic features of can now be inferred from Eqs. (5) and (6). First of all note that the excess free energy is such that and . Most importantly is a concave function of T-09 () which approaches for . Figure 1 provides a sketch of and the corresponding for the two classes A and B introduced above. The last two properties imply the existence of a threshold in : the infimum in Eq. (6) for is and consequently . The behavior of close to the threshold , instead, is determined by the one of for and in particular , while the approach to it is determined by the corrections in Eq. (4).
The universality of these finite-size corrections close to critical points G-09 () carries over into the large deviation statistics of for . Indeed, if the post-quench Hamiltonian is close to a quantum critical point the finite-size corrections to the free energy density of the (near-critical) classical -dimensional system, which are responsible for the so-called critical Casimir effect G-09 (), take the universal scaling form for , where is the correlation length and some microscopic length scale. The scaling function is universal in the sense of critical phenomena G-09 (), as it depends only on the universality class of the classical critical point. In addition, due to the presence of the boundaries, depends on their surface universality class D-86 () or, equivalently, on which among the few effective boundary (i.e., initial) states , flows to as the critical point is approached. Once the scaling function is known, the rate function is calculated via Eq. (6). In particular, if the post-quench Hamiltonian is critical, then and
with . While and the possible corrections depend on the specific parameters of the initial state, the leading dependence of on is universal and non-analytic. In the case of finite but large , the approach to the value is eventually controlled by where the universal constants , , and depend, along with , on the bulk and surface universality class of the transition, and they are known for a variety of universality classes G-09 () (e.g., , for a quench of the quantum Ising chain within the same phase GS-11 ()). In this case, one finds but with significant logarithmic corrections. Notice that though this leading order is independent of , a study of the rate function for larger values of provides information on the complete scaling function , and hence on the corresponding boundary universality class. Accordingly, not only the edge singularities of the extensive work studied in Ref. GS-11 () are determined by universal features of the system and of the quench, but also the large deviations of the intensive variable display universal properties in their rate function close to the threshold NotaBene ().
In order to illustrate the discussion above we focus on a free bosonic theory described by a Hamiltonian diagonalizable in independent momentum modes
where and the integral runs over the first Brillouin zone . We assume a relativistic dispersion relation and consider quenches of the mass from to CC-06 (); GSS-12 (). This Hamiltonian captures the low-energy properties of a number of physical systems, including the ideal harmonic chain, interacting fermions and bosons in one dimension Cazalilla2011 (), and it models the relative phase fluctuations of split one-dimensional condensates Gring2012 (). has a critical point at and the corresponding classical theory is that of a Gaussian field in spatial dimensions and mass . The quench is characterized by and from Eqs. (1), (2), and (5) one finds GSS-12 ()
which is defined for and, as anticipated, belongs to class B. This can be decomposed as in Eq. (4) and upon approaching the critical point , i.e., for sufficiently large and , the correction takes the (-independent) scaling form where is the scaling function of the critical Casimir effect for the classical field with boundaries belonging to the so-called ordinary surface universality class D-86 (), corresponding to Dirichlet boundary conditions for . can be read, e.g., in Eq. (6.6) of Ref. KD-92 (). Accordingly, upon approaching the critical point, the ground state of flows towards the fixed-point state corresponding to this surface universality class. However, as decreases, calculated from Eq. (9) is no longer independent of and corrections to the critical Casimir term arise. These corrections are partly but effectively accounted for by changing in the previous scaling form, where the so-called extrapolation length D-86 () takes here the value . This can be interpreted as the fact that the fixed-point Dirichlet boundary condition on is effectively realized on confining surfaces which are at a distance from the boundaries of the film of thickness , resulting in an effective film of thickness D-86 (); CC-06 (); CG-11 (). For films sufficiently thick and critical, with , this correction is unnecessary, while it becomes increasingly important as , , or decrease.
Figure 2 presents the rate function (solid line) in for a quench from a non-critical to the critical point. In panel (a) the thin vertical dashed line indicates the mean work . The dashed curve, instead, provides the quadratic approximation of around , which describes the Gaussian distribution of the small fluctuations. While additional features of this are rationalized further below, we focus in panel (b) on the region of small , where we expect universality to emerge. The dash-dotted line correspond to Eq. (7), with KD-92 (). This universal behavior sets in rather close to the threshold. However, the agreement between and Eq. (7) extends to a wider range by accounting for the correction due to (dashed curve). The features displayed in Fig. 2 for carries over to the case , which requires the knowledge of the full scaling function . For a fixed value of , instead, the corrections to the scaling behavior due to increase upon decreasing and eventually, after crossing the line of no quench, they lead to a change in the effective boundary state CG-11 (); GS-11 () for .
Quantum regime and condensation – Let us now consider the case of large work . Upon increasing further away from the threshold, the value of for which the infimum in Eq. (6) is attained — and which satisfies — decreases and so does the thickness of the corresponding film. The behavior of such a film is expected to become increasingly dominated by its microscopic details, with a generic lack of universality even close to the critical point. Correspondingly decreases because . For , and vanishes, while it grows again for , with (see Fig. 1).
The rate function for is thus determined by for ("negative" film thickness), which lacks a thermodynamic interpretation because the quantum-to-classical correspondence does not hold in this case. The qualitative behavior of depends crucially on the class the system belongs to. In Fig. 1 we report a sketch of (a) and (b) the associated rate function corresponding to classes A and B discussed above and characterized by (A) a bound (e.g., the quantum Ising chain) or (B) an unbound spectrum (e.g., free bosonic theory). In particular, in case A, diverges upon approaching , with as required by the fact that vanishes above the intensive threshold . In case B, instead, and therefore . This is seen in Fig. 2(a), though the asymptotic linear behavior for , with slope (indicated by the dashed line) is actually approached only for rather large values of . In general, is system-specific and depends on the parameters of the quench.
Even though the emergence of universality is apparently limited to , systems belonging to class B might display unexpected universal properties in the fully quantum regime . In order to demonstrate this, we focus again on the free bosonic theory in Eq. (8) and we show that for the statistics of the work displays a behavior analogous to the Bose-Einstein condensation of the ideal Bose gas in the grand canonical ensemble. This implies a transition in the large deviation statistics for from exponential to algebraic. In fact, we note that the excess free energy in Eq. (9) has the same form as half the scaled cumulant generating function of the fluctuations of the spatial density of ideal Bose particles (of mass ) within a large region of volume . At equilibrium in an ensemble with chemical potential (in units of temperature ) one finds , where with and the integral is over . Accordingly the plot of has the form B in Fig. 1(a), with . For the ideal Bose gas, the condensation occurs as : the asymptotic slope of the rate function vanishes together with the function itself (see Fig. 1(b)). The mean value above which this happens is the critical density for condensation H-87 (), which is finite only for , where is the thermal wavelength. vanishes for because the probability acquires an algebraic dependence on — due to the contributions of fluctuations in single-particle states with small — and indeed the momenta with diverge as ; e.g., where for .
For the statistics of the work, plays a role similar to , although the occupation of the energy levels is determined by the non-thermal distribution generated by the quench and not by the Bose statistics. In fact, both and determine the -dependence of and , respectively, on which the onset of the condensation depends. In the case of the intensive work, is the control parameter: for , is finite for with a corresponding "critical value" . The emergence of is due to the fact that the dependence of on crosses over from quadratic for to linear for . Analogous crossover occurs in the condensation of an ideal Bose gas with relativistic dispersion BKM-79 (). The rate function vanishes identically for and acquires an algebraic dependence on because of the slow asymptotic decay of the probability distribution of the work done on modes with small , which are mildly confined in the initial state with . As a result, moments with diverge in this limit with, e.g., in .
Conclusions – We discussed the qualitative features of the large deviation statistics of the work done during a quantum quench, highlighting the emergence of universality and, for bosonic systems, of a non-thermal condensation transition. Even though large fluctuations are exponentially rare as the system size increases, the value of the rate function can be reduced by a suitable choice of the quench parameters, making them observable by a post-selection of experimental data.
Acknowledgements – AG and AS are grateful to KITP for hospitality. This research was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. AG is supported by MIUR within "Incentivazione alla mobilità di studiosi stranieri e italiani residenti all’estero."
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