We develop a truncated Hamiltonian method to investigate the dynamics of the
(1+1)-dimensional ${\ensuremath{\phi}}^{4}$ theory following quantum quenches. The results are compared to two different semiclassical approaches, the self-consistent Gaussian approximation and the truncated Wigner approximation, and used to determine the range of validity of these widely used approaches. We show that the self-consistent approximation is strongly limited in comparison to the truncated Hamiltonian method which for larger cutoffs is practically exact for the parameter range studied. We find that the self-consistent approximation is only valid when the effective mass is in the vicinity of the renormalized mass. Similarly to the self-consistent approximation, the truncated Wigner approximation (TWA) is not able to capture the correct mass renormalization, and breaks down for strong enough interactions where the bare mass becomes negative. We attribute the failure of TWA to the presence of a classical symmetry-broken fixed point. Aside from establishing the truncated Hamiltonian approach as a powerful tool for studying the dynamics of the ${\ensuremath{\phi}}^{4}$ model, our results on the limitation of semiclassical approximations are expected to be relevant for modeling the dynamics of other quantum field theories.
Abstract. We consider quantum quenches to the sinh-Gordon integrable quantum field theory from a particular class of initial states. Our analysis includes the case of mass and interaction quenches starting from a non-interacting theory. By means of the recently developed quench action method, we fully characterize the stationary state reached at long times after the quench in terms of the corresponding rapidity distribution. We also provide exact results for the expectation values of arbitrary vertex operators in the post-quench stationary state by proposing a formula based on the analogy with the standard thermodynamic Bethe ansatz. Finally, we comment on the behavior of the post-quench stationary state under the mapping between the sinh-Gordon field theory and the one-dimensional Lieb–Liniger model.
We expand on the investigation of the universal scaling properties in the early time behaviour of fast but smooth quantum quenches in a general d-dimensional conformal field theory deformed by a relevant operator of dimension Δ with a time-dependent coupling. The quench consists of changing the coupling from an initial constant value λ 1 by an amount of the order of δλ to some other final value λ 2, over a time scale δt. In the fast quench limit where δt is smaller than all other length scales in the problem, δt ≪ λ 1 1/() , λ 2 1/() , δλ 1/(Δ − d), the energy (density) injected into the system scales as δℰ ∼ (δλ)2(δt) d− 2Δ . Similarly, the change in the expectation value of the quenched operator at times earlier than the endpoint of the quench scales as
⟨OΔ⟩∼δλ(δt)d−2Δ , with further logarithmic enhancements in certain cases. While these results were first found in holographic studies, we recently demonstrated that precisely the same scaling appears in fast mass quenches of free scalar and free fermionic field theories. As we describe in detail, the universal scaling refers to renormalized quantities, in which the UV divergent pieces are consistently renormalized away by subtracting counterterms derived with an adiabatic expansion. We argue that this scaling law is a property of the conformal field theory at the UV fixed point, valid for arbitrary relevant deformations and insensitive to the details of the quench protocol. Our results highlight the difference between smooth fast quenches and instantaneous quenches where the Hamiltonian abruptly changes at some time.
One of the fundamental principles of statistical physics is that only partial information about a system's state is required for its macroscopic description. This is not only true for thermal ensembles, but also for the unconventional ensemble, known as Generalized Gibbs Ensemble (GGE), that is expected to describe the relaxation of integrable systems after a quantum quench. By analytically studying the quench dynamics in a prototypical one-dimensional critical model, the massless free bosonic field theory, we find evidence of a novel type of equilibration characterized by the preservation of an enormous amount of memory of the initial state that is accessible by local measurements. In particular, we show that the equilibration retains memory of non-Gaussian initial correlations, in contrast to the case of massive free evolution which erases all such memory. The GGE in its standard form, being a Gaussian ensemble, fails to predict correctly the equilibrium values of local observables, unless the initial state is Gaussian itself. Our findings show that the equilibration of a broad class of quenches whose evolution is described by Luttinger liquid theory with an initial state that is non-Gaussian in terms of the bosonic field, is not correctly captured by the corresponding bosonic GGE, raising doubts about the validity of the latter in general one-dimensional gapless integrable systems such as the Lieb-Liniger model. We also propose that the same experiment by which the GGE was recently observed [Langen et al., Science 348 (2015) 207-211] can also be used to observe its failure, simply by starting from a non-Gaussian initial state.
In this Thesis, the problem of a quantum quench in quantum field theories is analyzed. This involves studying the real time evolution of observables in a theory that undergoes a change in one of its couplings. These quenches are then characterized by two parameters:
deltalambda, the magnitude of the quench and most importantly,
deltat, the quench duration. In contrast to previous studies of abrupt quenches in the condensed matter theory community, we will be interested in smooth quenches with a finite
deltat.\nMotivated by existing results in holographic theories, we studied the problem of a fast smooth quench in free field theories by quenching the mass of a free scalar and a free Dirac fermion. For specific mass profiles, exact analytic answers were found. We found that expectation values for the quenched operators obey universal scaling properties. We provide both numerical and analytic evidence for this scaling to hold in free field theories and argue that the same scaling properties should hold independently of the theory, the coupling or the quench profile. In fact, we show that the renormalized expectation value of an operator ${\\cal{O}}_\\Delta$ of dimension
Delta should scale as $\\vev{{\\cal{O}}_\\Delta}_{ren} \\sim \\delta \\lambda/(\\delta t)^{2\\Delta -d}$, when the quench rate is fast compared to the quench amplitude. This growth is further enhanced by a logarithmic factor in even dimensions.\nThis result suggests that, for
Delta>d/2, expectation values will diverge in the limit of
deltatto0, which seem to contradict previous studies of abrupt quenches. In this Thesis, we carefully analzye the relation between the two approaches and establish restrictions to the kind of objects that can be studied using the abrupt approximation.\nApart from the fast smooth quench, another regime of interest is the slow quench through a critical point. In free field theories we found that adiabatic behaviour breaks down when the system is close enough to the critical point and renormalized expectation values scale different, following expectations from the Kibble-Zurek argument. Given any finite fixed time, we are able to show how expectation values scale at any value of the quench duration
deltat.
Abstract In many quantum quench experiments involving cold atom systems the post-quench phase can be described by a quantum field theory of free scalars or fermions, typically in a box or in an external potential. We will study mass quench of free scalars in arbitrary spatial dimensions d with particular emphasis on the rate of relaxation to equilibrium. Local correlators expectedly equilibrate to GGE; for quench to zero mass, interestingly the rate of approach to equilibrium is exponential or power law depending on whether d is odd or even respectively. For quench to non-zero mass, the correlators relax to equilibrium by a cosine-modulated power law, for all spatial dimensions d, even or odd. We briefly discuss generalization to O(N ) models.