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arXiv:1512.02187

Thermalization in 2D critical quench and UV/IR mixing

Gautam Mandal, Shruti Paranjape, Nilakash Sorokhaibam

Published Dec 7, 2015• 4 citations
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Abstract

We consider quantum quenches in models of free scalars and fermions with a generic time-dependent mass $m(t)$ that goes from $m_0$ to zero. We prove that, as anticipated in MSS \cite{Mandal:2015jla}, the post-quench dynamics can be described in terms of a state of the generalized Calabrese-Cardy form $|ψ\rangle$= $\exp[-κ_2 H -\sum_{n>2}^\infty κ_n W_n]| \hbox{Bd} \rangle$. The $W_n$ ($n=2,3,...$, $W_2=H$) here represent the conserved $W_\infty$ charges and $| \hbox{Bd} \rangle$ represents a conformal boundary state. Our result holds irrespective of whether the pre-quench state is a ground state or a squeezed state, and is proved without recourse to perturbation expansion in the $κ_n$'s as in MSS. We compute exact time-dependent correlators for some specific quench protocols $m(t)$. The correlators explicitly show thermalization to a generalized Gibbs ensemble (GGE), with inverse temperature $β= 4κ_2$, and chemical potentials $μ_n=4κ_n$. In case the pre-quench state is a ground state, it is possible to retrieve the exact quench protocol $m(t)$ from the final GGE, by an application of inverse scattering techniques. Another notable result, which we interpret as a UV/IR mixing, is that the long distance and long time (IR) behaviour of some correlators depends crucially on all $κ_n$'s, although they are highly irrelevant couplings in the usual RG parlance. This indicates subtleties in RG arguments when applied to non-equilibrium dynamics.

DOI
10.48550/arxiv.1512.02187

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We consider quantum quenches in models of free scalars and fermions with a generic time-dependent mass m(t)m(t)m(t) that goes from m0m_0m0​ to zero. We prove that, as anticipated in MSS \cite{Mandal:2015jla}, the post-quench dynamics can be described in terms of a state of the generalized Calabrese-Cardy form ∣ψ⟩|ψ\rangle∣ψ⟩= exp⁡[−κ2H−∑n>2∞κnWn]∣Bd⟩\exp[-κ_2 H -\sum_{n>2}^\infty κ_n W_n]| \hbox{Bd} \rangleexp[−κ2​H−∑n>2∞​κn​Wn​]∣Bd⟩. The WnW_nWn​ (n=2,3,...n=2,3,...n=2,3,..., W2=HW_2=HW2​=H) here represent the conserved W∞W_\inftyW∞​ charges and ∣Bd⟩| \hbox{Bd} \rangle∣Bd⟩ represents a conformal boundary state. Our result holds irrespective of whether the pre-quench state is a ground state or a squeezed state, and is proved without recourse to perturbation expansion in the κnκ_nκn​'s as in MSS. We compute exact time-dependent correlators for some specific quench protocols m(t)m(t)m(t). The correlators explicitly show thermalization to a generalized Gibbs ensemble (GGE), with inverse temperature β=4κ2β= 4κ_2β=4κ2​, and chemical potentials μn=4κnμ_n=4κ_nμn​=4κn​. In case the pre-quench state is a ground state, it is possible to retrieve the exact quench protocol m(t)m(t)m(t) from the final GGE, by an application of inverse scattering techniques. Another notable result, which we interpret as a UV/IR mixing, is that the long distance and long time (IR) behaviour of some correlators depends crucially on all κnκ_nκn​'s, although they are highly irrelevant couplings in the usual RG parlance. This indicates subtleties in RG arguments when applied to non-equilibrium dynamics.
We consider quantum quenches in models of free scalars and fermions with a generic time-dependent mass m(t) that goes from m0 to zero. We prove that, as anticipated in MSS [1], the post-quench dynamics can be described in terms of a state of the generalized Calabrese-Cardy form |ψ〉 = exp[−κ2H − ∑ >2 ∞ κ n W n ]|Bd〉. The W n (n = 2, 3, . . ., W2 = H) here represent the conserved W∞ charges and |Bd〉 represents a conformal boundary state. Our result holds irrespective of whether the pre-quench state is a ground state or a squeezed state, and is proved without recourse to perturbation expansion in the κ n ’s as in MSS. We compute exact time-dependent correlators for some specific quench protocols m(t). The correlators explicitly show thermalization to a generalized Gibbs ensemble (GGE), with inverse temperature β = 4κ2, and chemical potentials μ n = 4κ n . In case the pre-quench state is a ground state, it is possible to retrieve the exact quench protocol m(t) from the final GGE, by an application of inverse scattering techniques. Another notable result, which we interpret as a UV/IR mixing, is that the long distance and long time (IR) behaviour of some correlators depends crucially on all κ n ’s, although they are highly irrelevant couplings in the usual RG parlance. This indicates subtleties in RG arguments when applied to non-equilibrium dynamics.
We study the unitary dynamics and the thermalization properties of free-fermion-like Hamiltonians after a sudden quantum quench, extending the results of S. Ziraldo et al. [Phys. Rev. Lett. 109, 247205 (2012)]. With analytical and numerical arguments, we show that the existence of a stationary state and its description with a generalized Gibbs ensemble (GGE) depend crucially on the observable considered (local versus extensive) and on the localization properties of the final Hamiltonian. We present results on two one-dimensional (1D) models, the disordered 1D fermionic chain with long-range hopping and the disordered Ising/XYXYXY spin chain. We analytically prove that, while time averages of one-body operators are perfectly reproduced by GGE (even for finite-size systems, if time integrals are extended beyond revivals), time averages of many-body operators might show clear deviations from the GGE prediction when disorder-induced localization of the eigenstates is at play.
In this article, we study the quantum field theoretic generalization of the Caldeira-Leggett model in general curved space-time considering interactions between two scalar fields in a classical gravitational background. The thermalization phenomena is then studied from the obtained de Sitter solution using quantum quench from one scalar field model obtained from path integrated effective action. We consider an instantaneous quench in the time-dependent mass protocol of the field of our interest. We find that the dynamics of the field post-quench can be described in terms of the state of the generalized Calabrese-Cardy (gCC) form and computed the different types of two-point correlation functions in this context. We explicitly found the conserved charges of W∞ algebra that represents the gCC state after a quench in de Sitter space and found it to be significantly different from the flat space-time results. We extend our study for the different two-point correlation functions not only considering the pre-quench state as the ground state, but also a squeezed state. We found that irrespective of the pre-quench state, the post quench state can be written in terms of the gCC state showing that the subsystem of our interest thermalizes in de Sitter space. Furthermore, we provide a general expression for the two-point correlators and explicitly show the thermalization process by considering a thermal Generalized Gibbs ensemble (GGE). Finally, from the equal time momentum dependent counterpart of the obtained results for the two-point correlators, we have studied the hidden features of the power spectra and studied its consequences for different choices of the quantum initial conditions.
In this article, we study the quantum field theoretic generalization of the Caldeira-Leggett model in general curved space-time considering interactions between two scalar fields in a classical gravitational background. The thermalization phenomena is then studied from the obtained de Sitter solution using quantum quench from one scalar field model obtained from path integrated effective action. We consider an instantaneous quench in the time-dependent mass protocol of the field of our interest. We find that the dynamics of the field post-quench can be described in terms of the state of the generalized Calabrese-Cardy (gCC) form and computed the different types of two-point correlation functions in this context. We explicitly found the conserved charges of algebra that represents the gCC state after a quench in de Sitter space and found it to be significantly different from the flat space-time results. We extend our study for the different two-point correlation functions not only considering the pre-quench state as the ground state, but also a squeezed state. We found that irrespective of the pre-quench state, the post quench state can be written in terms of the gCC state showing that the subsystem of our interest thermalizes in de Sitter space. Furthermore, we provide a general expression for the two-point correlators and explicitly show the thermalization process by considering a thermal Generalized Gibbs ensemble (GGE). Finally, from the equal time momentum dependent counterpart of the obtained results for the two-point correlators, we have studied the hidden features of the power spectra and studied its consequences for different choices of the quantum initial conditions.
At thermal equilibrium, the concept of effective central charge for massive deformations of two-dimensional conformal field theories (CFT) is well understood, and can be defined by comparing the partition function of the massive to that of a CFT. This temperature-dependent effective charge interpolates monotonically between the central charge values corresponding to the IR and UV fixed points at low and high temperatures, respectively. We propose a non-equilibrium, time-dependent generalization of the effective central charge for integrable models after a quantum quench, ceff(t)c_{\rm eff}(t)ceff​(t), obtained by comparing the return amplitude to that of a CFT quench. We study this proposal for a large mass quench of a free boson, where the charge is seen to interpolate between ceff=0c_{\rm eff}=0ceff​=0 at t=0t=0t=0, and ceff∼1c_{\rm eff}\sim 1ceff​∼1 at t→∞t\to\inftyt→∞, as is expected. We use our effective charge to define an to quench protocol, where the charge evolves from ceff=1/2c_{\rm eff}=1/2ceff​=1/2 at t=0t=0t=0, to ceff=7/10c_{\rm eff}=7/10ceff​=7/10 at t→∞t\to\inftyt→∞, the corresponding values of the first two unitary minimal CFT models. We then argue that the inverse Tricritical Ising to quench is impossible with our methods. These conclusions can be generalized for quenches between any two adjacent unitary minimal CFT models. We finally study a large mass quench into the model (sinh-Gordon with a particular complex coupling). At short times after the quench, the effective central charge increases in a discrete staircase structure, where the values of the charge at the steps can be computed in terms of the central charges of unitary minimal CFT models. When the initial state is a pure state, one always finds that ceff(t→∞)≥ceff(t=0)c_{\rm eff}(t\to\infty)\geq c_{\rm eff}(t=0)ceff​(t→∞)≥ceff​(t=0), though ceff(t)c_{\rm eff}(t)ceff​(t), generally oscillates at finite times. We explore how this constraint may be related to RG flow irreversibility.
At thermal equilibrium, the concept of effective central charge for massive deformations of two-dimensional conformal field theories (CFT) is well understood, and can be defined by comparing the partition function of the massive model to that of a CFT. This temperature-dependent effective charge interpolates monotonically between the central charge values corresponding to the IR and UV fixed points at low and high temperatures, respectively. We propose a non-equilibrium, time-dependent generalization of the effective central charge for integrable models after a quantum quench, ceff(t)c_{\rm eff}(t)ceff​(t), obtained by comparing the return amplitude to that of a CFT quench. We study this proposal for a large mass quench of a free boson, where the charge is seen to interpolate between ceff=0c_{\rm eff}=0ceff​=0 at t=0t=0t=0, and ceff∼1c_{\rm eff}\sim 1ceff​∼1 at t→∞t\to\inftyt→∞, as is expected. We use our effective charge to define an "Ising to Tricritical Ising" quench protocol, where the charge evolves from ceff=1/2c_{\rm eff}=1/2ceff​=1/2 at t=0t=0t=0, to ceff=7/10c_{\rm eff}=7/10ceff​=7/10 at t→∞t\to\inftyt→∞, the corresponding values of the first two unitary minimal CFT models. We then argue that the inverse "Tricritical Ising to Ising" quench is impossible with our methods. These conclusions can be generalized for quenches between any two adjacent unitary minimal CFT models. We finally study a large mass quench into the "staircase model" (sinh-Gordon with a particular complex coupling). At short times after the quench, the effective central charge increases in a discrete "staircase" structure, where the values of the charge at the steps can be computed in terms of the central charges of unitary minimal CFT models. When the initial state is a pure state, one always finds that ceff(t→∞)≥ceff(t=0)c_{\rm eff}(t\to\infty)\geq c_{\rm eff}(t=0)ceff​(t→∞)≥ceff​(t=0), though ceff(t)c_{\rm eff}(t)ceff​(t), generally oscillates at finite times. We explore how this constraint may be related to RG flow irreversibility.