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, c_{\rm eff}(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 effective charge is seen to interpolate between c_{\rm eff}=0 at t=0 t = 0 , and c_{\rm eff}\sim 1 at t\to\infty t → ∞ , as is expected. We use our effective charge to define an “Ising to Tricritical Ising" quench protocol, where the charge evolves from c_{\rm eff}=1/2 at t=0 t = 0 , to c_{\rm eff}=7/10 at t\to\infty t → ∞ , 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 c_{\rm eff}(t\to\infty)\geq c_{\rm eff}(t=0) , though c_{\rm eff}(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, c_{\rm eff}(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 effective charge is seen to interpolate between c_{\rm eff}=0 at t=0 t = 0 , and c_{\rm eff}\sim 1 at t\to\infty t → ∞ , as is expected. We use our effective charge to define an “Ising to Tricritical Ising" quench protocol, where the charge evolves from c_{\rm eff}=1/2 at t=0 t = 0 , to c_{\rm eff}=7/10 at t\to\infty t → ∞ , 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 c_{\rm eff}(t\to\infty)\geq c_{\rm eff}(t=0) , though c_{\rm eff}(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), 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=0 at t=0, and ceff∼1 at t→∞, as is expected. We use our effective charge to define an "Ising to Tricritical Ising" quench protocol, where the charge evolves from ceff=1/2 at t=0, to ceff=7/10 at t→∞, 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), though 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 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), 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=0 at t=0, and ceff∼1 at t→∞, as is expected. We use our effective charge to define an to quench protocol, where the charge evolves from ceff=1/2 at t=0, to ceff=7/10 at t→∞, 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), though ceff(t), generally oscillates at finite times. We explore how this constraint may be related to RG flow irreversibility.
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 \cite{Mandal:2015jla}, the post-quench dynamics can be described in terms of a state of the generalized Calabrese-Cardy form ∣ψ⟩= exp[−κ2H−∑n>2∞κnWn]∣Bd⟩. The Wn (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 κn'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 μn=4κn. 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 κn'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 a quantum quench in a finite system of length L described by a 1+1-dimensional conformal field theory (CFT), of central charge c, from a state with finite energy density corresponding to an inverse temperature $\ensuremath{\beta}\ensuremath{\ll}L$. For times t such that $\ensuremath{\ell}/2<t<(L\ensuremath{-}\ensuremath{\ell})/2$ the reduced density matrix of a subsystem of length $\ensuremath{\ell}$ is exponentially close to a thermal density matrix. We compute exactly the overlap F of the state at time t with the initial state and show that in general it is exponentially suppressed at large $L/\ensuremath{\beta}$. However, for minimal models with c<1 (more generally, rational CFTs), at times which are integer multiples of L/2 (for periodic boundary conditions, L for open boundary conditions) there are (in general, partial) revivals at which F is O(1), leading to an eventual complete revival with F=1. There is also interesting structure at all rational values of t/L, related to properties of the CFT under modular transformations. At early times $t\ensuremath{\ll}(L\ensuremath{\beta}{)}^{1/2}$ there is a universal decay $\mathcal{F}\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}(\ensuremath{\pi}c/3)L{t}^{2}/\ensuremath{\beta}({\ensuremath{\beta}}^{2}+4{t}^{2}))$. The effect of an irrelevant nonintegrable perturbation of the CFT is to progressively broaden each revival at t=nL/2 by an amount O(n1/2).
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.
In this thesis we study the time dependence of correlators such as Green function, entanglement (Renyi) entropy and mutual information following quantum quench in Conformal Field Theories (CFTs) and their holographic duals (AdS).
We develop a new recursive method to analytically study the thermalization of retarded Green function in CFTs holographically dual to thin shell AdS Vaidya space times. Using this method, analytic results for short time thermalization of the Green function are obtained; its long time behaviour is obtained by implementing this method numerically. This method is also used to study the late time behaviour of the minimally coupled scalar in AdS3 Vaidya and the shear vector mode in AdS5 Vaidya. Studying the latter, we obtain the universal ratio of the shear viscosity to entropy density from a time dependent process.
We study the time evolution of single interval Renyi and entanglement entropies following local quantum quenches in CFTs at finite temperature for which the locally excited states have a finite temporal width. We show that the time dependence of Renyi and entanglement entropies, at second order in width, is universal and is determined by the expectation value of the stress tensor in the replica geometry.
We study local quenches in two dimensional CFT at large central charge by operators carrying higher spin charge. The holographic description is used to obtain the single interval entanglement entropy, mutual information and scrambling time following the quench. We find that the change in entanglement entropy is finite (and real) only if the spin-three charge, q, is bounded by the energy of the perturbation, E, as |q|/c < E^2/c^2. For larger values of the spin-three charge, the scrambling time is shorter than for pure gravity and controlled by the spin-three Lyapunov exponent.