Mass quench in interacting finite-size quantum field system

The gist: This work theoretically investigates the non-equilibrium evolution of correlation functions for a quantum scalar field following a mass quantum quench in a finite-size system, revealing that at time approximately equal to half the system size, the effective mass experiences a strong disturbance due to interference between counter-propagating wave fronts of fluctuations induced by periodic boundary conditions.

Theoretical Framework and Formalism

The study employs the Keldysh–Schwinger formalism to describe post-quench evolution The Keldysh–Schwinger formalism is the theoretical foundation for describing post-quench evolution. An adequate description of such systems requires a formalism capable of simultaneously and self-consistently accounting for both the change in the energy spectrum of elementary excitations (spectral characteristics) and the evolution of the population of these modes (statistical characteristics). The present work considers a one-dimensional real scalar quantum field theory in a system of length L with periodic boundary conditions described by the Hamiltonian. The time dependence of the mass is given as a step function modeling the quench.

The evolution of the expectation value of an observable Oˆ = O(ˆφ) is given by the expression tr rˆ0OˆH(t) = tr rˆ0Uˆ† (t, 0)OˆUˆ(t, 0). The matrix elements of the evolution operators from 0 to t and back can be expressed as path integrals over the fields φ F (t, x) and φ B (t, x), where the indices F and B denote the forward and backward branches of the closed time path. For convenience, we introduce symmetrized (“classical”) φ cl and antisymmetrized (“quantum”) φ q integration variables in the spacetime functional integral (hereafter ħ = 1). The Keldysh action, defined as the difference of classical actions on the two branches of the closed time path, in this case takes the following form.

Correlation Functions and Equations of Motion

The main objects of research are two-point correlation functions. The Keldysh (statistical) Green’s function is given by the expression ⟨φ cl(t1, x1)φ cl(t2, x2)⟩ = F(t1, x1;t2, x2) = 1/2 tr rˆ0 φˆ(t1, x1), φˆ(t2, x2). The Green’s spectral function ρ(t1, x1;t2, x2), which determines the causal dynamics and commutation relations, is expressed as ⟨φ cl(t1, x1)φ q(t2, x2)⟩ = -iθ(t1 − t2)ρ(t1, x1;t2, x2) = θ(t1 − t2)tr rˆ0[ˆφ(t1, x1), φˆ(t2, x2)]. The Keldysh and spectral functions possess the following (anti)symmetry properties F(t1, x1;t2, x2) = F(t2, x2;t1, x1), ρ(t1, x1;t2, x2) = -ρ(t2, t x; t x).

For a spatially uniform distribution with a zero mean field (⟨φ⟩ = 0), applying the twoparticle-irreducible effective action formalism or, equivalently, the partial resummation procedure of diagrams entering the Dyson–Schwinger equations in the one-loop (Hartree–Fock) approximation leads to a closed system of differential equations. The explicit form of the Dyson equations in this approximation is = + ΣR + ΣA (11) = + ΣR (12) where ΣR = + o(λ) (13). Here, a straight line is a Keldysh function ⟨φ cl(t1, x1)φ cl(t2, x2)⟩, and an arrow line is a retarded function ⟨φ cl(t1, x1)φ q(t2, x2)⟩. The effective mass Meff(t) is determined self-consistently via the sum over all momentum modes of the Keldysh correlation function at equal times M2eff(t) = m21 + δm2 + λ/2L X q Fq(t, t).

Initial Conditions and Numerical Parameters

The initial conditions for the Keldysh function Fp(t1, t2) are determined by the Wigner functional of the initial state. In this case: Fp(0, 0) = 1/2ω0,p coth ω0,p 2T. The initial conditions for the spectral function ρp(t1, t2) are determined by the canonical commutation relations and do not depend on the initial state: ρp(0, 0) = 0.

In the calculations, we employed the following dimensionless system parameters: number of lattice sites N = 512, coupling constant λ = 10, pre-quench initial mass m0 = 1.5, post-quench final mass m1 = 1, and temperature T = 0.1. To investigate effects related to periodic boundary conditions, different values of the spatial system size L = 10, 30, 60 were analyzed.

Interference Effect and Light Cone Dynamics

The primary result of the numerical analysis of the system’s post-quench evolution is the discovery of a sharp jump in the behavior of the effective mass Meff(t) at time t ≈ L/2. To effectively explain the physical nature of this phenomenon, it is useful to move from the momentum representation to a spatial analysis of the retarded Green’s function on the ring.

Due to the periodic boundary conditions, the relation between the spectral function on the ring ρring j(t) = ρring(t, ja; 0, 0) (where j is the spatial lattice site index) and the function on an infinite line ρline j(t) can be expressed as an infinite sum over sectors ρring j(t) = X∞ n=−∞ ρline j+nN (t). Due to microcausality in QFT, the spectral function on the line strictly vanishes outside the light cone (in units where c = 1) ρline j+nN(t) = 0 for j + nN · a > t. As seen from Eq. (26), at t = L/2, the summation index n on the right-hand side takes not only the value zero, as was the case for t < L/2, but also n = ±1. The spacetime dependence of the retarded Green’s function (exhibiting the characteristic light cone structure) is shown in Fig. 2. A comparison between the results obtained on the ring and on the infinite line is shown in Fig. 3. The behavior of the retarded function on the line was simulated on a ring whose size was four times larger than the original one. As can be seen from the plots, the calculation results obtained by the two methods coincide with high precision. Thus, this effect is the result of interference between counter-propagating fluctuations that meet at the same point in space due to periodic boundary conditions.

Verification of Results

To verify that the above-described change in the behavior of the effective mass is not an artifact of the numerical computation scheme, we calculate the Keldysh function using a different method. This involves replacing the loop in the self-energy with this computed value, which we denote by a square in the diagrams.

1. A new AI system can accurately predict a sharp jump in the behavior of the effective mass Meff(t) at time t ≈ L/2 by modeling its dynamics using interference between counter-propagating wave fronts of fluctuations.

2. The improved system can simulate finite-size quantum field effects in non-equilibrium scenarios, allowing it to analyze how the relation between the spectral function on the ring ρ ring j(t) and the function on an infinite line ρ line j(t) can be expressed as an infinite sum over sectors.

3. The AI system can perform self-consistent calculations of correlation functions by utilizing a method that replaces the loop in the self-energy with this computed value to derive the desired expression for the Keldysh function Fp(t1, t2) with high precision.

4. The system can verify numerical results against theoretical frameworks by comparing two different calculation methods, showing that the solutions agree with very high precision.

In this work, we theoretically investigate the non-equilibrium evolution of the correlation functions of a quantum scalar field following a mass quantum quench in a finite-size system. The self-consistent equation for the Keldysh (statistical) correlation function is solved numerically in the one-loop approximation. A specific spatio-temporal effect is observed: at a point in time approximately equal to half the system size, the effective mass experiences a strong disturbance. It is shown that this effect is driven by the interference of counter-propagating wave fronts of fluctuations that propagate within the light cone and is induced by periodic boundary conditions.