Loschmidt amplitude and work distribution in quenches of the sine-Gordon model

Colin Rylands and Natan Andrei

Phys. Rev. B 99, 085133 – Published 22 February 2019

Abstract

The sine-Gordon—equivalently, the massive Thirring—Hamiltonian is ubiquitous in low-dimensional physics, with applications that range from cold atom and strongly correlated systems to quantum impurities. We study here its nonequilibrium dynamics using the quantum quench protocol—following the system as it evolves under the sine-Gordon Hamiltonian from initial Mott-type states with large potential barriers. By means of the Bethe ansatz, we calculate exactly the Loschmidt amplitude, the fidelity, and work distribution characterizing these quenches for different values of the interaction strength. Some universal features are noted as well as an interesting duality relating quenches in different parameter regimes of the model.

Received 29 September 2018

DOI:https://doi.org/10.1103/PhysRevB.99.085133

Physics Subject Headings (PhySH)

Integrability in field theory Quantum quench Work theorems

Nonequilibrium systems

Bethe ansatz

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Colin Rylands^* and Natan Andrei

Department of Physics, Rutgers University, Piscataway, New Jersey 08854, USA

^*rylands@physics.rutgers.edu

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Issue

Vol. 99, Iss. 8 — 15 February 2019

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Images

Figure 1
A typical quantum quench protocol occurring in cold atom systems. Atoms are initially held in a deep optical lattice. The lattice depth is then suddenly lowered and the system allowed to evolve. The sine-Gordon/massive Thirring model is an effective description of this process.
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Figure 2
The light cone lattice consists of left- and right-oriented diagonal lines representing the world lines of bare relativistic particles. The vertical direction is time and the horizontal is space with $δ$ being the lattice spacing in both directions. The number of intersections in the spacelike direction is $N$ while the number in the timelike direction is $M$ . To each intersection, we associate a matrix of transition amplitudes $R_{k, k + 1} (2 Θ)$ . We assume periodic boundary conditions in the spatial direction.
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Figure 3
(a)The Loschmidt amplitude as given in Eq. (13) takes the form of the partition function of a 2D classical lattice with the initial state appearing as boundaries along the top and bottom. Each horizontal line represents $τ (Θ)$ or $τ (η - Θ)$ . The trace present in $τ (u)$ , Eq. (10) represents periodic boundary conditions imposed in the spacelike direction. (b) Under a rotation, the amplitude is then the partition function of a different 2D classical lattice model with $N$ and $M$ exchanged and the initial conditions becoming spatial boundaries. Each horizontal line here represents $T (Θ)$ or $T (- Θ)$ , given by Eq. (15). Although the partition functions calculated using the viewpoint in (a) or (b) are the same, the transfer matrices in either case are not the same owing to the fact that the model is Lorentz invariant rather than Euclidean invariant.
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Figure 4
By choosing the initial state appropriately, the Loschmidt amplitude simplifies considerably. If $|Φ_{i}〉 = {|v〉}^{N}$ with $v$ given in Eq. (17), only two horizontal lines are coupled. The amplitude then reduces to the form Eq. (18).
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Figure 5
Plot of rescaled work distribution function at different values of $λ = 8 π / β^{2} - 1 = γ / (π - γ)$ in the region of the threshold for $ξ = i π / 2$ . The distribution has been rescaled by the fidelity and system size, $p (w) = 4 π P (W) / (m L F)$ , and is plotted as a function of work measured from $δ E$ in units of $m, w = (W - δ E) / m$ .
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Figure 6
Plot of rescaled work distribution function, $p (w)$ , at different values of $λ = 8 π / β^{2} - 1 = γ / (π - γ)$ in the region of the threshold for $ξ = 0$ . The distribution has been rescaled by the fidelity and system size, $p (w) = 4 π P (W) / (m L F)$ , and is plotted as a function of work measured from $δ E$ in units of $m, w = (W - δ E) / m$ . We see that the distribution exhibits an edge singularity with exponent $- 3 / 2$ and becomes more strongly peaked at the threshold as $λ \to 1 / 2$ .
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