Work and quantum phase transitions: Quantum latency

E. Mascarenhas, H. Bragança, R. Dorner, M. França Santos, V. Vedral, K. Modi, and J. Goold

Phys. Rev. E 89, 062103 – Published 3 June 2014

Abstract

We study the physics of quantum phase transitions from the perspective of nonequilibrium thermodynamics. For first-order quantum phase transitions, we find that the average work done per quench in crossing the critical point is discontinuous. This leads us to introduce the quantum latent work in analogy with the classical latent heat of first order classical phase transitions. For second order quantum phase transitions the irreversible work is closely related to the fidelity susceptibility for weak sudden quenches of the system Hamiltonian. We demonstrate our ideas with numerical simulations of first, second, and infinite order phase transitions in various spin chain models.

Received 26 March 2014

DOI:https://doi.org/10.1103/PhysRevE.89.062103

Authors & Affiliations

E. Mascarenhas¹, H. Bragança¹, R. Dorner^2,3, M. França Santos¹, V. Vedral^3,4,5, K. Modi⁶, and J. Goold⁷

¹Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil
²Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom
³Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
⁴Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543
⁵Department of Physics, National University of Singapore, 2 Science Drive 2, Singapore 117543
⁶School of Physics, Monash University, VIC 3800, Australia
⁷The Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy

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Vol. 89, Iss. 6 — June 2014

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Images

Figure 1
Schematic representation of a level crossing giving rise to a first-order QPT and an avoided crossing leading to a second-order QPT as discussed in the main text. The corresponding first- and second-order derivatives of the ground-state energy are also presented. For $ε = 0$ , the ground and excited states of the Landau-Zener Hamiltonian [Eq. (3)] exhibit a level crossing as the external field is tuned through the critical value $λ_{c} = Δ / (2 a)$ , while for $ε \neq 0$ , the energy levels exhibit an avoided crossing. In the case of the level crossing, both derivatives are discontinuous. For the avoided crossing, the first and second derivatives are continuous for finite $ε$ and become discontinuous in the limit $ε \to 0$ as the turning point in the ground-state energy becomes a kink. As an aside, we note that the Landau-Zener Hamiltonian is isomorphous to the Ising mean-field Hamiltonian, which is accurate for the infinitely connected many-body lattice [3].
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Figure 2
Density matrix renormalization group (DMRG) data (in units of $J = 1$ ) for the average work (filled blue circles) and irreversible work (empty green circles) done in a weak sudden quench of the $X X Z$ Hamiltonian with 112 sites and $δ λ = 10^{- 5}$ . Here, we assume the presence of a small local energy shift at one lattice site, which lifts the degeneracy in the ground state of the Hamiltonian. The data has a DMRG truncation error and energy convergence of $\approx 10^{- 9}$ . The quench protocol we consider is an instantaneous change of the $Z$ -coupling $δ λ = λ_{f} - λ_{i} ≪ 1$ , with the system initialized in the ground state of the initial Hamiltonian. Both the average and irreversible work display a discontinuity at the critical point of the first-order transition at $λ = - 2$ . On the left-hand side of the first-order transition, the gapped spectrum of the ferromagnetic phase enforces adiabaticity as the system cannot be taken out of equilibrium by weak quenches within the same phase. This means that the system is not excited by the quench and the sole contribution to the average work is the change in internal energy. The discontinuities at the critical point indicate work required to drive the system from the ferromagnetic phase to the Luttinger liquid phase. The new phase is characterized by a continuous energy spectrum and so subsequent quenches can excite the system, leading to the dissipation of work. In contrast, neither the average work nor the irreversible work indicate the Berezinskii-Kosterlitz-Thouless transition at $λ = 2$ .
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Figure 3
Exact numerical results for the average work and irreversible work induced by a sudden quench of the external magnetic field $h$ in the $X X$ model. This model incorporates a second-order QPT from a ferromagnetic phase for $h / J ≪ 1$ to a paramagnetic phase for $h / J ≫ 1$ , with the critical point occurring when the external field is two times the internal coupling $h = 2 J$ . The phase diagram exhibits a discontinuity in the average work and a divergent irreversible work at the critical point. This behavior reflects the relationship between the irreversible work and fidelity susceptibility discussed in the main text.
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