[Paper Review] Dynamical Quantum Phase transitions and Recurrences in the Non-Equilibrium BCS model

Colin Rylands, Victor Galitski|arXiv (Cornell University)|Jan 27, 2020

Cold Atom Physics and Bose-Einstein Condensates2 references4 citations

TL;DR

This paper analytically demonstrates that dynamical quantum phase transitions (DQPTs) in the non-equilibrium BCS model occur precisely at each local maximum of the superconducting order parameter's oscillations, which are soliton-like. These DQPTs are first-order and emerge only in the thermodynamic limit; when the order parameter reaches a steady, constant value, DQPTs vanish entirely.

ABSTRACT

Non-equilibrium aspects of the BCS model have fascinated physicists for decades, from the seminal works of Eliashberg to modern realizations in cold atom experiments. The latter scenarios have lead to a great deal of interest in the quench dynamics of fermions with pairing interactions. The recently introduced notion of a dynamical quantum phase transition is an attempt to classify the myriad of possible phenomena which can result in such far from equilibrium systems. These are defined as non-analytic points of the logarithm of the Loschmidt echo and are linked to oscillations in the dynamics a systems order parameter. In this work we analytically investigate the relation between DQPTs and oscillation of the superconducting order parameter in quenches of the BCS model. We find that each oscillation of the order parameter is accompanied by a DQPT which is first order in nature. We show this for a variety of initial states and furthermore find that when the order parameter attains a constant steady state then no DQPTS occur.

Motivation & Objective

• To establish a rigorous analytical connection between dynamical quantum phase transitions (DQPTs) and oscillatory behavior of the superconducting order parameter in the non-equilibrium BCS model.
• To investigate whether DQPTs emerge in systems with time-periodic order parameter oscillations, particularly in the context of quenched BCS Hamiltonians.
• To determine under what conditions DQPTs are absent, especially when the system evolves into a steady state with constant order parameter.
• To extend the understanding of DQPTs beyond numerical studies by providing exact analytical treatment in the thermodynamic limit.
• To clarify the role of initial state preparation (ground or excited states) in determining the presence or absence of DQPTs.

Proposed method

• Analytically compute the Loschmidt echo $ L(t) = |G(t)|^2 $, where $ G(t) = raket{\Psi_i|e^{-iH_f t}|\Psi_i} $, using exact solutions of the quenched BCS model.
• Employ the Lax vector formalism to solve the dynamics of the mean-field BCS Hamiltonian after a quench, mapping it to a classically integrable spin system.
• Derive the time evolution of the order parameter $ \Delta(t) $, showing it exhibits soliton-like oscillations with period $ t_{\text{DQPT}} $ when $ \Delta_- \ll \Delta_+ $.
• Analyze the analytic structure of $ G(t) $ in the complex plane, identifying zeros on the real axis as signatures of DQPTs.
• Use the condition $ \mathcal{K}(\epsilon_p) e^{-2it\sqrt{\epsilon_p^2 + \Delta_\infty^2}} = -1 $ to determine whether DQPTs can occur in the steady-state regime.
• Evaluate the magnitude of $ \mathcal{K}(\epsilon) $, proving $ |\mathcal{K}(\epsilon)| < 1 $, which rules out the existence of DQPTs in the constant-order-parameter limit.

Experimental results

Research questions

• RQ1 Does each oscillation of the superconducting order parameter in the quenched BCS model correspond to a dynamical quantum phase transition (DQPT)?
• RQ2 What is the nature (e.g., first- or second-order) of the DQPTs that occur during order parameter oscillations?
• RQ3 Under what conditions do DQPTs vanish, particularly when the system reaches a steady state with a constant order parameter?
• RQ4 How does the initial state (ground or excited) affect the occurrence and structure of DQPTs in the non-equilibrium BCS model?
• RQ5 Can the connection between DQPTs and order parameter dynamics be established analytically, rather than numerically, in the thermodynamic limit?

Key findings

• DQPTs occur exactly at each local maximum of the oscillating superconducting order parameter $ \Delta(t) $, with the period of these DQPTs matching the oscillation period of $ \Delta(t) $.
• The DQPTs are first-order in nature, as confirmed by the non-analytic behavior of $ \log L(t) $ at these points, and they only emerge in the thermodynamic limit.
• When the order parameter approaches a constant steady-state value $ \Delta_\infty $, no DQPTs occur, as the condition for a DQPT cannot be satisfied due to $ |\mathcal{K}(\epsilon)| < 1 $.
• The analytical expression for the Loschmidt echo $ L(t) $ in the steady-state regime confirms the absence of DQPTs, as the phase factor never reaches $ -1 $ for any $ t $.
• The results hold for a wide range of initial states, including both ground and excited states, demonstrating robustness of the DQPT-order parameter connection.
• The study provides the first analytical treatment of DQPTs in the BCS model, confirming earlier numerical observations and extending them to exact analytical results in the thermodynamic limit.