Quantum Quenches in Extended Systems (0704.1880v2)

Abstract: We study in general the time-evolution of correlation functions in a extended quantum system after the quench of a parameter in the hamiltonian. We show that correlation functions in d dimensions can be extracted using methods of boundary critical phenomena in d+1 dimensions. For d=1 this allows to use the powerful tools of conformal field theory in the case of critical evolution. Several results are obtained in generic dimension in the gaussian (mean-field) approximation. These predictions are checked against the real-time evolution of some solvable models that allows also to understand which features are valid beyond the critical evolution. All our findings may be explained in terms of a picture generally valid, whereby quasiparticles, entangled over regions of the order of the correlation length in the initial state, then propagate with a finite speed through the system. Furthermore we show that the long-time results can be interpreted in terms of a generalized Gibbs ensemble. We discuss some open questions and possible future developments.

• The paper reveals that post-quench dynamics are dominated by quasiparticle propagation, which triggers light-cone effects in correlation functions.
• The paper employs boundary conformal field theory and renormalization group techniques to map quantum quenches to classical critical phenomena in an extra dimension.
• The paper demonstrates that systems relax towards states describable by a generalized Gibbs ensemble, unifying thermal-like behavior across integrable and non-integrable regimes.

Quantum Quenches in Extended Systems

The paper "Quantum Quenches in Extended Systems" by Pasquale Calabrese and John Cardy investigates the time-evolution of quantum systems subjected to a sudden change, or quench, in a parameter of their Hamiltonian. This work focuses primarily on the behavior of correlation functions post-quench in d-dimensional systems and utilizes concepts from boundary critical phenomena applied in d+1 dimensions, with a particular emphasis on conformal field theory (CFT) for one-dimensional systems.

Analytical Approach and Key Findings

The analysis begins by considering an extended quantum system prepared in the ground state of a Hamiltonian $H_0$ at time $t = 0$, which is then evolved under a different Hamiltonian $H$. This transformation is treated as a quench, assuming it occurs rapidly compared to the system's mass gap. The authors employ boundary CFT techniques when the system post-quench is at or near criticality, facilitating the use of renormalization group insights into boundary conditions that map the quantum problem to a classical one in an extra dimension.

The authors extend their investigation into higher dimensions, focusing on a mean-field approximation that provides predictions and insights into the general behavior of quantum quenches. They compare and validate these predictions against exactly solvable models, including a chain of harmonic oscillators and the transverse-field Ising model. These models help verify that certain features of the evolution, such as the light-cone effect where correlations develop after a critical time related to their distance, remain valid even beyond critical dynamics.

A prominent conclusion of the paper is that such quenches are characterized by a propagation of quasiparticles, initially entangled over regions comparable to the initial state's correlation length. These quasiparticles move with finite speed, thereby instigating a caustic correlation structure that manifests as an emergent horizon effect, where correlations remain largely unchanged until a certain threshold time dictated by the quasiparticle velocity is reached.

Moreover, the long-time behavior of these systems can be understood within the framework of a generalized Gibbs ensemble (GGE). The GGE accounts for all integrals of motion in the system, suggesting that quenched systems tend towards thermal-like states characterized by effective temperatures proportional to the energy levels of these integrals, which potentially allows description even in non-integrable settings.

Implications and Future Directions

The methodologies and insights gained from this work provide a robust framework for understanding non-equilibrium phenomena in quantum systems. The implications extend across various domains, including quantum computing, condensed matter physics, and many-body quantum dynamics, where switching of parameters is a fundamental process.

The theoretical depth achieved by leveraging boundary critical phenomena and CFT opens up avenues for future theoretical and experimental validation. Further investigations could focus on non-integrable systems, exploring the role of finite temperature, disorder, and multi-quench dynamics. Moreover, numerical methods like time-dependent density matrix renormalization group (t-DMRG) could be further utilized to simulate more complex systems beyond one-dimensional setups.

In summary, the findings presented in "Quantum Quenches in Extended Systems" underscore the profound changes quenches induce in quantum systems, revealing how fundamental theoretical constructs such as quasiparticle propagation and generalized Gibbs ensembles can coherently describe these transformations across critical and non-critical landscapes. The paper sets a precedent for deeper exploration into quantum dynamic systems that straddle low-temperature physics and high-energy quantum field theory.