[Paper Review] Quantum phase transitions in transverse field spin models: from statistical physics to quantum information

Amit Dutta, G. Aeppli+4 more|arXiv (Cornell University)|Dec 3, 2010

Quantum many-body systemsPhysics and Astronomy37 references172 citations

TL;DR

This paper provides a comprehensive review of quantum phase transitions (QPTs) in transverse field spin models, linking statistical physics and quantum information. It details exact solutions, critical phenomena, and the role of entanglement, quantum discord, and fidelity near quantum critical points, while connecting non-equilibrium dynamics—especially Kibble-Zurek scaling and quantum quenching—to quantum information metrics and quantum annealing.

ABSTRACT

We review quantum phase transitions of spin systems in transverse magnetic fields taking the examples of the spin-1/2 Ising and XY models in a transverse field. Beginning with an overview of quantum phase transitions, we introduce a number of model Hamiltonians. We provide exact solutions in one spatial dimension connecting them to conformal field theoretical studies. We also discuss Kitaev models and some other exactly solvable spin systems. Studies of quantum phase transitions in the presence of quenched randomness and with frustrating interactions are presented in detail. We discuss novel phenomena like Griffiths-McCoy singularities. We then turn to more recent topics like information theoretic measures of the quantum phase transitions in these models such as concurrence, entanglement entropy, quantum discord and quantum fidelity. We then focus on non-equilibrium dynamics of a variety of transverse field systems across quantum critical points and lines. After mentioning rapid quenching studies, we dwell on slow dynamics and discuss the Kibble-Zurek scaling for the defect density following a quench across critical points and its modifications for quenching across critical lines, gapless regions and multicritical points. Topics like the role of different quenching schemes, local quenching, quenching of models with random interactions and quenching of a spin chain coupled to a heat bath are touched upon. The connection between non-equilibrium dynamics and quantum information theoretic measures is presented at some length. We indicate the connection between Kibble-Zurek scaling and adiabatic evolution of a state as well as the application of adiabatic dynamics as a tool of a quantum optimization technique known as quantum annealing. The final section is dedicated to a detailed discussion on recent experimental studies of transverse Ising-like systems.

Motivation & Objective

• To bridge quantum phase transitions in transverse field spin systems with concepts in quantum information theory.
• To analyze the interplay between quantum criticality, entanglement, and non-equilibrium dynamics in low-dimensional quantum systems.
• To connect theoretical predictions of defect generation during quenching with experimental observables and quantum information measures.
• To explore the role of disorder, frustration, and topology in modifying quantum critical behavior and information-theoretic quantities.
• To establish a framework linking Kibble-Zurek scaling, adiabatic evolution, and quantum annealing through fidelity and defect density scaling.

Proposed method

• Utilizes the quantum-classical correspondence to map quantum phase transitions in d-dimensional systems to classical phase transitions in d+1 dimensions.
• Applies exact solutions via Jordan-Wigner transformation and conformal field theory for one-dimensional models.
• Employs Kitaev's anyon model and bosonization techniques to study topological and gapless phases.
• Analyzes quench dynamics using Kibble-Zurek scaling, with defect density scaling as $\tau^{-\alpha}$ for quench rate $\tau$.
• Computes quantum information measures such as concurrence, entanglement entropy, quantum discord, and fidelity susceptibility near critical points.
• Models decoherence via Loschmidt echo and studies non-equilibrium dynamics under linear, nonlinear, and local quenching protocols.

Experimental results

Research questions

• RQ1 How do quantum phase transitions in transverse field Ising and XY models manifest in terms of entanglement and quantum information measures?
• RQ2 What is the scaling behavior of defect density during quenching across quantum critical points, and how does it depend on the quench rate and system dimensionality?
• RQ3 How do Griffiths-McCoy singularities and activated dynamics emerge in disordered transverse field Ising models?
• RQ4 What is the relationship between Kibble-Zurek scaling and adiabatic evolution in quantum annealing protocols?
• RQ5 How do quantum information metrics like fidelity and quantum discord behave near multicritical points and gapless phases?

Key findings

• Defect density after quenching scales as $\tau^{-1/(3-K)}$ in Tomonaga-Luttinger liquids, with $K$ as the Luttinger parameter, and $\tau^{-1}$ for $K>2$.
• Entanglement entropy diverges logarithmically near quantum critical points, with scaling governed by the central charge in conformal field theory.
• Quantum fidelity susceptibility diverges at Berezinskii-Kosterlitz-Thouless (BKT) transitions, signaling criticality in $XXZ$ chains.
• For nonlinear quenching with $q$-th order coupling, defect density scales as $\tau^{-q/(q+1)}$, generalizing the Kibble-Zurek scaling.
• Loschmidt echo decays exponentially in quantum critical environments, indicating strong decoherence of qubits coupled to critical spin chains.
• In disordered systems, Griffiths-McCoy singularities lead to power-law activated dynamics, with logarithmic scaling of relaxation times.