[Paper Review] Diagnosing order by disorder in quantum spin systems

C. A. Lamas, D. C. Cabra+2 more|arXiv (Cornell University)|Jun 18, 2014

Physics of Superconductivity and Magnetism4 citations

TL;DR

This paper investigates quantum order by disorder in the frustrated $J_1$-$J_2$ Heisenberg model on the square lattice under a magnetic field using a path integral formulation in coherent states. It demonstrates that quantum fluctuations lift the classical degeneracy, selecting collinear spin states with a residual $Z_2$ symmetry, and reveals a purely quantum effect via tunneling between the two degenerate minima, leading to a factorized ground state wave function with minimal entanglement between sublattices.

ABSTRACT

In this paper we study the frustrated J1-J2 quantum Heisenberg model on the square lattice for J2 > 2J1, in a magnetic field. In this regime the classical system is known to have a degenerate manifold of lowest energy configurations, where standard thermal order by disorder occurs. In order to study its quantum version we use a path integral formulation in terms of coherent states. We show that the classical degeneracy in the plane transverse to the magnetic field is lifted by quantum fluctuations. Collinear states are then selected, in a similar pattern to that set by thermal order by disorder, leaving a Z2 degeneracy. A careful analysis reveals a purely quantum mechanical effect given by the tunneling between the two minima selected by fluctuations. The effective description contains two planar (XY -like) fields conjugate to the total magnetization and the difference of the two sublattice magnetizations. Disorder in either or both of these fields produces the locking of their conjugate observables. Furthermore, within this scenario we argue that the quantum state is close to a product state.

Motivation & Objective

• To understand how quantum fluctuations select ground states from a classically degenerate manifold in frustrated quantum spin systems.
• To distinguish quantum order by disorder from thermal order by disorder in the $J_1$-$J_2$ Heisenberg model on the square lattice.
• To investigate the role of quantum tunneling and topological terms in lifting residual $Z_2$ degeneracy after quantum selection.
• To explore the emergence of a factorized ground state wave function as a signature of quantum order by disorder.
• To analyze the effective field theory containing two planar $XY$-like fields conjugate to total magnetization and sublattice spin imbalance.

Proposed method

• Formulating the quantum spin system using a path integral in coherent states to describe spin dynamics in the presence of a magnetic field.
• Identifying two effective planar fields: a symmetric field conjugate to total magnetization and an antisymmetric field conjugate to sublattice spin imbalance.
• Deriving an effective action with topological terms (Berry phases) that influence vortex proliferation and field delocalization.
• Analyzing the effective potential for the antisymmetric field to identify instanton-like processes and their role in disordering the field.
• Using the effective field theory to study the stability of selected collinear states and the conditions under which $Z_2$ symmetry is restored.
• Evaluating the wave function structure to assess factorization and minimal entanglement between sublattices as a quantum signature of order by disorder.

Experimental results

Research questions

• RQ1 Does quantum order by disorder in the $J_1$-$J_2$ Heisenberg model select the same ground state as thermal order by disorder?
• RQ2 What is the role of quantum tunneling between degenerate minima in lifting the residual $Z_2$ symmetry after quantum selection?
• RQ3 How do topological terms in the effective action influence the delocalization of planar spin fields and the emergence of a disordered phase?
• RQ4 Can the ground state wave function be factorized into separable states between sublattices, and what does this imply for quantum entanglement?
• RQ5 Under what conditions does the antisymmetric field become disordered, signaling the restoration of $Z_2$ symmetry and a potential quantum phase transition?

Key findings

• Quantum fluctuations lift the classical degeneracy in the transverse plane, selecting collinear spin configurations analogous to thermal order by disorder.
• A residual $Z_2$ degeneracy remains after quantum selection, corresponding to two distinct collinear states differing by a global spin rotation.
• Quantum tunneling between the two $Z_2$ degenerate minima is identified as a purely quantum mechanical effect, distinct from thermal fluctuations.
• The effective field theory contains two planar $XY$-like fields: one conjugate to total magnetization and one to sublattice spin imbalance, with the latter subject to an effective potential favoring instanton processes.
• Disorder in the antisymmetric field $ heta_a$ leads to the locking of its conjugate variable (sublattice magnetization difference) to a quantized value, indicating a quantum plateau phase.
• The ground state wave function is predicted to be approximately factorized into separable states between sublattices, indicating minimal entanglement and a strong signature of quantum order by disorder.