[Paper Review] Berry Phases and Quantum Phase Transitions

Alioscia Hamma|ArXiv.org|Feb 10, 2006

Topological Materials and Phenomena3 references17 citations

TL;DR

This paper establishes a topological criterion linking Berry phases to quantum phase transitions (QPTs) in many-body systems: if a sequence of Berry phases around shrinking loops in parameter space fails to converge to zero, the limit point is a quantum critical point. The key result is that non-contractible Berry phases—specifically in the thermodynamic limit—signal the presence of a QPT, demonstrated explicitly in the anisotropic XY spin chain where the XX critical region is detected via a nontrivial Berry phase of π in the ground state.

ABSTRACT

We study the connection between Berry phases and quantum phase transitions of generic quantum many-body systems. Consider sequences of Berry phases associated to sequences of loops in the parameter space whose limit is a point. If the sequence of Berry phases does not converge to zero, then the limit point is a quantum critical point. Quantum critical points are associated to failures of adiabaticity. We discuss the remarkable example of the anisotropic XY spin chain in a transverse magnetic field and detect the XX region of criticality.

Motivation & Objective

• To establish a general topological connection between Berry phases and quantum phase transitions in many-body systems.
• To address the challenge that quantum critical points are not isolated but form manifolds, complicating standard topological detection.
• To demonstrate that non-contractible sequences of Berry phases imply the existence of a quantum critical point.
• To analyze the anisotropic XY spin chain as a concrete example where the XX critical region is detected via Berry phase non-triviality.
• To show that the failure of adiabaticity at critical points is signaled by divergent Berry curvature, linked to non-zero Berry phases in the thermodynamic limit.

Proposed method

• Define a sequence of closed loops in the parameter space shrinking toward a candidate critical point λ₀.
• Compute the Berry phase acquired by the ground state as the system adiabatically traverses each loop in the sequence.
• Use the condition that if the Berry phases do not converge to zero, λ₀ is a quantum critical point—defining non-contractible sequences.
• Apply the formalism to the anisotropic XY spin chain with Hamiltonian parameters λ (anisotropy) and γ (transverse field), using Jordan-Wigner transformation to map to free fermions.
• Calculate the ground state wavefunction and Berry phase as a sum of single-qubit Berry phases over momentum modes k, with φ as the adiabatic parameter.
• Analyze the limit γ → 0 and M → ∞ (thermodynamic limit), showing that a single mode k₀ with cosθₖ₀ = 0 contributes π to the total Berry phase, yielding non-trivial phase.

Experimental results

Research questions

• RQ1 Can non-contractible sequences of Berry phases serve as a topological indicator of quantum critical points in many-body systems?
• RQ2 How does the Berry phase of the ground state behave near the critical point of the anisotropic XY spin chain?
• RQ3 What is the role of the thermodynamic limit in the emergence of non-trivial Berry phases at quantum criticality?
• RQ4 How is the failure of adiabaticity at critical points related to the divergence of Berry curvature and non-zero Berry phases?
• RQ5 Can the relative Berry phase between ground and first excited states detect critical regions, and how does it compare to the ground state phase?

Key findings

• A sequence of Berry phases that does not converge to zero as loops shrink to a point λ₀ implies that λ₀ is a quantum critical point.
• In the anisotropic XY spin chain, the ground state Berry phase around φ ∈ [0, π] is given by φ(M) = ∑ₖ φₖ, where φₖ = π(1 − cosθₖ).
• For finite M and γ > 0, lim_{γ→0} φₖ = 0 or 2π, so the total phase is trivial (0 or 2πM), but in the thermodynamic limit, φₖ₀ → π for the critical mode k₀.
• In the limit M → ∞ and γ → 0, the average Berry phase per mode satisfies lim_{γ→0} lim_{M→∞} (1/M)φ(M) ≠ 0, confirming non-contractibility.
• The critical point at λ = 0 (XX point) is detected by a non-trivial Berry phase of π from the single critical mode k₀, signaling a failure of adiabaticity.
• The result aligns with Carollo and Pachos' finding that the relative Berry phase between ground and first excited states converges to −π, confirming the topological nature of the critical region.