[Paper Review] On the dynamics of interfaces in the ferromagnetic XXZ chain under weak perturbations

Bruno Nachtergaele, Wolfgang L. Spitzer+1 more|ArXiv.org|Oct 8, 2002

Theoretical and Computational Physics9 references3 citations

TL;DR

This paper investigates the dynamics of domain walls (kinks) in the ferromagnetic XXZ spin chain under weak magnetic fields using a scaling limit where the field strength λ→0 and time t→∞ with τ=λt fixed. It identifies the leading-order time evolution as a reduced dynamics governed by the time-averaged field projected onto the ground state space, with a correction of order λ^{1−δ}. For uniform fields, the magnetization profile exhibits ballistic motion if the field has no z-component, but periodic evolution if the z-component is non-zero—revealing a fundamental dichotomy in domain wall response.

ABSTRACT

We study the time evolution of interfaces of the ferromagnetic XXZ chain in a magnetic field. A scaling limit is introduced where the strength of the magnetic field tends to zero and the microscopic time to infinity while keeping their product constant. The leading term and its first correction are determined and further analyzed in more detail for the case of a uniform magnetic field.

Motivation & Objective

• To understand the time evolution of kink (domain wall) states in the ferromagnetic XXZ chain under weak external magnetic fields.
• To derive a scaling limit where the magnetic field strength λ→0 and time t→∞ with τ=λt fixed, capturing effective long-time dynamics.
• To identify the leading-order and first correction terms in the time evolution of the magnetization profile.
• To analyze the reduced dynamics for uniform, time-independent magnetic fields, particularly distinguishing behavior based on the presence or absence of a z-component in the field.
• To establish conditions under which the spectral gap and absolute continuity of the spectrum lead to a λ-order correction term in the dynamics.

Proposed method

• Introduce a scaling limit with λ→0 and t→∞ such that τ=λt is held constant, enabling the study of long-time dynamics under weak perturbations.
• Use the unperturbed kink Hamiltonian as the reference system and apply a weak magnetic field as a perturbation.
• Derive the leading-order dynamics as a reduced dynamics determined by the time-averaged magnetic field projected onto the ground state space of the unperturbed model.
• Apply spectral theory to the unperturbed Hamiltonian, assuming a spectral gap and absolute continuity of the rest of the spectrum to control the first-order correction.
• Use q-deformed algebraic structures and q-binomial identities to compute matrix elements of spin operators in kink states.
• Analyze the spectrum of the Stark-Jacobi operator on ℤ^d to characterize the energy level structure under linear potentials, crucial for understanding the dynamics in the scaling limit.

Experimental results

Research questions

• RQ1 How does the magnetization profile of a kink state evolve under a weak, time-dependent magnetic field in the scaling limit λ→0, t→∞ with τ=λt fixed?
• RQ2 What is the structure of the leading-order time evolution of the kink state under such a scaling?
• RQ3 Under what conditions does the first correction to the leading-order dynamics scale as λ, and how does this relate to the spectral properties of the unperturbed Hamiltonian?
• RQ4 How does the presence or absence of a z-component in a uniform magnetic field affect the long-time dynamics of the magnetization profile?
• RQ5 What is the role of the spectral gap and absolute continuity of the spectrum in determining the nature of the correction term in the time evolution?

Key findings

• The leading-order time evolution is given by a reduced dynamics determined by the time-averaged magnetic field projected onto the ground state space, with an error term of order λ^{1−δ} for δ∈(0,1).
• When the unperturbed Hamiltonian has a spectral gap and the rest of the spectrum is absolutely continuous, the first correction to the dynamics is of order λ.
• For a uniform, time-independent magnetic field, the magnetization profile evolves ballistically if the field has no z-component, indicating a coherent, unidirectional motion of the domain wall.
• If the magnetic field has a non-vanishing z-component, the magnetization profile evolves periodically, indicating oscillatory dynamics of the domain wall.
• The dynamics for the z-component case is governed by a spectrum of the Stark-Jacobi operator that is dense pure point when the field components are incommensurate, leading to periodic behavior.
• The matrix elements of spin operators in kink states are computed exactly using q-binomial identities, yielding explicit expressions for the magnetization profile and its time evolution.