[Paper Review] Spectral density of the quantum Ising model in two fields: Gaussian and multi-Gaussian approximations

Y. Y. Atas, E. Bogomolny|arXiv (Cornell University)|Feb 27, 2014

Quantum many-body systems18 references3 citations

TL;DR

This paper investigates the spectral density of the quantum Ising model in two fields (transverse and longitudinal) for finite but large spin systems. It proposes Gaussian and multi-Gaussian approximations to describe spectral densities, showing that while bulk behavior follows a Gaussian form, strong degeneracies in certain coupling limits lead to multi-peak structures. The key contribution is a simple analytical method to approximate multi-peak spectral densities using sums of Gaussians, validated against numerical results.

ABSTRACT

Spectral density of quantum Ising model in two fields for large but finite number of spins $N$, is discussed in detail. When all coupling constants are of the same order, spectral densities in the bulk are well approximated by a Gaussian function which is typical behaviour for many-body models with short-range interactions. The main part of the paper is devoted to the investigation of a different characteristic case when spectral densities have peaks related with strong degeneracies of unperturbed states in certain limits of coupling constants. In the strict limit $N o\infty$, peaks overlap and disappear but for values of $N$ accessible in numerical calculations they often strongly influence spectral densities and other quantities as well. A simple method is developed which permits to find general approximation formulae for multi-peak structure of spectral density in good agreement with numerics.

Motivation & Objective

• To understand the spectral density of the quantum Ising model in two fields for large but finite N, especially when standard Gaussian approximations break down due to degeneracies.
• To identify and characterize non-universal corrections to the Gaussian spectral density arising from strong degeneracies in specific limits of coupling constants.
• To develop a simple analytical method that approximates multi-peak spectral densities without full diagonalization of the Hamiltonian.
• To validate the proposed approximation method against numerical calculations for various coupling constants and system sizes.

Proposed method

• Uses the quantum Ising Hamiltonian with transverse and longitudinal fields to model spin-1/2 systems under periodic boundary conditions.
• Applies perturbation theory in the small coupling limit (λ ≪ 1) to compute energy corrections from transitions between unperturbed states.
• Derives analytical expressions for the number of states with specific configurations (n, m, k) using combinatorial functions like binomial coefficients.
• Constructs the mean energy shift ΔE_n,k(α,λ) via second-order perturbation theory, incorporating contributions from single-spin flips and spin-pair flips.
• Proposes a multi-Gaussian approximation by summing Gaussians whose parameters (mean, variance) are derived from the unperturbed energy levels and transition counts.
• Validated the method numerically by comparing the analytical approximation to exact spectral densities obtained via full diagonalization for finite N.

Experimental results

Research questions

• RQ1 How does the spectral density of the quantum Ising model in two fields deviate from the universal Gaussian form at finite but large N?
• RQ2 What causes the emergence of pronounced peaks in the spectral density when coupling constants are tuned to specific values?
• RQ3 Can a simple analytical method be developed to approximate multi-peak spectral densities without full diagonalization of the Hamiltonian?
• RQ4 How do degeneracies in the unperturbed spectrum influence the spectral density in the thermodynamic limit and at finite N?
• RQ5 To what extent do perturbative corrections accurately describe the spectral structure in the presence of strong degeneracies?

Key findings

• For large but finite N, the spectral density in the bulk region of the quantum Ising model with all coupling constants of order unity is well approximated by a Gaussian function, consistent with universal behavior in many-body systems with short-range interactions.
• In certain limits of coupling constants, strong degeneracies in the unperturbed spectrum lead to pronounced peaks in the spectral density, which persist for accessible finite N despite vanishing in the strict N→∞ limit.
• The paper derives analytical expressions for the number of states with specific spin configurations (n, m, k), using combinatorial functions such as C_{n-1}^{k-1} and C_{m-1}^{k-1}, which are essential for computing energy level counts and transition probabilities.
• The mean energy shift ΔE_n,k(α,λ) is computed to second order in λ, with contributions from single-spin flips and spin-pair flips, and is expressed in terms of combinatorial sums involving n_j and m_j configurations.
• The multi-Gaussian approximation, formed by summing Gaussians with analytically derived parameters, shows excellent agreement with numerical spectral densities across various coupling constants and system sizes.
• The method successfully captures the structure of multi-peak spectral densities without requiring full diagonalization, enabling efficient prediction of spectral features from the Hamiltonian's structure alone.