[Paper Review] Double-peak specific heat anomaly and correlations in the Bose-Hubbard model

Eduardo O. Rizzatti, Marco Aurélio A. Barbosa+1 more|arXiv (Cornell University)|Oct 13, 2020

Cold Atom Physics and Bose-Einstein Condensates4 citations

TL;DR

This paper identifies a double-peak specific heat anomaly in the Bose-Hubbard model arising from energetic competition and ground-state degeneracy, not geometric frustration. Using self-energy functional theory and finite-temperature perturbation theory, it attributes the anomaly to residual entropy and decomposes the specific heat into particle and hole excitation contributions, revealing distinct origins for each peak.

ABSTRACT

Considering the thermodynamics of bosons in a lattice described by the Bose-Hubbard Hamiltonian, we report the occurrence of anomalous double peaks in their specific heat dependence on temperature. This feature, usually associated with a high geometrical frustration, can also be a consequence of a purely energetic competition. By employing self-energy functional calculations combined with finite-temperature perturbation theory, we propose a mechanism based on ground-state degeneracies expressed as residual entropies. A general decomposition of the specific heat regarding all possible transitions between the system's eingenvalues provides an insight into the nature of each maximum. Furthermore, we address how the model parameters modify the structure of these peaks based on its spectral properties and atom-atom correlation function.

Motivation & Objective

• To investigate the origin of anomalous double-peak specific heat anomalies in the Bose-Hubbard model beyond geometric frustration.
• To determine whether energetic competition and ground-state degeneracy can produce Schottky-type anomalies.
• To decompose the specific heat into particle and hole excitation contributions to identify the physical origin of each peak.
• To analyze the role of atom-atom correlations and spectral properties in shaping the specific heat structure.
• To establish a link between residual entropy and the emergence of multiple specific heat maxima.

Proposed method

• Employed self-energy functional theory (SFT) and finite-temperature perturbation theory (PT) to compute thermodynamic properties of the Bose-Hubbard model.
• Used the Lehmann representation to express the non-interacting Green's function and decomposed it into particle and hole excitation branches.
• Performed analytical continuation to obtain the retarded Green's function and derived the local spectral function as a sum of delta functions at transition energies.
• Introduced a λ-continuum interpolation between the atomic limit (H⁰) and full Hamiltonian (H¹) to compute free energy via path integration.
• Calculated the atom-atom correlation function Cᵢⱼ(λ) = ⟨bᵢ†bⱼ⟩ₗₐₘbda as a function of hopping strength λ to relate it to free energy variation.
• Integrated the correlation function over λ to obtain the free energy shift ΔΩ = −zJNₛ∫₀¹C(λ)dλ, linking correlations to thermodynamic response.

Experimental results

Research questions

• RQ1 Can a double-peak specific heat anomaly emerge in the Bose-Hubbard model due to energetic competition rather than geometric frustration?
• RQ2 What is the physical origin of each peak in the specific heat anomaly—particle or hole excitations?
• RQ3 How does the presence of residual entropy from ground-state degeneracy contribute to the specific heat anomaly?
• RQ4 How do model parameters such as tunneling amplitude and interaction strength affect the structure and separation of the two specific heat peaks?
• RQ5 What is the quantitative role of atom-atom correlations in generating the double-peak feature?

Key findings

• The double-peak specific heat anomaly arises from ground-state degeneracy and residual entropy, not geometric frustration, providing a new mechanism for Schottky-type anomalies.
• The first peak in specific heat is primarily due to particle excitations (n → n+1 transitions), while the second peak originates from hole excitations (n → n−1 transitions).
• The spectral function A(ω) is expressed as a sum of delta functions at transition energies ΔEₙ→ₙ₊₁, with contributions from both particle and hole channels.
• The free energy shift ΔΩ is directly proportional to the integral of the atom-atom correlation function C(λ) over the hopping interpolation parameter λ.
• The model parameters, particularly the tunneling amplitude J, modulate the separation and intensity of the two specific heat peaks via their influence on spectral properties and correlations.
• The decomposition of specific heat into particle and hole branches confirms that the two maxima are physically distinct and not artifacts of numerical noise or approximation.