[Paper Review] Low-temperature regimes and finite-size scaling in a quantum spherical model

Hassan Chamati, Daniel M. Danchev+2 more|ArXiv.org|Jul 28, 1997

Physics of Superconductivity and Magnetism7 references3 citations

TL;DR

This paper investigates finite-size scaling and low-temperature critical behavior in a d-dimensional quantum spherical model with a general geometry $L^{d-d'} \times \infty^{d'} \times L_\tau^z$, using rigorous analytical methods. It derives exact expressions for free energy, susceptibility, and equation of state in terms of special functions, revealing universal scaling behavior near zero-temperature quantum critical points for $1 < d < 3$ and $0 \leq d' \leq d$, with detailed analysis of the 2D case.

ABSTRACT

A $d$--dimensional quantum model in the spherical approximation confined to a general geometry of the form $L^{d-d^{\prime}} imes\infty^{d^{\prime}} imes L_τ^{z}$ ($L$--linear space size and $L_τ$--temporal size) and subjected to periodic boundary conditions is considered. Because of its close relation with the quantum rotors model it can be regarded as an effective model for studying the low-temperature behavior of the quantum Heisenberg antiferromagnets. Due to the remarkable opportunity it offers for rigorous study of finite-size effects at arbitrary dimensionality this model may play the same role in quantum critical phenomena as the popular Berlin-Kac spherical model in classical critical phenomena. Close to the zero-temperature quantum critical point, the ideas of finite-size scaling are utilized to the fullest extent for studying the critical behavior of the model. For different dimensions $1

Motivation & Objective

• To understand finite-size effects in quantum critical systems at zero temperature.
• To extend the applicability of the spherical model to quantum phase transitions by incorporating general spatial and temporal boundary conditions.
• To provide a rigorous analytical framework for studying critical behavior in quantum Heisenberg antiferromagnets via finite-size scaling.
• To derive exact expressions for thermodynamic quantities in terms of classical special functions across varying dimensions and geometries.
• To establish the role of the quantum spherical model as a rigorous analog to the classical Berlin-Kac model in quantum critical phenomena.

Proposed method

• The model is formulated on a lattice with geometry $L^{d-d'} \times \infty^{d'} \times L_\tau^z$, combining spatial and imaginary time dimensions.
• Periodic boundary conditions are imposed in all directions to ensure translational invariance and enable exact solution techniques.
• The spherical constraint is applied to enforce a fixed average spin length, enabling exact treatment of quantum fluctuations.
• Exact expressions for the free energy, susceptibility, and equation of state are derived using the method of functional integration and special functions such as the Riemann zeta and Hurwitz zeta functions.
• Finite-size scaling is applied near the zero-temperature quantum critical point by analyzing the dependence on $L$ and $L_\tau$, revealing universal scaling forms.
• The analysis is performed for arbitrary $d$ in the range $1 < d < 3$ and $0 \leq d' \leq d$, with special attention to the two-dimensional case.

Experimental results

Research questions

• RQ1 How do finite-size effects manifest in quantum critical systems at zero temperature in dimensions $1 < d < 3$?
• RQ2 What is the scaling behavior of thermodynamic quantities like free energy and susceptibility near the quantum critical point in a general geometry?
• RQ3 How does the interplay between spatial dimensions $d$, compactified dimensions $d'$, and imaginary time $L_\tau$ affect critical scaling?
• RQ4 What is the exact form of the equation of state in the quantum spherical model for $d=2$?
• RQ5 To what extent can the quantum spherical model serve as a rigorous analog to the classical Berlin-Kac model in the context of quantum criticality?

Key findings

• Exact analytical expressions for the free energy, susceptibility, and equation of state are derived in terms of special functions, valid for $1 < d < 3$ and $0 \leq d' \leq d$.
• The model exhibits universal finite-size scaling behavior near the zero-temperature quantum critical point, with scaling functions depending on the ratio $L_\tau / L^{d/d'}$.
• In the two-dimensional case ($d=2$), the critical behavior is governed by a universal scaling function that depends on the anisotropy between spatial and temporal directions.
• The susceptibility diverges at the quantum critical point with a power-law dependence on system size, confirming the presence of long-range quantum correlations.
• The equation of state is found to be universal in form, independent of microscopic details, and expressible in terms of the Riemann zeta and Hurwitz zeta functions.
• The results confirm the quantum spherical model as a powerful and rigorous tool for studying quantum critical phenomena, analogous to the classical spherical model in classical criticality.