Negative Spin Model Unlocks New Insights into High-Energy Particle Collisions

Researchers investigated the thermodynamics of the isotropic Heisenberg XXX spin chain with negative spin, a model possessing connections to both deep inelastic scattering in high-energy chromodynamics and the lattice nonlinear Schrödinger equation. Rong Zhong, Yang-Yang Chen, and Kun Hao, all from the Institute of Modern Physics at Northwest University, alongside Wen-li Yang from the same institution and Vladimir Korepin of Stony Brook University, demonstrate that this negative spin variant exhibits a markedly different behaviour compared to its conventional counterpart. Their analysis, utilising the thermodynamic Bethe Ansatz, reveals a unique vacuum structure and excitation spectrum, ultimately leading to unconventional low-temperature properties and a distinct thermodynamic landscape. This work establishes that while mathematically similar to the well-studied Lieb-Liniger Bose gas, the negative spin chain’s thermodynamics and scaling properties are fundamentally different, offering new insights into the behaviour of correlated quantum systems.

Negative Spin Heisenberg Chains and Modified Thermodynamic Properties exhibit unique magnetic behaviors

Scientists have uncovered unconventional thermodynamic behaviour within the isotropic Heisenberg XXX spin chain when configured with negative spin, specifically focusing on a spin value of -1. This model, mathematically equivalent to the lattice nonlinear Schrödinger (NLS) model, arises as a crucial effective theory in the analysis of deep inelastic scattering within high-energy chromodynamics.
Leveraging the model’s integrability and a consistent Bethe Ansatz description, researchers have meticulously examined its ground state, elementary excitations, and finite-temperature properties. The study reveals a distinct vacuum structure and excitation spectrum in the negative spin model, diverging significantly from its positive spin counterpart and leading to modified thermodynamic Bethe Ansatz equations.

Although the integral equations bear resemblance to those governing the Lieb-Liniger Bose gas, the resulting thermodynamics and scaling properties are qualitatively different, precluding a continuous connection between the two systems. Detailed calculations of the free energy, entropy, and specific heat have identified a quantum phase transition that demarcates separate thermodynamic regimes within the chain.

At zero temperature, the excitation spectrum exhibits a linear behaviour in the continuum limit, accurately described by a conformal field theory, establishing a connection to established theoretical frameworks. The findings demonstrate that the negative spin chain shares features with both interacting Bose gases and conventional Heisenberg models, while simultaneously possessing a unique mathematical structure and physical interpretation. Numerical solutions are stable and efficient due to the real and symmetrically distributed Bethe roots, offering a robust platform for further investigation.

Thermodynamic Bethe Ansatz analysis of the negative spin Heisenberg model reveals interesting ground state properties

The study centers on the isotropic Heisenberg XXX spin chain with negative spin, specifically focusing on the case of s = −1. This model exhibits equivalence to the lattice nonlinear Schrödinger (NLS) model and serves as an effective theory within deep inelastic scattering in high-energy chromodynamics.

Researchers employed the thermodynamic Bethe Ansatz to meticulously analyze the ground state, elementary excitations, and finite-temperature properties of this system. A key methodological innovation involves utilizing the algebraic Bethe Ansatz framework to formulate the model and determine both ground-state energy and momentum.

This approach allowed for the derivation of exact solutions by taking appropriate limits and applying the Yang, Yang thermodynamic formalism. The negative spin model’s unique vacuum structure and excitation spectrum necessitated modified TBA equations, revealing unconventional low-temperature behaviour distinct from the conventional positive spin chain.

Numerical solutions were facilitated by the fact that all Bethe roots remained real and were symmetrically distributed, enabling stable and efficient calculations. The research introduced quasiparticle and hole densities to derive thermodynamic equilibrium equations governing the system, building upon established Yang, Yang thermodynamics.

At zero temperature, the excitation spectrum became linear in the continuum limit, describable by a conformal field theory, and the low-temperature regime exhibited characteristics of a Luttinger liquid, unique to this negative spin chain. The absence of string solutions, unlike in positive-spin chains, simplified the thermodynamic analysis and provided a transparent model for exploring integrable quantum systems.

Thermodynamic properties of the negative spin isotropic Heisenberg XXX chain are presented

The research details a comprehensive thermodynamic analysis of the isotropic Heisenberg XXX spin chain with negative spin, specifically focusing on the case where spin s equals -1. This model, equivalent to the lattice nonlinear Schrödinger (NLS) model, is relevant to deep inelastic scattering in high-energy chromodynamics.

Through the thermodynamic Bethe Ansatz, the study meticulously examines the ground state, elementary excitations, and finite-temperature properties of this system. The derived free energy, entropy, and specific heat identify separating thermodynamic regimes within the model. At zero temperature, the excitation spectrum becomes linear in the continuum limit, describable by a conformal field theory, realising a Luttinger-liquid like behaviour with characteristics unique to the negative spin XXX chain.

The work establishes that the spin-(s, s) R-matrix, crucial for describing interactions, can be extended to generic values of spin through analytic continuation. For a spin of -1, the Casimir eigenvalue vanishes, demonstrating a mapping between deep inelastic scattering and this specific spin chain configuration.

The fundamental transfer matrix, τ(λ), is constructed, exhibiting commutation with other transfer matrices for differing spectral parameters, and enabling the construction of conserved quantities. The total Hamiltonian for the spin s = −1 model is obtained via differentiation of the transfer matrix, revealing its connection to the integrable NLS equation.

Bethe equations, derived for the quantum lattice NLS model, simplify under specific conditions, a coupling constant of 1 and ∆= 2, to describe the XXX spin chain with negative spin s = −1. This model demonstrates equivalence to the lattice nonlinear Schrödinger model and arises as an effective theory within deep inelastic scattering in high-energy chromodynamics.

Through the thermodynamic Bethe Ansatz, analysis of the ground state, elementary excitations, and finite-temperature properties has revealed a distinct vacuum structure and excitation spectrum differing from conventional positive spin chains. The investigation establishes that despite similarities in integral equations to the Lieb-Liniger Bose gas, the thermodynamics and scaling properties are qualitatively different and lack a continuous connection.

Derivation of the free energy, entropy, and specific heat identified separating thermodynamic regimes, with the zero-temperature excitation spectrum exhibiting linear behaviour and described by a conformal field theory. The low-temperature regime manifests as a Luttinger liquid, possessing unique characteristics specific to this negative spin XXX chain.

The authors acknowledge limitations in fully capturing the complexities of the quantum phase transition and scaling behaviour, suggesting further research is needed to refine these aspects. Future work could explore the implications of these findings for understanding strongly correlated systems and quantum criticality, as well as connections to high-energy QCD and the behaviour of reggeized gluon states.