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arXiv:2604.27885

Quantum integrable matrix models of spinor Bose gases in one spatial dimension

Hannes Köper, Thomas Gasenzer

Published Apr 30, 2026• 0 citations
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Abstract

Degenerate spinor Bose gases with repulsive density-density interaction and anti-ferromagnetic spin-spin coupling in one spatial dimension are shown to be described by a quantum integrable matrix extension of the nonlinear Schrödinger model, whose fundamental fields are described by an $m\,\times\,n$ matrix of bosonic field operators. The eigenstates of this model are constructed for arbitrarily sized matrix field operators by means of algebraic Bethe-ansatz techniques, and the corresponding Bethe equations governing the spectra of conserved quantities are derived. The approach thus generalizes previously chosen techniques to account for arbitrary spin multiplets and their spin-spin interaction. Focusing on the specific case of the $2\times2$ model, which is shown to correspond to a spin-$1$ Bose gas, a set of integral equations is derived, which describe its equilibrium thermodynamic properties. From these, the ground state phase diagram is computed both, numerically and analytically in the parameter plane spanned by the chemical potential and an external magnetic field. Furthermore, we establish a sufficient condition for interacting bosons in one dimension to obey the Pauli exclusion principle. In particular, we show that in the presence of paired bound states, no two quasiparticle rapidities can coincide, provided that the Lieb parameter satisfies $γ>4/3$.

DOI
10.48550/arxiv.2604.27885

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Degenerate spinor Bose gases with repulsive density-density interaction and anti-ferromagnetic spin-spin coupling in one spatial dimension are shown to be described by a quantum integrable matrix extension of the nonlinear Schrödinger model, whose fundamental fields are described by an m × nm\,\times\,nm×n matrix of bosonic field operators. The eigenstates of this model are constructed for arbitrarily sized matrix field operators by means of algebraic Bethe-ansatz techniques, and the corresponding Bethe equations governing the spectra of conserved quantities are derived. The approach thus generalizes previously chosen techniques to account for arbitrary spin multiplets and their spin-spin interaction. Focusing on the specific case of the 2×22\times22×2 model, which is shown to correspond to a spin-111 Bose gas, a set of integral equations is derived, which describe its equilibrium thermodynamic properties. From these, the ground state phase diagram is computed both, numerically and analytically in the parameter plane spanned by the chemical potential and an external magnetic field. Furthermore, we establish a sufficient condition for interacting bosons in one dimension to obey the Pauli exclusion principle. In particular, we show that in the presence of paired bound states, no two quasiparticle rapidities can coincide, provided that the Lieb parameter satisfies γ>4/3γ>4/3γ>4/3.
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We investigate magnetism and quantum phase transitions in a one-dimensional system of integrable spin-1 bosons with strongly repulsive density-density interaction and antiferromagnetic spin-exchange interaction via the thermodynamic Bethe ansatz method. At zero temperature, the system exhibits three quantum phases: (i) a singlet phase of boson pairs when the external magnetic field HHH is less than the lower critical field Hc1{H}_{c1}Hc1​; (ii) a ferromagnetic phase of atoms in the hyperfine state ∣F=1, mF=1⟩|F=1,\text{ }{m}_{F}=1⟩∣F=1, mF​=1⟩ when the external magnetic field exceeds the upper critical field Hc2{H}_{c2}Hc2​; and (iii) a mixed phase of singlet pairs and unpaired atoms in the intermediate region Hc1<H<Hc2{H}_{c1}<H<{H}_{c2}Hc1​<H<Hc2​. At finite temperatures, the spin fluctuations affect the thermodynamics of the model through coupling the spin bound states to the dressed energy for the unpaired mF=1{m}_{F}=1mF​=1 bosons. However, such spin dynamics is suppressed by a sufficiently strong external field at low temperatures. Thus the singlet pairs and unpaired bosons may form a two-component Luttinger liquid in the strong coupling regime.
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Quantum many-body physics has been studied for many decades. A lot of intriguing phenomena have been observed and theories have been developed. Yet many problems remain unsolved, which is largely due to a lack of general efficient classical computation method. One-dimensional (1D) systems have also drawn much attention over the past few decades. One reason is that many unique strongly correlated quantum phenomena only appear in low dimensions. Another reason is that many exact results can be obtained in 1D for cross benchmarking, such as Bethe Ansatz, Bosonization, Bose-Fermi mapping, etc. In addition, many numerical methods, such as Matrix-Product-State based and Monte Carlo methods, work most efficiently in 1D. In recent years, 1D quantum gases have been realized in many cold atom labs, providing experimental motivation for their studies. One of the most mysterious assumptions about a system of identical quantum particles is that the wavefunction must be symmetric (for bosons) or anti-symmetric (for fermions). This means that no two fermions can occupy the same state, while multiple occupancy is allowed for bosons. Bosons and fermions are therefore generally very different. But in 1D this distinction could become rather subtle, provided that the bosonic multiple occupancy is suppressed, which can happen if strong repulsion exists between them. However, when considering spin degrees of freedom, things become more complicated. In this work, we develop a generalized Bose-Fermi mapping theory, under which, the charge degrees of freedom is mapped to a spinless fermions, while the spin degrees of freedom to an effective spin chain. This mapping works for arbitrary spin, arbitrary trapping potentials, arbitrary spin-dependent interaction potential, and for either spinor bosons or fermions. In the strong interaction limit, the wavefunction of the system is represented by the strong coupling ansatz wavefunction (SCAW), which takes the form of a direct product of spinless fermions wavefunction and spin wavefunction. Using this mapping technique, we study the dynamics and collective modes of the system. Furthermore, we develop a very efficient method to calculate the one-body density matrix, from which we can calculate the momentum distribution of the system.
Chapter 1. Exact and explicit results are derived for the thermodynamic properties (isochores, isotherms, isobars, response functions, speed of sound) of a quantum gas in dimensions [special characters omitted] ≥ 1 and with fractional exclusion statistics 0 ≤ g ≤ 1 connecting bosons (g = 0) and fermions (g = 1). In [special characters omitted] = 1 the results are equivalent to those of the Calogero-Sutherland model, a gas with long-range two-body interaction. Emphasis is given to the crossover between boson-like and fermion-like features, caused by aspects of the statistical interaction that mimic long-range attraction and short-range repulsion. A phase transition along the isobar occurs at a nonzero temperature in all dimensions. The T-dependence of the speed of sound is in simple relation to isochores and isobars. The effects of soft container walls are accounted for rigorously for the case of a pure power-law potential. Chapter 2. The exact thermodynamics (isochores, isotherms, isobars, response functions, speed of sound) is worked out for a statistically interacting quantum gas in [special characters omitted] dimensions. The results in [special characters omitted] = 1 are those of the thermodynamic Bethe ansatz for the Nonlinear Schrödinger model, a gas with repulsive two-body contact potential. In all dimensions the ideal boson and fermion gases are recovered in the weak-coupling and strong-coupling limits, respectively. For all nonzero couplings ideal fermion gas behavior emerges for [special characters omitted] >> 1 and, in the limit [special characters omitted] → ∞, a phase transition occurs at T > 0. Significant deviations from ideal quantum gas behavior are found for intermediate coupling and finite [special characters omitted]. Chapter 3. Methodology previously developed in the framework of the coordinate Bethe ansatz applied to integrable quantum gas models is employed to calculate some ground-state properties and elementary excitations for quantum gas models in [special characters omitted] = 1 dimensions with statistical interactions that are not equivalent to dynamical interactions. The focus in this comparative study is on modifications of the Calogero-Sutherland and Nonlinear Schrödinger models. Statistical interactions cannot be chosen arbitrarily. They must satisfy certain physicality conditions, which will be discussed in some detail.