Thermodynamic root-density equation for the two-parameter Lieb–Liniger/Cheon–Shigehara interpolation beyond the non-interacting line

The two-parameter family of integrable one-dimensional continuum models introduced by Stouten, Claeys, Zvonarev, Caux and Gritsev (J. Phys. A 51, 485204 (2018)) interpolates between the Lieb–Liniger gas of delta-interacting bosons and its fermionic dual, the Cheon–Shigehara model. Its thermodynamic-limit ground state is currently described by a generalized Lieb (Love–Lieb) integral equation for the Bethe-root density that is valid only in the regime $c_0/c_1>|\Lambda|$ , where the coupling-dependent driving term is regular on the root support; outside this regime the driving term has a non-integrable pole at the image of the non-interacting line $k=-c_0/c_1$ , the equivalent kernel becomes singular, and the ground state reorders through level crossings. The open problem is a thermodynamic treatment of the ground-state root density, energy and particle density that is well posed for all couplings and verifiable against finite-size Bethe-ansatz solutions. The problem remains open.

Background

The Lieb–Liniger model describes $N$ bosons on a one-dimensional ring of length $L$ interacting through a repulsive point (Dirac-delta) potential. It is exactly solvable by the coordinate Bethe ansatz: the wavefunction is a superposition of plane waves labelled by rapidities (Bethe roots) $k_j$ obeying algebraic Bethe equations, and the ground state corresponds to real roots filling a symmetric interval $[-\Lambda,\Lambda]$ . In the thermodynamic limit ( $N,L\to\infty$ with $N/L$ fixed), Lieb and Liniger showed that the root density $\rho(k)$ , i.e. the density of roots per unit rapidity, satisfies a linear Fredholm integral equation of the second kind (the unknown function appears both inside the integral and as a plain term), from which the ground-state energy follows; the excitation spectrum was obtained by Lieb in a companion paper. In the mathematics literature this equation is known as the Love–Lieb equation. The Cheon–Shigehara model is the fermionic dual: spinless fermions interacting through the second derivative of a delta function, realized as a discontinuous boundary condition at particle coincidences, equivalent to the strongly interacting Lieb–Liniger gas under one-dimensional boson–fermion duality.

Stouten, Claeys, Zvonarev, Caux and Gritsev (2018) constructed a two-parameter family of integrable one-dimensional continuum models interpolating between these endpoints. The construction starts from a rational two-parameter solution of the functional equation for the two-body scattering function $F$ that underlies the Bethe ansatz, namely $F(k_i,k_j)=(c_0+c_1 k_i)(c_0+c_1 k_j)$ with couplings $c_0,c_1$ . In the bosonic sector the Bethe equations read $e^{ik_jL}=\prod_{l\ne j}\frac{k_j-k_l+i(c_0+c_1k_j)(c_0+c_1k_l)}{k_j-k_l-i(c_0+c_1k_j)(c_0+c_1k_l)}$ . At $c_1=0$ they reduce to Yang's form of the Lieb–Liniger model with coupling $c_0^2$ , and at $c_0=0$ to the Cheon–Shigehara model. The family carries an $SL(2,\mathbb{R})$ (Möbius) duality acting jointly on the rapidities and on the coupling vector $(c_0,c_1)$ , with the modular subgroup $SL(2,\mathbb{Z})$ connecting its special points. A distinctive feature is the existence of non-interacting modes: whenever a rapidity equals $k=-c_0/c_1$ , the scattering function vanishes, that mode decouples from all scattering, and it can only take the quantized values $k=2\pi n/L$ . As $c_1$ is varied from the Lieb–Liniger side toward the Cheon–Shigehara side, rapidities cannot cross this line, so the two known ground states are separated by level crossings, and finite-size numerics become unstable when roots approach the line.

Following Lieb's original route, the authors derive the thermodynamic-limit equation for the root density. After the $SL(2)$ change of variables $y=k/(1+(c_1/c_0)k)$ , the Bethe equations yield a generalized Lieb (Love–Lieb) equation $\rho(y)-\frac{1}{2\pi}\int_{-\Lambda}^{\Lambda}\rho(x)\frac{2c_0^2}{c_0^4+(x-y)^2}\,dx=\frac{1}{2\pi}\frac{1}{(1-(c_1/c_0)y)^2}$ , with normalization $\int_{-\Lambda}^{\Lambda}\rho(y)\,dy=N/L$ . The kernel is of Lieb–Liniger type, but the driving term is now coupling-dependent. In rescaled variables $y=\Lambda z$ with $z\in[-1,1]$ , $c_0=\Lambda^{1/2}\alpha$ , $c_0c_1=\beta^2$ , the driving term is $\frac{1}{2\pi}(1-(\beta^2/\alpha^2)z)^{-2}$ : for $\beta/\alpha\ge 1$ (equivalently $c_0/c_1\le|\Lambda|$ , i.e. when the non-interacting line lies inside the root support) it has a non-integrable second-order pole inside the integration interval, the kernel of the equivalent untransformed equation becomes singular, and the Taylor expansion on which the derivation rests loses its justification. The authors verified that within the regime $c_0/c_1>|\Lambda|$ the numerical solution of the equation reproduces the finite-size root density, but they were unable to continue outside this regime; the singular regime is precisely where the ground state reorders through level crossings at the non-interacting line. They explicitly leave a treatment valid for all parameter values as an open challenge.

Problem Statement

Construct a thermodynamic-limit description of the ground state of the two-parameter family of integrable one-dimensional models defined by the Bethe equations $e^{ik_jL}=\prod_{l\ne j}\frac{k_j-k_l+i(c_0+c_1k_j)(c_0+c_1k_l)}{k_j-k_l-i(c_0+c_1k_j)(c_0+c_1k_l)}$ (Stouten, Claeys, Zvonarev, Caux, Gritsev, 2018) that is valid for all real couplings $c_0,c_1$ . Specifically, replace the currently available generalized Lieb (Love–Lieb) equation for the root density, whose derivation and numerical solution are restricted to the regime $c_0/c_1>|\Lambda|$ (equivalently $\beta/\alpha<1$ ) in which the driving term $(1-(\beta^2/\alpha^2)z)^{-2}$ is regular on the root support, by a treatment that remains well posed when the non-interacting line $k=-c_0/c_1$ lies inside the rapidity support — where that driving term develops a non-integrable pole, the equivalent kernel becomes singular, and the ground state reorders through level crossings. The treatment should (i) be derived from the Bethe equations with explicitly stated assumptions on the root support and quantum numbers, (ii) reproduce the known Lieb–Liniger ( $c_1\to 0$ ) and Cheon–Shigehara ( $c_0\to 0$ ) thermodynamics, and (iii) deliver the ground-state root density, energy and particle density as functions of both couplings, in a form that can be checked against finite-size solutions of the Bethe equations on both sides of, and across, the non-interacting line.

References

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