Lieb-Liniger Model Overview

• Lieb-Liniger model is a precisely solvable framework for 1D bosons interacting via a delta potential, central to quantum integrable systems.
• Its Bethe ansatz solution provides clear insights into ground states, excitations, and correlation functions across weak and strong coupling regimes.
• Extensions incorporating anyonic statistics, finite-range interactions, and dissipative effects allow deeper study of non-equilibrium dynamics and quantum criticality.

The Lieb-Liniger model is a paradigmatic exactly solvable model describing one-dimensional (1D) bosons interacting via a contact (delta-function) potential. It holds a central position in the theory of quantum integrable systems, underpins the statistical mechanics and correlation theory of 1D quantum gases, and provides the theoretical foundation for a host of contemporary cold-atom experiments. Recent generalizations include anyonic statistics, finite-range interactions, dissipative channels, and coupling to external potentials, revealing an extraordinary range of quantum many-body phenomena.

1. Hamiltonian, Physical Setup, and Parameter Regimes

The Lieb-Liniger Hamiltonian for $N$ identical bosons of mass $m$ on a ring of circumference $L$ is given by

$$H = -\frac{\hbar^2}{2m} \sum_{i=1}^N \frac{\partial^2}{\partial x_i^2} + g \sum_{1 \leq i < j \leq N} \delta(x_i - x_j)$$

with periodic boundary conditions $x_i \equiv x_i + L$ and $g>0$ for repulsive interactions. The particle density is $n = N/L$, and the dimensionless interaction parameter is $\gamma = m g / (\hbar^2 n)$.

Notable parameter regimes and limits:

• Weak coupling ($\gamma \ll 1$): Quasi-condensate/Bogoliubov regime; large phase coherence; TLL with high Luttinger parameter.
• Strong coupling ($\gamma \gg 1$): Tonks–Girardeau limit; fermionization of bosons; energetics and correlations approach those of free fermions.
• Finite temperature: Physics controlled by competition between interaction, density, and thermal de Broglie length.

Experimental advances have enabled realization of the Lieb-Liniger gas in ultracold atom systems confined to quasi-1D geometries, providing direct access to all parameter regimes (Jiang et al., 2015, Kerr et al., 2024).

2. Integrability and Bethe Ansatz Solution

The model is exactly solvable by the coordinate Bethe ansatz. The $N$-particle wave function is a sum of plane waves, with amplitudes determined by scattering phase shifts from two-body $\delta$-interactions: $$\psi(x_1, ..., x_N) = \sum_{P \in S_N} A(P) \exp \left( i \sum_{j=1}^N k_{P_j} x_j \right)$$ with amplitudes related by two-body $S$-matrices: $$S(k_j, k_l) = \frac{k_j - k_l - i c}{k_j - k_l + i c},\qquad c = m g/\hbar^2$$ Periodic boundary conditions enforce the coupled Bethe equations for the quasimomenta $\{k_j\}$: $$e^{i k_j L} = \prod_{l \neq j} \frac{k_j - k_l + i c}{k_j - k_l - i c}$$ In the thermodynamic limit, these are recast as linear integral equations for the rapidity distribution $\rho(k)$. The ground-state and excitation spectra, as well as all local and nonlocal correlators, are uniquely determined via Bethe root densities (Jiang et al., 2015, Ristivojevic, 2014).

3. Excitations, Solitons, and Correlations

The Lieb-Liniger spectrum presents two branches of elementary excitations:

• Type I excitations: Particle-like (analogous to Bogoliubov quasiparticles/phonons at low $\gamma$).
• Type II excitations: Hole-like; identified as quantum dark solitons, with the quantum soliton–excitation correspondence made precise by matching the Type II dispersion to the mean-field soliton energy–momentum relation (Karpiuk et al., 2014, Golletz et al., 2019).

Correlation functions exhibit rich behavior:

• One-body density matrix: Algebraic decay set by Luttinger parameter $K$.
• Density–density correlations: Decay governed by temperature, $\gamma$, and show oscillatory crossovers at finite temperature. The leading correlation lengths are characterized by nonlinear integral equations and display regime-dependent crossovers, including critical oscillatory behavior as $T$ increases (Klumper et al., 2014).
• Pair correlator $g_2$: Analytical results available in both the weak and strong interaction limits, with suppression to zero in the impenetrable (Tonks-Girardeau) regime (Jiang et al., 2015, Kerr et al., 2024).

Recent advances allow for the direct computation and measurement of dynamical correlation functions, including finite-temperature Green's functions and spectral functions over the entire interaction range (Senese et al., 25 Aug 2025).

4. Thermodynamics, Criticality, and Hydrodynamics

The thermodynamics at finite temperature is exactly captured by the Yang–Yang thermodynamic Bethe ansatz (TBA), resulting in coupled integral equations for the dressed energy $\varepsilon(k)$. Analytic closed-form equations of state have been derived in six distinct asymptotic regimes, allowing for rapid and accurate computation of all thermodynamic quantities (pressure, entropy, compressibility, etc.) across both the ideal Bose and strongly correlated regimes (Kerr et al., 2024).

The model exhibits a zero-temperature quantum phase transition between a quasicondensate and a fermionized gas as $\gamma$ increases. At finite $T$, universal quantum critical scaling appears near the vacuum–TLL transition with characteristic exponents $z=2, \nu = 1/2$.

Hydrodynamic and collective mode properties (e.g., breathing mode frequencies) in trapped configurations can be directly derived from the equation of state, and the full local density profile is efficiently constructed using the local density approximation (LDA) with the analytic regime equations (Kerr et al., 2024).

5. Generalizations: Anyonic Statistics, Extended Interactions, and Dissipation

Anyonic Lieb-Liniger Model

A multicomponent anyonic generalization leads to a model where fields obey commutation relations interpolating between bosons and fermions, parameterized by mutual statistics $\kappa_{\alpha\beta}$. In the Tonks-Girardeau limit, the ground state structure displays robust statistics-driven quantum phase transitions: as $\kappa$ is tuned, transitions occur between ground states characterized by either a single or two distinct Dirac seas, with an extensive change in ground state energy (Santos et al., 2012). A full determinant formula for form factors is available for arbitrary $\kappa$, $c$, and $N$ (Piroli et al., 2020).

Finite-Range and Nonlocal Interactions

Replacing the contact potential with an exponentially-decaying interaction yields the extended Lieb-Liniger model (ELL). Continuous matrix product state (cMPS) approaches reveal a superfluid→super-Tonks-Girardeau→quasi-crystal crossover regime by adjusting interaction range and strength. The Luttinger parameter $K$ explores the full range $(0, \infty)$, not accessible in the contact model. The ELL extends the understanding of 1D bosons well beyond the standard LL paradigm (Rincon et al., 2015).

Generalized Contact Models

Introducing multi-delta potentials (e.g., three $\delta$-functions) yields a rich set of effective models where, in the dilute and low-energy limit, the system maps to a Lieb-Liniger form with renormalized effective coupling. Stability is governed by the sign of this effective coupling, and the Bethe ansatz solution straightforwardly generalizes (Veksler et al., 2015).

Dissipative Extensions

Incorporating strong inelastic two-body losses leads to the dissipative Lieb-Liniger model, governed by a non-Hermitian Hamiltonian or Lindblad master equation. Strong loss induces fermionization and drives the system to a Tonks-Girardeau gas even in absence of elastic repulsion, with a suppressed local pair correlation and loss rate scaling as $|g|^{-1}$ (0809.3696). This provides a mechanism for loss-induced quantum phases in cold-atom systems.

6. Quantum Quenches, Non-Equilibrium Dynamics, and Quantum Holonomy

Following a sudden interaction quench, the non-equilibrium time-evolution of local observables can be addressed via the Quench Action formalism, enabling systematic strong-coupling expansions and revealing universality in relaxation and emergence of steady-state correlations (Granet et al., 2021, Salvo et al., 2022). Exact overlap formulas for initial BEC states are available by coordinate Bethe ansatz techniques (Chen, 2020).

A unique feature of the Lieb-Liniger model is its quantum holonomy: under adiabatic cycles traversing the repulsive (Tonks-Girardeau) and attractive (super-Tonks-Girardeau) regimes, eigenstates permute nontrivially even though the final Hamiltonian is identical to the initial one. Spectral branches reveal “exotic quantum holonomy” associated with state permutation, clustering structure, and anholonomy in the spectrum (Yonezawa et al., 2013).

7. Experimental Relevance and Extensions

The predictions of the Lieb-Liniger model and its generalizations have been extensively tested in ultracold atom experiments:

• Measurement of ground-state energies, correlation functions, and collective modes.
• Observation of super-Tonks-Girardeau regimes and quantum solitons.
• Non-equilibrium Newton’s cradle setups validating generalized hydrodynamics and thermalization phenomena (Jiang et al., 2015).

Precise mapping between theoretical predictions (e.g., two- and three-body correlations, spectral functions, response functions) and experimental observables (e.g., absorption imaging, Bragg spectroscopy, time-of-flight) has been achieved, especially in regimes where exact analytic or efficient numerical results are available (Kerr et al., 2024, Senese et al., 25 Aug 2025, Baak et al., 2022).

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The Lieb-Liniger model, in both its original and extended forms, remains a cornerstone for understanding the interplay of integrability, quantum statistics, and nontrivial correlation effects in 1D many-body physics, and provides a rigorous framework for analyzing both equilibrium and non-equilibrium phenomena observed in ultra-cold atomic gases.