Repulsive Lieb–Liniger and ELL Models

• The repulsive Lieb–Liniger model is a quantum many-body system of interacting bosons in one dimension defined by both contact and exponential potentials.
• Variational methods like continuous matrix product states accurately capture its ground state, correlation functions, and phase transitions across superfluid, super-Tonks–Girardeau, and quasi-crystal regimes.
• Tuning dimensionless parameters in the ELL model provides a unified platform for exploring Luttinger-liquid behavior and various quantum phases.

The repulsive Lieb–Liniger model and its extensions form a foundational class of quantum many-body systems describing interacting bosons in one dimension. In the repulsive regime, these systems serve as paradigmatic integrable models exhibiting rich ground state and excitation properties, as well as a wide variety of quantum phases under modifications of the interaction potential. The repulsive Lieb–Liniger model with an exponentially-decaying two-body potential (the ELL model) generalizes the strictly local interactions of the original Lieb–Liniger model, enabling a unified phase diagram interpolating between superfluid, super-Tonks–Girardeau, and quasi-crystalline regimes within a Luttinger-liquid framework. The combination of analytical, exact, and variational tensor-network methods, especially continuous matrix product states (CMPS), has provided precise characterizations of the ground state, correlation functions, phase transitions, and excitation spectra across a broad parameter space (Rincon et al., 2015).

1. Hamiltonians: Contact and Exponentially-Decaying Potentials

The canonical repulsive Lieb–Liniger model describes bosons with purely local (contact) interactions, defined by the continuum Hamiltonian

$$H_{\rm LL} = \int_{-\infty}^{\infty} dx\, \Psi^\dagger(x) \left(-\frac{\hbar^2}{2m} \frac{d^2}{dx^2}\right)\Psi(x) + \frac{g}{2}\int_{-\infty}^{\infty} dx\, \Psi^\dagger(x)\Psi^\dagger(x)\Psi(x)\Psi(x),$$

where $\Psi(x)$ and $\Psi^\dagger(x)$ are canonical bosonic fields, $m$ the mass, and $g>0$ the contact repulsion strength.

In the exponentially-decaying interaction generalization (“ELL model”), the two-body potential is

$w_{\exp}(x-y) = g\,\frac{\eta}{2}e^{-\eta|x-y|},$

where $\eta$ parametrizes the interaction’s range, with $1/\eta$ setting the decay length. This yields the extended Hamiltonian

$$H_{\rm ELL} = \int dx\, \Psi^\dagger(x)\left(-\frac{\hbar^2}{2m}\partial_x^2\right)\Psi(x) + \frac{1}{2}\int dx\,dy\, w_{\exp}(x-y)\Psi^\dagger(x)\Psi^\dagger(y)\Psi(y)\Psi(x).$$

The ELL model recovers the contact case in the limit $\eta\to\infty$.

The physically relevant scales are the coupling $\Psi(x)$0, the interaction range $\Psi(x)$1, and the 1D density $\Psi(x)$2. Dimensionless couplings include $\Psi(x)$3 (analogous to the Lieb parameter) and $\Psi(x)$4 (dimensionless range).

2. Methodologies: Continuous Matrix Product States Approach

Exact Bethe ansatz methods provide complete solutions only for strictly local interactions. For the ELL model with nonlocal potentials, translation-invariant continuous matrix product state (CMPS) techniques are employed, utilizing a variational ansatz in the continuum: $\Psi(x)$5 with $\Psi(x)$6 as $\Psi(x)$7 matrices acting on an auxiliary space and $\Psi(x)$8 the Fock vacuum. The bond dimension $\Psi(x)$9 controls the variational accuracy.

Variational optimization proceeds by imaginary-time evolution using the time-dependent variational principle (TDVP). Correlation functions and observables are computed directly in the continuum (avoiding discretization artifacts) via transfer-matrix techniques, and the computational scaling is $\Psi^\dagger(x)$0. This approach robustly captures the ground-state physics and Luttinger-liquid correlations for a broad range of coupling and range parameters (Rincon et al., 2015).

3. Correlation Functions and Luttinger-Liquid Structure

The ELL model exhibits ground-state correlations characteristic of one-dimensional Luttinger liquids, characterized by a continuously tunable Luttinger parameter $\Psi^\dagger(x)$1 and sound velocity $\Psi^\dagger(x)$2. Key correlation functions are: $\Psi^\dagger(x)$3 Their long-distance asymptotics in a Luttinger liquid of parameter $\Psi^\dagger(x)$4 and $\Psi^\dagger(x)$5 are: $\Psi^\dagger(x)$6

$\Psi^\dagger(x)$7

with $\Psi^\dagger(x)$8 nonuniversal amplitudes set by the microscopic Hamiltonian. The single-particle correlator decays algebraically with exponent $\Psi^\dagger(x)$9, while density–density correlations have a $m$0 decay modulated by oscillations at $m$1 with decay exponent $m$2.

4. Phase Diagram and Regimes: Superfluid, Super-Tonks–Girardeau, and Quasi-Crystal

By extracting $m$3 from the long-distance decay of correlation functions using CMPS, the $m$4 phase diagram reveals three Luttinger-liquid regimes (Rincon et al., 2015):

• Superfluid (SF): $m$5. Slowest decay in $m$6, no strong density modulations, characteristic of weak-to-moderate coupling or negligible interaction range.
• Super-Tonks–Girardeau (sTG): $m$7. Strong repulsion induces fermion-like correlations, suppressed superfluidity, significant Friedel oscillations in $m$8.
• Quasi-crystal (QC): $m$9. Dominant density–density correlations, emergence of quasi-long-range order with $g>0$0 oscillatory correlations, akin to a Luttinger-liquid precursor of charge-density wave order.

The crossover lines correspond roughly to $g>0$1 (SF-sTG) and $g>0$2 (sTG-QC). At large $g>0$3 (short-range limit), the phase boundary to sTG/ QC shifts to larger $g>0$4, recovering features of the contact model; decreasing $g>0$5 (longer-range tail) shifts crossovers to smaller $g>0$6.

Regime	Luttinger Parameter $g>0$7	Physical Character
SF	$g>0$8	Slowly decaying $g>0$9, no strong $w_{\exp}(x-y) = g\,\frac{\eta}{2}e^{-\eta|x-y|},$0 oscillations
sTG	$w_{\exp}(x-y) = g\,\frac{\eta}{2}e^{-\eta|x-y|},$1	Suppressed SF, strong $w_{\exp}(x-y) = g\,\frac{\eta}{2}e^{-\eta|x-y|},$2 oscillations (Friedel)
QC	$w_{\exp}(x-y) = g\,\frac{\eta}{2}e^{-\eta|x-y|},$3	Power-law density order at $w_{\exp}(x-y) = g\,\frac{\eta}{2}e^{-\eta|x-y|},$4

5. Comparison with Contact and Long-Range Models: Tuning $w_{\exp}(x-y) = g\,\frac{\eta}{2}e^{-\eta|x-y|},$5

The extended Lieb–Liniger models are distinguished by the attainable range of the Luttinger parameter $w_{\exp}(x-y) = g\,\frac{\eta}{2}e^{-\eta|x-y|},$6:

• In the original (contact) Lieb–Liniger model, $w_{\exp}(x-y) = g\,\frac{\eta}{2}e^{-\eta|x-y|},$7 for all $w_{\exp}(x-y) = g\,\frac{\eta}{2}e^{-\eta|x-y|},$8, so only superfluid behavior is possible.
• For screened long-range interactions (e.g., decaying as $w_{\exp}(x-y) = g\,\frac{\eta}{2}e^{-\eta|x-y|},$9), $\eta$0, precluding a true superfluid regime at long range.
• The ELL model uniquely interpolates, admitting $\eta$1 as $\eta$2 and $\eta$3 are varied, thus accessing all three physical regimes (SF, sTG, QC) continuously (Rincon et al., 2015).

This property makes the ELL model a minimal theoretical platform for realizing and tuning all Luttinger-liquid regimes in a single microscopic Hamiltonian.

6. Physical Insights and Excitation Spectrum

At weak coupling ($\eta$4), the ground state is always a standard superfluid Luttinger liquid ($\eta$5), closely matching Lieb–Liniger behavior except for short-distance corrections. At strong coupling ($\eta$6), finite-range interactions ($\eta$7) begin to suppress superfluidity; for intermediate $\eta$8, the system enters the sTG regime characterized by strong fermion-like correlations; for smaller $\eta$9 (longer-range), quasi-crystalline order emerges ($1/\eta$0).

The excitation spectrum retains a phonon-like linear dispersion at low momenta ($1/\eta$1), but the finite-exponential range leads to nontrivial modifications at intermediate $1/\eta$2, shifting spectral weight and enhancing roton-like features as $1/\eta$3 falls.

Continuous tuning of $1/\eta$4 from infinity to zero captures physics of both short-range superfluids and long-range-ordered density waves not accessible in the pure contact case.

7. Summary and Impact

The repulsive Lieb–Liniger model with an exponentially-decaying two-body interaction realizes a comprehensive, exactly solvable platform for exploring one-dimensional quantum liquid physics beyond the paradigms set by purely contact or long-range interactions. The ELL model’s phase diagram embodies the full landscape—superfluid, super-Tonks–Girardeau, and quasi-crystallinity—parameterized by the Luttinger parameter $1/\eta$5, and enables the study of crossovers, phase transitions, and correlation regimes in a single, controlled theoretical setting. State-of-the-art CMPS techniques not only provide accurate quantitative access to these regimes, but also establish the broader utility of tensor-network variational methods for continuous, non-integrable models and long-range-interacting quantum fluids (Rincon et al., 2015).