---
title: Repulsive Lieb–Liniger and ELL Models
url: https://www.emergentmind.com/topics/repulsive-lieb-liniger-model
type: topic
---

# Repulsive Lieb–Liniger and ELL Models

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 [1508.04779].

## 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 $g$, the interaction range $1/\eta$, and the 1D density $n = \langle \Psi^\dagger(x)\Psi(x)\rangle$. Dimensionless couplings include $\gamma \equiv g/n$ (analogous to the Lieb parameter) and $\eta/n$ (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:
\[
|\Psi\rangle = \operatorname{Tr}_{\rm aux}\left[\,\mathcal{P}\exp\int_{-\infty}^{\infty} dx~(Q\otimes \mathbb{1} + R\otimes \Psi^\dagger(x))\right]|0\rangle,
\]
with $Q,R$ as $D\times D$ matrices acting on an auxiliary space and $|0\rangle$ the Fock vacuum. The bond dimension $D$ 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 $\mathcal{O}(D^3)$. This approach robustly captures the ground-state physics and Luttinger-liquid correlations for a broad range of coupling and range parameters [1508.04779].

## 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 $K$ and sound velocity $v$. Key correlation functions are:
\[
g_1(r) = \frac{\langle \Psi^\dagger(0)\Psi(r)\rangle}{n},\qquad
g_2(r) = \frac{\langle n(0)n(r)\rangle}{n^2} - 1,\quad n(x) = \Psi^\dagger(x)\Psi(x).
\]
Their long-distance asymptotics in a Luttinger liquid of parameter $K$ and $nr\gg1$ are:
\[
g_1(r) \simeq \frac{1}{(nr)^{1/(2K)}}\left[B_0 + B_1 \frac{\cos(2\pi nr)}{(nr)^{2K}}\right],
\]
\[
g_2(r) \simeq -\frac{K}{2\pi^2} \frac{1}{(nr)^2} + A_1\frac{\cos(2\pi nr)}{(nr)^{2K}},
\]
with $A_1,B_0,B_1$ nonuniversal amplitudes set by the microscopic Hamiltonian. The single-particle correlator decays algebraically with exponent $1/(2K)$, while density–density correlations have a $1/r^2$ decay modulated by oscillations at $2\pi n$ with decay exponent $2K$.

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

By extracting $K$ from the long-distance decay of correlation functions using CMPS, the $(\gamma, \eta/n)$ phase diagram reveals three Luttinger-liquid regimes [1508.04779]:
- **Superfluid (SF)**: $K>1$. Slowest decay in $g_1(r)$, no strong density modulations, characteristic of weak-to-moderate coupling or negligible interaction range.
- **Super-Tonks–Girardeau (sTG)**: $1/2 < K < 1$. Strong repulsion induces fermion-like correlations, suppressed superfluidity, significant Friedel oscillations in $g_2(r)$.
- **Quasi-crystal (QC)**: $K < 1/2$. Dominant density–density correlations, emergence of quasi-long-range order with $2\pi n$ oscillatory correlations, akin to a Luttinger-liquid precursor of charge-density wave order.

The crossover lines correspond roughly to $K=1$ (SF-sTG) and $K=1/2$ (sTG-QC). At large $\eta/n$ (short-range limit), the phase boundary to sTG/ QC shifts to larger $\gamma$, recovering features of the contact model; decreasing $\eta/n$ (longer-range tail) shifts crossovers to smaller $\gamma$.

| Regime        | Luttinger Parameter $K$ | Physical Character                      |
|---------------|:----------------------:|-----------------------------------------|
| SF            | $K > 1$                | Slowly decaying $g_1$, no strong $g_2$ oscillations  |
| sTG           | $1/2 < K < 1$          | Suppressed SF, strong $g_2$ oscillations (Friedel)   |
| QC            | $K < 1/2$              | Power-law density order at $2\pi n$     |

## 5. Comparison with Contact and Long-Range Models: Tuning $K\in(0,\infty)$

The extended Lieb–Liniger models are distinguished by the attainable range of the Luttinger parameter $K$:
- In the original (contact) Lieb–Liniger model, $K\in[1,\infty)$ for all $\gamma$, so only superfluid behavior is possible.
- For screened long-range interactions (e.g., decaying as $1/|x|^\alpha$), $K\in(0,1]$, precluding a true superfluid regime at long range.
- The ELL model uniquely interpolates, admitting $K\in(0,\infty)$ as $\gamma$ and $\eta/n$ are varied, thus accessing all three physical regimes (SF, sTG, QC) continuously [1508.04779].

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 ($\gamma\ll1$), the ground state is always a standard superfluid Luttinger liquid ($K>1$), closely matching Lieb–Liniger behavior except for short-distance corrections. At strong coupling ($\gamma\gg1$), finite-range interactions ($1/\eta$) begin to suppress superfluidity; for intermediate $\eta/n$, the system enters the sTG regime characterized by strong fermion-like correlations; for smaller $\eta/n$ (longer-range), quasi-crystalline order emerges ($K<1/2$).

The excitation spectrum retains a phonon-like linear dispersion at low momenta ($v \propto n\sqrt{K}$), but the finite-exponential range leads to nontrivial modifications at intermediate $k$, shifting spectral weight and enhancing roton-like features as $K$ falls.

Continuous tuning of $K$ 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 $K\in(0,\infty)$, 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 [1508.04779].

Source: https://www.emergentmind.com/topics/repulsive-lieb-liniger-model