[Paper Review] The Lieb-Liniger model at the critical point as toy model for Black Holes

Mischa Panchenko|arXiv (Cornell University)|Oct 15, 2015

Relativity and Gravitational Theory4 citations

TL;DR

This paper proposes the attractive Lieb-Liniger model at its quantum critical point as a laboratory-accessible toy model for black holes, leveraging its emergent gapless spectrum and Bose-Einstein condensate-like behavior. Using a novel diagonalization technique in the low-energy sector dominated by modes $a_0, a_{\pm1}$, the authors analytically derive a $N^{-1/3}$ scaling of the energy gap and show complete ground state depletion in the $N \to \infty$ limit, confirming black hole-like quantum criticality.

ABSTRACT

In a previous series of papers it was proposed that black holes can be understood as Bose-Einstein condensates at the critical point of a quantum phase transition. Therefore other bosonic systems with quantum criticalities, such as the Lieb-Liniger model with attractive interactions, could possibly be used as toy models for black holes. Even such simple models are hard to analyse, as mean field theory usually breaks down at the critical point. Very few analytic results are known. In this paper we present a method of studying such systems at quantum critical points analytically. We will be able to find explicit expressions for the low energy spectrum of the Lieb-Liniger model and thereby to confirm the expected black hole like properties of such systems. This opens up an exciting possibility of constructing and studying black hole like systems in the laboratory.

Motivation & Objective

• To establish the attractive Lieb-Liniger model at its quantum critical point as a tractable, analytically solvable toy model for black hole physics.
• To overcome the breakdown of mean-field and Bogoliubov theories at the critical point by developing a new diagonalization method for finite-dimensional Hamiltonians.
• To analytically determine the low-energy spectrum, energy gap scaling, and ground state properties of the system at critical coupling $\alpha = 1$.
• To confirm the emergence of gapless modes and critical scaling behavior consistent with black hole analogues.
• To enable quantitative predictions for future laboratory experiments on quantum critical bosonic systems.

Proposed method

• Restrict the Hamiltonian to the low-energy sector dominated by the $a_0, a_{\pm1}$ modes, forming a finite-dimensional matrix Hamiltonian.
• Develop a new perturbation theory in $1/N$ to diagonalize the interacting Hamiltonian at critical coupling $\alpha = 1$.
• Use a continuous field representation to map the three-mode system to a 2D isotropic quartic oscillator with potential $V(r) = \frac{7\alpha}{32N}r^4$ at $\alpha=1$.
• Apply Bogoliubov transformations to the non-interacting part and analyze the critical point where the gap vanishes in the $N \to \infty$ limit.
• Demonstrate that higher momentum modes decouple in the large $N$ limit, justifying the truncation to the low-energy sector.
• Cross-validate analytical results with numerical calculations to confirm scaling behavior and eigenstate structure.

Experimental results

Research questions

• RQ1 Does the attractive Lieb-Liniger model at the critical point $\alpha = 1$ exhibit a gapless low-energy spectrum consistent with black hole analogues?
• RQ2 What is the analytic scaling of the energy gap with particle number $N$ in the critical regime?
• RQ3 How does the ground state depletion scale with $N$, and does it approach full depletion in the thermodynamic limit?
• RQ4 Can a new perturbative diagonalization method yield exact expressions for the low-energy spectrum at criticality?
• RQ5 Is the low-energy dynamics of the system equivalent to a 2D purely quartic oscillator, and what does this imply for its spectral properties?

Key findings

• Only the modes $a_0, a_{\pm1}$ contribute to the low-energy spectrum at the critical point, justifying the low-dimensional truncation.
• The energy gap between adjacent low-energy eigenstates scales as $N^{-1/3}$, confirming the system becomes gapless in the $N \to \infty$ limit.
• The ground state is completely depleted in the $N \to \infty$ limit, with depletion scaling as $N^{1/3}$.
• An explicit analytic expression for the ground state is obtained to first order in $1/N$ using a new perturbation theory.
• The system at criticality is equivalent to a 2D isotropic purely quartic oscillator, revealing its spectral properties via this mapping.
• Higher momentum modes decouple in the large $N$ limit, validating the low-energy effective description.