Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1211.3267 · Physical Review A

Universal local pair correlations of Lieb-Liniger bosons at quantum criticality

Ming‐Ming Wang, Jiahao Huang, Chaohong Lee, X.-G. Yin, Xin Guan, Murray T. Batchelor

Published Apr 29, 2013• 14 citations
arXivHTMLPDF

Abstract

The one-dimensional Lieb-Liniger Bose gas is a prototypical many-body system featuring universal Tomonaga-Luttinger liquid (TLL) physics and free fermion quantum criticality. We analytically calculate finite temperature local pair correlations for the strong-coupling Bose gas at quantum criticality using the polylog function in the framework of the Yang-Yang thermodynamic equations. We show that the local pair correlation has the universal value ${g}^{(2)}(0)\ensuremath{\approx}2p/(n\ensuremath{\varepsilon})$ in the quantum critical regime, the TLL phase, and the quasiclassical region, where $p$ is the pressure per unit length rescaled by the interaction energy $\ensuremath{\varepsilon}=\frac{{\ensuremath{\hbar}}^{2}}{2m}{c}^{2}$ with interaction strength $c$ and linear density $n$. This suggests the possibility to test finite temperature local pair correlations for the TLL in the relativistic dispersion regime and to probe quantum criticality with the local correlations beyond the TLL phase. Furthermore, thermodynamic properties at high temperatures are obtained by both high temperature and virial expansion of the Yang-Yang thermodynamic equation.

DOI
10.1103/physreva.87.043634

Related papers

Polylogs, thermodynamics and scaling functions of one-dimensional quantum many-body systems

Xi‐Wen Guan, Murray T. Batchelor

We demonstrate that the thermodynamics of one-dimensional Lieb–Liniger bosons can be accurately calculated in analytic fashion using the polylog function in the framework of the thermodynamic Bethe ansatz. The approach does away with the need to numerically solve the thermodynamic Bethe ansatz (Yang–Yang) equation. The expression for the equation of state allows the exploration of Tomonaga–Luttinger liquid physics and quantum criticality in an archetypical quantum system. In particular, the low-temperature phase diagram is obtained, along with the scaling functions for the density and compressibility. It has been shown recently by Guan and Ho (arXiv:1010.1301) that such scaling can be used to map out the criticality of ultracold fermionic atoms in experiments. We show here how to map out quantum criticality for Lieb–Liniger bosons. More generally, the polylog function formalism can be applied to a wide range of Bethe ansatz integrable quantum many-body systems which are currently of theoretical and experimental interest, such as strongly interacting multi-component fermions, spinor bosons and mixtures of bosons and fermions.

arXiv:1010.4842•Journal of Physics A Mathematical and Theoretical•Feb 16, 2011•32 citations

Thermodynamics and Tomonaga-Luttinger liquid behavior of the quantum one-dimensional hard-rod model

Shengjie Yu, Zhaoxuan Zhu, Laurent Sanchez-Palencia

The one-dimensional hard rod model describes impenetrable bosons with finite diameter, extending the Lieb-Liniger model to systems with excluded volume interactions. Here, we investigate the thermodynamics of quantum HRs using Yang-Yang theory, path integral quantum Monte-Carlo calculations, and Luttinger liquid theory. We first discuss the behavior of characteristic thermodynamic quantities, exhibiting deviations to the Lieb-Liniger model for sufficiently high densities, with excellent agreement between analytical and numerical results. We then show that the hard rod model exhibits Tomonaga-Luttinger liquid behavior across a wide range of parameters, at zero and finite temperature, as unveiled by correlation functions. The Tomonaga-Luttinger parameter and thermal length can be extracted by fitting correlation functions to Tomonaga-Luttinger liquid theory, hence demonstrating a robust method for thermometry. This work provides a comprehensive study of strongly correlated hard rod systems at finite temperatures, with applications to quantum wires, spin chains, and ultracold atoms.

arXiv:2505.20376•Physical Review A•Feb 4, 2026•1 citation

Universality class of quantum criticality for strongly repulsive spin-1 bosons with antiferromagnetic spin-exchange interaction

C. C. N. Kuhn, Xin Guan, Angela Foerster +1 more

Using the thermodynamic Bethe ansatz equations we study the quantum phase diagram, thermodynamics, and criticality of one-dimensional (1D) spin-1 bosons with strongly repulsive density-density and antiferromagnetic spin-exchange interactions. We analytically derive a high-precision equation of state from which the Tomonaga-Luttinger-liquid physics and quantum critical behavior of the system are computed. We obtain explicit forms for the scaling functions near the critical points yielding the dynamical exponent $z=2$ and correlation length exponent $\ensuremath{\nu}=1/2$ for the quantum phase transitions driven by either the chemical potential or the magnetic field. Consequently, we further demonstrate that quantum criticality of the system can be mapped out from the finite temperature density and magnetization profiles of the 1D trapped gas. Our results provide the physical origin of quantum criticality in a 1D many-body system beyond the Tomonaga-Luttinger-liquid description.

arXiv:1201.6456•Physical Review A•Apr 9, 2012•9 citations

Quantum criticality and the Tomonaga-Luttinger liquid in one-dimensional Bose gases

Bing Yang, Yang-Yang Chen, Yong-Guang Zheng +5 more

We experimentally investigate the quantum criticality and Tomonaga-Luttinger liquid (TLL) behavior within one-dimensional (1D) ultracold atomic gases. Based on the measured density profiles at different temperatures, the universal scaling laws of thermodynamic quantities are observed. The quantum critical regime and the relevant crossover temperatures are determined through the double-peak structure of the specific heat. In the TLL regime, we obtain the Luttinger parameter by probing sound propagation. Furthermore, a characteristic power-law behavior emerges in the measured momentum distributions of the 1D ultracold gas, confirming the existence of the TLL.

arXiv:1611.00426•Physical Review Letters•Oct 18, 2017•94 citations

Universal low-energy physics in 1D strongly repulsive multi-component Fermi gases

Yuzhu Jiang, Peng He, Xi‐Wen Guan

Abstract It has been shown (Yang and You 2011 Chin. Phys. Lett. 28 020503 ) that at zero temperature the ground state of the one-dimensional (1D) w -component Fermi gas coincides with that of the spinless Bose gas in the limit . This behavior was experimentally evidenced through quasi-1D tightly trapping ultracold 173 Yb atoms in a recent paper (Pagano et al 2014 Nat. Phys. 10 198 ). However, understanding of low-temperature behavior of Fermi gases with a repulsive interaction requires spin-charge separated conformal field theories of an effective Tomonaga–Luttinger liquid and an antiferromagnetic SU ( w ) Heisenberg spin chain. Here we analytically derive universal thermodynamics of 1D strongly repulsive fermionic gases with SU ( w ) symmetry via the Yang–Yang thermodynamic Bethe ansatz method. The analytical free energy and magnetic properties of the systems at low temperature in a weak magnetic field are obtained through the Wiener–Hopf method. In particular, the free energy essentially manifests the spin-charge separated conformal field theories for high-symmetry systems with arbitrary repulsive interaction strength. We also find that the sound velocity of the Fermi gases in the large w limit coincides with that for the spinless Bose gas, whereas the spin velocity vanishes quickly as w becomes large. This indicates strong suppression of the Fermi exclusion statistics by the commutativity feature among the w -component fermions with different spin states in the Tomonaga–Luttinger liquid phase. Moreover, the equations of state and critical behavior of physical quantities at finite temperature are analytically derived in terms of the polylogarithm functions in the quantum critical region.

arXiv:1602.02507•Journal of Physics A Mathematical and Theoretical•Mar 18, 2016•7 citations

Analytic thermodynamic properties of the Lieb-Liniger gas

Matthew L. Kerr, Giulia De Rosi, Karen V. Kheruntsyan

We present a comprehensive review on the state-of-the-art of the approximate analytic approaches describing the finite-temperature thermodynamic quantities of the Lieb-Liniger model of the one-dimensional (1D) Bose gas with contact repulsive interactions. This paradigmatic model of quantum many-body-theory plays an important role in many areas of physics-thanks to its integrability and possible experimental realization using, e.g., ensembles of ultracold bosonic atoms confined to quasi-1D geometries. The thermodynamics of the uniform Lieb-Liniger gas can be obtained numerically using the exact thermal Bethe ansatz (TBA) method, first derived in 1969 by Yang and Yang. However, the TBA numerical calculations do not allow for the in-depth understanding of the underlying physical mechanisms that govern the thermodynamic behavior of the Lieb-Liniger gas at finite temperature. Our work is then motivated by the insights that emerge naturally from the transparency of closed-form analytic results, which are derived here in six different regimes of the gas and which exhibit an excellent agreement with the TBA numerics. Our findings can be further adopted for characterising the equilibrium properties of inhomogeneous (e.g., harmonically trapped) 1D Bose gases within the local density approximation and for the development of improved hydrodynamic theories, allowing for the calculation of breathing mode frequencies which depend on the underlying thermodynamic equation of state. Our analytic approaches can be applied to other systems including impurities in a quantum bath, liquid helium-4, and ultracold Bose gas mixtures.

arXiv:2404.06092•SciPost Physics Core•Jul 30, 2024•14 citations
The one-dimensional Lieb-Liniger Bose gas is a prototypical many-body system featuring universal Tomonaga-Luttinger liquid (TLL) physics and free fermion quantum criticality. We analytically calculate finite temperature local pair correlations for the strong-coupling Bose gas at quantum criticality using the polylog function in the framework of the Yang-Yang thermodynamic equations. We show that the local pair correlation has the universal value g(2)(0)\ensuremath≈2p/(n\ensuremathε){g}^{(2)}(0)\ensuremath{\approx}2p/(n\ensuremath{\varepsilon})g(2)(0)\ensuremath≈2p/(n\ensuremathε) in the quantum critical regime, the TLL phase, and the quasiclassical region, where ppp is the pressure per unit length rescaled by the interaction energy \ensuremathε=\ensuremathℏ22mc2\ensuremath{\varepsilon}=\frac{{\ensuremath{\hbar}}^{2}}{2m}{c}^{2}\ensuremathε=2m\ensuremathℏ2​c2 with interaction strength ccc and linear density nnn. This suggests the possibility to test finite temperature local pair correlations for the TLL in the relativistic dispersion regime and to probe quantum criticality with the local correlations beyond the TLL phase. Furthermore, thermodynamic properties at high temperatures are obtained by both high temperature and virial expansion of the Yang-Yang thermodynamic equation.
We demonstrate that the thermodynamics of one-dimensional Lieb–Liniger bosons can be accurately calculated in analytic fashion using the polylog function in the framework of the thermodynamic Bethe ansatz. The approach does away with the need to numerically solve the thermodynamic Bethe ansatz (Yang–Yang) equation. The expression for the equation of state allows the exploration of Tomonaga–Luttinger liquid physics and quantum criticality in an archetypical quantum system. In particular, the low-temperature phase diagram is obtained, along with the scaling functions for the density and compressibility. It has been shown recently by Guan and Ho (arXiv:1010.1301) that such scaling can be used to map out the criticality of ultracold fermionic atoms in experiments. We show here how to map out quantum criticality for Lieb–Liniger bosons. More generally, the polylog function formalism can be applied to a wide range of Bethe ansatz integrable quantum many-body systems which are currently of theoretical and experimental interest, such as strongly interacting multi-component fermions, spinor bosons and mixtures of bosons and fermions.
The one-dimensional hard rod model describes impenetrable bosons with finite diameter, extending the Lieb-Liniger model to systems with excluded volume interactions. Here, we investigate the thermodynamics of quantum HRs using Yang-Yang theory, path integral quantum Monte-Carlo calculations, and Luttinger liquid theory. We first discuss the behavior of characteristic thermodynamic quantities, exhibiting deviations to the Lieb-Liniger model for sufficiently high densities, with excellent agreement between analytical and numerical results. We then show that the hard rod model exhibits Tomonaga-Luttinger liquid behavior across a wide range of parameters, at zero and finite temperature, as unveiled by correlation functions. The Tomonaga-Luttinger parameter and thermal length can be extracted by fitting correlation functions to Tomonaga-Luttinger liquid theory, hence demonstrating a robust method for thermometry. This work provides a comprehensive study of strongly correlated hard rod systems at finite temperatures, with applications to quantum wires, spin chains, and ultracold atoms.
Using the thermodynamic Bethe ansatz equations we study the quantum phase diagram, thermodynamics, and criticality of one-dimensional (1D) spin-1 bosons with strongly repulsive density-density and antiferromagnetic spin-exchange interactions. We analytically derive a high-precision equation of state from which the Tomonaga-Luttinger-liquid physics and quantum critical behavior of the system are computed. We obtain explicit forms for the scaling functions near the critical points yielding the dynamical exponent z=2z=2z=2 and correlation length exponent $\ensuremath{\nu}=1/2$ for the quantum phase transitions driven by either the chemical potential or the magnetic field. Consequently, we further demonstrate that quantum criticality of the system can be mapped out from the finite temperature density and magnetization profiles of the 1D trapped gas. Our results provide the physical origin of quantum criticality in a 1D many-body system beyond the Tomonaga-Luttinger-liquid description.
We experimentally investigate the quantum criticality and Tomonaga-Luttinger liquid (TLL) behavior within one-dimensional (1D) ultracold atomic gases. Based on the measured density profiles at different temperatures, the universal scaling laws of thermodynamic quantities are observed. The quantum critical regime and the relevant crossover temperatures are determined through the double-peak structure of the specific heat. In the TLL regime, we obtain the Luttinger parameter by probing sound propagation. Furthermore, a characteristic power-law behavior emerges in the measured momentum distributions of the 1D ultracold gas, confirming the existence of the TLL.
Abstract It has been shown (Yang and You 2011 Chin. Phys. Lett. 28 020503 ) that at zero temperature the ground state of the one-dimensional (1D) w -component Fermi gas coincides with that of the spinless Bose gas in the limit . This behavior was experimentally evidenced through quasi-1D tightly trapping ultracold 173 Yb atoms in a recent paper (Pagano et al 2014 Nat. Phys. 10 198 ). However, understanding of low-temperature behavior of Fermi gases with a repulsive interaction requires spin-charge separated conformal field theories of an effective Tomonaga–Luttinger liquid and an antiferromagnetic SU ( w ) Heisenberg spin chain. Here we analytically derive universal thermodynamics of 1D strongly repulsive fermionic gases with SU ( w ) symmetry via the Yang–Yang thermodynamic Bethe ansatz method. The analytical free energy and magnetic properties of the systems at low temperature in a weak magnetic field are obtained through the Wiener–Hopf method. In particular, the free energy essentially manifests the spin-charge separated conformal field theories for high-symmetry systems with arbitrary repulsive interaction strength. We also find that the sound velocity of the Fermi gases in the large w limit coincides with that for the spinless Bose gas, whereas the spin velocity vanishes quickly as w becomes large. This indicates strong suppression of the Fermi exclusion statistics by the commutativity feature among the w -component fermions with different spin states in the Tomonaga–Luttinger liquid phase. Moreover, the equations of state and critical behavior of physical quantities at finite temperature are analytically derived in terms of the polylogarithm functions in the quantum critical region.
We present a comprehensive review on the state-of-the-art of the approximate analytic approaches describing the finite-temperature thermodynamic quantities of the Lieb-Liniger model of the one-dimensional (1D) Bose gas with contact repulsive interactions. This paradigmatic model of quantum many-body-theory plays an important role in many areas of physics-thanks to its integrability and possible experimental realization using, e.g., ensembles of ultracold bosonic atoms confined to quasi-1D geometries. The thermodynamics of the uniform Lieb-Liniger gas can be obtained numerically using the exact thermal Bethe ansatz (TBA) method, first derived in 1969 by Yang and Yang. However, the TBA numerical calculations do not allow for the in-depth understanding of the underlying physical mechanisms that govern the thermodynamic behavior of the Lieb-Liniger gas at finite temperature. Our work is then motivated by the insights that emerge naturally from the transparency of closed-form analytic results, which are derived here in six different regimes of the gas and which exhibit an excellent agreement with the TBA numerics. Our findings can be further adopted for characterising the equilibrium properties of inhomogeneous (e.g., harmonically trapped) 1D Bose gases within the local density approximation and for the development of improved hydrodynamic theories, allowing for the calculation of breathing mode frequencies which depend on the underlying thermodynamic equation of state. Our analytic approaches can be applied to other systems including impurities in a quantum bath, liquid helium-4, and ultracold Bose gas mixtures.