1
|
Luo JJ, Pu H, Guan XW. Exact results of the one-dimensional repulsive Hubbard model. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2024; 87:117601. [PMID: 39284352 DOI: 10.1088/1361-6633/ad7b70] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/29/2023] [Accepted: 09/16/2024] [Indexed: 09/20/2024]
Abstract
We present analytical results of the fundamental properties of the one-dimensional (1D) Hubbard model with a repulsive interaction. The new model results with arbitrary external fields include: (I) using the exact solutions of the Bethe ansatz equations of the Hubbard model, we first rigorously calculate the gapless spin and charge excitations, exhibiting exotic features of fractionalized spinons and holons. We then investigate the gapped excitations in terms of the spin string and thek-Λstring bound states at arbitrary driving fields, showing subtle differences in spin magnons and chargeη-pair excitations. (II) For a high-density and high spin magnetization region, i.e. near the quadruple critical point, we further analytically obtain the thermodynamic properties, dimensionless ratios and scaling functions near quantum phase transitions. (III) Importantly, we give the general scaling functions at quantum criticality for arbitrary filling and interaction strength. These can directly apply to other integrable models. (IV) Based on the fractional excitations and the scaling laws, the spin-incoherent Luttinger liquid (SILL) with only the charge propagation mode is elucidated by the asymptotic of the two-point correlation functions with the help of conformal field theory. We also, for the first time, obtain the analytical results of the thermodynamics for the SILL. (V) Finally, to capture deeper insights into the Mott insulator and interaction-driven criticality, we further study the double occupancy and propose its associated contact and contact susceptibilities, through which an adiabatic cooling scheme based upon quantum criticality is proposed. In this scenario, we build up general relations among arbitrary external- and internal-potential-driven quantum phase transitions, providing a comprehensive understanding of quantum criticality. Our methods offer rich perspectives of quantum integrability and offer promising guidance for future experiments with interacting electrons and ultracold atoms, both with and without a lattice.
Collapse
Affiliation(s)
- Jia-Jia Luo
- Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, People's Republic of China
- University of Chinese Academy of Sciences, Beijing 100049, People's Republic of China
| | - Han Pu
- Department of Physics and Astronomy, Rice University, Houston, TX 77251-1892, United States of America
| | - Xi-Wen Guan
- Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, People's Republic of China
- Hefei National Laboratory, Hefei 230088, People's Republic of China
- NSFC-SPTP Peng Huanwu Center for Fundamental Theory, Xi'an 710127, People's Republic of China
- Department of Fundamental and Theoretical Physics, Research School of Physics, Australian National University, Canberra ACT 0200, Australia
| |
Collapse
|
2
|
Rakhimov A, Abdurakhmonov T, Tanatar B. Critical behavior of Tan's contact for bosonic systems with a fixed chemical potential. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 33:465401. [PMID: 34407517 DOI: 10.1088/1361-648x/ac1ec6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/03/2021] [Accepted: 08/18/2021] [Indexed: 06/13/2023]
Abstract
The temperature dependence of Tan's contact parameterCand its derivatives for spin gapped quantum magnets are investigated. We use the paradigm of Bose-Einstein condensation (BEC) to describe the low temperature properties of quasiparticles in the system known as triplons. Since the number of particles and the condensate fraction are not fixed we use theμVTensemble to calculate the thermodynamic quantities. The interactions are treated at the Hartree-Fock-Bogoliubov approximation level. We obtained the temperature dependence ofCand its derivative with respect to temperature and applied magnetic field both above and belowTcof the phase transition from the normal phase to BEC. We have shown thatCis regular, while its derivatives are discontinuous atTcin accordance with Ehrenfest's classification of phase transitions. Moreover, we have found a sign change in ∂C/∂Tclose to the critical temperature. As to the quantum critical point,Cand its derivatives are regular as a function of the control parameterr, which induces the quantum phase transition. At very low temperatures, one may evaluateCsimply from the expressionC=m2μ2/a¯4, where the only parameter effective mass of quasiparticles should be estimated. We propose a method for measuring of Tan's contact for spin gapped dimerized magnets.
Collapse
Affiliation(s)
| | | | - B Tanatar
- Department of Physics, Bilkent University, Bilkent, 06800 Ankara, Turkey
| |
Collapse
|
3
|
Tan's two-body contact across the superfluid transition of a planar Bose gas. Nat Commun 2021; 12:760. [PMID: 33536418 PMCID: PMC7858573 DOI: 10.1038/s41467-020-20647-6] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/28/2020] [Accepted: 12/11/2020] [Indexed: 11/11/2022] Open
Abstract
Tan’s contact is a quantity that unifies many different properties of a low-temperature gas with short-range interactions, from its momentum distribution to its spatial two-body correlation function. Here, we use a Ramsey interferometric method to realize experimentally the thermodynamic definition of the two-body contact, i.e., the change of the internal energy in a small modification of the scattering length. Our measurements are performed on a uniform two-dimensional Bose gas of 87Rb atoms across the Berezinskii–Kosterlitz–Thouless superfluid transition. They connect well to the theoretical predictions in the limiting cases of a strongly degenerate fluid and of a normal gas. They also provide the variation of this key quantity in the critical region, where further theoretical efforts are needed to account for our findings. Here the authors use Ramsey interferometry to study Tan’s contact in uniform two-dimensional Bose gas of 87Rb atoms across the Berezinskii–Kosterlitz–Thouless superfluid transition. They find that the two-body contact is continuous across the critical point.
Collapse
|
4
|
Yao H, Clément D, Minguzzi A, Vignolo P, Sanchez-Palencia L. Tan's Contact for Trapped Lieb-Liniger Bosons at Finite Temperature. PHYSICAL REVIEW LETTERS 2018; 121:220402. [PMID: 30547613 DOI: 10.1103/physrevlett.121.220402] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/11/2018] [Indexed: 06/09/2023]
Abstract
The universal Tan relations connect a variety of microscopic features of many-body quantum systems with two-body contact interactions to a single quantity, called the contact. The latter has become pivotal in the description of quantum gases. We provide a complete characterization of the Tan contact of the harmonically trapped Lieb-Liniger gas for arbitrary interactions and temperature. Combining thermal Bethe ansatz, local-density approximation, and exact quantum Monte Carlo calculations, we show that the contact is a universal function of only two scaling parameters, and determine the scaling function. We find that the temperature dependence of the contact, or equivalently the interaction dependence of the entropy, displays a maximum. The presence of this maximum provides an unequivocal signature of the crossover to the fermionized regime and it is accessible in current experiments.
Collapse
Affiliation(s)
- Hepeng Yao
- CPHT, Ecole Polytechnique, CNRS, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau, France
| | - David Clément
- Laboratoire Charles Fabry, Institut d'Optique, CNRS, Université Paris-Saclay, 2 avenue Augustin Fresnel, F-91127 Palaiseau cedex, France
| | - Anna Minguzzi
- Univ. Grenoble-Alpes, CNRS, LPMMC, F-38000 Grenoble, France
| | - Patrizia Vignolo
- Université Côte d'Azur, CNRS, Institut de Physique de Nice, 1361 route des Lucioles, 06560 Valbonne, France
| | - Laurent Sanchez-Palencia
- CPHT, Ecole Polytechnique, CNRS, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau, France
| |
Collapse
|
5
|
Pâţu OI, Klümper A, Foerster A. Universality and Quantum Criticality of the One-Dimensional Spinor Bose Gas. PHYSICAL REVIEW LETTERS 2018; 120:243402. [PMID: 29956958 DOI: 10.1103/physrevlett.120.243402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/21/2017] [Indexed: 06/08/2023]
Abstract
We investigate the universal thermodynamics of the two-component one-dimensional Bose gas with contact interactions in the vicinity of the quantum critical point separating the vacuum and the ferromagnetic liquid regime. We find that the quantum critical region belongs to the universality class of the spin-degenerate impenetrable particle gas which, surprisingly, is very different from the single-component case and identify its boundaries with the peaks of the specific heat. In addition, we show that the compressibility Wilson ratio, which quantifies the relative strength of thermal and quantum fluctuations, serves as a good discriminator of the quantum regimes near the quantum critical point. Remarkably, in the Tonks-Girardeau regime, the universal contact develops a pronounced minimum, reflected in a counterintuitive narrowing of the momentum distribution as we increase the temperature. This momentum reconstruction, also present at low and intermediate momenta, signals the transition from the ferromagnetic to the spin-incoherent Luttinger liquid phase and can be detected in current experiments with ultracold atomic gases in optical lattices.
Collapse
Affiliation(s)
- Ovidiu I Pâţu
- Institute for Space Sciences, Bucharest-Măgurele, R 077125 Romania
| | - Andreas Klümper
- Fakultät für Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, 42097 Wuppertal, Germany
| | - Angela Foerster
- Instituto de Física da UFRGS, Avenida Bento Gonçalves 9500, Porto Alegre, Rio Grande do Sul, Brazil
| |
Collapse
|
6
|
He M, Zhang S, Chan HM, Zhou Q. Concept of a Contact Spectrum and Its Applications in Atomic Quantum Hall States. PHYSICAL REVIEW LETTERS 2016; 116:045301. [PMID: 26871339 DOI: 10.1103/physrevlett.116.045301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/19/2015] [Indexed: 06/05/2023]
Abstract
A unique feature of ultracold atoms is the separation of length scales, r_{0}≪k_{F}^{-1}, where k_{F} and r_{0} are the Fermi momentum characterizing the average particle distance and the range of interaction between atoms, respectively. For s-wave scattering, Shina Tan discovered that such diluteness leads to universal thermodynamic relations governed by contact. Here, we show that the concept of contact can be generalized to an arbitrary partial-wave scattering. Contact of all partial-wave scatterings forms a contact spectrum, which establishes universal thermodynamic relations with notable differences from those in the presence of s-wave scattering alone. Such a contact spectrum is particularly useful for characterizing many-body correlations in atomic quantum Hall states (QHSs). It has an interesting connection with a special bipartite entanglement spectrum of QHSs and enables an intrinsic probe of atomic QHSs using short-range two-body correlations.
Collapse
Affiliation(s)
- Mingyuan He
- Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong
| | - Shaoliang Zhang
- Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong
| | - Hon Ming Chan
- Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong
| | - Qi Zhou
- Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong
| |
Collapse
|