Although the one-dimensional repulsive Fermi-Hubbard model has been intensively studied over many decades, a rigorous understanding of many aspects of the model is still lacking. In this work, based on the solutions to the thermodynamic Bethe ansatz equations, we provide a rigorous study on the following. (1) We calculate the fractional excitations of the system in various phases, from which we identify the parameter regime featuring the spin-incoherent Luttinger liquid (SILL). We investigate the universal properties and the asymptotic of correlation functions of the SILL. (2) We study the interaction-driven phase transition and the associated criticality, and build up an essential connection between the contact susceptibilities and the variations of density, magnetization, and entropy with respect to the interaction strength. As an application of these concepts, which hold true for higher-dimensional systems, we propose a quantum cooling scheme based on the interaction-driven refrigeration cycle.
Although the one-dimensional repulsive Fermi-Hubbard model has been intensively studied over many decades, a rigorous understanding of many aspects of the model is still lacking. In this work, based on the solutions to the thermodynamic Bethe ansatz equations, we provide a rigorous study on the following. (1) We calculate the fractional excitations of the system in various phases, from which we identify the parameter regime featuring the spin-incoherent Luttinger liquid (SILL). We investigate the universal properties and the asymptotic of correlation functions of the SILL. (2) We study the interaction-driven phase transition and the associated criticality, and build up an essential connection between the contact susceptibilities and the variations of density, magnetization, and entropy with respect to the interaction strength. As an application of these concepts, which hold true for higher-dimensional systems, we propose a quantum cooling scheme based on the interaction-driven refrigeration cycle.
We present analytical results of fundamental properties of one-dimensional (1D) Hubbard model with a repulsive interaction, ranging from fractional excitations to universal thermodynamics, interaction-driven criticality, correlation functions, Contact susceptibilities and quantum cooling. 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. Based on the analysis on the fractional charge and spin excitations, the spin-incoherent Luttinger liquid with only the charge propagation mode is elucidated by the asymptotic of the two-point correlation functions with the help of the conformal field theory. Near quadruple critical point, we then further analytically obtain the thermodynamical properties, dimensionless ratios and scaling functions near quantum phase transitions in terms of chemical potential, magnetic field and interaction. In particular, we determine additivity rules of spin and charge susceptibilities, and derive explicit forms of thermodynamics of spin-incoherent Luttinger liquid. Finally, in order to capture deeper insight into the Mott insulator and interaction driven criticality, we further study the double occupancy and its associated Contact and Contact susceptibilities through which an adiabatic cooling scheme upon the quantum criticality is introduced.
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 the k − Λ string bound states at arbitrary driving fields, showing subtle differences in spin magnons and charge η -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.
The one-dimensional (1D) Hubbard model, describing electrons on a lattice with an on-site repulsive interaction, provides a paradigm for the physics of quantum many-body phenomena. Here, by solving the thermodynamic Bethe ansatz equations, we study the universal thermodynamics, quantum criticality, and magnetism of the 1D attractive Hubbard model. We show that the compressibility and the susceptibility of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-like state obey simple additivity rules at low temperatures, indicating an existence of two free quantum fluids. The magnetic properties, such as magnetization and susceptibility, reveal three physical regions: quantum fluids at low temperatures, a non-Fermi liquid at high temperatures, and the quantum fluid to non-Fermi liquid crossover in between. The lattice interaction is seen to significantly influence the nature of the FFLO-like state in 1D. Furthermore, we show that the dimensionless Wilson ratio provides an ideal parameter to map out the various phase boundaries and to characterize the two free fluids of the FLLO-like state. The quantum scaling functions for the thermal and magnetic properties yield the same dynamic critical exponent z=2 and correlation critical exponent $\ensuremath{\nu}=1/2$ in the quantum critical region whenever a phase transition occurs. Our results provide a rigorous understanding of quantum criticality and free fluids of many-body systems on a 1D lattice.
Bethe ansatz and bosonization procedures are used to describe the thermodynamics of the strong-coupled Hubbard chain in the spin-incoherent Luttinger liquid (LL) regime: $J(\ensuremath{\equiv}4{t}^{2}/U)\ensuremath{\ll}{k}_{B}T\ensuremath{\ll}{E}_{F}$, where t is the hopping amplitude, $U(\ensuremath{\gg}t)$ is the repulsive on-site Coulomb interaction, and ${k}_{B}T({E}_{F}\ensuremath{\sim}t)$ is the thermal (Fermi) energy. We introduce a fractional Landau LL approach, whose $U=\ensuremath{\infty}$ fixed point is exactly mapped onto an ideal gas with two species obeying the Haldane-Wu exclusion fractional statistics. This phenomenological approach sheds light on the behavior of several thermodynamic properties in the spin-incoherent LL regime: specific heat, charge compressibility, magnetic susceptibility, and Drude weight. In fact, besides the hopping (mass) renormalization, the fractional Landau LL parameters, due to quasiparticle interaction, are determined and relationships with velocities of holons and spinons are unveiled. The specific heat thus obtained is in very good agreement with previous density matrix renormalization group (DMRG) simulations of the $t\text{\ensuremath{-}}J$ model in the spin-incoherent regime. A phase diagram is provided and two thermodynamic paths to access this regime clarifies both the numerical and analytical procedures. Further, we show that the high-T limit of the fractional Landau LL entropy and chemical potential exhibit the expected results of the $t\text{\ensuremath{-}}J$ model, under the condition $U\ensuremath{\gg}{k}_{B}T$. Lastly, finite-temperature Lanczos simulations of the single-particle distribution function confirm the characteristics of the spin-incoherent regime and the high-T limit observed in previous DMRG studies.
One-dimensional conductors are a long-standing topic of research with direct applications to organic conductors and mesoscopic rings. The discovery of the ceramic high-temperature superconductors has revitalized the interest in low-dimensional charge and spin fluctuations of highly correlated electron systems. Several mechanisms proposed to explain the high-T c superconductors invoke properties of the two-dimensional Hubbard model, but probably also some one-dimensional aspects are relevant. Numerous one-dimensional models for correlated electrons have been studied with various approximate, asymptotically exact and exact methods. These results lead to the concept of Luttinger liquid for interacting electron gases without excitation gaps (metallic systems). Characteristic of Luttinger liquids are the charge and spin separation, marginal Fermi liquid properties, e.g. the absence of quasiparticles in the vicinity of the Fermi surface, nonuniversal power-law singularities in the one-particle spectral function and the related absence of a discontinuity in the momentum distribution at the Fermi level, the power-law decay of correlation functions for long times and large distances, persistent currents in finite rings, etc. Due to the peculiarities of the phase space in one dimension some of the models have sufficient conserved currents to be completely integrable. We review exact results derived within the framework of Bethe's ansatz for integrable one-dimensional models of correlated electrons. The Bethe-ansatz method is presented by explicitly showing the steps leading to the solution of the N-component electron gas interacting via a δ-function potential (repulsive and attractive interaction), which is probably the simplest model of correlated electrons. Emphasis is given to the procedure to extract the groundstate properties, the classification of states, the excitation spectrum, the thermodynamics and finite size effects, such as critical exponents of correlation functions and persistent currents. The method is then applied to numerous other models, e.g. (i) a two-band model involving attractive and repulsive potentials and crystalline fields splitting the bands, (ii) the traditional Hubbard chain with attractive and repulsive U, (iii) the degenerate Hubbard model with repulsive U, which displays a metal–insulator transition at a finite U, (iv) a two-band Hubbard model with repulsive U, (v) the traditional supersymmetric t–J model (vi) a two-band supersymmetric t–J model with band-splitting and (vii) the N-component supersymmetric t–J model. Finally, results for models with long-range interactions, in particular r-2 and sinh -2(r) potentials, are briefly reviewed.
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.