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The centrally extended superalgebra psu(2|2)xR^3 was shown to play an important role for the integrable structures of the one-dimensional Hubbard model and of the planar AdS/CFT correspondence. Here we consider its quantum deformation U_q(psu(2|2)xR^3) and derive the fundamental R-matrix. From the latter we deduce an integrable spin chain Hamiltonian with three independent parameters and the corresponding Bethe equations to describe the spectrum on periodic chains. We relate our Hamiltonian to a two-parametric Hamiltonian proposed by Alcaraz and Bariev which can be considered a quantum deformation of the one-dimensional Hubbard model.

We study the charge ordering (CO) in the one-dimensional (1D) extended Hubbard model at quarter filling where the nearest-neighbor Coulomb repulsion and dimerization in the hopping parameters are included. Using the cluster mean-field approximation to take into account the effect of quantum fluctuations, we determine the CO phase boundary of the model in the parameter space at T=0 K. We thus find that the dimerization suppresses the stability of the CO phase strongly, and in consequence, the realistic parameter values for quasi-1D organic materials such as (TMTTF)$_2$PF$_6$ are outside the region of CO. We suggest that the long-range Coulomb interaction between the chains should persist to stabilize the CO phase.

In one-dimensional systems a twisted superfluid phase is found which is induced by a spontaneous breaking of the time-reversal symmetry. Using the density-matrix renormalization group allows us to show that the excitation energy gap closes exponentially causing a quasi-degenerate ground state. The two degenerate ground states are connected by the time-reversal symmetry which manifests itself in an alternating complex phase of the long-range correlation function. The quantum phase transition to the twisted superfluid is driven by pair tunneling processes in an extended Bose-Hubbard model. The phase boundaries of several other phases are discussed including a supersolid, a pair superfluid, and a pair supersolid phase as well as a highly unconventional Mott insulator with a degenerate ground state and a staggered pair correlation function.

We investigate the metal-insulator transition of the one-dimensional SU(N) Hubbard model for repulsive interaction. Using the bosonization approach a Mott transition in the charge sector at half-filling (k_F=\pi/Na_0) is conjectured for N > 2. Expressions for the charge and spin velocities as well as for the Luttinger liquid parameters and some correlation functions are given. The theoretical predictions are compared with numerical results obtained with an improved zero-temperature quantum Monte Carlo approach. The method used is a generalized Green's function Monte Carlo scheme in which the stochastic time evolution is partially integrated out. Very accurate results for the gaps, velocities, and Luttinger liquid parameters as a function of the Coulomb interaction U are given for the cases N=3 and N=4. Our results strongly support the existence of a Mott-Hubbard transition at a {\it non-zero} value of the Coulomb interaction. We find $U_c \sim 2.2$ for N=3 and $U_c \sim 2.8$ for N=4.

It is found that the finite-energy spectral properties of the one-dimensional Hubbard model are controlled by the scattering of charged $\eta$-spin-zero $2\nu$-holon composite objects, spin-zero $2\nu$-spinon composite objects, and charged $\eta$-spin-less and spin-less objects, rather than by the scattering of independent $\eta$-spin 1/2 holons and spin 1/2 spinons. Here $\nu =1,2,...$. The corresponding $S$ matrix is calculated and its relation to the spectral properties is clarified.

We review fundamental issues arising in the exact solution of the one-dimensional Hubbard model. We perform a careful analysis of the Lieb-Wu equations, paying particular attention to so-called `string solutions'. Two kinds of string solutions occur: $\Lambda$ strings, related to spin degrees of freedom and $k-\Lambda$ strings, describing spinless bound states of electrons. Whereas $\Lambda$ strings were thoroughly studied in the literature, less is known about $k-\Lambda$ strings. We carry out a thorough analytical and numerical analysis of $k-\Lambda$ strings. We further review two different approaches to the thermodynamics of the Hubbard model, the Yang-Yang approach and the quantum transfer matrix approach, respectively. The Yang-Yang approach is based on strings, the quantum transfer matrix approach is not. We compare the results of both methods and show that they agree. Finally, we obtain the dispersion curves of all elementary excitations at zero magnetic field for the less than half-filled band by considering the zero temperature limit of the Yang-Yang approach.

One-band Hubbard model with hopping parameter $t$ and Coulomb repulsion $U$ is considered at half filling. By means of the Schwinger bosons and slave Fermions representation of the electron operators and integrating out the spin-singlet Fermi fields an effective Heisenberg model with antiferromagnetic exchange constant is obtained for vectors which identifies the local orientation of the spin of the itinerant electrons. The amplitude of the spin vectors is an effective spin of the itinerant electrons accounting for the fact that some sites, in the ground state, are doubly occupied or empty. Accounting adequately for the magnon-magnon interaction the N\'{e}el temperature is calculated. When the ratio $\frac tU$ is small enough ($\frac tU\leq 0.09$) the effective model describes a system of localized electrons. Increasing the ratio increases the density of doubly occupied states which in turn decreases the effective spin and N\'{e}el temperature. The phase diagram in plane of temperature $\frac {T_N}{U}$ and parameter $\frac tU$ is presented. The quantum critical point ($T_N=0$) is reached at $\frac tU=0.9$. The magnons in the paramagnetic phase are studied and the contribution of the magnons' fluctuations to the heat capacity is calculated. At N\'{e}el temperature the heat capacity has a peak which is suppressed when the system approaches quantum critical point.

We consider charge and spin transport in the one-dimensional Hubbard model at infinite temperature, half-filling and zero magnetization. Implementing matrix-product-operator simulations of the non-equilibrium steady states of boundary-driven open Hubbard chains for up to 100 sites we find clear evidence of diffusive transport for any (non-zero and finite) value of the interaction U.

A detailed study of the one-dimensional ionic Hubbard model with interaction $U$ is presented. We focus on the band insulating (BI) phase and the spontaneously dimerized insulating (SDI) phase which appears on increasing $U$. By a recently introduced continuous unitary transformation [Krull et al.\ Phys. Rev. B {\bf 86}, 125113 (2012)] we are able to describe the system even close to the phase transition from BI to SDI although the bare perturbative series diverges before the transition is reached. First, the dispersion of single fermionic quasiparticles is determined in the full Brillouin zone. Second, we describe the binding phenomena between two fermionic quasiparticles leading to an $S=0$ and to an $S=1$ exciton. The latter corresponds to the lowest spin excitation and defines the spin gap which remains finite through the transition from BI to SDI. The former becomes soft at the transition indicating that the SDI corresponds to a condensate of these $S=0$ excitons. This view is confirmed by a BCS mean field theory for the SDI phase.

We carry out an extensive study of the phase diagram of the extended Bose Hubbard model, with a mean filling of one boson per site, in one dimension by using the density matrix renormalization group and show that it can have Superfluid (SF), Mott-insulator (MI), density-wave (DW) and Haldane-insulator (HI) phases depending on the precise value of filling and how edge states are handled. We show that the critical exponents and central charge for the HI-DW, MI-HI and SF-MI transitions are consistent with those for models in the two-dimensional Ising, Gaussian, and Berezinskii-Kosterlitz-Thouless (BKT) universality classes, respectively; and we suggest that the SF-HI transition may be more exotic than a simple BKT transition.

Using quantum Monte Carlo simulations, results of a strong-coupling expansion, and Luttinger liquid theory, we determine quantitatively the ground state phase diagram of the one-dimensional extended Hubbard model with on-site and nearest-neighbor repulsions U and V. We show that spin frustration stabilizes a bond-ordered (dimerized) state for U appr. V/2 up to U/t appr. 9, where t is the nearest-neighbor hopping. The transition from the dimerized state to the staggered charge-density-wave state for large V/U is continuous for U up to appr. 5.5 and first-order for higher U.

We discuss a formulation of the fusion procedure for integrable models which is suitable for application to non-standard R-matrices. It allows for construction of bound state R-matrices for AdS/CFT worldsheet scattering or equivalently for the one-dimensional Hubbard model. We also discuss some peculiar cases that arise in these models.

In PRB 68, 153101 (2003), Guoping Zhang presented density-matrix renormalization group (DMRG) results which contradict my DMRG calculations and Hirsch's quantum Monte Carlo (QMC) simulations for the charge-density-wave (CDW) phase boundary in the one-dimensional extended Hubbard model at half filling. In this Comment I show that Zhang's results are inaccurate and that his criticism of my work is groundless.

We used exact diagonalization of small clusters and exact results in various limits to determine the phase diagram and the critical exponents of the one dimensional (1D) $U-V$ model at quarter-filling. We found an instability of the Luttinger liquid to a charge-density wave (CDW) insulator across a boundary going from ($U$,$V$) =($+\infty$,$2t$) to ($4t$,$+\infty$). In the metallic phase, the dominant fluctuations are superconducting if $V$ is large enough and spin-density wave (SDW) otherwise. In the latter case, the critical exponent $\alpha$ of the momentum distribution does not exceed 9/16. The relevance of these results to 1D organic conductors is discussed.

We propose a scheme for investigating the quantum dynamics of interacting electron models by means of time-dependent variational principle and spin coherent states of space lattice operators. We apply such a scheme to the one-dimensional hubbard model, and solve the resulting equations in different regimes. In particular, we find that at low densities the dynamics is mapped into two coupled nonlinear Schroedinger equations, whereas near half-filling the model is described by two coupled Josephson junction arrays. Focusing then to the case in which only the phases of the spin variables are dynamically active, we examine a number of different solutions corresponding to the excitations of few macroscopic modes. Based on fixed point equation of the simpler among them, we show that the standard one-band ground state phase space is found.

We propose a scheme for investigating the quantum dynamics of interacting electron models by means of time-dependent variational principle and spin coherent states of space lattice operators. We apply such a scheme to the one-dimensional hubbard model, and solve the resulting equations in different regimes. In particular, we find that at low densities the dynamics is mapped into two coupled nonlinear Schroedinger equations, whereas near half-filling the model is described by two coupled Josephson junction arrays. Focusing then to the case in which only the phases of the spin variables are dynamically active, we examine a number of different solutions corresponding to the excitations of few macroscopic modes. Based on fixed point equation of the simpler among them, we show that the standard one-band ground state phase space is found.

The one dimensional Hubbard model with nearest and (negative) next-nearest neighbour hopping has been studied with the density-matrix renormalization group (DMRG) method. A large region of ferromagnetism has been found for finite density and finite on-site interaction.

We compute the phase diagram of the one-dimensional Bose-Hubbard model with a quasi-periodic potential by means of the density-matrix renormalization group technique. This model describes the physics of cold atoms loaded in an optical lattice in the presence of a superlattice potential whose wave length is incommensurate with the main lattice wave length. After discussing the conditions under which the model can be realized experimentally, the study of the density vs. the chemical potential curves for a non-trapped system unveils the existence of gapped phases at incommensurate densities interpreted as incommensurate charge-density wave phases. Furthermore, a localization transition is known to occur above a critical value of the potential depth V_2 in the case of free and hard-core bosons. We extend these results to soft-core bosons for which the phase diagrams at fixed densities display new features compared with the phase diagrams known for random box distribution disorder. In particular, a direct transition from the superfluid phase to the Mott insulating phase is found at finite V_2. Evidence for reentrances of the superfluid phase upon increasing interactions is presented. We finally comment on different ways to probe the emergent quantum phases and most importantly, the existence of a critical value for the localization transition. The later feature can be investigated by looking at the expansion of the cloud after releasing the trap.

We study the quantum phase transition between the superfluid and the Mott insulator in the one-dimensional (1D) Bose-Hubbard model. Using the time-evolving block decimation method, we numerically calculate the tunneling splitting of two macroscopically distinct states with different winding numbers. From the scaling of the tunneling splitting with respect to the system size, we determine the critical point of the superfluid to Mott insulator transition for arbitrary integer filling factors. We find that the critical values versus the filling factor in 1D, 2D, and 3D are well approximated by a simple analytical function. We also discuss the condition for determining the transition point from a perspective of the instanton method.

We study finite size effects for the gap of the quasiparticle excitation spectrum in the weakly interacting regime one-dimensional Hubbard model with on-site attraction. Two type of corrections to the result of the thermodynamic limit are obtained. Aside from a power law (conformal) correction due to gapless excitations which behaves as $1/N_a$, where $N_a$ is the number of lattice sites, we obtain corrections related to the existence of gapped excitations. First of all, there is an exponential correction which in the weakly interacting regime ($|U|\ll t$) behaves as $\sim \exp (-N_a \Delta_{\infty}/4 t)$ in the extreme limit of $N_a \Delta_{\infty} /t \gg 1$, where $t$ is the hopping amplitude, $U$ is the on-site energy, and $\Delta_{\infty}$ is the gap in the thermodynamic limit. Second, in a finite size system a spin-flip producing unpaired fermions leads to the appearance of solitons with non-zero momenta, which provides an extra (non-exponential) contribution $\delta$. For moderate but still large values of $N_a\Delta_{\infty} /t$, these corrections significantly increase and may become comparable with the $1/N_a$ conformal correction. Moreover, in the case of weak interactions where $\Delta_{\infty}\ll t$, the exponential correction exceeds higher order power law corrections in a wide range of parameters, namely for $N_a\lesssim (8t/\Delta_{\infty})\ln(4t/|U|)$, and so does $\delta$ even in a wider range of $N_a$. For sufficiently small number of particles, which can be of the order of thousands in the weakly interacting regime, the gap is fully dominated by finite size effects.

In this thesis, we present a comprehensive study of chaos and thermalization of the one-dimensional Bose-Hubbard Model (BHM) within the classical field approximation. Two quantitative measures are compared: the ensemble-averaged Finite-time Maximal Lyapunov exponent, a measures of chaos and the normalized spectral entropy, a measure of the distance between the numerical time-averaged momentum distribution and the one predicted by thermodynamics. A threshold for chaos is found, which depends on two parameters, the nonlinearity and the total energy-per-particle. Below the threshold, the dynamics are regular, while far above the threshold, complete thermalization is observed, as measured by the normalized spectral entropy. We study individual resonances in the Bose-Hubbard model to determine the criterion for chaos. The criterion based on Chirikov's method of overlapping resonances diverges in the thermodynamic limit, in contrast to the criterion parameters inferred from numerical calculations, signifying the failure of the standard Chirikov's approach. The Ablowitz-Ladik lattice is one of several integrable models that are close to the BHM. We outline the method of Inverse Scattering Transform and generate the integrals of motion of the Ablowitz-Ladik lattice. Furthermore, we discuss the possible role of these quantities in the relaxation dynamics of the BHM.

We consider excitations with the quantum numbers of a hole in the one dimensional Hubbard model below half-filling. We calculate the finite-size corrections to the energy. The results are then used to determine threshold singularities in the single-particle Green's function for commensurate fillings. We present the analogous results for the Yang-Gaudin model (electron gas with delta-function interactions).

We present a multi-reference configuration mixing scheme for describing ground and excited states, with well defined spin and space group symmetry quantum numbers, of the one-dimensional Hubbard model with nearest-neighbor hopping and periodic boundary conditions. Within this scheme, each state is expanded in terms of non-orthogonal and variationally determined symmetry-projected configurations. The results for lattices up to 30 and 50 sites compare well with the exact Lieb-Wu solutions as well as with results from other state-of-the-art approximations. In addition to spin-spin correlation functions in real space and magnetic structure factors, we present results for spectral functions and density of states computed with an ansatz whose quality can be well-controlled by the number of symmetry-projected configurations used to approximate the systems with $N_{e}$ and $N_{e} \pm 1$ electrons. The intrinsic symmetry-broken determinants resulting from the variational calculations have rich structures in terms of defects that can be regarded as basic units of quantum fluctuations. Given the quality of the results here reported, as well as the parallelization properties of the considered scheme, we believe that symmetry-projection techniques, which have found ample applications in nuclear structure physics, deserve further attention in the study of low-dimensional correlated many-electron systems.

We investigate the quantum phases of hard-core bosonic atoms in an extended Hubbard model where particles interact via soft-shoulder potentials in one dimension. Using a combination of field-theoretical methods and strong-coupling perturbation theory, we demonstrate that the low-energy phase can be a conformal cluster Luttinger liquid (CLL) phase with central charge $c=1$, where the microscopic degrees of freedom correspond to mesoscopic ensembles of particles. Using numerical density-matrix-renormalization-group methods, we demonstrate that the CLL phase, first predicted in [Phys. Rev. Lett. 111, 165302 (2013)], is separated from a conventional Tomonaga-Luttinger liquid by an exotic critical point with central charge $c=3/2$. The latter is expression of an emergent conformal supersymmetry, which is not present in the original Hamiltonian. We discuss the observability of the CLL phase in realistic experimental settings with weakly-dressed Rydberg atoms confined to optical lattices. Using quantum Monte-Carlo simulations, we show that the typical features of CLLs are stable up to comparatively high temperatures. Using exact diagonalizations and quantum trajectory methods, we provide a protocol for adiabatic state preparation as well as quantitative estimates on the effects of particle losses.

We show that the one-dimensional extended Hubbard model has saturated ferromagnetic ground states with the spin-triplet electron pair condensation in a certain range of parameters. The ground state wave functions with fixed electron numbers are explicitly obtained. We also construct two ground states in which both the spin-rotation and the gauge symmetries are broken, and show that these states are transferred from one to the other by applying the edge operators. The edge operators are reduced to the Majorana fermions in a special case. These symmetry breaking ground states are shown to be stabilized by a superconducting mean field Hamiltonian which is related to the Kitaev chain with the charge-charge interaction.

We reconsider the quantum inverse scattering approach to the one-dimensional Hubbard model and work out some of its basic features so far omitted in the literature. It is our aim to show that $R$-matrix and monodromy matrix of the Hubbard model, which are known since ten years now, have good elementary properties. We provide a meromorphic parametrization of the transfer matrix in terms of elliptic functions. We identify the momentum operator for lattice fermions in the expansion of the transfer matrix with respect to the spectral parameter and thereby show the locality and translational invariance of all higher conserved quantities. We work out the transformation properties of the monodromy matrix under the su(2) Lie algebra of rotations and under the $\h$-pairing su(2) Lie algebra. Our results imply su(2)$\oplus$su(2) invariance of the transfer matrix for the model on a chain with an even number of sites.

We reconsider the quantum inverse scattering approach to the one-dimensional Hubbard model and work out some of its basic features so far omitted in the literature. It is our aim to show that $R$-matrix and monodromy matrix of the Hubbard model, which are known since ten years now, have good elementary properties. We provide a meromorphic parametrization of the transfer matrix in terms of elliptic functions. We identify the momentum operator for lattice fermions in the expansion of the transfer matrix with respect to the spectral parameter and thereby show the locality and translational invariance of all higher conserved quantities. We work out the transformation properties of the monodromy matrix under the su(2) Lie algebra of rotations and under the $\h$-pairing su(2) Lie algebra. Our results imply su(2)$\oplus$su(2) invariance of the transfer matrix for the model on a chain with an even number of sites.

Phase diagram of the Hubbard-Holstein model in the coexistence of electron-electron and electron-phonon interactions has been theoretically obtained with the density-matrix renormalization group method for one-dimensional (1D) systems, where an improved warm-up (the recursive sweep) procedure has enabled us to calculate various correlation functions. We have examined the cases of (i) the systems half-filled by electrons for the full parameter space spanned by the electron-electron and electron-phonon coupling constants and the phonon frequency, (ii) non-half-filled system, and (iii) trestle lattice. For (i), we have detected a region where both the charge and on-site pairing correlations decay with power-laws in real space, which suggests a metallic behavior. While pairing correlations are not dominant in (i), we have found that they become dominant as the system is doped in (ii), or as the electronic band structure is modified (with a broken electron-hole symmetry) in (iii) in certain parameter regions.

The spin gap of a one-dimensional repulsive Hubbard model is numerically calculated with the density matrix renormalization group, with a special emphasis on the effect of a next-nearest neighbor hopping (t') and the nearest-neighbor ferromagnetic exchange (J) interaction. At half-filling, a significant spin gap opens if |t'| \simeq |t| and J=0, in agreement with the weak coupling theory, while the gap is strongly suppressed by the introduction of J. On the other hand, the quarter-filled system has very small spin gaps regardless of the values of t' and J. Implications for the CuO_2 chain in Sr_{14}Cu_{24}O_{41} and related materials are discussed.

We calculate the charge and spin Drude weight of the one-dimensional extended Hubbard model with on-site repulsion $U$ and nearest-neighbor repulsion $V$ at quarter filling using the density-matrix renormalization group method combined with a variational principle. Our numerical results for the Hubbard model (V=0) agree with exact results obtained from the Bethe ansatz solution. We obtain the contour map for both Drude weights in the $UV$-parameter space for repulsive interactions. We find that the charge Drude weight is discontinuous across the Kosterlitz-Thouless transition between the Luttinger liquid and the charge-density-wave insulator, while the spin Drude weight varies smoothly and remains finite in both phases. Our results can be generally understood using bosonization and renormalization group results. The finite-size scaling of the charge Drude weight is well fitted by a polynomial function of the inverse system size in the metallic region. In the insulating region we find an exponential decay of the finite-size corrections with the system size and a universal relation between the charge gap $\Delta_c$ and the correlation length $\xi$ which controls this exponential decay.

We explore the pairing properties of the one-dimensional attractive Hubbard model in the presence of finite spin polarization. The correlation exponents for the most important fluctuations are determined as a function of the density and the polarization. We find that in a system with spin population imbalance, Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-type pairing at wavevector Q=|k_{F,\uparrow}-k_{F,\downarrow}| is always dominant and there is no Chandrasekhar-Clogston limit. We then investigate the case of weakly coupled 1D systems and determine the region of stability of the 1D FFLO phase. This picture is corroborated by density-matrix-renormalization-group (DMRG) simulations of the spatial noise correlations in uniform and trapped systems, unambiguously revealing the presence of fermion pairs with nonzero momentum Q. This opens up an interesting possibility for experimental studies of FFLO states.

We study the threshold for chaos and its relation to thermalization in the 1D mean-field Bose-Hubbard model, which in particular describes atoms in optical lattices. We identify the threshold for chaos, which is finite in the thermodynamic limit, and show that it is indeed a precursor of thermalization. Far above the threshold, the state of the system after relaxation is governed by the usual laws of statistical mechanics.

We investigate polaron formation in a many-electron system in the presence of a local repulsion sufficiently strong to prevent local-bipolaron formation. Specifically, we consider a Hubbard-Holstein model of interacting electrons coupled to dispersionless phonons of frequency $\omega_0$. Numerically solving the model in a small one-dimensional cluster, we find that in the nearly adiabatic case $\omega_0 < t$, the necessary and sufficient condition for the polaronic regime to occur is that the energy gain in the atomic (i.e., extremely localized) regime ${\cal E}_{pol}$ overcomes the energy of the purely electronic system $ {\cal E}_{el}$. In the antiadiabatic case, $\omega_0 > t$, polaron formation is instead driven by the condition of a large ionic displacement $g/\omega_0 >1$ ($g$ being the electron-phonon coupling). Dynamical properties of the model in the weak and moderately strong coupling regimes are also analyzed.

The low-temperature thermoelectric power of the repulsive-interaction one-dimensional Hubbard model is calculated using an asymptotic Bethe ansatz for holons and spinons. The competition between the entropy carried by the holons and that carried by the backflow of the spinons gives rise to an unusual temperature and doping dependence of the thermopower which is qualitatively similar to that observed in the normal state of high-$T_{c}$ superconductors.

Using the time-dependent density matrix renormalization group method and exact diagonalization, we study the non-equilibrium dynamics of the one-dimensional Fermi-Hubbard model following a quantum quench or a ramp of the onsite interaction strength. For quenches from the non-interacting to the attractive regime, we investigate the dynamical emergence of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) correlations, which at finite spin polarizations are the dominant two-body correlations in the ground state, and their signatures in the pair quasi-momentum distribution function. We observe that the post-quench double occupancy exhibits a maximum as the interaction strength becomes of the order of the bandwidth. Finally, we study quenches and ramps from attractive to repulsive interactions, which imprint FFLO correlations onto repulsively bound pairs. We show that a quite short ramp time is sufficient to wipe out the characteristic FFLO features in the post-quench pair momentum distribution functions.

The thermodynamics of the inhomogeneous one-dimensional repulsive fermionic Hubbard model with parabolic confinement is studied by a density-functional theory approach, based on Mermin's generalization to finite temperatures. A local-density approximation (LDA), based on exact results for the homogeneous model, is used to approximate the correlation part in the Helmholtz free-energy, comprising the thermodynamic Bethe ansatz LDA (TBALDA). The general presentation of the method is given and some properties of the homogeneous model that are relevant to the DFT approach are analyzed. Extensive comparison between TBALDA and numerical exact diagonalization results for thermodynamic properties of small inhomogeneous chains is discussed. In the remaining, a classical thermodynamic treatment of the confined system is developed with the focus on global properties of large systems. A unusual behavior under isentropic expansion is found and discussed.

We use a stochastic series expansion quantum Monte Carlo method to study the phase diagram of the one-dimensional extended Hubbard model at half filling for small to intermediate values of the on-site (U) and nearest-neighbor (V) repulsions. We confirm the existence of a novel, long-range-ordered bond-order-wave (BOW) phase recently predicted by Nakamura (J. Phys. Soc. Jpn. 68, 3123 (1999)) in a small region of the parameter space between the familiar charge-density-wave (CDW) state for V > U/2 and the state with dominant spin-density-wave (SDW) fluctuations for V < U/2. We discuss the nature of the transitions among these states and evaluate some of the critical exponents. Further, we determine accurately the position of the multi-critical point, (U_m,V_m)= (4.7 +/- 0.1, 2.51 +/- 0.04) (in energy units where the hopping integral is normalized to unity), above which the two continuous SDW-BOW-CDW transitions are replaced by one discontinuous (first-order) direct SDW-CDW transition. We also discuss the evolution of the CDW and BOW states upon hole doping. We find that in both cases the ground state is a Luther-Emery liquid, i.e., the spin gap remains but the charge gap existing at half-filling is immediately closed upon doping. The charge and bond-order correlations decay with distance r as r^{-K_rho}, where K_rho is approximately 0.5 for the parameters we have considered. We also discuss advantages of using parallel tempering (or exchange Monte Carlo) -- an extended ensemble method that we here combine with quantum Monte Carlo -- in studies of quantum phase transitions.

We study the thermodynamics of the one-dimensional extended Hubbard model at half-filling using a density-matrix renormalization group method applied to transfer matrices. We show that the various phase transitions in this system can be detected by measuring usual thermodynamic quantities like the isothermal compressibility and the uniform magnetic susceptibility. For the isothermal compressibility we show that universal crossing points exist which allow to accurately determine the line where the charge gap vanishes. By studying in addition several correlation functions, we confirm the existence of a phase with long-range dimer order (bond order) which has been a matter of debate for several years. According to our calculations this phase is located in a narrow region between the spin-density and charge-density wave phases up to a tricritical point which we estimate to be at $U_t=6.7\pm 0.2$, $V_t =3.5\pm 0.1$. Our results for the phase diagram are in good agreement with the most recent zero-temperature density-matrix renormalization group study, however, they disagree in some important aspects from the most recent Quantum-Monte-Carlo study.

We study the phase diagram of the zero-temperature, one-dimensional Bose-Fermi-Hubbard model for fixed fermion density in the limit of small fermionic hopping. This model can be regarded as an instance of a disordered Bose-Hubbard model with dichotomic values of the stochastic variables. Phase boundaries between compressible, incompressible (Mott-insulating) and partially compressible phases are derived analytically within a generalized strong-coupling expansion and numerically using density matrix renormalization group (DMRG) methods. We show that first-order correlations in the partially compressible phases decay exponentially, indicating a glass-type behaviour. Fluctuations within the respective incompressible phases are determined using perturbation theory and are compared to DMRG results.

We consider a fermionic Hubbard chain with an additional next-to-nearest neighbor hopping term. We study the thermalization rates of the quasi-momentum distribution function within a quantum Boltzmann equation approach. We find that the thermalization rates are proportional to the square of the next-to-nearest neighbor hopping: Even weak next-to-nearest neighbor hopping in addition to nearest neighbor hopping leads to thermalization in a two-particle scattering quantum Boltzmann equation in one dimension. We also investigate the temperature dependence of the thermalization rates, which away from half filling become exponentially small for small temperature of the final thermalized distribution.

The density-matrix renormalization group is used to study the phase diagram of the one-dimensional half-filled Hubbard model with on-site (U) and nearest-neighbor (V) repulsion, and hopping t. A critical line V_c(U) approximately equal to U/2 separates a Mott insulating phase from a charge-density-wave phase. The formation of bound charge excitations for V > 2t changes the phase transition from continuous to first order at a tricritical point U_t = 3.7t, V_t=2t. A frustrating effective antiferromagnetic spin coupling induces a bond-order-wave phase on the critical line V_c(U) for U_t < U < 7-8 t.

In their Comment (see cond-mat/0301237), Sandvik, Sengupta, and Campbell present some numerical evidences to support the existence of an extended bond-order-wave (BOW) phase at couplings (U,V) weaker than a tricritical point (U_t,V_t) in the ground state phase diagram of the one-dimensional half-filled U-V Hubbard model. They claim that their results do not agree with the phase diagram proposed in my Letter (cond-mat/0204244), which shows a BOW phase for couplings stronger than the critical point only. However, I argue here that their results are not conclusive and do not refute the phase diagram described in the Letter.

In order to study consequences of the differences between the ionic-to-neutral and neutral-to-ionic transitions in the one-dimensional extended Peierls-Hubbard model with alternating potentials for the TTF-CA complex, we introduce a double pulse of oscillating electric field in the time-dependent Schr\"odinger equation and vary the interval between the two pulses as well as their strengths. When the dimerized ionic phase is photoexcited, the interference effect is clearly observed owing to the coherence of charge density and lattice displacements. Namely, the two pulses constructively interfere with each other if the interval is a multiple of the period of the optical lattice vibration, while they destructively interfere if the interval is a half-odd integer times the period, in the processes toward the neutral phase. The interference is strong especially when the pulse is strong and short because the coherence is also strong. Meanwhile, when the neutral phase is photoexcited, the interference effect is almost invisible or weakly observed when the pulse is weak. The photoinduced lattice oscillations are incoherent due to random phases. The strength of the interference caused by a double pulse is a key quantity to distinguish the two transitions and to evaluate the coherence of charge density and lattice displacements.

We use a density matrix renormalization group method to study quantitatively the phase diagram of a one-dimensional extended Hubbard model at half-filling by investigating the correlation functions and structure factors. We confirm the existence of a novel narrow region with long-rang bond-order-wave order which is highly controversial recently between the charge-density-wave phase and Mott insulator phase. We determined accurately the position of the tricritical point $U_t\simeq 7.2t$, $V_t\simeq 3.746t$ which is quite different from previous studies.

We study the one-dimensional Anderson-Hubbard model using the density-matrix renormalization group method. The influence of disorder on the Tomonaga-Luttinger liquid behavior is quantitatively discussed. Based on the finite-size scaling analysis of density-density correlation functions, we find the following results: i) the charge exponent is significantly reduced by disorder at low filling and near half filling, ii) the localization length decays as $\xi \sim \Delta^{-2}$, where $\Delta$ is the disorder strength, independently of the on-site Coulomb interaction as well as band filling, and iii) the localization length is strongly suppressed by the on-site Coulomb interaction near half filling in association with the formation of the Mott plateaus.

We present the zero-temperature phase diagram of the one-dimensional t2g-orbital Hubbard model, obtained using the density-matrix renormalization group and Lanczos techniques. Emphasis is given to the case for the electron density n=5 corresponding to five electrons per site, of relevance for some Co-based compounds. However, several other cases for electron densities between n=3 and 6 are also studied. At n=5, our results indicate a first-order transition between a paramagnetic (PM) insulator phase and a fully-polarized ferromagnetic (FM) state by tuning the Hund's coupling. The results also suggest a transition from the n=5 PM insulator phase to a metallic regime by changing the electron density, either via hole or electron doping. The behavior of the spin, charge, and orbital correlation functions in the FM and PM states are also described in the text and discussed. The robustness of these two states varying parameters suggests that they may be of relevance in more realistic higher dimensional systems as well.

We investigate the dynamics of the one-dimensional strongly repulsive spin-1/2 Bose-Hubbard model for filling $\nu\le1.$ While at $\nu=1$ the system is a Hubbard-Mott insulator exhibiting dynamical properties of the Heisenberg ferromagnet, at $\nu

We formulate in terms of the quantum inverse scattering method the algebraic Bethe ansatz solution of the one-dimensional Hubbard model. The method developed is based on a new set of commutation relations which encodes a hidden symmetry of 6-vertex type.

We study the strong correlation effects in the vicinity of the Mott metal-insulator transition using coupled clean or disordered Hubbard chains with a infinitely large coordinate number $D_{\perp}\to\infty$ in the direction perpendicular to the chains and with a long-range transverse hopping. Strong electron correlation effects are treated partially non-perturbatively with the use of the exact results for the 1D Hubbard model. In the case of clean systems, the thermodynamic and transport quantities which characterize the Mott transition from the Fermi liquid state, such as the specific heat coefficient, the Drude weight, and the compressibility, are obtained as functions of hole-doping $\delta=1-n$ ($n$, electron density) by the systematic expansion in terms of the inverse of the transverse hopping range $l_{\perp}$. We find that the $\delta$-dependence of these quantities shows non-universal behaviors with exponents depending on the strength of electron-electron interaction. In the presence of disorder, it is shown that the frequency-dependence of the dynamical conductivity obeys the Mott's law, implying the possibility of {\it Mott glass state} for $\delta \to 0$, provided that there exists a finite range interaction.

We investigate the momentum distribution function near the Mott-Hubbard transition in the one-dimensional t1-t2 Hubbard model (the zig-zag Hubbard chain), with the density-matrix renormalization-group technique. We show that for strong interactions the Mott-Hubbard transition occurs between the metallic-phase and an insulating dimerized phase with incommensurate spin excitations, suggesting a decoupling of magnetic and charge excitations not present in weak coupling. We illustrate the signatures for the Mott-Hubbard transition and the commensurate-incommensurate transition in the insulating spin-gapped state in their respective ground-state momentum distribution functions.