We study the dimensional crossover from one to two dimensions in half-filled and lightly doped, weakly interacting N-leg Hubbard ladders. In this case, the Hubbard ladders are equivalent to an N-band model. Using renormalization-group techniques, we find that the half-filled ladders exhibit (in the spin sector) an odd-even effect only below a crossover energy ${E}_{c}\ensuremath{\propto}\mathrm{exp}[\ensuremath{-}\ensuremath{\alpha}\mathrm{exp}(\ensuremath{\gamma}N)]$ $(\ensuremath{\alpha}\ensuremath{\ll}1$ and $\ensuremath{\gamma}\ensuremath{\sim}1$ depend on the interaction strength and on the hopping matrix elements): Below
Ec, the dominant interactions take place within band pairs $(j,N+1\ensuremath{-}j)$ [and within the band
(N+1)/2 for N odd], such that even-leg ladders are an insulating spin liquid, while odd-leg ladders have one gapless spinon mode. In contrast, above the energy scale
Ec, all bands are interacting with each other and the system is a two-dimensional- (2D)-like (insulating) antiferromagnet; we obtain an analytical expression for the Hamiltonian which is similar to the 2D Heisenberg antiferromagnet. Bosonization techniques show that in the charge sector the Mott insulator is as well below and above
Ec of the same type as in
N/2 half-filled two-leg ladders. Doping away from half-filling, we find that the effect of an increasing doping $\ensuremath{\delta}$ is very similar to decreasing the number of legs N: In both cases interactions between unpaired bands are suppressed and thereby the antiferromagnetic correlations reduced. The resulting band pairs form then insulating spin liquids and when doped, there is a spin gap, but phase coherence exists only within the band pairs. At higher doping levels ${\ensuremath{\delta}}_{c}={\ensuremath{\delta}}_{c}(N),$ phase coherence between all band pairs sets in and the system becomes a 2D-like d-wave superconductor $[{\ensuremath{\delta}}_{c}(N)\ensuremath{\rightarrow}0$ for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}].$
The ground-state phase diagram of the half-filled two-leg Hubbard ladder with intersite Coulomb repulsions and exchange coupling is studied by using the strong-coupling perturbation theory and the weak-coupling bosonization method. Considered here as possible ground states of the ladder model are four types of density-wave states with different angular momentum
(s-density-wave state, p-density-wave state, d-density-wave state, and f-density-wave state) and four types of quantum disordered states, i.e., Mott insulating states
(S-Mott, D-Mott, ${S}^{\ensuremath{'}}$-Mott, and ${D}^{\ensuremath{'}}$-Mott states, where S and D stand for s- and d-wave symmetry). The s-density-wave state, the d-density-wave state, and the D-Mott state are also known as the charge-density-wave state, the staggered-flux state, and the rung-singlet state, respectively. Strong-coupling approach naturally leads to the Ising model in a transverse field as an effective theory for the quantum phase transitions between the staggered-flux state and the D-Mott state and between the charge-density-wave state and the S-Mott state, where the Ising ordered states correspond to doubly degenerate ground states in the staggered-flux or the charge-density-wave state. From the weak-coupling bosonization approach it is shown that there are three cases in the quantum phase transitions between a density-wave state and a Mott state: the Ising
(Z2) criticality, the
SU(2)2 criticality, and a first-order transition. The quantum phase transitions between Mott states and between density-wave states are found to be the U(1) Gaussian criticality. The ground-state phase diagram is determined by integrating perturbative renormalization-group equations. It is shown that the S-Mott state and the staggered-flux state exist in the region sandwiched by the charge-density-wave phase and the D-Mott phase. The p-density-wave state, the ${S}^{\ensuremath{'}}$-Mott state, and the ${D}^{\ensuremath{'}}$-Mott state also appear in the phase diagram when the next-nearest-neighbor repulsion is included. The correspondence between Mott states in extended Hubbard ladders and spin-liquid states in spin ladders is also discussed.
Abstract We present a T = 0 K renormalization group (RG) phase diagram for the electronic Hubbard model in two dimensions on the square lattice at half filling. The RG procedure treats quantum fluctuations in the single particle occupation number nonperturbatively via the unitarily decoupling of one electronic state at every RG step. The resulting phase diagram thus possesses the quantum fluctuation energy scale (ω) as one of its axes. A relation is derived between ω and the effective temperature scale upto which gapless, as well as emergent gapped phases can be obtained. We find that the normal and insulating phases of the half-filled Hubbard model correspond, for any on-site repulsion, to a marginal Fermi liquid normal phase and a topologically-ordered gapped Mott insulating liquid respectively. The marginal Fermi liquid is found to arise from singular forward scattering in directions normal to the nested Fermi surface, while singular backscattering events lead to Mott liquid state. The transition between these two phases involves passage through a pseudogapped phase bookended by Fermi surface topology-changing Lifshitz transitions. The pseudogap phase is observed to arise from the electronic differentiation encoded within the nested Fermi surface, and involves the gradual gapping of the Fermi surface from antinodes to nodes via charge and spin excitations that are mutually entangled. We obtain effective Hamiltonians for various phases, as well as wavefunctions for the low-energy many-body eigenstates of the Mott liquid. Benchmarking of the ground-state energy per particle and the double-occupancy fraction for the Mott liquid against existing numerical results yields excellent agreement. Presence of a Néel ordering symmetry-breaking perturbation in the RG leads to an antiferromagnetic spin-ordered charge insulating Mott state. Our results thus offer novel insights on a variety of aspects of the Mott–Hubbard problem, and can be extended to the doped system.
The Hubbard model on a two-leg ladder structure has been studied by a combination of series expansions at T = 0 and the density-matrix renormalization group. We report results for the ground-state energy E 0 and spin gap Δ s at half-filling, as well as dispersion curves for one- and two-hole excitations. For small U both E 0 and Δ s show a dramatic drop near t / t ⊥ ~0.5, which becomes more gradual for larger U . This represents a crossover from a `band insulator' phase to a strongly correlated spin liquid. The various features are collected in a `phase diagram' for the model.
We study the effect of doping away from half-filling in weakly (but finitely) interacting N-leg Hubbard ladders using renormalization-group and bosonization techniques. For a small on-site repulsion U, the N-leg Hubbard ladders are equivalent to an N-band model, where at half-filling the Fermi velocities are ${v}_{1}{=v}_{N}<{v}_{2}{=v}_{N\ensuremath{-}1}<\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}.$ We then obtain a hierarchy of energy scales, where the band pairs $(j,N+1\ensuremath{-}j)$ are successively frozen out. The low-energy Hamiltonian is then the sum of
N/2 [or $(N\ensuremath{-}1)/2$ for N odd] two-leg ladder Hamiltonians without gapless excitations (plus a single chain for N odd with one gapless spin mode)---similar to the N-leg Heisenberg spin ladders. The energy scales lead to a hierarchy of gaps. Upon doping away from half-filling, the holes first enter the band(s) with the smallest gap: For odd N, the holes enter first the nonbonding band
(N+1)/2 and the phase is a Luttinger liquid, while for even N, the holes enter first the band pair
(N/2,N/2+1) and the phase is a Luther-Emery liquid, similar to numerical treatments of the $t\ensuremath{-}J$ model, i.e., at and close to half-filling, the phases of the Hubbard ladders for small and large U are the same. For increasing doping, hole pairs subsequently enter at critical dopings the other band pairs $(j,N+1\ensuremath{-}j)$ (accompanied by a diverging compressibility): The Fermi surface is successively opened by doping, starting near the wave vector $(\ensuremath{\pi}/2,\ensuremath{\pi}/2).$ Explicit calculations are given for the cases
N=3 and 4.
We study the ground state of the two-dimensional Anderson-Hubbard model using a quantum real space renormalization group method. We obtain the phase diagram near half filling. The system is always insulating with disorder. At half filling, the system undergoes a transition from a gapless (Anderson) insulator to an incompressible (Mott-Hubbard) insulator as the interaction
U reaches a critical value
Uc. Away from half filling, the insulating phase is always gapless and is found to be controlled by the Anderson fixed point at half filling. This result is similar to that obtained in the corresponding one dimensional system and suggests strongly the importance of the electron-electron correlation in this gapless insulating phase.