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arXiv:cond-mat/0002445 · Physical review. B, Condensed matter

Successive opening of the Fermi surface in dopedN-leg Hubbard ladders

Urs Ledermann, Karyn Le Hur, Thomas Maurice Rice

Published Dec 15, 2000• 36 citations
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Abstract

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.

DOI
10.1103/physrevb.62.16383

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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 v1=vN<v2=vN\ensuremath−1<\ensuremath⋅\ensuremath⋅\ensuremath⋅.{v}_{1}{=v}_{N}<{v}_{2}{=v}_{N\ensuremath{-}1}<\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}.v1​=vN​<v2​=vN\ensuremath−1​<\ensuremath⋅\ensuremath⋅\ensuremath⋅. We then obtain a hierarchy of energy scales, where the band pairs (j,N+1\ensuremath−j)(j,N+1\ensuremath{-}j)(j,N+1\ensuremath−j) are successively frozen out. The low-energy Hamiltonian is then the sum of N/2N/2N/2 [or (N\ensuremath−1)/2(N\ensuremath{-}1)/2(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(N+1)/2(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)(N/2,N/2+1)(N/2,N/2+1) and the phase is a Luther-Emery liquid, similar to numerical treatments of the t\ensuremath−Jt\ensuremath{-}Jt\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)(j,N+1\ensuremath{-}j)(j,N+1\ensuremath−j) (accompanied by a diverging compressibility): The Fermi surface is successively opened by doping, starting near the wave vector (\ensuremathπ/2,\ensuremathπ/2).(\ensuremath{\pi}/2,\ensuremath{\pi}/2).(\ensuremathπ/2,\ensuremathπ/2). Explicit calculations are given for the cases N=3N=3N=3 and 4.
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,{E}_{c},Ec​, the dominant interactions take place within band pairs $(j,N+1\ensuremath{-}j)$ [and within the band (N+1)/2(N+1)/2(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,{E}_{c},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{E}_{c}Ec​ of the same type as in N/2N/2N/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 two-leg $t\ensuremath{-}J$ ladder forms a spin liquid at half filling that evolves to a Luther-Emery liquid upon doping. Our aim is to obtain a complete phase diagram for isotropic coupling (i.e., rungs and legs equal) as a function of electron density n and the ratio J/t.J/t.J/t. Two known limiting cases are n<12,n<\frac{1}{2},n<21​, which is a single band Luttinger liquid, and small hole doping $\ensuremath{\delta}\ensuremath{\ll}1$ for $J/\stackrel{\ensuremath{\rightarrow}}{t}0,$ which is a Nagaoka ferromagnet. Using Lanczos techniques we examine the region between the Nagaoka and Luther-Emery phases for 1>n>12.1>n>\frac{1}{2}.1>n>21​. We find evidences for gapless behavior in both spin and charge channels for J/t<0.3,J/t<0.3,J/t<0.3, consistent with Luttinger liquids in both bonding and antibonding bands (i.e., C2S2).C2S2).C2S2). This proposal is based on the behavior of spin and charge correlation functions. For example, the hole-hole correlation function that displays hole pairing at larger J/tJ/tJ/t shows hole-hole repulsion in this region. As a further test, we examined the dependence of the energy on a relative phase shift between bonding and antibonding bands. For J/t<0.3J/t<0.3J/t<0.3 this is very weak, indicating a lack of pairing between these channels.
The three-leg ladder has one odd-parity and two even-parity channels. At low doping these behave quite differently. Numerical calculations for a $t\ensuremath{-}J$ model show that the initial phase upon hole doping has two components---a conducting Luttinger liquid in the odd-parity channel, coexisting with an insulating (i.e., undoped) spin-liquid phase in the even-parity channels. This phase has a partially truncated Fermi surface and violates the Luttinger theorem. This coexistence of conducting fermionic and insulating paired bosonic degrees of freedom is similar to the recent proposal of Geshkenbein, Ioffe, and Larkin [Phys. Rev. B 55, 3173 (1997)] for the underdoped spin-gap normal phase of the cuprates. A mean-field approximation is derived which has many similarities to the numerical results. One difference however is an induced hole pairing in the odd-parity channel at arbitrary small dopings, similar to that proposed by Geshkenbein, Ioffe, and Larkin for the two-dimensional case. At higher dopings, we propose that a quantum phase transition will occur as holes enter the even-parity channels, resulting in a Luther-Emery liquid with hole pairing with essentially ddd-wave character. In the mean-field approximation a crossover occurs which we interpret as a reflection of this quantum phase transition deduced from the numerical results.
We revisit the ground state of the Hubbard model on two-legged ladders in this paper. We perform density matrix renormalization group (DMRG) calculations on large system sizes with large numbers of kept states and perform extrapolation of DMRG results with truncation errors in the converged region. We find that the superconducting correlation exponent Ksc{K}_{sc}Ksc​ extracted from the pair-pair correlation is very sensitive to the position of the reference bond, reflecting a huge boundary effect on it. By systematically removing the effects from boundary conditions, finite sizes, and truncation errors in DMRG, we obtain more accurate values of Ksc{K}_{sc}Ksc​ and ${K}_{\ensuremath{\rho}}$ with DMRG. With these exponents, we confirm that the two-legged Hubbard model is in the Luther-Emery liquid phase with ${K}_{sc}\ifmmode\cdot\else\textperiodcentered\fi{}{K}_{\ensuremath{\rho}}=1$ from tiny doping near half filling to 1/81/81/8 hole doping. When the doping is increased to $\ensuremath{\delta}⪆1/6$, the behaviors of charge, pairing, and spin correlations do not change qualitatively, but the relationship ${K}_{sc}\ifmmode\cdot\else\textperiodcentered\fi{}{K}_{\ensuremath{\rho}}=1$ is likely to be violated. With the further increase of the doping to $\ensuremath{\delta}=1/3$, the quasi-long-ranged charge correlation turns into a true long-ranged charge order, and the spin gap is closed, while the pair-pair correlation still decays algebraically. Our work provides a standard way to analyze the correlation functions when studying systems with open boundary conditions.
Holes weakly doped into a four-leg t-JJJ ladder bind in pairs. At dopings exceeding a critical doping of ${\ensuremath{\delta}}_{c}\ensuremath{\simeq}\frac{1}{8}$ four-hole clusters are observed to form in density-matrix renormalization-group calculations. The symmetry of the ground-state wave function does not change and we are able to reproduce this behavior qualitatively with an effective bosonic model in which the four-leg ladder is represented as two coupled two-leg ladders and hole pairs are mapped on hard-core bosons moving along and between these ladders. At lower dopings, $\ensuremath{\delta}<{\ensuremath{\delta}}_{c},$ a one-dimensional bosonic representation for hole pairs works and allows us to calculate accurately the Luttinger liquid parameter ${K}_{\ensuremath{\rho}}$ that takes the universal value ${K}_{\ensuremath{\rho}}=1$ as half-filling is approached.
A lightly doped single-band Hubbard model on a two-leg ladder exhibits a Luther-Emery phase, whereas the three-band Hubbard ladder behaves as a Luttinger liquid upon hole doping. To understand this discrepancy, we present a systematic density-matrix renormalization group study of the three-band Hubbard model on two-leg cylinders with further-neighbor particle hoppings. The inclusion of the longer-range hopping is motivated by the studies of the single-band Hubbard model in which the further-neighbor hopping terms are suggested to be crucial for the unconventional superconductivity. When the longer-range hopping parameters are small, the ground state is a Luttinger liquid having mutually commensurate superconducting, charge, and spin density wave correlations. Increasing the longer-range hopping drives a transition into a Luther-Emery phase with quasi-long-ranged superconducting and charge orders but short-ranged spin-spin correlations. By down-folding the three-band Hubbard model into an effective $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J\text{\ensuremath{-}}{J}^{\ensuremath{'}}$ model, we find that in the Luther-Emery phase, both the nearest- and second-neighbor kinetic energies are enhanced due to an effective increase of copper-oxygen hybridization. Amplifying intercell oxygen orbital hopping mirrors the benefits of reducing the charge transfer energy, causing doped holes to favor oxygen orbitals and strengthening superconducting pairing.