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arXiv:2007.00661 · Physical review. B./Physical review. B

Hubbard ladders at small U revisited

Yuval Gannot, Yi‐Fan Jiang, Steven Allan Kivelson

Published Sep 16, 2020• 38 citations
arXivHTMLPDF

Abstract

We re-examine the zero-temperature phase diagram of the two-leg Hubbard ladder in the small $U$ limit, both analytically and using density-matrix renormalization group (DMRG). We find a ubiquitous Luther-Emery phase, but with a crossover in behavior at a characteristic interaction strength, ${U}^{★}$; for $U\ensuremath{\gtrsim}{U}^{★}$, there is a single emergent correlation length $ln[\ensuremath{\xi}]\ensuremath{\sim}1/U$, characterizing the gapped modes of the system, but for $U\ensuremath{\lesssim}{U}^{★}$, there is a hierarchy of length scales, differing parametrically in powers of $U$, reflecting a two-step renormalization group flow to the ultimate fixed point. Finally, to illustrate the versatility of the approach developed here, we sketch its implications for a half-filled triangular lattice Hubbard model on a cylinder and find results in conflict with inferences concerning the small $U$ phase from recent DMRG studies of the same problem.

DOI
10.1103/physrevb.102.115136

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We re-examine the zero-temperature phase diagram of the two-leg Hubbard ladder in the small UUU limit, both analytically and using density-matrix renormalization group (DMRG). We find a ubiquitous Luther-Emery phase, but with a crossover in behavior at a characteristic interaction strength, U★{U}^{★}U★; for U\ensuremath≳U★U\ensuremath{\gtrsim}{U}^{★}U\ensuremath≳U★, there is a single emergent correlation length ln[\ensuremathξ]\ensuremath∼1/Uln[\ensuremath{\xi}]\ensuremath{\sim}1/Uln[\ensuremathξ]\ensuremath∼1/U, characterizing the gapped modes of the system, but for U\ensuremath≲U★U\ensuremath{\lesssim}{U}^{★}U\ensuremath≲U★, there is a hierarchy of length scales, differing parametrically in powers of UUU, reflecting a two-step renormalization group flow to the ultimate fixed point. Finally, to illustrate the versatility of the approach developed here, we sketch its implications for a half-filled triangular lattice Hubbard model on a cylinder and find results in conflict with inferences concerning the small UUU phase from recent DMRG studies of the same problem.
The phase diagram of the two-leg $t\text{\ensuremath{-}}{J}_{z}$ ladder is explored, using the density-matrix renormalization group method. Results are obtained for energy gaps, electron density profiles, and correlation functions for the half filled and quarter filled cases. The effective Lagrangian velocity parameter ${v}_{\ensuremath{\rho}}$ is shown to vanish at half filling. The behavior of the one-hole gap in the Nagaoka limit is investigated, and found to disagree with theoretical predictions. A tentative phase diagram is presented, which is quite similar to the full $t\text{\ensuremath{-}}J$ ladder, but scaled up by a factor of about 2 in coupling. Near half filling a Luther-Emery phase is found, which may be expected to show superconducting correlations, while near quarter filling the system appears to be in a Tomonaga-Luttinger phase.
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.
Motivated by recent experiments on ultracold fermionic spin-1/2 6^66Li atoms in a Lieb lattice at various Hubbard repulsion UUU and filling fractions nnn (Lebrat et al., arXiv:2404.17555), we conduct a density matrix renormalization group (DMRG) analysis of the Hubbard model on a two-leg Lieb ladder. At half-filling, we find a ferrimagnetic Mott insulating ground state, consistent with Lieb's theorem. Away from half-filling, a state with finite total spin S⃗2≠0\vec{S}^2 \neq 0S2=0 and vanishing charge gap persists down to filling nc≈2/3n_c \approx 2/3nc​≈2/3. For lower, incommensurate fillings, the system is described by a Luttinger liquid with one charge and one spin mode. Intriguingly, in a narrow window near nc=2/3n_c = 2/3nc​=2/3, close to the onset of ferromagnetic order, we identify a superconducting Luther-Emery phase with dominant sxys_{xy}sxy​-wave pairing.
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.
Inspired by the observation of a robust ddd-wave superconducting phase driven by tuning the next-nearest-neighbor (NNN) electron hopping in recent density matrix renormalization group (DMRG) studies of six- and eight-leg $t\text{\ensuremath{-}}J$ model, we systematically study the phase diagram of the two-leg $t\text{\ensuremath{-}}J$ ladder with the NNN couplings (including NNN hopping and spin interaction) in a large region of doping level, by means of the DMRG calculations. Upon doping from half filling, we identify the Luther-Emery liquid (LEL) phase, which can be distinguished as the pairing-dominant and charge density-dominant regime by comparing the Luttinger parameter ${K}_{\ensuremath{\rho}}$. With the growing NNN couplings, pairing correlations are enhanced and correspondingly ${K}_{\ensuremath{\rho}}$ increases, driving the system from the density-dominant to the pairing-dominant regime. In the Tomonaga-Luttinger liquid (TLL) phase in the larger doping region, we identify two TLL regimes with different features of charge density correlation. At the quarter filling (1/21/21/2 doping level), we find that the strong dimer orders of bond energy in the open system actually decay algebraically and thus do not indicate a spontaneous translational symmetry breaking. Our results show that in the LEL phase of the two-leg ladder, the NNN couplings seem to play the similar role as that on the wider $t\text{\ensuremath{-}}J$ cylinder, and studies on this more accessible system can be helpful toward understanding the emergence of the remarkable ddd-wave superconducting phase on the wider system.
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/2N/2N/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(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 $t\ensuremath{-}J$ 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=3N=3N=3 and 4.