Bipairing and the stripe phase in four-leg Hubbard ladders

Ming-Shyang Chang and Ian Affleck

Phys. Rev. B 76, 054521 – Published 29 August 2007

Abstract

Density matrix renormalization group (DMRG) calculations on four-leg $t − J$ and Hubbard ladders have found a phase exhibiting “stripes” at intermediate doping. Such behavior can be viewed as generalized Friedel oscillations, with wavelength equal to the inverse hole density, induced by the open boundary conditions. So far, this phase has not been understood using the conventional weak-coupling bosonization approach. Based on studies of a general bosonization proof, finite size spectrum, an improved analysis of weak-coupling renormalization group equations, and the decoupled two-leg ladder limit, we here find new types of phases of four-leg ladders, which exhibit stripes. They also inevitably exhibit “bipairing,” meaning that there is a gap to add one or two electrons (but not four) and that both single electron and electron pair correlation functions decay exponentially, while correlation functions of charge-4 operators exhibit a power-law decay. Whether or not bipairing occurs in the stripe phase found in DMRG is an important open question.

Received 25 June 2007

DOI:https://doi.org/10.1103/PhysRevB.76.054521

Authors & Affiliations

Ming-Shyang Chang and Ian Affleck

Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1

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Vol. 76, Iss. 5 — 1 August 2007

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Images

Figure 1
(Color online) The topography of RG flows near the fixed ray with $λ ⩽ 0$ , $0 < λ ⩽ 1$ , and $λ > 1$ . It is clear that the deviation is irrelevant for $λ ⩽ 0$ and relevant for $λ > 1$ . The analysis of RG flow is more subtle for $0 < λ ⩽ 1$ . In this case, although the deviation from the fixed ray is growing, the fixed ratios still remain. Therefore, RG still flows onto the fixed ray, but the phase is not only determined by the fixed-ray couplings.Reuse & Permissions
Figure 2
(Color online) This is the $\log [∣ g_{i} (l) ∣]$ vs $\log [(l_{d} - l)]$ plot for several typical couplings of stripe fixed ray [Eq. (4.13)]. The slopes give us the divergent exponent of each coupling. The solid (red) lines are the numerical solutions of the RG equations. The dashed lines are pure straight lines as reference for the predicted $λ_{i}^{\max}$ : 1 [pink: $c_{14}^{σ}$ (a), $f_{23}^{σ}$ (b)], $5 ∕ 8$ [blue: $b_{12}^{ρ}$ (c), $c_{34}^{σ}$ (d)], $1 ∕ 2$ [green: $c_{11}^{σ}$ (e)], and 0 [yellow: $f_{13}^{σ}$ (f)]. In this case, the numerical solutions agree very well with the prediction for all the couplings.Reuse & Permissions
Figure 3
(Color online) $\log [∣ g_{i} (l) ∣]$ vs $\log [(l_{d} - l)]$ plot for several typical couplings. The parameters are chosen as $t = t_{i, ⊥} = 1$ and $U = 0.01$ , and the hole doping is 0.135. Its slope gives us the divergent exponent of each coupling. The solid (red) lines are the numerical solutions of the RG equations. The dashed lines are pure straight lines as reference for the predicted slopes 1 [pink: $c_{44}^{σ}$ (a), $c_{14}^{σ}$ (b)], $1 ∕ 2$ [blue: $f_{14}^{σ}$ (c)], $15 ∕ 16$ [green: $c_{12}^{σ}$ (d)], and 0 [yellow: $f_{13}^{σ}$ (e), $c_{23}^{σ}$ (f)]. As one can see, the numerical solution of $c_{23}^{σ}$ (f) does not agree with its predicted exponent. The number of couplings, whose slopes do not agree with the stability analysis, depends on the initial conditions. With the initial conditions used here, there are a total of 11 couplings, predicted with zero slope near the fixed ray, but have nonzero slopes in the numerical solution. However, these new terms do not change the pinned bosons, and the final phase is the same as that when these terms are irrelevant.Reuse & Permissions
Figure 4
(Color online) In the limit that $t_{2, ⊥}$ and $V_{2, ⊥}$ are much smaller than the minimum gap of the bosons, $Δ$ , each two-leg ladder is well described by the C1S0 phase, which has pairing and $4 {\bar{k}}_{F}$ density oscillation. The direct electron hopping $t_{2, ⊥}$ becomes an irrelevant process, yet pair hopping $t^{'}$ generated by the higher order process will appear and the $4 {\bar{k}}_{F} - 4 {\bar{k}}_{F}$ component $V^{'}$ is the lowest order relevant term in the interaction $V_{2, ⊥}$ . The phase is determined by the competition between $t^{'}$ and $V^{'}$ . If $t^{'}$ dominates, the system has pairing (boson superfluid) and $8 {\bar{k}}_{F}$ $(4 π ρ_{0})$ density oscillation in fermion (boson) language, where $ρ_{0}$ is the average boson density. If $V^{'}$ dominates, the system has bipairing (boson-pair superfluid) and $4 {\bar{k}}_{F}$ $(2 π ρ_{0})$ density.Reuse & Permissions

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