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arXiv:cond-mat/9912352 · Physical review. B, Condensed matter

Phase separation in t−J ladders

Stefan Rommer, Steven R. White, Douglas J. Scalapino

Published May 15, 2000• 29 citations
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Abstract

The phase separation boundary of isotropic $t\ensuremath{-}J$ ladders is analyzed using density matrix renormalization group techniques. The complete boundary to phase separation as a function of $J/t$ and doping is determined for a chain and for ladders with two, three, and four legs. Six-chain ladders have been analyzed at low hole doping. We use a direct approach in which the phase separation boundary is determined by measuring the hole density in the part of the system which contains both electrons and holes. In addition, we examine the binding energy of multihole clusters. An extrapolation in the number of legs suggests that the lowest $J/t$ for phase separation to occur in the two-dimensional $t\ensuremath{-}J$ model is $J/t\ensuremath{\sim}1.$

DOI
10.1103/physrevb.61.13424

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The phase separation boundary of isotropic t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J ladders is analyzed using density matrix renormalization group techniques. The complete boundary to phase separation as a function of J/tJ/tJ/t and doping is determined for a chain and for ladders with two, three, and four legs. Six-chain ladders have been analyzed at low hole doping. We use a direct approach in which the phase separation boundary is determined by measuring the hole density in the part of the system which contains both electrons and holes. In addition, we examine the binding energy of multihole clusters. An extrapolation in the number of legs suggests that the lowest J/tJ/tJ/t for phase separation to occur in the two-dimensional t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model is J/t\ensuremath∼1.J/t\ensuremath{\sim}1.J/t\ensuremath∼1.
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.
Using density-matrix renormalization-group techniques, we have studied the ground states of the four-leg t-J ladder doped near half-filling. Depending upon J/t and the hole doping x, three characteristic types of ground states are found: (1) a state containing ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ pairs, (2) a striped charge-density wave (CDW) domain-wall state, and (3) a phase-separated regime. The CDW domain walls consist of fluctuating hole pairs and this state has significant ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ pair field correlations.
Accurate numerical simulations of a doped $t\ensuremath{-}J$ model on a two-leg ladder are presented for the particle number, chemical potential, magnetic susceptibility, and entropy in the limit of large exchange coupling on the rung using a finite-temperature density matrix renormalization group method. This required an improved algorithm to achieve numerical stability down to low temperatures. The thermal dissociation of hole pairs and of the rung singlets are separately observed, and the evolution of the hole pair binding energy and magnon spin gap with hole doping is determined.
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.
Quasiparticle collapsing is a central issue in the study of strongly correlated electron systems. In the one-dimensional case, the quasiparticle collapsing in a form of spin-charge separation has been well established, but the problem remains elusive in dimensions higher than one. By using the density matrix renormalization group (DMRG) algorithm, we show that in an anisotropic two-leg $t\ensuremath{-}J$ ladder, an injected single hole behaves like a well-defined quasiparticle in the strong rung limit but undergoes a ``phase transition'' with the effective mass diverging at a quantum critical point (QCP) towards the isotropic limit. After the transition, the quasiparticle collapses into a loosely bound object of a charge (holon) and a spin-1/2 (spinon) accompanied by an unscreened phase string as well as a substantially enhanced binding energy between two doped holes. A phase diagram of multileg ladders is further obtained, which extrapolates the QCP towards the two-dimensional limit. The underlying mechanism generic for any dimensions is also discussed.
Spin-charge separation is a hallmark of one-dimensional fermionic systems, yet its realization in higher dimensions remains an open question. To address this issue, we investigate a two-leg t-J ladder using the density matrix renormalization group (DMRG) method and its time-dependent extension. By analyzing ground-state correlations and single-particle removal spectra, we systematically examine the effects of plaquette diagonal hopping, spin exchange, and hole doping. Within appropriate parameter regimes, these factors drive the system from the well-known Luther Emery phase, with gapped spin and gapless charge modes, into a Luttinger liquid phase characterized by gapless spin and charge excitations, where signatures of spin-charge separation emerge. In combination with previous studies using exact diagonalization, our results provide evidence that spin-charge separation may persist in wider ladder systems.