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arXiv:cond-mat/9812144 · Physical Review Letters

Thermodynamics of the t−J Ladder: A Stable Finite-Temperature Density Matrix Renormalization Group Calculation

Beat Ammon, Matthias Troyer, Thomas Maurice Rice, Naokazu Shibata

Published May 10, 1999• 22 citations
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Abstract

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.

DOI
10.1103/physrevlett.82.3855

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Accurate numerical simulations of a doped t\ensuremath−Jt\ensuremath{-}Jt\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 effect of a ring exchange on doped two-leg ladders is investigated combining exact diagonalization (ED) and density matrix renormalization group (DMRG) computations. We focus on the nature and weights of the low-energy magnetic excitations and on superconducting pairing. The stability with respect to this cyclic term of a remarkable resonant mode originating from a hole pair-magnon bound state is examined. We also find that, near the zero-doping critical point separating rung-singlet and dimerized phases, doping reopens a spin gap.
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/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 $t\ensuremath{-}J$ model is $J/t\ensuremath{\sim}1.$
In this paper, we have systematically studied the single-hole problem in two-leg Hubbard and $t\ensuremath{-}J$ ladders by large-scale density-matrix renormalization-group calculations. We found that the doped holes in both models behave similarly, while the three-site correlated hopping term is not important in determining the ground-state properties. For more insights, we have also calculated the elementary excitations, i.e., the energy gaps to the excited states of the system. In the strong-rung limit, we found that the doped hole behaves as a Bloch quasiparticle in both systems where the spin and charge of the doped hole are tightly bound together. In the isotropic limit, while the hole still behaves like a quasiparticle in the long-wavelength limit, our results show that its spin and charge components are only loosely bound together inside the quasiparticle, whose internal structure can lead to a visible residual effect which dramatically changes the local structure of the ground-state wave function.
A chemical potential difference between the legs of a two-leg ladder is found to be harmful for the Cooper pairing. The instability of superconductivity in such systems is analyzed by compairing results of various analytical and numerical methods. Within a strong-coupling approach for the $t\ensuremath{-}J$ model, supplemented by exact numerical diagonalization, hole binding is found unstable beyond a finite, critical chemical potential difference. The spinon-holon mean field theory for the $t\ensuremath{-}J$ model shows a clear reduction of the BCS gaps upon increasing the chemical potential difference leading to a breakdown of superconductivity. Based on a renormalization group approach and an Abelian bosonization, the doping dependent phase diagram for the weakly interacting Hubbard model with different chemical potentials was determined.
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 numerical results from density matrix renormalization group (DMRG) calculations for the t-J model, on systems as large as 10\ifmmode\times\else\texttimes\fi{}7, we examine the structure of the one and two hole ground states in ladder systems and in two dimensional clusters. A simple theoretical framework is used to explain why holes bind in pairs in two-dimensional antiferromagnets. For the case J/t=0.5, which we have studied, the hole pairs reside predominantly on a 2\ifmmode\times\else\texttimes\fi{}2 core plaquette with the probability that the holes are on diagonal sites greater than nearest-neighbor sites. There is a strong singlet bond connecting the spins on the two remaining sites of the plaquette. We find that a general characteristic of dynamic holes in an antiferromagnet is the presence of frustrating antiferromagnetic bonds connecting next-nearest-neighbor sites across the holes. Pairs of holes bind in order to share the frustrating bonds. At low doping, in addition to hole pairs, there are two additional low-energy structures which spontaneously form on certain finite systems. The first is an undoped L\ifmmode\times\else\texttimes\fi{}2 spin-liquid region, or ladder. The second is a hole moving along a one dimensional chain of sites. At higher doping we expect that hole pairing is always favored.