© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:cond-mat/9501093 · Physical review. B, Condensed matter

Spin and charge modes of thet-Jladder

Didier Poilblanc, Douglas J. Scalapino, Werner R. Hanke

Published Sep 1, 1995• 45 citations

Abstract

The spin and charge excitations of the t-J ladder are studied by exact diagonalization techniques for several electron densities. The various modes are classified according to their spin (singlet or triplet excitations) and their parity under a reflection with respect to the symmetry axis along the chains and a finite-size scaling of the related gaps is performed. At low doping, formation of hole pairs leads to a spin gap for all J/t ratios. This phase is characterized by (at least) one vanishing energy mode only in the charge bonding channel when ${\mathit{K}}_{\mathit{x}}$\ensuremath{\rightarrow}0 consistent with the existence of superconducting pairing correlations. At larger doping the spin gap disappears. Although the antibonding modes remain gapped, low-energy ${\mathit{K}}_{\mathit{x}}$\ensuremath{\sim}0 and ${\mathit{K}}_{\mathit{x}}$\ensuremath{\sim}2${\mathit{k}}_{\mathit{F}}$ spin and charge bonding modes are found consistent with a single-band Luttinger scenario. At sufficient low electron density and above a critical value of J/t we also expect another phase of electron pairs with gapped spin excitations.

DOI: 10.1103/physrevb.52.6796

Related papers

Pairing and excitation spectrum in dopedt-Jladders

Hirokazu Tsunetsugu, Matthias Troyer, Thomas Maurice Rice

Exact-diagonalization studies for a doped t-J ladder (or double chain) show hole pairing in the ground state. The excitation spectrum separates into a limited number of quasiparticles which carry charge +\ensuremath{\Vert}e\ensuremath{\Vert} and spin 1/2 and a triplet mode. At half-filling the former vanish but the latter evolves continuously into the triplet band of the spin liquid. At low doping the quasiparticles form a dilute Fermi gas with a strong attraction but simultaneously the Fermi wave vector, as would be measured in photoemission, is large.

arXiv:cond-mat/9401050•Physical review. B, Condensed matter•Jun 1, 1994•140 citations

Spin gaps in coupledt-Jladders

Didier Poilblanc, Hirokazu Tsunetsugu, Thomas Maurice Rice

Spin gaps in coupled t-J ladders are investigated by exact diagonalization of small clusters up to 4\ifmmode\times\else\texttimes\fi{}8 sites. At half-filling, the numerical results for the triplet excitation spectrum are in very good agreement with a second-order perturbation expansion in terms of small interladder and intraladder exchange couplings between rungs (J/J'0.25). The band of local triplet excitations moving coherently along the ladder (with momenta close to \ensuremath{\pi}) is split by the interladder coupling. For intermediate couplings, finite-size scaling is used to estimate the spin gap. In the isotropic infinite four-chain system (two coupled ladders) we find a spin gap of 0.245J, roughly half of the single-ladder spin gap. When the system is hole doped, bonding and antibonding bound pairs of holes can propagate coherently along the chains and the spin gap remains finite.

arXiv:cond-mat/9405045•Physical review. B, Condensed matter•Sep 1, 1994•49 citations

Phase diagram of coupled ladders

T. F. A. Müller, Thomas Maurice Rice

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 $J/t.$ Two known limiting cases are $n<\frac{1}{2},$ 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 $1>n>\frac{1}{2}.$ We find evidences for gapless behavior in both spin and charge channels for $J/t<0.3,$ consistent with Luttinger liquids in both bonding and antibonding bands (i.e., $C2S2).$ 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 $J/t$ 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 $J/t<0.3$ this is very weak, indicating a lack of pairing between these channels.

arXiv:cond-mat/9802297•Physical review. B, Condensed matter•Aug 1, 1998•18 citations

Luttinger-liquid behavior and superconducting correlations int-Jladders

C A Hayward, Didier Poilblanc

The low-energy behavior of the isotropic t-J ladder system is investigated using exact diagonalization techniques, specifically finding the Drude weight, the charge velocity, and the compressibility. By applying the ideas of Luttinger-liquid theory, we determine the correlation exponent ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$ which defines the behavior of the long-range correlations in the system. The boundary to phase separation is determined and a phase diagram is presented. At low electron density, a Tomonaga-Luttinger-like phase is stabilized while at higher electron densities a gapped phase with power law pairing correlations is stabilized: A large region of this gapped phase is found to exhibit dominant superconducting correlations. \textcopyright{} 1996 The American Physical Society.

arXiv:cond-mat/9509123•Physical review. B, Condensed matter•May 1, 1996•81 citations

Ground-state properties of the doped three-leg t - J ladder

Steven R. White, Douglas J. Scalapino

Results for a doped three-leg $t$-$J$ ladder obtained using the density-matrix renormalization group are reported. At low hole doping, the holes form a dilute gas with a uniform density. The momentum occupation of the odd band shows a sharp decrease at a large value of ${k}_{F}$ similar to the behavior of a lightly doped $t$-$J$ chain, while the even modes appear gapped. The spin-spin correlations decay as a power law consistent with the absence of a spin gap, but the pair-field correlations are negligible. At larger doping we find evidence for a spin gap and as $x$ increases further we find three-hole diagonal domain walls. In this regime there are significant pair-field correlations and the internal pair orbital has ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-like symmetry.

arXiv:cond-mat/9706250•Physical review. B, Condensed matter•Feb 1, 1998•56 citations

Exact ground and excited states of at-Jladder doped with two holes

Indrani Bose, Saurabh Gayen

A two-chain ladder model is considered that is described by the strong-coupling t - t ´- J - J ´ Hamiltonian. For the case of two holes moving in a background of antiferromagnetically interacting spins, exact, analytical results are derived for the ground-state energy and low-lying excitation spectrum. The ground state is a bound state of two holes with total spin S = 0. The charge excitation is gapless and the spin excitation has a gap. The corresponding wave functions are also exactly determined. The bound hole pair is found to have symmetry of the d-wave type. In the limit of strong rung coupling, the model maps onto an effective hard-core-boson model which exhibits dominant superconducting pairing correlations.

arXiv:cond-mat/9812140•Journal of Physics Condensed Matter•Aug 6, 1999•9 citations

The spin and charge excitations of the t-J ladder are studied by exact diagonalization techniques for several electron densities. The various modes are classified according to their spin (singlet or triplet excitations) and their parity under a reflection with respect to the symmetry axis along the chains and a finite-size scaling of the related gaps is performed. At low doping, formation of hole pairs leads to a spin gap for all J/t ratios. This phase is characterized by (at least) one vanishing energy mode only in the charge bonding channel when

{\mathit{K}}_{\mathit{x}}

\ensuremath{\rightarrow}0 consistent with the existence of superconducting pairing correlations. At larger doping the spin gap disappears. Although the antibonding modes remain gapped, low-energy

{\mathit{K}}_{\mathit{x}}

\ensuremath{\sim}0 and

{\mathit{K}}_{\mathit{x}}

\ensuremath{\sim}2

{\mathit{k}}_{\mathit{F}}

spin and charge bonding modes are found consistent with a single-band Luttinger scenario. At sufficient low electron density and above a critical value of J/t we also expect another phase of electron pairs with gapped spin excitations.

Exact-diagonalization studies for a doped t-J ladder (or double chain) show hole pairing in the ground state. The excitation spectrum separates into a limited number of quasiparticles which carry charge +\ensuremath{\Vert}e\ensuremath{\Vert} and spin 1/2 and a triplet mode. At half-filling the former vanish but the latter evolves continuously into the triplet band of the spin liquid. At low doping the quasiparticles form a dilute Fermi gas with a strong attraction but simultaneously the Fermi wave vector, as would be measured in photoemission, is large.

Spin gaps in coupled t-J ladders are investigated by exact diagonalization of small clusters up to 4\ifmmode\times\else\texttimes\fi{}8 sites. At half-filling, the numerical results for the triplet excitation spectrum are in very good agreement with a second-order perturbation expansion in terms of small interladder and intraladder exchange couplings between rungs (J/J'0.25). The band of local triplet excitations moving coherently along the ladder (with momenta close to \ensuremath{\pi}) is split by the interladder coupling. For intermediate couplings, finite-size scaling is used to estimate the spin gap. In the isotropic infinite four-chain system (two coupled ladders) we find a spin gap of 0.245J, roughly half of the single-ladder spin gap. When the system is hole doped, bonding and antibonding bound pairs of holes can propagate coherently along the chains and the spin gap remains finite.

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.

Two known limiting cases are

n<\frac{1}{2},

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>\frac{1}{2}.

We find evidences for gapless behavior in both spin and charge channels for

J/t<0.3,

consistent with Luttinger liquids in both bonding and antibonding bands (i.e.,

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/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<0.3

this is very weak, indicating a lack of pairing between these channels.

The low-energy behavior of the isotropic t-J ladder system is investigated using exact diagonalization techniques, specifically finding the Drude weight, the charge velocity, and the compressibility. By applying the ideas of Luttinger-liquid theory, we determine the correlation exponent ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$ which defines the behavior of the long-range correlations in the system. The boundary to phase separation is determined and a phase diagram is presented. At low electron density, a Tomonaga-Luttinger-like phase is stabilized while at higher electron densities a gapped phase with power law pairing correlations is stabilized: A large region of this gapped phase is found to exhibit dominant superconducting correlations. \textcopyright{} 1996 The American Physical Society.

Results for a doped three-leg

t

-

J

ladder obtained using the density-matrix renormalization group are reported. At low hole doping, the holes form a dilute gas with a uniform density. The momentum occupation of the odd band shows a sharp decrease at a large value of

{k}_{F}

similar to the behavior of a lightly doped

t

-

J

chain, while the even modes appear gapped. The spin-spin correlations decay as a power law consistent with the absence of a spin gap, but the pair-field correlations are negligible. At larger doping we find evidence for a spin gap and as

x

increases further we find three-hole diagonal domain walls. In this regime there are significant pair-field correlations and the internal pair orbital has ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-like symmetry.

A two-chain ladder model is considered that is described by the strong-coupling t - t ´- J - J ´ Hamiltonian. For the case of two holes moving in a background of antiferromagnetically interacting spins, exact, analytical results are derived for the ground-state energy and low-lying excitation spectrum. The ground state is a bound state of two holes with total spin S = 0. The charge excitation is gapless and the spin excitation has a gap. The corresponding wave functions are also exactly determined. The bound hole pair is found to have symmetry of the d-wave type. In the limit of strong rung coupling, the model maps onto an effective hard-core-boson model which exhibits dominant superconducting pairing correlations.