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Confinement in weakly-coupled double Hubbard chains, at half-filling

Karyn Le Hur

Published Jun 14, 2000• 0 citations
OpenAlex

Abstract

We study two (very) weakly-coupled Hubbard chains in the half-filled case, and especially the situation where the intrachain Mott scale $m$ is much larger than the bare single-electron interchain hopping $t_{\perp}$. First, we find that the divergence of the intrachain Umklapp channel at the Mott transition results in the complete vanishing of the single-electron interchain hopping: This is significant of a strong confinement of coherence along the chains. Excitations are usual charge fermionic solitons and spinon pairs of the Heisenberg chain. Then, we show rigorously how the tunneling of spinon-pairs produces a magnetic interchain exchange $J_{\perp}={t_{\perp}}^2/m>0$. The result is an insulating state with spin-gapped excitations as in the delocalized limit (i.e. for rather large interchain hoppings), where the two-leg ladder is in the well-known D-Mott phase. Unlike for Bechgaard salts, the confinement/deconfinement transition at absolute zero is here a simple crossover: no metallic phase is found in undoped two-leg ladders. This statement might be generalized for N-leg ladders with N=3,4... (but not too large).

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Weakly coupled Hubbard chains at half-filling and confinement

Karyn Le Hur

We study two (very) weakly coupled Hubbard chains in the half-filled case, and especially the situation where the intrachain Mott scale m is much larger than the (bare) single-electron interchain hopping ${t}_{\ensuremath{\perp}}.$ First, we find that the divergence of the intrachain umklapp channel at the Mott transition results in the complete vanishing of the single-electron interchain hopping: this is significant of a strong confinement of coherence along the chains. Excitations are usual charge fermionic solitons and spinon-(anti)spinon pairs of the Heisenberg chain. Then, we show rigorously how the tunneling of spinon-(anti)spinon pairs produces an antiferromagnetic interchain exchange of the order of ${J}_{\ensuremath{\perp}}{=t}_{\ensuremath{\perp}}^{2}/m.$ In the ``confined'' phase and in the far infrared, the system behaves as a pure spin ladder. The final result is an insulating ground state with spin-gapped excitations exactly as in the opposite ``delocalized'' limit (i.e., for rather large interchain hoppings) where the two-leg ladder is in the well-known insulating $D\ensuremath{-}\mathrm{Mott}$ phase. Unlike materials with an infinite number of coupled chains (Bechgaard salts), the confinement/deconfinement transition at absolute zero is here a simple crossover: no metallic phase is found in undoped two-leg ladders. This statement might be generalized for N-leg ladders with $N=3,4,\dots{}$ (but not too large).

arXiv:cond-mat/0006224•Physical review. B, Condensed matter•Apr 4, 2001•24 citations

Spinless fermion ladders at half filling

P. Donohue, Masahisa Tsuchiizu, Thierry Giamarchi +1 more

We study a half-filled ladder of spinless fermions. We show that contrary to a single chain, the ladder becomes a Mott insulator for arbitrarily small repulsive interactions. We obtain the full phase diagram and physical quantities such as the charge gap. We show that there is only a single insulating phase for repulsive interactions, regardless of the strength of the interchain hopping and single chain Mott gap. There is thus no confinement-deconfinement transition in this system but a simple crossover. Using bosonization with the renormalization group we show that upon doping the system becomes a Luttinger liquid with a universal parameter $K=1/2$ different from the one of the single chain $(K=1/4).$

arXiv:cond-mat/0007336•Physical review. B, Condensed matter•Jan 9, 2001•18 citations

Dimensional crossover in half-filled and lightly dopedN-leg Hubbard ladders

Urs Ledermann

We study the dimensional crossover from one to two dimensions in half-filled and lightly doped, weakly interacting N-leg Hubbard ladders. In this case, the Hubbard ladders are equivalent to an N-band model. Using renormalization-group techniques, we find that the half-filled ladders exhibit (in the spin sector) an odd-even effect only below a crossover energy ${E}_{c}\ensuremath{\propto}\mathrm{exp}[\ensuremath{-}\ensuremath{\alpha}\mathrm{exp}(\ensuremath{\gamma}N)]$ $(\ensuremath{\alpha}\ensuremath{\ll}1$ and $\ensuremath{\gamma}\ensuremath{\sim}1$ depend on the interaction strength and on the hopping matrix elements): Below ${E}_{c},$ the dominant interactions take place within band pairs $(j,N+1\ensuremath{-}j)$ [and within the band $(N+1)/2$ for N odd], such that even-leg ladders are an insulating spin liquid, while odd-leg ladders have one gapless spinon mode. In contrast, above the energy scale ${E}_{c},$ all bands are interacting with each other and the system is a two-dimensional- (2D)-like (insulating) antiferromagnet; we obtain an analytical expression for the Hamiltonian which is similar to the 2D Heisenberg antiferromagnet. Bosonization techniques show that in the charge sector the Mott insulator is as well below and above ${E}_{c}$ of the same type as in $N/2$ half-filled two-leg ladders. Doping away from half-filling, we find that the effect of an increasing doping $\ensuremath{\delta}$ is very similar to decreasing the number of legs N: In both cases interactions between unpaired bands are suppressed and thereby the antiferromagnetic correlations reduced. The resulting band pairs form then insulating spin liquids and when doped, there is a spin gap, but phase coherence exists only within the band pairs. At higher doping levels ${\ensuremath{\delta}}_{c}={\ensuremath{\delta}}_{c}(N),$ phase coherence between all band pairs sets in and the system becomes a 2D-like d-wave superconductor $[{\ensuremath{\delta}}_{c}(N)\ensuremath{\rightarrow}0$ for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}].$

arXiv:cond-mat/0103159•Physical review. B, Condensed matter•Nov 8, 2001•7 citations

Spinon confinement: Dynamics of weakly coupled Hubbard chains

Marcin Raczkowski, Fakher F. Assaad

Using large-scale determinant quantum Monte Carlo simulations in combination with the stochastic analytical continuation, we study two-particle dynamical correlation functions in the anisotropic square lattice of weakly coupled one-dimensional (1D) Hubbard chains at half filling and in the presence of weak frustration. The evolution of the static spin structure factor upon increasing the interchain coupling is suggestive of the transition from the power-law decay of spin-spin correlations in the 1D limit to long-range antiferromagnetic order in the quasi-1D regime and at $T=0$. In the numerically accessible regime of interchain couplings, the charge sector remains gapped. The low-energy momentum dependence of the spin excitations is well described by the linear spin-wave theory with the largest intensity located around the antiferromagnetic wave vector. This magnon mode corresponds to a bound state of two spinons. At higher energies the spinons deconfine and we observe signatures of the two-spinon continuum which progressively fade away as a function of interchain hopping.

arXiv:1306.5775•Physical Review B•Aug 21, 2013•24 citations

Charge gaps at fractional fillings in boson Hubbard ladders

Tao Ying, G. G. Batrouni, G. X. Tang +2 more

The Bose-Hubbard Hamiltonian describes the competition between superfluidity and Mott insulating behavior at zero temperature and commensurate filling as the strength of the on-site repulsion is varied. Gapped insulating phases also occur at noninteger densities as a consequence of longer ranged repulsive interactions. In this paper we explore the formation of gapped phases in coupled chains due instead to anisotropies ${t}_{x}\ensuremath{\ne}{t}_{y}$ in the bosonic hopping, extending the work of Crepin et al. [Phys. Rev. B 84, 054517 (2011).] on two coupled chains, where a gap was shown to occur at half filling for arbitrarily small interchain hopping ${t}_{y}$. Our main result is that, unlike the two-leg chains, for three- and four-leg chains, a charge gap requires a finite nonzero critical ${t}_{y}$ to open. However, these finite values are surprisingly small, well below the analogous values required for a fermionic band gap to open.

arXiv:1403.0249•Physical Review B•May 21, 2014•2 citations

Collective excitations and confinement in the excitation spectra of the spinless fermion model on a ladder

Kenji Yonemitsu

Intrachain and interchain local charge-transfer excitation spectra and the single-particle density of states are calculated in the spinless fermion model on a ladder with varying intrachain nearest-neighbor repulsion and interchain transfer integral at and near half filling by using the finite-temperature density-matrix renormalization-group method. Collective excitations are found to govern the low-energy intrachain spectra, while only individual local excitations are present in the interchain spectra. For strong intrachain repulsion, the low-energy motion of fermions is confined within a chain. The interchain motion of fermions is not bandlike but incoherent. As a consequence, the low-energy intrachain spectra are sensitive at half filling to the interchain transfer integral that weakens the density-density correlation along the chains. Similarly, the low-energy intrachain spectra are sensitive near half filling to the chemical potential that reduces the effect of the umklapp process. Similarities are pointed out between these findings and the experimentally observed, optical conductivity spectra in the quasi-one-dimensional organic conductors $(\mathrm{TMTSF}{)}_{2}X.$

Physical review. B, Condensed matter•Jul 31, 2002•0 citations
We study two (very) weakly-coupled Hubbard chains in the half-filled case, and especially the situation where the intrachain Mott scale mmm is much larger than the bare single-electron interchain hopping t⊥t_{\perp}t⊥​. First, we find that the divergence of the intrachain Umklapp channel at the Mott transition results in the complete vanishing of the single-electron interchain hopping: This is significant of a strong confinement of coherence along the chains. Excitations are usual charge fermionic solitons and spinon pairs of the Heisenberg chain. Then, we show rigorously how the tunneling of spinon-pairs produces a magnetic interchain exchange J⊥=t⊥2/m>0J_{\perp}={t_{\perp}}^2/m>0J⊥​=t⊥​2/m>0. The result is an insulating state with spin-gapped excitations as in the delocalized limit (i.e. for rather large interchain hoppings), where the two-leg ladder is in the well-known D-Mott phase. Unlike for Bechgaard salts, the confinement/deconfinement transition at absolute zero is here a simple crossover: no metallic phase is found in undoped two-leg ladders. This statement might be generalized for N-leg ladders with N=3,4... (but not too large).
We study two (very) weakly coupled Hubbard chains in the half-filled case, and especially the situation where the intrachain Mott scale m is much larger than the (bare) single-electron interchain hopping ${t}_{\ensuremath{\perp}}.$ First, we find that the divergence of the intrachain umklapp channel at the Mott transition results in the complete vanishing of the single-electron interchain hopping: this is significant of a strong confinement of coherence along the chains. Excitations are usual charge fermionic solitons and spinon-(anti)spinon pairs of the Heisenberg chain. Then, we show rigorously how the tunneling of spinon-(anti)spinon pairs produces an antiferromagnetic interchain exchange of the order of ${J}_{\ensuremath{\perp}}{=t}_{\ensuremath{\perp}}^{2}/m.$ In the ``confined'' phase and in the far infrared, the system behaves as a pure spin ladder. The final result is an insulating ground state with spin-gapped excitations exactly as in the opposite ``delocalized'' limit (i.e., for rather large interchain hoppings) where the two-leg ladder is in the well-known insulating $D\ensuremath{-}\mathrm{Mott}$ phase. Unlike materials with an infinite number of coupled chains (Bechgaard salts), the confinement/deconfinement transition at absolute zero is here a simple crossover: no metallic phase is found in undoped two-leg ladders. This statement might be generalized for N-leg ladders with N=3,4,…N=3,4,\dots{}N=3,4,… (but not too large).
We study a half-filled ladder of spinless fermions. We show that contrary to a single chain, the ladder becomes a Mott insulator for arbitrarily small repulsive interactions. We obtain the full phase diagram and physical quantities such as the charge gap. We show that there is only a single insulating phase for repulsive interactions, regardless of the strength of the interchain hopping and single chain Mott gap. There is thus no confinement-deconfinement transition in this system but a simple crossover. Using bosonization with the renormalization group we show that upon doping the system becomes a Luttinger liquid with a universal parameter K=1/2K=1/2K=1/2 different from the one of the single chain (K=1/4).(K=1/4).(K=1/4).
We study the dimensional crossover from one to two dimensions in half-filled and lightly doped, weakly interacting N-leg Hubbard ladders. In this case, the Hubbard ladders are equivalent to an N-band model. Using renormalization-group techniques, we find that the half-filled ladders exhibit (in the spin sector) an odd-even effect only below a crossover energy ${E}_{c}\ensuremath{\propto}\mathrm{exp}[\ensuremath{-}\ensuremath{\alpha}\mathrm{exp}(\ensuremath{\gamma}N)]$ $(\ensuremath{\alpha}\ensuremath{\ll}1$ and $\ensuremath{\gamma}\ensuremath{\sim}1$ depend on the interaction strength and on the hopping matrix elements): Below Ec,{E}_{c},Ec​, the dominant interactions take place within band pairs $(j,N+1\ensuremath{-}j)$ [and within the band (N+1)/2(N+1)/2(N+1)/2 for N odd], such that even-leg ladders are an insulating spin liquid, while odd-leg ladders have one gapless spinon mode. In contrast, above the energy scale Ec,{E}_{c},Ec​, all bands are interacting with each other and the system is a two-dimensional- (2D)-like (insulating) antiferromagnet; we obtain an analytical expression for the Hamiltonian which is similar to the 2D Heisenberg antiferromagnet. Bosonization techniques show that in the charge sector the Mott insulator is as well below and above Ec{E}_{c}Ec​ of the same type as in N/2N/2N/2 half-filled two-leg ladders. Doping away from half-filling, we find that the effect of an increasing doping $\ensuremath{\delta}$ is very similar to decreasing the number of legs N: In both cases interactions between unpaired bands are suppressed and thereby the antiferromagnetic correlations reduced. The resulting band pairs form then insulating spin liquids and when doped, there is a spin gap, but phase coherence exists only within the band pairs. At higher doping levels ${\ensuremath{\delta}}_{c}={\ensuremath{\delta}}_{c}(N),$ phase coherence between all band pairs sets in and the system becomes a 2D-like d-wave superconductor $[{\ensuremath{\delta}}_{c}(N)\ensuremath{\rightarrow}0$ for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}].$
Using large-scale determinant quantum Monte Carlo simulations in combination with the stochastic analytical continuation, we study two-particle dynamical correlation functions in the anisotropic square lattice of weakly coupled one-dimensional (1D) Hubbard chains at half filling and in the presence of weak frustration. The evolution of the static spin structure factor upon increasing the interchain coupling is suggestive of the transition from the power-law decay of spin-spin correlations in the 1D limit to long-range antiferromagnetic order in the quasi-1D regime and at T=0T=0T=0. In the numerically accessible regime of interchain couplings, the charge sector remains gapped. The low-energy momentum dependence of the spin excitations is well described by the linear spin-wave theory with the largest intensity located around the antiferromagnetic wave vector. This magnon mode corresponds to a bound state of two spinons. At higher energies the spinons deconfine and we observe signatures of the two-spinon continuum which progressively fade away as a function of interchain hopping.
The Bose-Hubbard Hamiltonian describes the competition between superfluidity and Mott insulating behavior at zero temperature and commensurate filling as the strength of the on-site repulsion is varied. Gapped insulating phases also occur at noninteger densities as a consequence of longer ranged repulsive interactions. In this paper we explore the formation of gapped phases in coupled chains due instead to anisotropies ${t}_{x}\ensuremath{\ne}{t}_{y}$ in the bosonic hopping, extending the work of Crepin et al. [Phys. Rev. B 84, 054517 (2011).] on two coupled chains, where a gap was shown to occur at half filling for arbitrarily small interchain hopping ty{t}_{y}ty​. Our main result is that, unlike the two-leg chains, for three- and four-leg chains, a charge gap requires a finite nonzero critical ty{t}_{y}ty​ to open. However, these finite values are surprisingly small, well below the analogous values required for a fermionic band gap to open.
Intrachain and interchain local charge-transfer excitation spectra and the single-particle density of states are calculated in the spinless fermion model on a ladder with varying intrachain nearest-neighbor repulsion and interchain transfer integral at and near half filling by using the finite-temperature density-matrix renormalization-group method. Collective excitations are found to govern the low-energy intrachain spectra, while only individual local excitations are present in the interchain spectra. For strong intrachain repulsion, the low-energy motion of fermions is confined within a chain. The interchain motion of fermions is not bandlike but incoherent. As a consequence, the low-energy intrachain spectra are sensitive at half filling to the interchain transfer integral that weakens the density-density correlation along the chains. Similarly, the low-energy intrachain spectra are sensitive near half filling to the chemical potential that reduces the effect of the umklapp process. Similarities are pointed out between these findings and the experimentally observed, optical conductivity spectra in the quasi-one-dimensional organic conductors (TMTSF)2X.(\mathrm{TMTSF}{)}_{2}X.(TMTSF)2​X.