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arXiv:cond-mat/0503600 · Physical Review B

Single-site entanglement at the superconductor-insulator transition in the Hirsch model

Alberto Anfossi, C. Degli Esposti Boschi, Arianna Montorsi, F. Ortolani

Published Feb 16, 2006• 45 citations
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Abstract

We investigate the transition to the insulating state in the one-dimensional Hubbard model with bond-charge interaction $x$ (Hirsch model), at half-filling and $T=0$. By means of the density-matrix renormalization group algorithm the charge gap closure is examined by both standard finite-size scaling analysis and looking at singularities in the derivatives of single-site entanglement. The results of the two techniques show that a quantum phase transition takes place at a finite Coulomb interaction ${u}_{\mathrm{c}}(x)$ for $x\ensuremath{\gtrsim}0.5$. The region $0<u<{u}_{\mathrm{c}}$ turns out to have a superconducting nature, at least for not too large $x>{x}_{\mathrm{c}}$.

DOI
10.1103/physrevb.73.085113

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We investigate the transition to the insulating state in the one-dimensional Hubbard model with bond-charge interaction xxx (Hirsch model), at half-filling and T=0T=0T=0. By means of the density-matrix renormalization group algorithm the charge gap closure is examined by both standard finite-size scaling analysis and looking at singularities in the derivatives of single-site entanglement. The results of the two techniques show that a quantum phase transition takes place at a finite Coulomb interaction uc(x){u}_{\mathrm{c}}(x)uc​(x) for x\ensuremath≳0.5x\ensuremath{\gtrsim}0.5x\ensuremath≳0.5. The region 0<u<uc0<u<{u}_{\mathrm{c}}0<u<uc​ turns out to have a superconducting nature, at least for not too large x>xcx>{x}_{\mathrm{c}}x>xc​.
We investigate numerically at various fillings the ground state of the one-dimensional Hubbard model with correlated hopping xxx (Hirsch model). It is found that, for a large range of filling values nnn around half filling, and for repulsive Coulomb interaction $u\ensuremath{\le}{u}_{c}(x,n)$, phase separation at a nanoscale (NPS phase) between two conducting phases at different densities occurs when $x\ensuremath{\gtrsim}2/3$. The NPS phase is accompanied by the opening of a spin gap and the system behaves as a Luther-Emery liquid with dominant superconducting correlations. Close to half filling, an anomalous peak emerges in the charge structure factor related to the density of doubly occupied sites, which determines the size of the droplets in the NPS phase. For $1/2\ensuremath{\lesssim}x\ensuremath{\lesssim}2/3$ a crossover to a homogeneous phase, still superconducting, takes place.
We examine the phase diagram of the half-filled one-dimensional extended Hubbard model using quantum information entropies within the density-matrix renormalization group. It is well known that there is a charge-density-wave phase at large nearest-neighbor and a small on-site Coulomb repulsion and a spin-density wave at small nearest-neighbor and large on-site Coulomb repulsion. At intermediate Coulomb interaction strength, we find an additional narrow region of a bond-order phase between these two phases. The phase transition line for the transition out of the charge-density-wave phase changes from first order at strong coupling to second order in a parameter regime where all three phases are present. We present evidence that the additional phase-transition line between the spin-density-wave and bond-order phases is infinite order. While these results are in agreement with recent numerical work, our study provides an independent unbiased means of determining the phase boundaries by using quantum information analysis, yields values for the location of some of the phase boundaries that differ from those previously found, and provides insight into the limitations of numerical methods in determining phase boundaries, especially those of infinite-order transitions.
We study a one-dimensional SU(N) Hubbard model with an attractive on-site interaction and N>2N>2N>2 at half-filling on the bipartite lattice using density-matrix renormalization-group method and a perturbation theory. We find that the ground state of the SU(N) Hubbard model is a charge density wave state with two-fold degeneracy. All the excitations are found to be gapful, resulting in an insulating ground state, on contrary to that in the SU(2) case. Moreover, the charge gap is equal to the Cooperon gap, which behaves as −2Nt2/(N−1)U-2Nt^2/(N-1)U−2Nt2/(N−1)U in the strong coupling regime. However, the spin gap ΔsΔ_{s}Δs​ and the quasiparticle gap Δ1Δ_{1}Δ1​ as well open exponentially in the weak coupling region, while in the strong coupling region, they linearly depend on UUU such that Δs∼−U(N−1)Δ_{s}\sim -U(N-1)Δs​∼−U(N−1) and Δ1∼−U(N−1)/2Δ_{1}\sim -U(N-1)/2Δ1​∼−U(N−1)/2.
We study the phase diagram of a half-filled one-dimensional superlattice Hubbard model with an alternating orbital energy ± ϵ/2 and on-site Coulomb repulsion U(A) and U(B) using the density-matrix renormalization group and finite-size scaling analysis. For U(A) >> U(B), we observe an incommensurate charge correlation phase. For U(B) = 0, we find a transition from the band insulating phase to the correlated metallic phase at the critical point ϵ = U(A)/2. For U(B) > 0, we find three insulating phases as in the ionic Hubbard model. In the gapless region, the correlation functions exhibit the asymptotic behavior of power-law decay as in the Tomonaga-Luttinger liquid.
The one-dimensional SU(n)\mathrm{SU}(n)SU(n) Hubbard model is investigated numerically for n=2n=2n=2, 3, 4, and 5 at half-filling and 1∕n1∕n1∕n filling using the density-matrix renormalization-group method. The energy gaps and various quantum information entropies are calculated. In the half-filled case, finite spin and charge gaps are found for arbitrary positive UUU if n>2n>2n>2. Furthermore, it is shown that the transition to the gapped phase at Uc=0{U}_{c}=0Uc​=0 is of Kosterlitz-Thouless type and is accompanied by a bond dimerization both for even and odd nnn. In the 1∕n1∕n1∕n-filled case, the transition has similar features as the metal-insulator transition in the half-filled SU(2) Hubbard model. The charge gap opens exponentially slowly for U>Uc=0U>{U}_{c}=0U>Uc​=0, the spin sector remains gapless, and the ground state is nondimerized.
By using the density matrix renormalization group technique, the phase diagram of the half-filling extended Hubbard model is investigated. The conventional order parameter, the two-site entanglement entropy, and the block-block entanglement entropy are analyzed in detail. According to the numerical results, in the weak coupling region, an intermediate bond-order-wave (BOW) phase is shown to exist indeed between the charge-density-wave (CDW) and the spin-density-wave (SDW) phases. The critical phase transition points are determined by the singularity of the first order derivative of two-site entanglement entropy. In strong coupling region, a direct phase transition occurs from SDW phase to CDW phase, and shows discontinuous (first order) character accompanied with energy level crossing. The numerical results support the phase diagram proposed previously by some authors Sengupta et al., (2002); Sandvik et al., (2004); Zhang, (2004). Therefore, the quantum entanglement is a sensitive tool to describe quantum phase transitions in strongly correlated electron systems.