Numerical Study of the 1D Asymmetric Hubbard Model Cristian Degli Esposti Boschi CNR, Unità di ricerca CNISM di Bologna and Dipartimento di Fisica, Università di Bologna Marco Casadei and Fabio Ortolani Dipartimento di Fisica, Università di Bologna and Sezione INFN di Bologna Italian Quantum Information Science Conference, Camerino, 24 th -29 th october 2008

The model: notations and optical lattices implementation density “magnetization” or imbalance “up” and “down” notation will be used to indicate the two species of fermions also in the case of optical lattices. [fine structure splitting of excited states giving rise to spin-dependent Stark shift; Liu, Wilczek & Zoller, PRA 70, (2004)]

Full particle-hole transformation property still valid In the asymmetric case SU(2) symmetries (spin and pseudospin) are lost; however, for every value of the asymmetry factor the number of particles in each specie is separately conserved. Is it also fixed in experiments? We can restrict to n ≤ 1

- The usual symmetric Hubbard model has been solved via Bethe- ansatz by Lieb and Wu [PRL 20, 1445 (1968)] - The Falicov-Kimball limit (z = 1) il also nontrivial. Inhomogeous domains (phase separation/segregation) are formed in every dimension for n ≠ 1 and large U. [Freericks, Lieb & Ueltschi, Comm. Math. Phys. 227, 243 (2002) and PRL 88, (2002)] Known results It can be proven that the same occurs for z ~ 1 [Ueltschi, J. Stat. Phys. 116, 681 (2004)]. Effective attractive interaction between “light electrons” mediated by “heavy electrons” [Domański & co-workers, JMMM 140, 1205 (1995) in 1D, JPC 8, L261 (1996) in 2D]. - Thermodyn. and PS [Macedo & de Souza, PRB 65, (2002)]

Roughly speaking, the light electrons need to occupy connected large regions in order to lower their kinetic energy. For N free fermions on an interval of size A the energy increases if one makes k > 1 “cuts”

Almost all the recent papers focus on the case n 0 Bosonization and some numerics at S z = 0, away from n = 1: RG for the spin gap (?) and crossovers at U < 0 [Cazalilla, Ho & Giamarchi, PRL 95, (2005)] - also dual condition n = 1, S z ≠ 0 and Bose-Bose and Bose-Fermi mixtures [Mathey, PRB 75, (2007)] - phase separation occurs when the velocity of the decoupled mode vanishes [Wang, Chen & Gu, PRB 75, (2007)] - absence of phase separation for small enough z [Gu, Fan & Lin, PRB 76, (2007)] - spatial configurations and correlations (also 2D) [Farkašovský, PRB 77, (2008)] Effect of the confining harmonic trap: [Silva-Valencia, Franco & Figueira, JMMM 320, e431 (2008)] also Xianlong et al., PRL 98, (2007) for the Luther-Emery phase at effective negative U.

DMRG & large-U expansion at n = 1. [Fath, Domański & Lemański, PRB 52, (1995)] In the region z, U > 0 the system behaves effectively as Néel antiferromagnet with nonvanishing spin gap (that vanishes for z = 0). Possible gapless neighbourhood of z = 1, U = 0. Our analysis: Lanczos method and DMRG with: - periodic boundary conditions (no Friedel oscillations or midgap states) - up to 1300 optimized states - up to 7 finite-system sweeps to achieve sufficient accuracy. Charge excitations are gapless! Energy relative error: < for L < 30, ~10 -5 for L < 50

(qualitative) Phase diagram for n < 1 Fig. 1 in PRL 95, Depends on the filling (PRB 77, ) Absence of PS for small enough z (PRB 76, ) ? Focus here (no numerical studies). Attractive U conceivable with cold atoms

Charge versus Pair correlations with Using periodic boundary conditions, they should depend only on r, but the DMRG truncation breaks translational invariance; the std dev obtained by varying j is a measure of the error on the correlation functions (grows with L). From bosonization:

for models with SU(2) spin symmetry and gapless spin excitations for models with nonvanishing spin gap for models with SU(2) spin symmetry When the charge gap is closed, as in this case, pair or singlet superconducting (SS) correlations dominate over the charge ones when (U SS = 0 in the Hubbard model) We compute through the charge structure factor Sandvik, Balents & Campbell, PRL 92, , (2004)

In general these correlation functions turn out to be weakly dependent on z. Log-log plots... Difficult to decide whether CDW or SS dominate for negative U.

slope

L fixed, linear fit in q Re-entrance into a charge density wave (CDW) phase unexpected ?!

The error on induced by the DMRG seems to grow with z. An alternative (more rigorous) way is to perform a finite-size scaling. Using the minimum possible momentum with PBC Not-too-large sizes can be used in order to avoid propagation of the DMRG error. red: SS blue: CDW Extrapolations Shrinking of SS phase for z ≥ 0.5 small slope

“Spin Gap” ? Depending on the physical conditions of the experiment, is it necessary or even correct to perform bosonization starting from equally populated noninteracting bands for the two species?

Minima at

(minima at m = 0 for n = 1)

“Spin Gap”

For more informations about our activities “Breached pairing” to BCS ( = 2) transition by varying the effective magnetic field (detuning): Uniform mixture of normal and paired particles [Liu, Wilczek & Zoller, PRA 70, (2004)] Is the mean field (Hartree Fock) reliable in 1D ?