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, 24th-29th october 2008
The model: notations and optical lattices implementation L H AHM = j=1 t c =, n j, =c j, c j 1, hc Un j, n j, 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, 033603 (2004)] j, c j, N N z S= 2 N n = L magnetization or imbalance N n= L density z S m= L
In the asymmetric case SU(2) symmetries (spin and pseudospin) are lost; however, for every value of the asymmetry factor t t z= t t t 0, 0 z 1 the number of particles in each specie is separately conserved. Is it also fixed in experiments? t U t Full particle-hole transformation property still valid E 2 n, m =E n, m UL n 1 We can restrict to n 1 n 1 n
Known results - 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, 106401 (2002)] 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, 153109 (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 [ 2 N 1 sin 2 L 1 1 E A, N = 1 2 sin 2 L 1 increases if one makes k > 1 cuts E A, n A k E A/ k, n A/ k ]
Almost all the recent papers focus on the case n < 1, U > 0 Bosonization and some numerics at Sz = 0, away from n = 1: RG for the spin gap (?) and crossovers at U < 0 [Cazalilla, Ho & Giamarchi, PRL 95, 226402 (2005)] - also dual condition n = 1, Sz 0 and Bose-Bose and BoseFermi mixtures [Mathey, PRB 75, 144510 (2007)] - phase separation occurs when the velocity of the decoupled mode vanishes [Wang, Chen & Gu, PRB 75, 165111 (2007)] - absence of phase separation for small enough z [Gu, Fan & Lin, PRB 76, 125107 (2007)] - spatial configurations and correlations (also 2D) [Farkašovský, PRB 77, 085110 (2008)] Effect of the confining harmonic trap: [Silva-Valencia, Franco & Figueira, JMMM 320, e431 (2008)] also Xianlong et al., PRL 98, 030404 (2007) for the Luther-Emery phase at effective negative U.
DMRG & large-u expansion at n = 1. [Fath, Domański & Lemański, PRB 52, 13910 (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: < 10-6 for L < 30, ~10-5 for L < 50
(qualitative) Phase diagram for n <1 Absence of PS for small enough z (PRB 76, 125107) Depends on the filling (PRB 77, 085110)? Fig. 1 in PRL 95, 226402 Focus here (no numerical studies). Attractive U conceivable with cold atoms
Charge versus Pair correlations C r = n j n j r n j n j r j P r = j r j =c j, c j, 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: K cos 2 k F r C r ~ 2 2 const K K r r const P r ~ K 1/ K r
K =1 for models with SU(2) spin symmetry and gapless spin excitations K =0 for models with nonvanishing spin gap When the charge gap is closed, as in this case, pair or singlet superconducting (SS) correlations dominate over the charge ones when K 1 (USS = 0 in the Hubbard model) We compute K through the charge structure factor S q = r exp iqr [ n j n j r n ] 2 S 0 =0 q S q ~K q 0 Sandvik, Balents & Campbell, PRL 92, 236401, (2004)
u 0 Log-log plots... Difficult to decide whether CDW or SS dominate for negative U. u 0 In general these correlation functions turn out to be weakly dependent on z.
u=u / t 0 slope K 2 k F / =n
u 0
L fixed, linear fit in q Re-entrance into a charge density wave (CDW) phase unexpected?!
The error on K induced by the DMRG seems to grow with z. An alternative (more rigorous) way is to perform a finitesize scaling. Using the minimum possible momentum with PBC 2 q1 = L L K L = S q 1 2 Not-too-large sizes can be used propagation of the DMRG error. Extrapolations u/z 0,2 0,5 0,8-0,5 1,04035 1,01679 0,94000-3,0 1,21780 0,95100 0,46500 Shrinking of SS red: SS phase for z 0.5 blue: CDW in order small slope to avoid
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 tan [ m n/2 ]= 1 z / 1 z sin n cot n u=0
u= 3
u= 3 (minima at m = 0 for n = 1)
Spin Gap u=0 : =±zt cos n 2
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, 033603 (2004)] Is the mean field (Hartree Fock) reliable in 1D? For more informations about our activities cristian.degliesposti@unibo.it http://www.df.unibo.it/fismat/theory/