Phase separation and pair correlation functions Phase separation and pair correlation functions in the Hubbard model with in the Hubbard model with density-dependent hoppingdensity-dependent hopping

Cristian Degli Esposti Boschi

CNISM and Dipartimento di Fisica, Università di Bologna

cristian.degliesposti@unibo.it

www.df.unibo.it/fismat/theory

with Arianna Montorsi and Alberto Anfossi, Politecnico di Torino

XV Convegno Nazionale di Fisica Statistica e dei Sistemi ComplessiParma, 21-23 giugno 2010

Application of the Hubbard model to materials with extended orbitals: the charge localised in the bonds affects the screening of the effective potential between the valence electrons, the extension of the Wannier orbitals and the hopping between them. Relevant for hole superconductivity (x' = 0, Hirsch and co-workers, 1989).

Hubbard model with bond charge/correlated hopping

ii jj

Hu=u∑ini ni

H=−t∑⟨ij ⟩ {ci

c j [1−x ni n j x ' ni n j ] }Hu

● More recently, mixtures of two species of cold atoms can be embedded in optical lattices of different effective dimensionality, including 1D. Typically the atoms are bosonic but also Bose-Fermi or even Fermi-Fermi mixtures can be trapped [for instance Jördens et al., Nature 455, 204 (2008); Bloch, Dalibard and Zwerger, RMP 80, 885 (2008)]. See also Barbiero's poster.

● By exploiting the phenomenon of Feshbance resonance, the effective interaction between species can be tuned both on the attractive and on the repulsive side. The effect of different hopping rates (e.g. masses) for the different species can also be studied [arXiv:1001.5226→PRB and refs. therein].

● Unbalanced populations: Enhancement or suppression of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) behaviour depending on x and x' [Wang and Duan, PRA 79, 043612 (2009), U < 0 and x' = 2x].

Symmetries

● SU(2) ⊗ U(1) (rotation in spin space and particle number conservation)

● Particle-hole symmetry is lost, except for x' = 2x

x' = 0 Additional SU(2) symmetry (pseudospin) at different wavenumbers

x = 0 x = 1

[H−u2N−

12 ,P=]=0

P=∑ j j exp−i P⋅R j j=c j c j

[P ,P ]=−2z [P ,z ]=P

z=N−L

2

[H−u2N−

12 ,P=0 ]=0

1D exactly solvable cases:

● x = x'= 0: Integrable; Bethe-ansatz solution [Lieb and Wu]

● x = 1, x' = 0: Integrable [Arrachea and Aligia, PRL 73, 2240 (1994), Schadschneider, PRB 51, 10386 (1995), de Boer, Korepin and Schadschneider, PRL 74, 789 (1995)].

● x = 1, any x': The number of doubly occupied sites is conserved. [Aligia, Arrachea and Gagliano, PRL 51, 13774 (1995); Montorsi, JSTAT, L09001 (2008)].

[ H , Hu ]=0

x' = 0, x ≠0, 1: Nonintegrable. Limit to x < 1 and use a u,x transformation otherwise

H x ,u ,∣2 x−1∣H sign2 x−1x ,u ,u−

p=p

∣1−2 x∣p=x ,u ,

Chemical potential

● For x = 0 bosonisation yields two separated (charge/spin) sine-Gordon models at the Berezinskii-Kosterlitz-Thouless point for . Gapped charge sector (insulator) and gapless spin excitations for u > 0; interchanged role for u < 0. BKT point:

● For x > 0 bosonisation yields

uc , s=0

H SG=12∂2g cos 2=8

ueffective=u8 x cos /2n

1−n x

● Entanglement helps to clarify the nature of the various transitions in the case x = 1. Analytical study of multipartite entanglement in (U,n) space [Anfossi et al., PRL 95, 056402 (2005); PRB 75, 165106 (2007)].

● Anfossi et al., PRB 78, 144519 (2008): Inspection of bi- and multipartite entanglement for electrons and for -pairs, both in momentum and in direct space using various measures (entropy, negativity, Meyer-Wallach's).

Half-filling n=NL=1

Balanced species n=n

(here, 2007)

● Our numerics: Density-matrix renormalisation group (code developed in Bologna by F. Ortolani)

- Spin Density Wave: no spin gapand nonzero charge gap

- Bond-Order Wave: fully gappedwith order parameter

Bosonisation starts to fail at x > ½. [Anfossi, et al., PRB 73, 085113 (2006); Aligia et al., PRL 99, 206401 (2007)].

● Same scenario predicted for x = 0 but for x = 1 the exact solution gives a transition to a phase with gapless charge modes at u

c = 4

c=E L N=L2E L N=L−2−2 E L N=L

4=L /N 2

s=EL Sz=1−EL S

z=0

OBOW=L−1∑ j−1 j ⟨c j

c j1hc ⟩

Incommensurability of one point (with open boundary conditions) and two-points correlation functions, similarly to what happens in the t-t'-U Hubbard model with next-to-nearest neighbours hopping (Japaridze, Noack and Baeriswyl, PRB 76, 115118 (2007))

X = 0.8t U = 1.5t

Singlet superconducting (SS) correlations dominate over the charge ones when .

Charge vs pair correlations

C r =⟨n j n jr ⟩−⟨n j⟩ ⟨n jr ⟩ P r =⟨ j jr ⟩

From bosonisation:

C r ~−K

2 r2constcos 2 k F r

r KK P r ~ const

r K 1 /K

K=1

K=0

for models with SU(2) spin symmetry andgapless spin excitations

for models with nonvanishing spin gap

K1

ICSS phase: Nonzero spin gap, dominant pairing correlations. Anomalous flux quantisation and infinite compressibility .

We compute either by direct fits or through the charge structure factor

K

S q=∑rexp iqr [⟨n j n jr ⟩−⟨n ⟩2

]

S 0=0 S q~K

q

q0

[Sandvik, Balentsand Campbell, PRL 92, 236401,(2004)]

P(r)

C(r)

x = 0.8u = 1.0 K ≃1.3

Peaks at a Fermi momentum 2nd, related to double

occupations and not to the total densities as in the usual Hubbard model.

SS

x = 0.8 u = 0

Away from Half-filling n=NL≠1

with balanced species n=n

Hubbard operators: X =∣ ⟩ ⟨∣

H=H 01H 12H mix

H 01=−t∑⟨ ij ⟩ X i

0 X j0hc

H 12=−t x∑⟨i j ⟩ X i

2 X j 2hc u∑i

X i22

t x=t 1−2 x−x '

H mix=−sx∑⟨ ij ⟩ X i

2 X j0 hc s x=t 1−x

When x = 1 the Hamiltonian splits into two terms acting on two separate Hilbert subspaces with no doubly occupied sites (S

01) and no empty sites (S

12) respectively.

Insensitive to spin degrees of freedom.

So-called Simon-Aligiamodel

x = 1: Grand-canonical construction (Montorsi, 2008)

E gs=min∣ ⟩∈S 01S 12⟨∣H∣ ⟩

∣ ⟩=a∣01N 0' ⟩1−a2∣12N 2

' ⟩

N 0=a2N 0

' N 2=1−a2N 2' N 0N 1N 2=L

N 12N 2=N

Ground states in S01

and S12

● Hole-rich and pair-rich coexisting conducting phases, with different Fermi wavenumbers. Incommensurate phase separation. Insensitive to spin degrees of freedom.● Only the hole-rich state is conducting while the pair-rich becomes an insulator (conventional phase separation)

● Transition to complete insulator

n0'n2

'1

n0'n2

'=1

n0'=n2

'=1

x < 1: Searching for ICSS phase with DMRG

1=n2 ∂

2 e∂ n2

Divergingcompressibility:Phase coexistenceof different fillingsn

l ≤ n ≤ n

h

Phase separation and diagrams at varying x or varying u

Very close to the analyticalresult for x = 1 [Montorsi, JSTAT 2008]

SC-LEL: supeconducting Luther-Emery Liquid (with nonvanishing spin gap)

SC-LEL phase

● the spin gap is opened; the spin of singly occupied sites in the two conducting phases – that by themselves have essentially spinless features - rearranges so to lower the energy and establish a phase coexistence.

● charge correlations decay faster than pair correlations

● Pairs carry no relative momentum. No FFLO behaviour.

Spin gap (u = 1,x = 0.8)

K 1

Near half-filling the width of the peak is related to the difference of Fermi momenta for the low and high density components. Its center marks the inverse of the characteristic size of micro/nanoscopic “islands” in a phase separated state. Microscale phase separation disappears for x < 2/3 (where a competition effect in the Simon-Aligia model takes place; t

x=-s

x).

L=120

S

Outside the SC-LELat large or small totalfilling n the profileis like the one ofspinless fermionswith a cusp at 4k

F=2n

u = 1x = 0.8

u = 1x = 0.8

u = 1x = 0.6

Onset of “superconducting” behaviour

Essentially same intervals (with numerical uncertainty) found for SC-LEL using chemical potential plateauxfor x ≥ 0.7.

Unbalanced species(magnetic field)

nn

mz≡n−n

2

● When a magnetic field is introduced (or the net magnetisatonis nonzero) starting from the SC-LEL phase at x > 2/3, for small intensities the number of double occupations is constant. Then some pairs start to breach at a critical field where the phase separation is extended over macroscopic regions.

● Qualitatively, this behaviour is seen experimentally in novel superconducting iron-based materials called pnictides.

u = 0x = 0.8

Complete phasediagram at finite u in Anfossi et al., PRA 80, 043602 (2009)

[p = 2mz]

u = 0x = 0.8

open b.c.

Densitiesprofiles

Various types of pairing correlators

Singlet

or

Triplet

onadjacentsites

Distance R = 1 and starting site i along the chain

u = 1x = 0.6

open b.c.Nounpaireddownparticles

A cartoon for FFLO behaviour

Pairing like in Bardeen-Cooper-Schrieffer theory but with nonzero relative momentum; can be seen as a peak in pair-pair correlation functions.

FFLO peak

u = 0x = 0.6

As a function of distance Rfor selected starting sites i

Pairing correlations dominateover large distances ( )K 1

On average the FFLO effect disappears by increasing m

z or by

increasing the repulsion u.It is mantained withinthe low-density regions.

u = 0x = 0.6

u = 1x = 0.6

A “superconductor-ferrinsulator” transition in 1D

● At h > hc1

there are no more minority electrons in the high-density domains and the superconducting correlations of the original SC-LEL state are now limited to low-density domains, while the high-density ones support itinerant ferromagnetism. Globally the system behaves as a normal metal with a finite spin gap. The spatial extension of the phase separation pattern becomes macroscopic, at variance with the microscopic character at zero field.

● Further, if the field is pushed beyond a second value hc2, the

fully polarized high-density domains become highly localised and globally the system is an insulator made of ferromagnetic domains alternating with superconducting islands, in which a certain fraction of pairs are breached in order to follow the magnetic field.

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