New Vistas on Quantum Matter Opened by Dipolar Fermions

ISIS Facility, STFC School of
Rutherford Appleton Laboratory Physical Sciences

New Vistas on Quantum Matter
Opened by Dipolar Fermions
Jorge Quintanilla
University of Kent
& Rutherford Appleton Laboratory

Collaborators: Sam T. Carr (Karlsruhe)
Joseph J. Betouras (Loughborough)
Andy J. Schofield (Birmingham)
Masud Haque (MPI Dresden)
Chris Hooley (St. Andrews)
Ben J. Powell (Queensland)

Funding: STFC, SEPNet
2010 Annual Meeting of the UK Cold-atom/Condensed Matter Physics Network, St. Andrews, 9th-10th Sept. 2010

St. Andrews, 9 September 2010 blogs.kent.ac.uk/strongcorrelations

SEPNet


ISIS


STRONG CORRELATIONS


Hubbard Model


Hubbard Model
Proceedings of the Royal Society of London. Series A, Mathematical and
Physical Sciences, Vol. 276, No. 1365 (Nov. 26, 1963), pp. 238-257


Hubbard Model
Proceedings of the Royal Society of London. Series A, Mathematical and
Physical Sciences, Vol. 276, No. 1365 (Nov. 26, 1963), pp. 238-257
“A theory of correlations [...] will
be mainly concerned with
understanding [...] the balance
between band-like and atomic-like
behaviour.”


Strongly correlated quantum
matter
2
p
V r  r'
2
p
 2m  2m  V r  r'
many many pairs of pairs of
particles particles particles particles

kinetic energy interaction energy

 


matter
2
p
V r  r'
2
p
 2m  2m  V r  r'


λ ~ rs ~ Å
>
pz  
Fermi
surface

py px


matter
2
p
V r  r'
2
p
 2m  2m  V r  r'


λ ~ rs ~ Å
>
pz   Wigner
Fermi crystal
surface

Mott
py px
insulator


matter
2
p
V r  r'
2
p
 2m  2m  V r  r'


λ ~ rs ~ Å
>
Fermi liquid theory:
pz   Wigner
Fermi crystal
surface
•Effective mass m*
•Fermi momentum pF
Mott
py px •Landau parameters
insulator
F0, F1, F2, …


matter
2
p
V r  r'
2
p
 2m  2m  V r  r'


λ ~ rs ~ Å
>
pz  STRONGLY  Wigner
Fermi crystal
surface CORRELATED
ELECTRON
SYSTEMS Mott
py px
insulator


Cuprates
La2CuO4
Parameter:
Cu O Cu
O O 1 x  Number of
electrons per
Cu O Cu CuO2 plaquette
( E.g. La2-xSrxCuO4 )

25-30% holes / CuO2  1 electron / CuO2

pz •High-temperature Antiferromagnetic
Fermi superconductivity, Mott insulator
liquid
•stripes,
[Tranquada et al., Nature (1995)]

py px •Non-Fermi liquid,

[Hussey et al.]
•pseudo-gap,…


Two-dimensional quantum wells
Parameter:

B   n electrons / n magnetic field
lines

= hcn el /eB
ν >> 1 ν << 1

Two-dimensional
Fermi liquid 
•quantum Hall effect, Wigner
• fractional quantum Hall crystal
effect,
py px •anisotropic state.
[ M.B. Santos et al.,
[ M.P. Lilly et al., PRL (1999) ] Phys.Rev.Lett. 68, 1188
[ V. Senz et al., PRL (2000); (1992) ]
Y. Y. Proskuryakov et al., PRL (2001) ]


Exact solution of the
Hubbard model
Consider the Hubbard model in D = 1:

Phase diagram known exactly... U/t

Mott insulator
Elliott H. Lieb and F. Y. Wu,
Phys. Rev. Lett. 20, 1445 Luttinger liquid
(1968); 21, 192 (1968).
f
0 1 2
But an exact solution is not available for 1<D<∞.


SOFT QUANTUM MATTER
AND THE POMERANCHUK
INSTABILITY


Soft quantum matter
[ S. A. Kivelson, E. Fradkin, V. J. Emery, Nature 393, 550 (1998) ]
classical
temperature

ideal normal liquid solid
gas liquid crystals state
quantum

STRONGLY
Fermi Fermi CORRELATED
Wigner
gas liquid ELECTRON crystal/Mott
SYSTEMS insulator

correlations


Soft quantum matter
/sci/physics/theory/research/simulation/ ]
[http://www2.warwick.ac.uk/fac
Pictures: Mike Allen

Fermi liquid Wigner
crystal


Soft quantum matter

Fermi liquid Wigner
“stripes”
crystal


Soft quantum matter

Fermi liquid nematic Fermi Wigner
“stripes”
liquid crystal


The Pomeranchuk instability
[ Pomeranchuk (1958) ]

l0 (s-wave)
Stoner Magnetism

l 1 (p-wave)
 standing current
…

 l  2 (d-wave)
l3 nematic
(f-wave)




Instability condition
[ Pomeranchuk (1958) ] n(k)  0
Arbitrary Fermi surface deformation:
n(k)  0
Quasiparticle energy:




Landau parameters: f k,k'  hv F  Fl cosl k   k '  (2D)
l 0
Pomeranchuk Instability condition:

E  0  Fl   2



MEAN FIELD THEORIES OF
THE POMERANCHUK
INSTABILITY


What is the order parameter?
Many ordered states of electrons can be
l q
described in terms of pair formation...

... and condensation.


l q
particle-particle particle-hole

q=0 l=0 s-wave Stoner ferromagnet,
superconductor gas-liquid transition

q≠0 l=0 FFLO spin- and charge-
state density waves

q=0 l≠0 unconventional pairing Pomeranchuk
superconductor instability

q≠0 l≠0 FFLO + “d”-density waves
unconventional pairing


l q
particle-particle particle-hole

q=0 l=0 s-wave Stoner ferromagnet,
superconductor gas-liquid transition
The order parameter is *
q≠0 l=0 FFLO spin- and charge-
∫dθp cos(lθp) n(p)
state density waves
l = 1,2,3,...
q=0 l≠0 unconventional pairing Pomeranchuk
* Expression for the case of a 2D continuum
superconductor instability

q≠0 l≠0 FFLO + “d”-density waves
unconventional pairing


Microscopic description
[ J. Quintanilla & A. J. Schofield, Physical Review B 74, 115126 (2006) ]
[ J. Quintanilla, M. Haque & A. J. Schofield, Physical Review B 78, 035131 (2008) ]
1) Microscopic model: free fermions + isotropic interaction

 p 2 
H     V r  r'
2m  all pairs of
many
particles particles

2) Trial ground state:   c
ˆ 
k, 0
 k 0
  variationally:
3) Determine (k)  H   minimum



Instability condition
[ J. Quintanilla & A. J. Schofield, Physical Review B 74, 115126 (2006) ]
[ J. Quintanilla, M. Haque & A. J. Schofield, Physical Review B 78, 035131 (2008) ]

4 v F
E  0  Vl kF ,kF   (2D)
kF

 



Topological Fermi surface
transitions
[ J. Quintanilla y A. J. Schofield, Physical Review B 74, 115126 (2006) ]
Same recipe: interactions with sharp length scale r0 > rs : ~
V(r) V(r)

r r
r0 r0
g/r0ε0

kFr0 kFr0


POMERANCHUK INSTABILITY
AND DECONFINEMENT


Pomeranchuk on a lattice
[ JQ, C. Hooley, B.J. Powell, A.J. Schofield & M. Haque, Physica B, 403, 1279-1281 (2008). ]

Theory can be generalised to crystal lattices:
t2
u0 u2
t1 t3 … +
u1 u2 …

Interactions beyond on-site ⇒ band-structure renormalisation:

t1 , t2 , t3 ,… → t1* , t2* , t3* ,…


Pomeranchuk on a lattice
[ JQ, C. Hooley, B.J. Powell, A.J. Schofield & M. Haque, Physica B, 403, 1279-1281 (2008). ]
Example: square lattice with t1 ≠0 , u1 ≠0 (u0 won’t do!)

Empty Half-filled
band band Band
V1   filling
large
V1

 Order parameter:
t * /V1
c 1c j
j
small
V1





Confinement
C.f. the “confinement hypothesis”

1D 2D
t’ / t
0 ( t’ / t )crit

[ David G. Clarke, S. P. Strong, and P. W. Anderson, Phys.
t’ Rev. Lett. 72, 3218 (1994) ]

t


Confinement
C.f. the “confinement hypothesis”

1D 2D
t’ / t
0 ( t’ / t )crit

[ David G. Clarke, S. P. Strong, and P. W. Anderson, Phys.
t’ Rev. Lett. 72, 3218 (1994) ]

t

Latest evidence:
functional RG ( t → 0 limit )
[Sascha Ledowski and Peter Kopietz (2007) ]


1D → 2D
[ J. Quintanilla, S.T. Carr, J.J. Betouras, PRA 79, 031601(R) (2009) ]

What about the opposite: can interactions restore 2D behaviour?

1D 2D
V
0 Vcrit
V
t
Model:


MAKING SOFT QUANTUM
MATTER


Soft quantum matter
with dipolar fermions
Use dipolar fermions (e.g. 40K87Rb molecules* or 161/163Dy**).
* K.-K. Ni et al., Science 322, 231 (2008).
** M. Lu, S. H. Youn, & B. L. Lev, PRL 104, 063001 (2010).

Applied field polarises the fermions.

Load onto quasi-1D optical lattice.

Align chains at the “magic
angle” to the applied field.

J. Quintanilla, S.T. Carr, J.J. Betouras, PRA(R) 79, 031601 (2009)


Soft quantum matter

By tuning the ratio of the lattice constants, a/b, can
make in-plane interaction strongly anisotropic:



Soft quantum matter

By tuning the ratio of the lattice constants, a/b, can
make in-plane interaction strongly anisotropic:

V(k) 2Vcos(ky)



Phase diagram, large a >> b




meta-nematic transition




crystallisation



Phase diagram, a ~ b
crystallisation
  ,   



stripes
 0,   




A controlled realisation of the
soft quantum matter scenario



A controlled realisation of the
soft quantum matter scenario

Weak-
coupling



Nature of the meta-nematic
transition

S.T. Carr, J. Quintanilla & J.J. Betouras, PRB 82, 045110 (2010)


transition

Vol. 11, pp. 1130-1135 (1960)



transition
The “2+½-order” Lifshitz
transition is the quantum
critical endpoint of the
1st-order meta-nematic
transition.



transition
The “2+½-order” Lifshitz
transition is the quantum
critical endpoint of the
1st-order meta-nematic
transition.

It is a non-analytic
transition (in the sense
of BCS theory):

C.f. Y. Yamaji, T. Misawa & M.
Imada, JPSJ 75, 094719
(2006) (t-t’ Hubbard model).



Finite temperature



Effect of the trapping potential



CONCLUSIONS
•Soft quantum matter can be realised, in a controlled
way (both theoretically and experimentally), using
ultra-cold dipolar fermions in a suitable optical lattice.

•This should enable us to establish the extent to
which soft quantum matter can be a useful framework
for understanding strongly-correlated materials.

•As always, we learn more than we expected as we
go along.


THANKS!

New Vistas on Quantum Matter Opened by Dipolar Fermions

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