() U ()qpt.physics.harvard.edu/talks/mott.pdf · Mott insulator in a strong electric field-S....
Transcript of () U ()qpt.physics.harvard.edu/talks/mott.pdf · Mott insulator in a strong electric field-S....
( ) ( )† † †1 ; ; Vary 2ij i j j i i i i i i
i j i
UH w c c c c n n n c c U wα α α α α α<
= − + + − =� �
A. S=1/2 fermions (Si:P , 2DEG)Metal-insulator transitionEvolution of magnetism across transition.Phil. Trans. Roy. Soc. A 356, 173 (1998) (cond-mat/9705074)Pramana 58, 285 (2002) (cond-mat/0109309)
B. S=0 bosons (ultracold atoms in an optical lattice)Superfluid-insulator transitionMott insulator in a strong electric field -S. Sachdev, K. Sengupta, S.M. Girvin, cond-mat/0205169
Ea U≈
Talk online at http://pantheon.yale.edu/~subir
ReznikovSarachik
Transport in coupled quantum dots150000 dots
A. S=1/2 fermionsU w small ; charge transport “metallic”
Magnetic properties of a single impurity
t
w
LT χ
LT χ
Low temperature magnetism dominated by such impurities
M. Milovanovic, S. Sachdev, R.N. Bhatt, Phys. Rev. Lett. 63, 82 (1989).R.N. Bhatt and D.S. Fisher, Phys. Rev. Lett. 68, 3072 (1992).
t/w
U/w
A. S=1/2 fermionsU w large ; charge transport “insulating”
R.N. Bhatt and P.A. Lee, Phys. Rev. Lett. 48, 344 (1982).B. Bernu, L. Candido, and D.M. Ceperley, Phys. Rev. Lett. 86, 870 (2002).G. Misguich, B. Bernu, C. Lhuillier, and C. Waldtmann, Phys. Rev. Lett. 81, 1098 (1998).
( ); ~ exp /ij i j ij i ji j
H J S S J a<
= ⋅ − −�� �
r r
Strong disorderRandom singlet phase
( )12
= ↑↓ − ↓↑Spins pair up in singlets
( ) ~ ~ ; 0.8P J J Tα αχ α− −� ≈
Weak disorderSpin gap
( )~ exp Tχ ∆−
Spin glass order
0iS ≠�
Higher density of moments + longer range of exchange interaction induces spin-glass order. Also suggested by strong-coupling flow of triplet interaction amplitude in Finkelstein’s (Z. Phys. B 56, 189 (1984)) renormalized weak-disorder expansion.
S. Sachdev, Phil. Trans. Roy. Soc. 356A, 173 (1998) (cond-mat/9705074); S. Sachdev, Pramana. 58, 285 (2002) (cond-mat/0109309).
U w
“Metallic”Local moments
“Insulating”Random singlets
/spin gap
Metal-insulator transition
Theory for spin glass transition in metalS. Sachdev, N. Read, R. Oppermann, Phys. Rev B 52, 10286 (1995).A.M. Sengupta and A. Georges, Phys. Rev B 52, 10295 (1995).
A. S=1/2 fermions
Glassy behavior observed by S. Bogdanovich and D. Popovic, cond-mat/0106545.
Effect of a parallel magnetic field on spin-glass state.
B
M
Singular behavior at a critical field at T=0Possibly related to observations of
S. A. Vitkalov, H. Zheng, K. M. Mertes, M. P. Sarachik, and T. M. Klapwijk, Phys. Rev. Lett. 87, 086401 (2001).
S. Sachdev, Pramana. 58, 285 (2002) (cond-mat/0109309).N. Read, S. Sachdev, and J. Ye, Phys. Rev. B 52, 384 (1995);
Spin glass order in plane orthogonal to B�
No order
Bσ
Mσ /d z B BM M TT
σσ ϕ −� �− = � �
� �
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
Related earlier work by C. Orzel, A.K. Tuchman, M. L. Fenselau, M. Yasuda, and M. A. Kasevich, Science 291, 2386 (2001).
Superfluid-insulator transition of 87Rb atoms in a magnetic trap and an optical lattice potential
B. S=0 bosons
Detection method
Trap is released and atoms expand to a distance far larger than original trap dimension
( ) ( ) ( )2
, exp ,0 exp ,02 2
where with = the expansion distance, and position within trap
m mmT i i iT T T
ψ ψ ψ� �� � ⋅
= ≈ +� �� �� � � �
=
0 0� � �
20 0
0 0
R R rRR
R = R + r, R r
In tight-binding model of lattice bosons bi ,
detection probability ( )( )† 0
,
exp with i j i ji j
mb b i
T∝ ⋅ − =�
�
Rq r r q
Measurement of momentum distribution function
Schematic three-dimensional interference pattern with measured absorption images taken along two orthogonal directions. The absorption images were obtained after ballistic
expansion from a lattice with a potential depth of V0 = 10 Er and a time of flight of 15 ms.
Superfluid state
V0=10 Ere coil ττττperturb = 2 m s V0= 13 Ere coil ττττperturb = 4 m s
V0= 16 Ere coil ττττperturb = 9 m s V0= 20 Ere coil ττττperturb = 20 m s
What is the quantum state
here ?
( ) ( )† †
†
12
i j j i i i i iij i i
i i i
UH w b b b b n n n
n b b
= − + + − − ⋅
=
� � �E r
, ,U E w E U− �
Describe spectrum in subspace of states resonantly coupled to the Mott insulator
Effective Hamiltonian for a quasiparticle in one dimension (similar for a quasihole):
( )† † †eff 1 13 j j j j j j
jH w b b b b Ejb b+ +
� �= − + +� ��
( ) ( )Exact eigenvalues ;Exact eigenvectors 6 /
m
m j m
Em mj J w E
εψ −
= = −∞ ∞=
�
All charged excitations are strongly localized in the plane perpendicular electric field.Wavefunction is periodic in time, with period h/E (Bloch oscillations)
Quasiparticles and quasiholes are not accelerated out to infinity
dk Edt
=
Semiclassical picture
k
Free particle is accelerated out to infinityIn a weak periodic potential, escape to infinity occurs via Zener tunneling across band gaps
Experimental situation: Strong periodic potential in which there is negligible Zener tunneling, and the
particle undergoes Bloch oscillations
Creating dipoles on nearest neighbor links creates a state with relative energy U-2E ; such states are not
part of the resonant manifold
Nearest-neighbor dipole
Important neutral excitations (in one dimension)
Nearest-neighbor dipoles
Dipoles can appear resonantly on non-nearest-neighbor links.Within resonant manifold, dipoles have infinite on-link
and nearest-link repulsion
Nearest neighbor dipole
A non-dipole state
State has energy 3(U-E) but is connected to resonant state by a matrix element smaller than w2/U
State is not part of resonant manifold
Hamiltonian for resonant dipole states (in one dimension)
( )
†
† †
† † †1 1
Creates dipole on link
6 ( )
Constraints: 1 ; 0
d
d
H w d d U E d d
d d d d d d+ +
�
= − + + −
≤ =
� �
�
� � � �
� �
� � � � � �
�
Note: there is no explicit dipole hopping term.
However, dipole hopping is generated by the interplay of terms in Hd and the constraints.
Determine phase diagram of Hd as a function of (U-E)/w
Weak electric fields: (U-E) w
Ground state is dipole vacuum (Mott insulator)
�
† 0d�First excited levels: single dipole states
0
Effective hopping between dipole states † 0d�
† † 0md d�
† 0md0
If both processes are permitted, they exactly cancel each other.The top processes is blocked when are nearest neighbors
2
A nearest-neighbor dipole hopping term ~ is generatedwU E−
�
, m�
w w
w w
Strong electric fields: (E-U) w�
Ground state has maximal dipole number.
Two-fold degeneracy associated with Ising density wave order:† † † † † † † † † † † †1 3 5 7 9 11 2 4 6 8 10 120 0d d d d d d or d d d d d d� � � �
Ising quantum critical point at E-U=1.08 w
-1.90 -1.88 -1.86 -1.84 -1.82 -1.800.200
0.204
0.208
0.212
0.216
0.220
πS /N3/4
λ
N=8 N=10 N=12 N=14 N=16
Equal-time structure factor for Ising order
parameter
Resonant states in higher dimensionsResonant states in higher dimensions
Quasiparticles
Quasiholes
Dipole states in one dimension
Quasiparticles and quasiholes can move
resonantly in the transverse directions in higher
dimensions.
Constraint: number of quasiparticles in any column = number of quasiholes in column to its left.
Hamiltonian for resonant states in higher dimensions
( )
( )( )
† †1, , 1, ,
,
† †,
† † † †, , , , , , , ,
, , ,,
† †, , , ,
,
6
( ) 2
2 3 H.c.
1 ; 1 ; 0
ph n n n nn
n n n nn
n m
n n n n n n n
n mnm
n
H w p h p h
U E p p h h
w h h p
p p h h p p h h
p
+ += − +
≤ ≤ =
−+ +
− + +
�
�
�
� � � �
�
� � � �
�
� �
� � � �
�
�
� �
�
� �
†,
†,
Creates quasiparticle in column and transverse position
Creates quasihole in column and transverse position n
n
p n
h n
�
�
�
�
�
�
Terms as in one dimension
Transverse hopping
Constraints
New possibility: superfluidity in transverse direction (a smectic)
Ising density wave order
( ) /U E w= −
( ) /U E w= −
Transverse superfluidity
Possible phase diagrams in higher dimensions
Implications for experiments
•Observed resonant response is due to gapless spectrum near quantum critical point(s).
•Transverse superfluidity (smectic order) can be detected by looking for “Bragg lines” in momentum distribution function---bosons are phase coherent in the transverse direction.
•Present experiments are insensitive to Ising density wave order. Future experiments could introduce a phase-locked subharmonicstanding wave at half the wave vector of the optical lattice---this would couple linearly to the Ising order parameter. The AC stark shift of the atomic hyperfine levels would differ between adja-cent sites. The relative strengths of the split hyperfine absorption lines would then be a measure of the Ising order parameter.