l.n. bulaevskii et al- electronic orbital currents and polarization in mott insulators
TRANSCRIPT
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Electronic Orbital Currents and Polarizationin Mott Insulators
L.N. Bulaevskii, C.D. Batista, LANLM. Mostovoy, Groningen Un.
D. Khomskii, Koln Un.
Introduction: why electrical properties of Mott
insulators differ from those of band insulators.
Magnetic states in the Hubbard model.
Orbital currents.
Electronic polarization.
Low frequency dynamic properties.Conclusions.
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Difference between Mott and band insulators
Only in the limit electrons are localized on sites.
At electrons can hop between sites.
Orbital current
U
2( ) ( 1) ,
2ij i j j i i
ij i
U H t c c c c n
+ += + + 1.in =
2
1 2
4( 1/ 4).S
t H S S
U=
r r
/ 0t U
1 2
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Definition of magnetic states for Hubbard Hamiltonian
ground state spin-degenerate, S-subspace.
Excited (polar) states are separated by the Hubbard gap,
increases, degeneracy is adiabatically lifted magnetic states
Magnetic state energies
Any other electron operator
are expressed in terms of spin operators
U
2( ) ( 1) ,2
ij i j j i i
ij i
U H t c c c c n
+ += + + 1.in =
ijt
2N
.U
( )
0.
S te
=
0,
* *0 0| | | |S S H e He = .S SS H Pe He P=
*| |O .
S S
SO Pe Oe P=
,S SH O .i i iS c c
+=
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Spin Hamiltonian
For square lattice
2
,
,
4( 1/ 4) ( 1/ 4)
( 1/ 4)( 1/ 4),
S i j ik i k k
ij ik
ij kl i j k l
ij kl
t H S S J S S
U
J S S S S
= +
+
r r r r
r r r r
Only even number of spin operators.
Some excited magnetic states (collective modes) lay below
the Hubbard gap resulting in low-T thermodynamic and low-frequency dynamic properties of Mott insulators.
il
j k
4 3
,, / .ij ij kl J J t U
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Spin current operator and scalar spin chirality
Current operator for Hubbard Hamiltonian on bond ij:
12 23 31
,12 1 2 32
24
(3) [ ] .
ij
Sij
r e t t t
I S S Sr U=
rr r rr
h
Projected current operator: odd # of spin operators, scalar in
spin space. For smallest loop, triangle,
Current via bond 23
On bipartite nn lattice is absent.
( ).ij ijij i j j iij
iet r I c c c cr
+ +=
rr
h
1
2
3
42
1 3
,23 ,23 ,23(1) (4).S S S I I I = +SI
Bogolyubov, 1949.
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Orbital currents in the spin ordered ground state
Necessary condition for orbital currents is nonzero averagechirality
It may be inherent to spin ordering or induced by magnetic field.
0iS r
12,3 1 2 3[ ] ,S S S =
r r r
Triangles with chirality
On tetrahedron chirality maybe nonzero but orbitalcurrents absent.
, 0.ij k
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Chirality in the ground state without magnetic ordering
Geometrically frustrated 2d system Mermin-Wagner
theorem
State with maximum entropy may be with broken discretesymmetry
Example: model on kagome lattice:
12,3 1 2 3[ ] 0,S S S = r r r
0.iS =r
0.iS =r
12,30.
1 2J J
1 2.
S i j i
ij ij
H J S S J S S= + kr r r r
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Ordering in model on kagome lattice at T=0
Phase diagram for classical
order parameter
3 3q =
Q=0 Neel
Antiferro a)
Neel Antiferro b)
1
Cuboc Antiferro
Ferro
J
2J
1 2J J
For S=1/2 for low T cuboc is
chiral with orbital currents b)
and without spin ordering.
Monte Carlo results by Domenge et al., 2007.
0T b)
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Boundary and persistent current
1
2
3
4 6
5
1 2 3
4 5 6
const =
Boundary current in
gaped 2d insulator
x yz
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Spin dependent electronic polarization
Charge operator on site i:
Projected charge operator
Polarization on triangle
Charge on site i is sum over triangles at site i.
.i i iQ e c c
+=
,,
S S
S i in Pe n e P
=
12 23 31
,1 1 2 3 2 32
8(2, 3) 1 [ ( ) 2 ].St t tn S S S S S
U= + r r r r r
1
2
3
123 ,
1,2,3
,S i ii
P e n r =
= r
r,
3.S ii
n =
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Electronic polarization on triangle
Magnitostriction results in similar dependenceof polarization on spins.
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Charges on kagome lattice
Current ordering
induced by perp.magnetic field
Charge ordering for
spins 1/2 in magneticfield at /3.zS S=
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Low frequency dynamic properties: circular dichroism
Responses to ac electric and ac magnetic field arecomparable for 100 K:
Compare and
Nonzero orbital moment
Rotation of electric field polarization in chiral groundstate.
0 , ,
0 2 2
0
0 | | | | 08
( ) ,
n S i S k
ik ik
n n
P n n P
V i
= + +0 0
,n n
E E = h .S n H n E n=
J
3 38 / ,P eat U .Bg S
0 | | , 0 | | cos , sin 0x yP n P n
/z
L
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Low frequency dynamic properties: negativerefraction index
Responses toac electric and ac magnetic field arecomparable for 100 K:
Spin-orbital coupling may lead to common poles inand
Negative refraction index if dissipation is weak.
0 , ,
0 2 2
0
0 | | | | 08( ) ,
n S i S k
ik ik
n n
P n n P
V i
= +
+
J
( )ik
( )ik
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Low frequency dynamic properties of isolatedtriangles
Trinuclear spin complex V15
Energy levels of the HeisenbergHamiltonian:
groundstate:quartetS=1/2, (pseudospin),
quartetS=3/2 separatedby3J/2.
Polarization and orbital current in low-energy subspace:
,,S x xP CT= , ,S y yP CT=
6 15 6 42 2 2[ ( )] 8 .
IVK V As O H O H O
1,z
T = =
( / ) ,S za U I T =h
3.12 3 ( / )C ea t U =
1/2,z
S =
1 2 2 3 1 3( 3 / 4) :SH J S S S S S S= + +
r r r r r r
S=1/2
S=3/2
E
V
V
V
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Isolated triangle: accounting for DM interaction
DM coupling:
For V15
Splits lowest quartet into 2 doublets
and separated by energy
Ac electric field induces transitions between
Ac magnetic field induces transitions between
. DM ij i jij
H D S S= r r
. DM z z z H D L S
,+
,+ .zD =1. =
1/2.zS =
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Isolated triangle with DM interaction
.zD =
Near resonance strong rotation of electric field polarization.Below resonance at B=0 both and
are negative if dissipation is low enough.
( )ik
( )i k
( )ik
Frequnces of EPR and microwave absorption differ in field.
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Conclusions
Dissipationless orbital microscopic currents,macroscopic boundary currents,macroscopic current around the ring like insuperconductor.
Spin-dependent electronic polarization andmultiferroic behavior.
Chirality results in circular dichroism, option toprobe chirality when magnetic ordering is absent.
Crystal with negative refraction index ?