L.N. Bulaevskii et al- Electronic Orbital Currents and Polarization in Mott Insulators

8/3/2019 L.N. Bulaevskii et al- Electronic Orbital Currents and Polarization in Mott Insulators

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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


3/18

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 ?

L.N. Bulaevskii et al- Electronic Orbital Currents and Polarization in Mott Insulators

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