USiRu

Hidden Order in URu2Si2: can we now solve this riddle ?

Gabriel Kotliar

Work in collaboration with Kristjan Haule

K. Haule and G. Kotliar EPL 89 57006(2010)K. Haule and G. Kotliar Nat Phys 5:637 641(2009)‐Recent work with A. Toth

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At the Informal Seminaire Physique Quantique Hors Equilibrium Paris Tuesday Sept 21 (2010)

URu2Si2: a typical problem in the theory of correlated electron materials

• A non-historical review of some important experimental facts about URu2Si2.

• URu2Si2 a good test of the LDA+ DMFT strategy. New insights into a very old problem.

• Comparison with some experiments• Revisiting experiments in dilute systems.• Outlook and Conclusions : key open questions and some more general perspectives on strongly correlated materials.

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Dynamical Mean Field Theory. Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992).

†

0 0 0

( )[ ( ' ] ( '))o o o oc c U n nb b b

s st m tt

t t ­ ¯

¶+ D-

¶- +òò ò

,ij i j i

i j i

J S S h S- -å å eMF offhH S=-† †

, ,

( )( )ij ij i j j i i ii j i

t c c c c U n n

*

( )V Va a

a a

ww e

D =-å

† † † † †Anderson Imp 0 0 0 0 0 0 0

, , ,

( +c.c). H c A A A c c UcV c c c

A(w)

10

( )wD ®

latt ( ,1

G [ ]( ) [( ) ]

)[ ]n imp

nn

ik ii

ktw m

ww+ + - S

DD

=

latt( ) G ([ [)] ] ,imp n nk

G i i kw wD D=å 8

[ ]ijij

jm mJth hb= +å1

( )[ ]( )

( )[ ]imp n

n n ni i iG iw

w w wD +DD

- S -B

atomic levels

Quantifying the degree oflocalization/delocalization

( ),

( ),

( )[ ]n

imp n

A

G i

i

w

w

wS D

®

Impurity Solver

Machine for summing all local diagrams in PT in U to all orders.

latt ( ,1

G [ ]( ) [( ) ]

)[ ]n imp

nn

ik ii

ktw m

ww+ + - S

DD

=

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DMFT picture. Atom in a medium obeying a self consistency condition, Impurity Model. Simplified reference frame for describing correlated solids New concepts, more precise language to quantify the degree of itineracy.Generalizations to cluster and to realistic modelling of materialsMany technical advances over the past decade.

Dynamical Mean Field Picture A. Georges and G. Kotliar PRB 45, 6479 (1992).

1( , )

( ) ( )G k i

i t k i

| 0 ,| , | , | | ... JLSJM g> ­> ¯> ­ ¯> >®, ,

,

[ ] [ ]( )

[ ] [ ]spd sps spd f

f spd ff

H k H kt k

H k H k

æ ö÷ç ÷ç ÷ç ÷çè ø®

Determine energy and and S self consistently from extremizing a functional : the spectral density functional . Chitra and Kotliar (2001) . Savrasov and Kotliar (2001) Full self consistent implementation . Review: Kotliar et.al. RMP (2006)

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Spectra=- Im G(k,w)

LDA+DMFT. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). Lichtenstein and Katsnelson (1998) LDA++

0 0

0 ff Edc

æ ö÷ç ÷S ç ÷ç ÷ç S -è ø®

,[ ] [ , ]dft lda dmf loct G Ur r+G ¾¾®G

abcdU U®

'. ' * 2

'

1[ ] [ / 2 [ ]]R R

ik R RKS

R R

H k e VNs

ab a bc r c-

-

= < - Ñ + >å

Main DMFT Concepts in electronic structure.

Valence Histograms. Describes the

Probability of finding the correlated site in the solid in a given atomic state J. H. Shim, K. Haule, and G. Kotliar, Nature London 446, 513 (2007).

*

( )a b

ab V Va a

a a

ww e

D =-å

Weiss Weiss field, collective hybridization function, quantifies the degree of localization

Functionals of density and spectra, total energies: spectral density functional.

1( , )

( )k

G k ii H i

Local Self Energies and Correlated BandsOrbitally Resolved Spectral FunctionsTransfer of spectral weight.

Review: Realistic DMFT. Rev. Mod. Phys. G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti Rev. Mod. Phys. 78,865 (2006).

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Tunnelling: Orbitally resolved DOS High temperature. Fano-shapes first observed by S. Davis group

spd DOS small changes

only f DOS is gapped [no Kondo peak!!]

Kondo effect arrested by the splitting of thetwo singlets (which is the consequence of the bare small crystal field and the hexadecapolar order ).

Single particle gap~7 mev

Just like T0, it should decrease with increasing magnetic field. [ prediction]

Notice BCS-like coherence peaks in f DOS when hidden order gap forms.

K. Haule and G. Kotliar Nat Phys 5:637 641(2009)‐14

URu2Si2:­DMFT­allows­two­broken­translational­symmetry­states­at­low­T

Moment free phase:

Large moment phase:

tetragonal symmetry broken->these terms nonzero

Density matrix for U 5f state the J=5/2 subspace

J=5/2

J=5/2

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f2, “Kondo “ limit, Hunds. S=1, L=5, J=4. Crystal fields two low lying singlets

Therefore there are two singlets relevant at low energies but they are not non Kramer doublets. Conspiracy between cubic crystal field splittings and tetragonal splittings bring these two states close. This is why only URu2Si2 is different from thousands of U based heavy fermions. Does not arise in one particle crystal field scheme. It is not forced by symmetry.

.

URu2Si2 Valence histogram.

•Under reflections x -x or y -y (x+iy)4 (x-iy)4

•[0> - [0> (odd ) and [1> [1> (even)

J=4

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Order parameter:

Different orientation gives different phases: “adiabatic continuity” explained!

In the atomic limit:

DMFT­order­parameter.­Approximate­X-Y­symmetry

Does not break the time reversal, nor C4 symmetry. It breaks inversion symmetry.

Moment only in z-direction!

X01 =[0><1]

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XY-Ising

crystal field: z direction

Magnetic moment: y-direction

Hexadecapole: x-direction

A toy model

The­two­broken­symmetry­states

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Mean field Exp. by E. Hassinger et.al. PRL 77, 115117 (2008)

HO­&­AFM­in­magnetic­field

Only four fitting parameters: Jeff1 , Jeff

2

determined by exp. transition temperature,and pressure dependence .

Notice that T0decreases with Increasing magnetic field but mangetic field stabilizes hidden order.

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Key­experiment:­Neutron­scattering

The low energy resonance

A.Villaume, F. Bourdarot, E. Hassinger, S. Raymond, V. Taufour, D. Aoki, and J. Flouquet,PRB 78, 012504 (2008)

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hexadecapole

Goldstone mode

Symmetry is approximate“Pseudo-Goldstone” mode

Fluctuation of m - finite mass

The exchange constants J are slightly different in the two phases (~6%)

AFM moment AFM

moment

“Pseudo Goldstone” mode

Interpretation­of­Neutron­scattering­experiments

K. Haule and G. Kotliar EPL 89 57006(2010)

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Contrast this with the tunnelling gap, or the optical gap or the gap in the neutron scattering at (1.4,0,0) which decreases with incresing magnetic field.

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Fermi­surface­nesting,­Reconstruction­below­Tc

2 incommensurate peaks (0.6,0,0), (1.4,0,0)

Nesting 0.6a* and 1.4a*

T>T0 T<T0

Wiebe et.al. 2008Fermi surface reconstruction 23

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Visualizing the Formation of the Kondo Lattice and the Hidden Order inURu2Si2 Pegor Aynajian, Eduardo H. da Silva Neto, Colin V. Parker, Yingkai Huang, Abhay Pasupathy, John Mydosh, Ali Yazdani

arXiv:1003.5259 [pdf]

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THE STATE UNIVERSITY OF NEW JERSEY

RUTGERS

Functional formulation. Chitra and Kotliar Phys. Rev. B 63, 115110 (2001) Ambladah et. alInt. Jour Mod. Phys. B 13, 535 (1999) .

1 †1( ) ( , ') ( ') ( ) ( ) ( )

2Cx V x x x i x x xff f y y-+ +òò ò

† ' ( ') ( ')locG R Rr r y r y r=- < > ( ' ') ( ) ( ' ') ( )R R R R Wf r f r f r f r< >- < >< >=

Ir>=|R, r>

[ , ] [ , , 0, 0]EDMFT loc loc nonloc nonlocG W G W G W

1 1 1 10

1 1[ , ] [ ] [ ] [ , ]

2 2 C hartreeG W TrLnG Tr G G G TrLnW Tr V W W E G W

Double loop in Gloc and Wloc

•Full implementation in the context of a a one orbital lattice model. P Sun and G. Kotliar Phys. Rev. B 66, 85120 (2002). After finishing the loop one can treat the graphs involving Gnonloc Wnonloc in perturbation theory. . Phys. Rev. Lett. 92, 196402 (2004) Limiting case (perturbation theory as solvers) Zeyn and Antropov. N. E. Zein and V. P. Antropov, J. Appl. Phys. 89, 7314 (2001), Phys. Rev. Lett. 89, 126402 (2002)

• Application to semiconductors N. Zeyn S. Savrasov and G. Kotliar PRL 96, 226403, 2006

EDMFT loop Chitra and Kotliar Phys. Rev. B 63, 115110 (2001). G. Kotliar and S. Savrasov in New Theoretical Approaches to Strongly Correlated Systems, A. M. Tsvelik Ed. 2001 Kluwer Academic Publishers. 259-301 . cond-mat/0208241 S. Y. Savrasov, G. Kotliar, Phys. Rev. B 69, 245101 (2004)

1

1

( ) ,

1

( ) ,

lock

loc loc

locq

C loc loc

GH k G W

G

WV q G W

W

P

, ,

,

[ ] [ ]( )

[ ] [ ]abcd

0 0

0 Uloc spd sps loc spd f

locloc f spd loc ff

W WW i

W Ww

æ ö÷ç ÷=ç ÷ç ÷çè ø

é ùê ú®ê úë û

Determine energy and and S self consistently from extremizing a functional : the spectral density functional . Chitra and Kotliar (2001) . R. Chitra and G. Kotliar, Phys. Rev. B 63, 115110(2001). Savrasov and Kotliar (2001) Full self consistent implementation . Review: Kotliar et.al. RMP (2006)

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LDA+DMFT. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997).

, ,

,

0 0 [ ] [ ]

0 [ ] [ ]spd sps spd f

ff f spd ff

Vxc k Vxc k

Edc Vxc k Vxc k

æ ö æ ö÷ ÷ç ç÷ ÷S +ç ç÷ ÷ç ç÷ ÷ç çS -è ø è ø®

,[ ] [ , ]dft lda dmf loct G Ur r+G ¾¾®G

U is parametrized in terms of Slater integrals F0 F2 F4 ….

rdr WPW

vvPW1

1

]1[

,]1[

1( ) [1 ( )]

( ) ( )rest rest

rest

W vP v

W U

Effective interaction among electrons. Constrained RPA (cRPA) Ferdi Ariasetiwan ,A, M Imada, A Georges, G Kotliar, S Biermann, AI Lichtenstein, PRB 70, 195104 (2004)

energy-dependent effective interaction between the 3d electrons

Can be used to extract a screened U

rW

loc restP P P Identity:

J.H.Shim,­KHaule,­G.Kotliar,Science­318,­1615­(2007)

Protracted screening and multiple hybridization Gaps in Ce115’s.

K. Burch et.al.

Quasiparticle multiplets in Plutonium and its Compounds.

J.H.Shim,­KHaule,­G.Kotliar,­Nature­446,­513­(2007).

Hidden Order in URu2Si2, Kondo effect and hexadecapole order.

KHaule­,andG.­Kotliar,­Nature­Physics­5,­796­-­799­(2009).

Strong Correlations without high energy satellites in Ba Fe2As2

Origin of the particle-hole assymetry between LaSrCuO4 and NdCuO4

C. Weber K Haule and G. Kotliar submitted to Nature Physics

A. Kutepov K. Haule S. Savrasov and G. Kotliar to be submitted to PRL

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