USiRu

The mystery of hidden order in URuSi2.

Gabriel Kotliar

K. Haule and G. Kotliar EPL 89 57006(2010)K. Haule and G. Kotliar Nature Physics 5:637 641(2009)‐

1

104 statistical mechanics meeting December 19-21 (2010)

An application of statistical mechanical ideas and methods to quantum materials

Strongly Correlated Electron Materials.

U

SiRu

• Do remarkable things •Under intense investigation •Not well described by band theory or by atomic physics. Intermediate strength of correlations -localization-delocalization•new organizing principles and new techniques for their description

Examples:.........……….

BaFe2As2

VO2

2

Realistic implementations of Dynamical Mean Field Theory as a tool for a) thinking about strong correlations, b) for system specific study of materials and c) to start organizing our knowledge of the space of materials

Mean Field Theory Methods in classical stat mech• Zeroth order description•Reference system [ departures from MFT]

3

DMFT views a solid as a collection of atoms in a medium

† †

, ,

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

t c c c c U n n

† † † † †Anderson Imp 0 0 0 0 0 0 0

, , ,

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

4

[ ][ ]

*1

1(

( ) ( ),

,)n n

n nk

i ii t

V

Vi

V

kVa a

aaaa aa

ew e

w m ww w em

-é ùê ú+ - S = ê ú+ - - Sêë û

-- ú

å åA. Georges and G. Kotliar PRB 45,

6479 (1992).

5

• Simple extensions to phases with LRO• Locality: simple extensions to cluster of sites. • Rapid advances in impurity solvers [Montecarlo based ED based

]• Surprisingly accurate for models [ B. Svitunov’s talk ]• Interface with electronic structure [also using stat mech

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

• Impurity model provides us with a simple way to think about materials, local spectral functions, weiss fields, valence histograms, functionals of spectra ……..

REVIEWS: A. Georges W. Krauth G. Kotliar and M. Rozenberg RMP 68,13(1996) G. Kotliar S. Savrasov K. Haule O Parcollet V. Oudovenko and C. Marianetti RMP 78,865(2006)

Hidden Order The CMT dark matter problem.

URu2Si2: T. T. M. Palstra, A. A. Menovsky, J. van den Berg, A. J. Dirkmaat, P. H. Kes, G. J. Nieuwenhuys and J. A. Mydosh Physical Review Letters 55, 2727 (1985)

U

SiRu

Entropy Loss at the transition: 1/5 Log[2]

6

•Similar T0 and TN

•Almost identical thermodynamic quantities ( e.g. jump in Cv) and similar

oscillation frequencies.

“Adiabatic continuity” between HO & AFM phase

E. Hassinger et.al. PRL 77, 115117 (2008)

“Adiabatic “is a misnomer. Need a better term.

7

Basic Questions• What orders ?• Nature of the gap(s) in the order phase. What is their origin.• Nearby phases what is their relation to the HO phase• Degree of correlation/localization [ Fermi surface

reconstructions – itinerant ph excitations ]• What are the basic elementary excitations and how the show up in experiments ?

• What happens when you perturb: response to strain impurities, fields, inhomgeneities

………………………..8

•Lev. P. Gorkov: 1996:

-Three point spin correlators.

• Chandra and Coleman ., Nature’02 - Incommensurate Orbital Antiferromagnetism (ddw)

• Mineev & Zhitomirsky, PRB ’05 - SDW (with tiny moment, moment cancellation)

• Varma & Zhu, PRL’06 - Helical Order, Pomeranchuk instability of the Fermi surface time reversal

breaking)

• Elgazaar, & Oppeneer, Nature Materials’08- DFT: antiferromagnetic order parameter, fluctuations.

• Santini and Amoretti PRL 04

-Quadrupolar ordering.

• Fazekas and Kiss PRB 07

-Octupolar ordering……………………A Balatzky inconmensurate CDW order

Haule and Kotliar : hexadecapolar order.

Some proposals for the hidden order in the literature(disagreement aboutbasic aspects, Kondo physics valence etc)

9

Itinerant or localized ?URu2Si2 (CTQMC) Valence Histogram

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

•[0> - [0> (odd ) and [1> [1> (even) 10

URu2Si2: DMFT two broken translational symmetry (A-B sublatice) states at low T

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

11

Order parameter:

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

In the atomic limit:

DMFT order parameter. Approximate X-Y symmetry

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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Extreme Anisotropy.

Magnetic susceptibility

0

2

4

6

8

10

12

0 100 200 300 400

c (1

0 -3 e

mu

/ mol

)

T (K)

URu2Si2

H // c

H // a

To

mzeff ~ 2.2 mB

13

XY-Ising

crystal field: z direction

Magnetic moment: y-direction

Hexadecapole: x-direction

A toy model

The two broken symmetry states

14

Low energy model Exp. by E. Hassinger et.al. PRL 77, 115117 (2008)

HO & AFM in magnetic fieldNotice that T0decreases with Increasing magnetic field but mangetic field stabilizes hidden order.

15

Key experiment: Neutron scattering

The low energy resonance

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

16

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

Interpretation of Neutron scattering experiments

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

17

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)‐18

Pseudo-gap opens at Tc. URu2Si2 measured through optical conductivity, D. A. Bonn et al. PRL (1988). [Missing Drude peak found by D. VanDerMarel et. al. and R. Lobo et. al. independently (2010)]

7.5 mev

19

0

100

200

300

400

500

1 10 100 1000

T.T.M. Palstra et al.(1985)W. Schlabitz et al.(1986) M.B. Maple et al.(1986)

To ~ 17.5 K

I // a

I // c

T (K)

Tc ~ 1.2 Kr (m

Wcm

)

Resistivitykeeps decreasing with decreasing T

Heavy fermion at high T,low T HO + SC

20

Visualizing the Formation of the Kondo Lattice and the Hidden Order inURu2Si2 Pegor Aynajian et. al. PNAS 2010 http://www.pnas.org/content/early/2010/05/18/1005892107 arXiv:1003.5259

21

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

Arrested Kondo effect and hidden order in URu2Si2, Kristjan Haule and Gabriel Kotliar, Nature Physics 5, 796 - 799 (2009). Theoretical STM with LDA+DMFT

Physical Insight: as the temperature is lowered the f degrees of freedom begins to absorbed in the formation of heavy quasi-particles.

But this process gets arrested by the hexadecapolar order

Nature 465, 570 ( 2010)

URu2Si2 • Hidden U(1) symmetry links the staggered

hexadecapolar order (“hidden order “) and antiferromagnetism.

• Root of the deep similarity of the high pressure and low pressure phase in this system.

• Almost Goldstone mode of magnetic character as a fingerprint of hexadecapolar condensation. . Contact with numerous experimental Spectroscopies : Inelastic Neutron Scattering and Scanning Tunneling Microscopy

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

23

Conclusions: Somewhat Broader ViewURu2Si2 has many things in common with many other strongly correlated electron systems.Hidden orderPseudogapFermi Surface reconstructionNon Fermi liquid behaviorCoherence Incoherence crossoverUnconventional SuperconductivityMultiple gapsItineracy and localization.

Good illustration of general concepts in the theory of strongly correlated electron systems , and the need for systemspecific studies Progress in the theory of strongly correlated electronsfor material exploration

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Thank you for your attention!

Experimental Consequences. For the two gaps.

Magnitude of the Neutron gap at (0 0 1) should decrease with increasing pressure [PREDICTION]

and should increase with magnetic field[OBSERVED].

•Magnitude of the Neutron gap at (0 0 1.4) should increase with increasing pressure and should decrease with magnetic field[PREDICTIONS].

Magnitude of the optical gap , tunnelling gap and specific heat gap should increase with increasing pressure and should decrease with magnetic field[PREDICTIONS].

URu2Si2 Stress in ab plane

Large moment when stress in ab planeNo moment when stress in c plane

M Yokoyama, JPSJ 71, Supl 264 (2002).

Further Japanese work showed that NMR in unstrained samples did not broaden below T0

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HO & AFM under stress

J’s sensitive to compression (strain), modeled by:

Very different effect of in plane stress

and uniaxial stress

In plane stress favors AFM state

c-axis stress favors HO

M Yokoyama, JPSJ 71, Supl 264 (2002). 27

Lattice response

28

24

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

Experiments : A. R. Schmidt et.

al. Nature 465, 570 ( 2010)

Realistic implementations of Dynamical Mean Field Theory as tool for material exploration. Theoretical spectroscopy

Starting from high and intermediate energiesProvide some guidance to low energy treatments.

U

SiRu

Where we are we going to be ?

We are we now ?

Examples:.........……….

PuCoGa5

•Access low energy physics•Catalog instabilites , potential ordred states.•Compute parameters for effective actions. •Cluster DMFT short range order•Always linked to structure •Many Many More Theoretical Spectroscopies <neutrons>•Short vs Long Wavelength Physics•Fluctuations around LDA+CDMFT •Superconductivity in the vicinity of a localization-delocalization crossover

Molecular Dynamics +LDA+DMFT Kinetics.From Material Exploration to Material Design, using correlated electron systems.

XY-Ising

crystal field: z direction

Magnetic moment: y-direction

Hexadecapole: x-direction

A toy model

The two broken symmetry states

18

URu2Si2 • Hidden U(1) symmetry links the staggered

hexadecapolar order (“hidden order “) and antiferromagnetism.

• Root of the deep similarity of the high pressure and low pressure phase in this system.

• Almost Goldstone mode of magnetic character as a fingerprint of hexadecapolar condensation. Spectroscopies : Inelastic Neutron Scattering

and Scanning Tunneling Microscopy K. Haule and G. Kotliar Nat Phys 5:637 641(2009)‐K. Haule and G. Kotliar EPL 89 57006(2010)

Dynamical Mean Field Theory

• The spectra of correlated materials contain atomic like features and band like features

• DMFT treats both on the same footing.• Technique designed to treat correlation

functions, one electron spectral function (measured in photoemission)

• Technique designed to treat finite electronic temperatures.

DMFT for model Hamiltonians Hubbard model

,[ ] ~ [ ] [ ]ii ij o iidmft i atomG G G G

0[ , ] [ ] [ ] [ ]ii ijG M Trln i t M Tr MG G

1

[ ( ) ]ii kii

Gi t k M

[ ]ii atomM Gii

Gii

† †

, ,

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

t c c c c U n n

Baym Kadanoff Functional, sum over two particle irreducible graphs.

Paradigm Shift: partial selection of graphs sum ALL LOCAL graphs

A.Georges, G. K., W. Krauth and M. J. Rozenberg, R. M.P. 68, 13 (1996).

Embedding ˆ HHS ® S

1ˆ ˆˆ ˆ( ) ( )ˆ ˆˆ ˆ( ) ( )

n k n

n k n

i O H k E ii O H k E i

w ww w

- - - S ®- - - S

0 0ˆ0

loc HHHH

G GG

é ùê ú= ®ê úë û

ˆ LL LH

HL HH

H HH

H H

é ùê ú=ê úë û

22

11

0 0 0ˆ 0 0HH

HH

é ù é ùSê ú ê úS = ® S =ê ú ê úS Së û ë û

1 10 ( ) ( )HH HHn nG i G iw w- -= +S

Integrating over BZ

Truncation

1ˆ ( )ˆ ˆˆ ˆ( ) ( )

loc n

n k nk

G ii O H k E i

ww w

=- - - Så

Inversion

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

• The light, SP (or SPD) electrons are extended, well described by LDA .The heavy, D (or F) electrons are localized treat by DMFT.

• LDA Kohn Sham Hamiltonian already contains an average interaction of the heavy electrons, subtract this out by shifting the heavy level (double counting term)

The U matrix can be estimated from first principles of viewed as parameters. Solve resulting model using DMFT.

See also LDA++. A Lichtenstein and M. Katsnelson PRB 57, 6884 (1988).

, ,

,

[ ] [ ]( )

[ ] [ ]spd sps spd f

f spd ff

H k H kt k

H k H k Edc

æ ö÷ç ÷ç ÷ç ÷ç -è ø®

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

12

1( , )

( ) ( )G k i

i t k i

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

0 0

0 ff

æ ö÷ç ÷S ç ÷ç ÷ç Sè ø®

abcdU U®

[ ] [ ( ) ]F J S JAZ e d d e y yy y+- + - += =ò

[ ]F

J A aJ

dd

=< >=

[ ] [ [ ] ] [ ]a F J a aJ aG = -

Spectral density functional. Effective action construction.e.g Fukuda et.al

hartree xcDG=G +G

0 intS S Sl= +

0 1J J Jl= + +L

[ ]aG = 0G1l+ G +L[ ]a+DG0 0 0[ ]F J aJ-

1

int

0

[ ] ( , ( , ))a d S J al l lDG = < >ò

0[ ]J aa

ddDG

=00

[ ]F

J aJ

dd

=

In practice we need good approximations to the exchange correlation, in DFT LDA. In spectral density functional theory, DMFT. Review: Kotliar et.al. Rev. Mod. Phys. 78, 865 (2006)

Kohn Sham equations

Remarks:• Exact functionals of an observable A,• In practice approx are needed• Many a’s many theories.

[ ]exact aG

[ ] [ ]mft exacta aG G:

• Introduction of a reference system. Separation into “free part” and exchange+ correlation. •Formal expression for the correlation part of the exact functional as a coupling constant integration. •Good approximate functionals obtained by approximating the xc part. [ small parameter helps!]• While the construction aims to calculate <A>=a, other quantities, e.g. correlation functions, emerge as a byproduct [bands, correlation functions…... ]

Crucial Role of the constraining field

• Different functionals (self energy functional, BK functional, Harris Foulkes functional, etc )

0[ ]J a

0 0[ ], [ , ], [ , ], [ ]a a J a J JG G G G

Different reference systems [ e.g. band limit or atomic limit ] define different constraining fields.

Different methods differ by the choice of variable a used.

• DFT

• Spin and Density FT

*a=G ( , ', ) ( ') ( ) ( )loc Rb Ra ababRr r r r G

a= (r)

a= (r), (r)

Spectral Density Functional R. Chitra and G.K Phys. Rev. B 62, 12715 (2000). S. Savrasov and G.K PRB (2005)

Density functional Kohn Sham equations

2 / 2 ( ) KS kj kj kjV r y e y- Ñ + =

( ')( )[ ( )] ( ) ' [ ]

| ' | ( )xc

KS ext

ErV r r V r dr

r r r

drr r

dr= + +

-ò

2( ) ( ) | ( ) |kj

kj kjr f rr e y=å

200 0[ , ] [ ] [ ]extii iiG M TrLn i V J Tr J

, , [ ]XC XC lda uniform elecron gas

DMFT: functional construction. Hubbard model

, , [ ]iiXC XC dmft i XCatom G

0[ , ] [ ] [ ]ii ii ij iiG M TrLn i t M Tr MG

1

[ ( ) ]ii kii

Gi t k M

[ ]ii atomM Gii

Gii

† †

, ,

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

t c c c c U n n

Hidden Order The CMT dark matter problem.

URu2Si2: T. T. M. Palstra, A. A. Menovsky, J. van den Berg, A. J. Dirkmaat, P. H. Kes, G. J. Nieuwenhuys and J. A. Mydosh Physical Review Letters 55, 2727 (1985)

U

SiRu

Entropy Loss at the transition: 1/5 Log[2]

3

•Similar T0 and TN

•Almost identical thermodynamic quantities ( e.g. jump in Cv) and similar

De Haas Van Alfen oscillation frequencies [ E. Hassinger et. al. (2010)

“Adiabatic “continuity between HO & AFM phase

E. Hassinger et.al. PRL 77, 115117 (2008)

“Adiabatic “is a misnomer. Need a better term.

5

U

SiRu

Order parameter:

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

In the atomic limit:

Accidental X-Y symmetry (not protected hence approximate ) Unique to URu2Si2

Two low temp DMFT solutions !

Moment only in z-direction!

X01 =[0><1]

17

Key experiment: Neutron scattering

The low energy resonance

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

20

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

Interpretation of Neutron scattering experiments

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

21

Experimental Consequences: two gaps [ p.h continuum gap –U(1) anisotropy gap] with opposite pressure and magnetic field dependence. Neutrons probe both. Magnitude of the Neutron gap at (0 0 1) should decrease with increasing pressure [PREDICTION] Should increase with magnetic field[OBSERVED].

•Magnitude of the Neutron gap at (0 0 1.4) should increase with increasing pressure and should decrease with magnetic field[PREDICTIONS].

Magnitude of the optical gap , tunnelling gap and specific heat gap should increase with increasing pressure and should decrease with magnetic field[PREDICTIONS].

Neutron intensity present in two regions, around (1,0,0) and around (1.4, 0,0)And (.6, 0, 0)

Wiebe, C. et al. Nature Phys. 3, 96–100 (2007).

Inelastic Neutron Scattering

9

Hall effect as function of temperature in different external fields, Y.S. Oh et al. PRL 98, 016401(2007).

•Fermi surface reconstruction in zero and small magnetic fields•Very large fields metamagnetic transition to polarized Fermi liquid.

10

Arrested Kondo effect and hidden order in URu2Si2, Kristjan Haule and Gabriel Kotliar, Nature Physics 5, 796 - 799 (2009). Theoretical STM with LDA+DMFT

Physical Insight: as the temperature is lowered the f degrees of freedom begins to absorbed in the formation of heavy quasi-particles.

But this process gets arrested by the hexadecapolar order

Nature 465, 570 ( 2010)

24

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

Experiments : A. R. Schmidt et.

al. Nature 465, 570 ( 2010)