Sleuthing Hidden Order in URu2Si2

collaborator, Kristjan Haule

Colloquim, Hebrew University , Jerusalem Novem-ber 7th 2011

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

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Thanks to :

NSF DMR 0906943The Lady Davis Trust and the Hebrew University

Maybe you can’t see it, but there’s ‘Hidden Order’ here!

U

SiRu

from Prof. Matsuda’s talk

Correlated electron materials. A condensed matter physics problem

•Intermediate length scales•Emergent Phenomena• Advances thru interplay of theory and experiment. • Basic fundamental research problems-applications • Requires “Model” simplified Hamiltonians • Requires material specific calculations.

Works well for weakly Correlated Materials. Success based on having a good reference system to describe the relevant physics / materials.

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“Standard Model” of solids devel-oped in the twentieth century

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Band Theory. Fermi Liquid Theory (Landau 1957). Density Functional Theory (Kohn Sham 1964)

2 / 2 ( )[ ] KS kj kj kjV r r y e y- Ñ + = Reference Frame for Weakly Correlated Systems.

Starting point for perturbation theory in the screened Coulomb interactions (Lars Hedin 1965)

M. VanSchilfgaarde Phys. Rev. Lett. 93, 126406 (2004)

0( ) *( ) ( )

kjkj kjr r r

er y y

<= å

+ [ - ]KSV10KSG 1G

Many other properties can be com-puted, transport, optics, phonons,

etc…4

Nobel Laureate 1962

Nobel laureate 1998

The “free electron” reference frame does not work for strongly correlated materials.

• Interesting systems, remarkable properties under in-tensive investigation. Standard model fails.

• High temperature superconductivity, metal to insulator transitions, exotic ordered phases, heavy fermion behavior, anomlous transport and optical proper-ties (giant and colossal magnetoresistance, thermoelec-tricity……..). Heavy fermion behavior Results of serendipity• Theoretical challenge:Restore atomic character to the

solid state description (Mott) .

5 Nobel Laureate (1977)

Albert Fert and Peter Grün-bergNobel Laureate 2007

Nobel Laureate 1987

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

A non-historical review of some experimental

facts about URu2Si2 [1986-2011] URu2Si2 a good test of the LDA+ DMFT (Georges,

Kotliar, Metzner, Vollhardt) strategy.

New insights into an old problem. Comparison with some experiments. Outlook and Conclusions and some more general

perspectives on correlated materials.

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U

SiRu

URu2Si2 - heavy fermion - hidden order

Moment screened

Elec. cv: g ~ 70 mJ/mol K2 T*

Coherence: T*~70K

Second order phase transition

SOPT: T0~17.8K

N. H. van Dijk, PRB 56, 14493 (1997).

mzeff ~ 2.2 mB

URu2Si2: T. T. M. Palstra et. al. PRL 55, 2727 (1985). W. Schlabitz .B. Maple et al.(1986) Curie-Weiss: mz

eff ~ 2.2 mB

USiRu

Entropy Loss 1/5 Log[2]

WHY ?WHO ?

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Phase diagram T vs P based upon resistivity and calorimetric experiments under pressure. E. Hassinger et al. PRB 77, 115117(2008).

Very similar to earlier Amitsuka’s T – P phase diagram

Consistency among experimental groups. First order phase transi-tion. 8

LMAF

Small parasitic moment in the HO phase < .o1

mB

Hidden Order

Phase

• Similar T0 and TN

• Almost identical thermodynamic quantities ( e.g. jump in Cv) • Similar oscillation frequencies.

“Adiabatic continuity” between HO & AFM phase

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

“Adiabatic “is a mis-nomer.

Need a better term. 9

? LMAF

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

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Pseudo-gap opens at Tc. URu2Si2 measured through optical conductiv-ity, D. A. Bonn et al. PRL (1988). VanderMarel (2011)

7.5 mev

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• 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 )

• Santini and Amoretti PRL 04

-Quadrupolar ordering• Fazekas and Kiss PRB 07

-Octupolar ordering…

• Elgazaar Oppeneer and Mydosh Nature Materials’08- DFT: no order parameter, slow spins

• ………………………………..

Haule and Kotliar (2009): hexadecapolar order.

Some proposals for the hidden order in the literature(disagreement also about everything else …. )

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Replace a many body problem by a single site in an effective medium. DMFT Reference frame for correlated materials

,ij i j i

i j i

J S S h S- -å å eMF offhH S=-

DMFT

A. Georges and G. Kotliar PRB 45, 6479 (1992).

DMFT self consistency:makes medium and site equivalent. Exact DMFT: reproduce the exact local spectral function of the problem.

Effective medium: quanti-fieds the notion of “ metal-licity” or itineracy

† †

, ,

( )( )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 Uc c c

V

c

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

Katsenelson and Lichtenstein PRB (1998) Sum all “LOCAL “ Feyman graphs on top to a “traditional”

mean field self energy for the less correlated orbitals. “LOCAL” refers to a predetermined d or f orbitals (projector)

on a given site or “cluster” of sites. Screened Coulomb interactions [ orbitally dependent concept] Total energy functional of density AND spectra. Spectral den-

sity functional. Dual description: atoms –bands, real –momentum space. Bands in a frequency dependent “ LOCAL “ potential. Atomic shell embedded in a medium-Valence Histograms. Many advances in implementation by many groups….. Reviews : K. Held (1997) G. Kotliar S. Savrasov K. Haule O

Parcollet V. Oudovenko and C. Marianetti RMP (1996) 14

URu2Si2 CTQMC valence histogram

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

• [0> - [0> (odd ) and [1> [1> (even)

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

The DMFT equations have TWO different solutions at low temperatures. They break translational symmmetry with a staggered pattern along c axis.

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Order parameter:

Rotation of U(1) relates different phases: “adiabatic continuity” explained!

In the atomic limit:

DMFT order parameter. Approximate X-Y symmetry

Hexadecapole does not break the time reversal, nor C4 symmetry.

Moment only in z-direction-magnetic anisotropy explained!

X01 =[0><1]

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XY-Ising

crystal field: z direction

Magnetic moment: y-direction

Hexadecapole: x-direction

A toy model

The two broken symmetry states

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

HO & AFM in magnetic field

Only four fitting parameters: Jeff1 , Jeff

2

determined by exp. transition temper-ature,

and pressure dependence .

Notice that T0decreases with Increasing magnetic field but mangetic field stabilizes hid-den order.

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Key experiment: Neutron scattering

The low energy resonance at 1.5 mev is a fingerprint of the HO stateA.Villaume, F. Bourdarot, E. Hassinger, S. Raymond, V. Taufour, D. Aoki, and J. Flou-

quet,PRB 78, 012504 (2008)

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hexadecapole

Goldstone mode

Symmetry is approximate“Pseudo-Goldstone” modeFluctuation 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)

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Tunnelling: Orbitally resolved DOS • High temperature. Fano-shapes first observed by S. Davis group. spd DOS changes weakly with T. Explains resisitivity. • f DOS T dependent and is gapped

[no Kondo peak!!] ARRESTED KONDO EFFECT

Kondo effect arrested by the splitting of the two singlets (which is the conse-quence of the bare small crystal field and the hexadecapolar order ).• f Single particle gap~7-10 mev Just like T0, it should decrease with in-creasing magnetic field. [ prediction]BCS-like coherence peaks in f

DOS when hidden order gap forms.

K. Haule and G. Kotliar Nat Phys 5:637‐641(2009)21

DMFT A(k,w) vs ARPES

Very good agreement, except at X point

Off resonance

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Surface Slab CalculationLDA+DMFT - bulk

• Hole pocket surface state appears at X-point!

LDA+DMFT - Si-terminated surface slab

• No hole-pocket at the X-point.

Z X Z X

Surface origin of pocket at X point

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Layer resolved spectra

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Conclusions URu2Si2, at very high temperatures garden variety

heavy fermion. Low temperatures, Kondo effect ar-rested by x-field splitting. HO ordered states.

HO= hexadecapolar order of in the U 5f2 shell. Search using RIXS in a magnetic field.

HO converts to LMAF by pressure through a first or-der line.

HO and LMAF are remarkably similar (“Mydosh’s adi-abatic continuity”) because they are related by U(1) symmetry.

Longitudinal pseudo-gap. optics, tunneling, pho-toemission

HO signature pseudo-Goldstone mode in INS modes: (1,0,0)@1.5 meV .

Longitudinal gap shows up in INS at(1.4,0,0)@5meV .

HO involves an electronic topological reconstruction of the fermi surface . Low temperatures compensated semimetal. Large Hall Effect and Nernst Effect.

HO (but not LAMF) turns into (time-reversal breaking ) superconductivity at 1.7 K. 22

Conclusions: Somewhat Broader ViewURu2Si2 has many things in common with many other strongly correlated electron systems (say cuprates, or Ce or Pu 115’s, or iron pnictides……) Hidden orderPseudogapFermi Surface reconstructionNon Fermi liquid behaviorCoherence Incoherence crossoverUnconventional SuperconductivityMultiple (pseudo) gapsItineracy and localization.• Good illustration of general concepts in the theory of strongly

correlated electron systems.• Requires care in interpretation in a material specific con-

text. • Pushing the quantitative aspects of theory to its current

limits. On URu2Si2 the jury is out. In simpler materials LDA+DMFT has passed many expt tests with flying colors. Future is bright.

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Thank you for your attention!