Uncover the mystery of strange metal state in

Chung-Hou Chung 仲崇厚

Department of Electrophysics, National Chiao Tung University, Hsinchu, Taiwan

NTU, Nov 3, 2020

Uncover the mystery of strange metal state in

correlated electron systems

In memory of Prof. Pauchy Huang (黃偉彥 教授)

I was Prof. Huang’s Master degree student during 1991-1993 in NTU.

• Strange metal phenomena in correlated electron systems

• Strange metal in heavy fermion metals/superconductors

Heavy-fermion metal: Ge-substituted YbRh2Si2

Heavy-fermion superconductors CeMIn5, M=Co, Rh, Ir Mechanism: Kondo vs. AF RKKY

• Paramagnetic heavy-fermion metal on frustrated lattice

• Summary

Outlines

Elementary excitations in fermionic solid state systems: quasiparticles

quasi-particles:

weakly interacting electron-hole pairs

T

Landou’s Fermi -liquid theory: normal metals

Enrico Fermi States of Fermi-liquid described by quasi-particle distributions

Normal states of most metals

Lev Landou

electrons dressed by density fluctuations

ρ(T)=ρ(0)+aT²

Electrical

resistivity:

T-linear specific heat T² -resistivity

Strongly correlated electron systems

Transition metal compounds

• x=0, Large Coulomb repulsion U--> Mott Insulators +Heisenberg anti-ferromagnet

• x > xc, holes destroy AF order- normal Fermi liquid metal

Cu: d-orbitals

Strongly correlated quantum many-body systems

U

http://qcmd.mpsd.mpg.de/files/qcmd-theme/research/science/Mott/mott-diagr-for-web-2014-dbb-

https://www.psi.ch/swissfel/OrigInsTransEN/igp_1024x640%3E_V_11.png

YBa

Cu

O

High-Tc cuprate superconductors

= hole doping (x)

M. Ainslie, PhD Thesis, Cambridge U. , 2012

Competing ground state: AF vs. FL

https://upload.wikimedia.org/wikipedia/commons/thumb/0/05/Spinon_movin png/130px-Spinon_moving.png

Strange Metal Pseudo-gap

AF Normal Metal

AF insulator

Strange metal near edge of AF pseudogap and Fermi liquid phases

Quantum phase transitions

c

Τ

g g

True level crossing: Usually a first-order transition Avoided level crossing which becomes sharp in the infinite volume limit: Second-order transition

• Critical point is a novel state of matter

• Critical excitations control dynamics in the wide quantum-critical region at non-zero temperatures

• Quantum critical region exhibits universal power-law behaviors: Non-Fermi liquid

Sachdev, quantum phase transitions, Cambridge Univ. press, 1999

Competing Quantum Ground States

Non-analyticity in ground state properties as a function of some control parameter g

k_BΤ> |g-g_c|^νz Τ^∗

QF~TF

QF >TF

TF: thermal fluctuations QF: quantum fluctuations

Phase I Phase II

QF >TF

δ = |g-g

|

k_BΤ< |g-g_c|^νz k_BΤ< |g-g_c|^νz

Universal quantum critical behaviours:

Fractal Cauliflower, self-similarity --- Quantum Criticality

Same correlations at ALL length scale ! Dynamical scaling form near QCP:

Sondhi et al, RMP 1997

<S(0) S(r)>~ G(r)~ exp(-r / )

Vojta, RPP, 2003

Quantum phase transition (QPT) & universal scaling

Hyperscalings: for d+z < 4

➼ Relations between various exponents.

Effective dimension :

d + z

Near QCP r

universal scaling

<S(0) S(r)>~ G(r)~ exp(-r / )

Strange Metal: linear-T resistivity

L. Taillefer, Ann Rev. 2010

Strange Metal: linear-T resistivity

L. Taillefer, Ann Rev. 2010

Generic, Ubiquitous across various correlated materials near phase transitions

Strange Metal: T-logarithmic specific heat coefficient Cv/T

L. Taillefer, Ann Rev. 2010

Signature of QCP?

L. Taillefer, Ann Rev. 2010

SM phase (new ground state) SM region (single QCP ) Debatable Open Question !

Strange Metal Behaviours near Quantum Phase Transitions and superconductivity:

High-Tc cuprate superconductors

Origin of Strange Metal ?

Strong correlations--Kondo Effect in metals with magnetic impurity

Anti-ferromagnetic spin-exchange between conduction electrons and local impurity spins

https://upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Kscheme.jpg/320px-Kscheme.jpg

Kondo effect in metals with magnetic impurities

For T<Tk (Kondo Temperature), spin-flip scattering off impurities enhances Ground state is

Resistance increases as T is lowered

electron-impurity spin-flip scattering

logT

(Kondo, 1964)

(Glazman et al. Physics world 2001)

Antiferromagnetic RKKY coupling Kondo effect on a lattice: Kondo lattice

heavy fermi-liquid via Kondo effect J

P. Coleman, Magnetism and Advanced Magnetic Materials,95-148 (2007).

Matsuda, AAPPS Bulletin 2017

d-electron

dilute magnetic metal ions, the oscillatory RKKY  “spin glass”.

dense systems, the RKKY ordered antiferromagnetic state

Kondo Hybridization in heavy-fermion systems

Flat band:

local moment f-electrons

Dispersive band: conduction d-electrons

hybridized band

hybridized band

P. Coleman, Electrons at the edge of magnetism, Handbook of Magnetism and Advanced Magnetic Materials, Wiley, 2007

Strange Metal Behaviours near Quantum Phase Transitions and superconductivity:

Heavy-fermion metals/superconductors

http://inac.cea.fr/Images/astImg/522_1.png

Strange Metal

What are the key quantum critical fluctuations?

To address the strange metal physics, we must find out :

Bosonic Kondo (charge) fluctuations

Bosonic RVB sin liquid (made of fermionic spinons)

In heavy fermion systems, they are:

Chung-Hou Chung

Department of Electrophysics, National Chiao Tung University, Hsinchu, Taiwan

Mechanism of strange metal state near a heavy-fermion quantum critical point

Collaborators:

Yung-Yeh Chang (NCTU, Taiwan)

Silke Paschen (TU Vienna, Austria)

PRB 97, 035156 (2018)

Strange Metal (SM) near a AF quantum critical point (QCP)

heavy fermion Kondo lattice systems YbRh2Si2

quantum critical non-Fermi-liquid

Anti-ferromagnetic Fermi liquid

Paramagnetic heavy Fermi liquid

SM

Yb: 4f, 5d Rh:4d

T-linear resistivity: YbRh2Si2

O. Trovarelli et al. PRL 2000

Specific heat coefficient: T-logarithmic YbRh2Si2

O. Trovarelli et al. PRL 2000

Log(T)

Divergence of A-coefficient and effective mass near QCP

O. Trovarelli et al. PRL 2000

A~(m*)

~1/|B-Bc|

A~ quasiparticle–quasiparticle scattering cross-section

R_H: Hall coefficient~1/V_FS

S. Paschen et al., Nature (2004)

Jump in Fermi surface volume at QCP for T0 for YRS

Phase diagram -

Ge-doped YbRh

Si

Field-tuned ➸

quantum critical point

Custers, Nature, 2003, Custers, PRL, 2010

S. Wirth, JPCM, 2012

Heavy fermions :

Yb: 4f, 5d Rh:4d

YRS Ge-YRS

T_N ~18mK

B

~0.3T

B

~0.66T

Non-Fermi Liquid Strange Metal Behaviors: Ge-YRS

Power-law (T^-𝛼𝛼) + ln (T0/T) — Specific heat coefficient

Linear-in-T Resistivity

Custers, et al., Nature, 2003 Custers, et al., PRL, 2010

breakup of quasi-particles spin (f)-charge (d) separation

T<0.3K T<10K

2D SDW

10mK<T<10K

T<0.3 K

P. Coleman’s talk in NCTU, 2016

Kondo breakdown and Quantum Criticality in Heavy-fermions

Doniach phase diagram

P. Coleman,

Magnetism and Advanced Magnetic Materials,

95-148 (2007).

QCP

Frustrated Kondo lattice

J. Custers et al. Phys. Rev. Lett. 104, 186402 (2010)

B-field

Ge doping induces disorder ~ frustration

New Kondo breakdown scenario

AF RKKY + disorder induced frustration:

Fractionized Fermi liquid (FL*) RVB spin-liquid metal

Kondo effect

Coleman et al., J Low Temp Phys 161, No1-2, 182-202 (2010)

P. Coleman,

Magnetism and Advanced Magnetic Materials, 95-148 (2007).

VS.

Phase diagram for Ge-YRS

Large-N (Sp(N))Mean-field Kondo-Heisenberg Model

(conduction electrons) (localized electrons)

➔ RVB spin-singlet bond (Characterize the spin-liquid)

➔ Kondo hybridisation (Characterize the Kondo phase)

Fractionalized

Fermi-liquid (FL*)

Kondo

Heavy Fermi Liquid

B field suppresses

superconducting phase

V.S.

Custers, Nature, 2003 Custers, PRL, 2010

Proposed phase diagram

Effective Action—Mean-field

Effective action—amplitude (Gaussian) fluctuation

boson-fermion Yukawa coupling

new scaling!

Beyond Ginzburg-Landau theory of phase transitions

quasi-2d:

d=z+η, z=2, 0<η<<1

Crossover scales

☞ B suppresses J _𝛷𝛷 but keep J _𝜒𝜒 nearly at J _𝜒𝜒 *

T ^{FL* : J}

^{> J}

^{*, J}

is marginal.

T ^{LFL : J}

^{< J}

^{*, J}

is marginal, 𝜒𝜒 is relevant relative to marginal J

T

^: 𝜒𝜒 (J

)

is marginal but J

is irrelevant.

Divergence of A-coefficient in FL phase

theory prediction:

O. Trovarelli et al. PRL 2000

Specific heat coefficient

fitting parameter:

Critical bosonic RVB fluctuations

Specific heat coefficient

Theory Experiment

T

~ | J

-J

| ~ | B-B

|

anomalous exponent 2d bosonic fluctuations quantum critical

Linear-T Resistivity

Conduction electron T-matrix

Critical Kondo fluctuations (bosonic charge)

Linear-T Resistivity

𝜏𝜏(k) : life-time of c-electrons.

Conductivity

T-linear Resistivity:

Custers, et al., PRL, 2010

𝛼𝛼 : constant Friedeman et al., PNAS, 2010

T > 0 T = 0

Jump in Fermi surface volume

Strange superconductivity near heavy-fermion quantum critical point

:

PHYSICAL REVIEW B 99, 094513 (2019)

Chung-Hou Chung 仲崇厚

Department of Electrophysics,

National Chiao Tung University, Hsinchu, Taiwan

Collaborators:

Yung-Yeh Chang (NCTU), Feng Hsu (NTHU),

S. Kirchner (Zhejiang U., China), C. Y. Mou (NTHU) T. K. Lee (Academia Sinica)

Acknowledgement:

J. D. Thompson (LANL), Piers Coleman (Rutgers)

CeCoIn5: Lattice Structure

Tetragonal

Matsuda, AAPPS Bulletin 2017

Tetragonal

• Discovered in 2001 by Fisk et al., heavy-fermion analogue of cuprate (LaCu2O4) superconductor

CeIn3

• quasi-2D structure + proximity to magnetic order, favorable for unconventional superconductivity

Co²⁺ (3d⁷)

• local-moment 4f electron on Ce + itinerant 5d (Ce) and 3d (Co) electrons

Kondo hybridization between f and d electrons, Anti-ferromagnetism (Ce)

Superconductivity at the boarder (quantum critical point QCP) of anti-ferromagnetism

CoIn2

strange superconductivity in CeCoIn5:

Non-Fermi liquid (strange metal) normal state

anomalous power-law magnetic susceptibility

Tc ~ 2.3K

// c

// ab

Magnetic susceptibility

C. Petrovic, JPCM. 13, L337-L342 (2001).

d

wave gap

Global Phase Diagrams of CeCoIn5

J.D. Thompson et al. Phys. Rev. Lett. 106, 087003 (2011)

quantum critical lin

SM

Phase diagram for CeRhIn5

Co-existing AFM+SC

First-order transition

2 peaks: Tc and T

J.D. Thompson et al. New Jouranl of Physics 11, 055062 (2009)

Spiral (incommensurate) spin order (SDW)

W. Bao, PRB 2000

P2: QCP in the absence of SC

SM

Kondo breakdown QCP for CeRhIn5

Small Fermi surface

Kondo destruction) Large Fermi surface

(Kondo)

T. Park, et al., Nature 440, 65 (2006).

Q. Si et al. Science 329, 1161 (2010)

S

J. Thompson et al. arXiv:0910.2287

Sub-linear-T resistivity (non-Fermi liquid)

Open issues on mechanism of strange superconductivity in CeMIn5

• How do (f) electrons incorporate in the superconducting state?

• How does a strange metal turn into a superconductor?

• What are the links among SM, Kondo coherence,

superconductivity, and QCP?

Anderson’s RVB spin-liquid for cuprate supeconductors

Resonating Valence Bond (RVB) spin-liquid

Kondo stablized spin-liquid close to magnetic instability (phase transition)

Escape of RVB singlets into conduction sea

Bose condensing Cooper pairing-

superconductivity Andrei, Coleman JPCM 1989

inter-layer proximity

SM

SC

Large-N Mean-field phase diagram

RVB spin-liquid

LFL

CeCoIn5

Superconductivity = co-existence btw Kondo and RVB spin-liquid

Kondo + RVB

Strange metal (SM), superconductivity and quantum criticality

SM

SM

(B = magnetic field)

Kondo Heavy Fermi Liquid

Fractionalized

Fermi-liquid (FL*)

Coexisting superconducting

FL*

SM  QCP by Suppressing SC QCP

QCP

SM

Effective field theory beyond mean-field

Y. Chang et al PRB 2018

Effective action beyond mean-field —amplitude (Gaussian) fluctuation

Competition Kondo (S

) vs. RVB (S

)

+

Composite Cooper pairing:

via higher order collaborations btw Kondo and RVB

Cooper instability: RG analysis near g

and g

Near g

: phase diagram of CeCoIn5 (Kondo dominated)

SC

QCP

RVB spin-liquid breakdown QCP

CeCoIn5

Strong Kondo, weak RKKY

linear crossover (via RVB breakdown under RG)

Near g

: phase diagram of CeRhIn5 and CeCoIn5

(strong AF RKKY limit)

SC + Kondo breakdown

- SC+AF SC + Kondo

Kondo breakdown QCP (g)

CeRhIn5

Strong RKKY, weak Kondo

CeCoIn5

weaker RKKY

CeRhIn5

stronger RKKY

Outstanding puzzles:

 How do the f-electrons incorporate in the superconducting state? Kondo?

 How does superconductivity emerge from the strange-metallic (SM) normal state?

 What are the links among SM, Kondo coherence, superconductivity, and QCP?

strange superconductivity near heavy-fermion quantum critical point

CeCoIn5 CeRhIn5

QCP QCP

Chung-Hou Chung (仲崇厚)

Department of Electrophysics, NCTU, Hsinchu, Taiwan

Collaborators

Jiangfan Wang (NCTU, Taiwan & IoP, CAS, China ) Yung-Yeh Chang (NCTS & NCTU, Taiwan)

Strange metal state in paramagnetic Kondo lattice:

dynamical large-N Fermionic multichannel pseudo fermion

approach (arXiv: 2005.03427)

CePdAl under B, p

Paramagnetic heavy-fermion metal

Ce: 5d, 4f

Crystal structure: Kagome Kondo lattice

H. v. Lohneysen et al., PRB, 2014

Peijie Sun et al., PRB, 2018 Peijie Sun et al., Nature Phys., 2019

T-quasi-linear Resistivity

AF

FL

Strange metal phase: new!

paramagnetic spin-liquid

LFL

quantum critical strange metal phase

FS reconstruction /Kondo breakdown

AF AF ^LFL

Peijie Sun et al., Nature Phys, 2019

Non-Fermi liquid strange metal resistivity

B=0 under pressure

Kondo breakdown and Fermi surface crossover-line B*(T)

Peijie Sun et al., PRB, 2018

Sharp jump in Fermi surface volume at Kondo breakdown QCP

Q. Si et. al. Science, 329, 1161 (2010)

Pauli spin susceptibility T0

 Fermionic excitations

No pressure, paramagnetic fermionic metallic spin-liquid (state “P”)

Peijie Sun et al., PRB, 2018

𝜒𝜒^ac increases upon cooling

and saturates at low temperature→

Spin liquid

Frustrated Kondo Lattice

J. Custers et al., PRL, 2010

B-field or doping

Peijie Sun et al., PRB, 2018

Peijie Sun et al., Nature, Phys.

2019

Frustrated Kondo Lattice

Quantum phase transition between

a paramagnetic spin-liquid NFL phase and a heavy FL phase

Peijie Sun et al., Nature, Phys.

2019

RG Phase diagram for Ge-YRS

YY Chang et al PRB 2018 gaped spin-liquid

Fermionic Multichannel dynamical large-N 2D Kondo-Heisenberg (KH) Lattice model

(conduction electrons) (localized electrons)

P. Coleman et al., PRL 2018

square lattice

Hubbard-Stratonovich transformation & order parameters

Mean-field order parameters

Fermionic RVB spin-singlet bond

Bosonic Kondo correlation

RVB: resonating valence bond

Bosonic fields

Sp(N) sym.

Dynamical large-N self consistent NCA equations

Saddle-point eqs.

F : free energy

Large-

N

local bath approximation ~DMFT

Quantum phase transition & critical spin liquid strange metal phase for 𝝹𝝹 = ½

𝜿𝜿 = 1/2

• Particle-hole symmetry for 𝝹𝝹 = 1/2.

• Region I.: Quantum critical strange metal phase:

Spinons and holons show critical (gapless) power-law spectral functions.

• g^QC becomes a QCP (continuous transition):

• Region II. NFL SM becomes truly quantum critical region

Spectral weight

NFL properties of critical spin liquid

Power-law T-matrix

& scaling

Entropy &

specific heat coefficient

The critical spinon and holon give rise to NFL behavior

in various observables

C

/T _∝ -ln(T)

S _∝ T-T ln (T)

Peijie Sun et al., Nature, 2019

~T NFL SM resistivity

𝜿𝜿 = 1/2 𝜿𝜿 = 0.3

Summary

•

Strange metal state are generic non-Fermi liquid properties in correlated electron systems near quantum phase transitions

• Kondo in competition with RVB spin-liquid provides an excellent description on the mechanism of strange metal behaviors observed in quasi-2D heavy-fermion metals and superconductors

• Critical Kondo (bosonic charge) fluctuations lead to T-linear resistivity

• Critical bosonic RVB spin-liquid fluctuations (made of fermionic spinons) lead to T-logarithmic singularity in specific heat

coefficient

Joe Thompson, LANL

Acknowledgement

Frank Steglich, Max-Planck, Dresden

Experimentalists Theorists

Piers Coleman, Rutgers U.

Matthias Vojta, TU Dresden Stefan Kirchner

Zhejiang U.

Qimiao Si Rice U.

S. Sachdev’s Onsarger Prize Talk APS March Meeting 2018

Perturbative renormalization group (RG)

Feynman diagrams ➠ (one-loop)

⬅⬅ Bare Green functions

Wave-function + coupling constant renormalizations

Correlation length ξ:

Crossover scale T

RG relative to fixed J

⇒

quasi-2d: d=z+η, z=2, 0<η<<1

RG equations and RG flows

The Gaussian fluctuation of RVB singlets dominate the specific heat.

rescaling of T

(Hertz-Millis theory)

m

m

O

e

T

T

T

Millis, PRB, 1993

Specific heat coefficient

Anomalous Scaling in Free Energy and Hyperscaling Violation.

Conventional Hyperscaing

hyperscaling violation due to boson-fermion coupling:

The Gaussian fluctuation of RVB singlets dominate the Gaussian Free energy (spin)

Specific heat coefficient

fitting parameter:

Open issues

• Microscopic mechanism of SM (NFL) properties

Due to QCP? What are the competing states?

Nature of the transition?

• Role of magnetic field?

• How to explain exotic scaling behaviors in SM state?

Mean-field phases diagram

(B = magnetic field)

Fractionalized

Fermi-liquid (FL*)

Coexisting

superconducting

Kondo

Heavy Fermi Liquid

FL*

• SM in Ge-YRS can be explained by a quantum critical region due to a single QCP at gc within Kondo breakdown scenario.

• The magnetic field mainly suppresses the RVB term, while the Kondo term stays nearly critical. YRS has spatial dimension d = 2 + 𝜂𝜂, 𝜂𝜂  0.

• Remarkable agreements between our theory and experiments on Ge-YRS.

The specific heat is dominated by the RVB (spinon) fluctuation

Kondo fluctuation contributes to the electrical (charge) transport.

• Hyperscaling violation

Anomalous exponent in specific heat coefficient is explained

• The dynamical ω/T scaling exists even for d+z > 4 due to the Kondo breakdown

Summary

Intertwine between dynamics and thermodynamics

Partition function (thermodynamics)

Imaginary-time Feynmann’s path integral (dynamics)

Imaginary-time

Sondhi et al, RMP 1997

g

Power-law divergent correlation lengths

g-g^c t=

<S(0) S(r)>~ G(r)~ exp(-r / )

Τ

g^c

QF~TF

https://upload.wikimedia.org/wikipedia/commons/thumb/0/05/Spinon_moving.

png/130px-Spinon_moving.png

NFL SM behaviors in other heavy-fermion compounds

H. v. Löhneysen, PRL, 1994 H. v. Löhneysen, JPCM, 1996

CeCu

Au

QPT by

doping

P. Coleman,

Magnetism and Advanced Magnetic Materials, 95-148 (2007).

CeCoIn5:

d-wave nodal superconducting quasi-particle scattering nodal gap

STM QPI

d_x2-y2 wave gap

d

wave gap

Ali Yazdani et al. Nature Physics 9, 474–479 (2013

)

J.C. Davis et al., Nature Physics 9, 468-473 (2013)

Anderson’s RVB spin liquid

The Green function

𝜔𝜔/T scaling:

- ^scaling

usually exists at d+z < 4 G-L theory

Even for d+z > 4,

𝜔𝜔/T still exists due to boson-fermion interactions.

Vojta, RPP, 2003