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Strong coupling approach to critical quasiparticles

Jörg Schmalian Institute for Theory of Condensed Matter

Institute for Solid State Physics Karlsruhe Institute of Technology

Strange Metals workshop High Field Magnet Laboratory, Nijmegen, Jan. 2018

Collaborators

Elihu Abrahams (UCLA) Peter Wölfle (KIT)

Strong coupling theory of heavy fermion criticality, Elihu Abrahams, Jörg Schmalian, Peter Wölfle, Phys. Rev. B 90, 045105 (2014) Strong coupling theory of heavy fermion criticality II, Peter Wölfle, Jörg Schmalian, Elihu Abrahams Rep. Prog. Phys. 80 044501 (2017)

Motivation: AF - quantum critical points

H. v. Löhneysen, A. Rosch, M. Vojta, and P. Wölfle, RMP (2007)

correlation length of a soft, bosonic degree of freedom diverges

à motivation for purely bosonic theories (Hertz, Moriya, Lonzarich, Moriya...)

But all degrees of freedom should be critical.

⇠ / r�⌫ , T�1/z

� (Q+ q) / 1

⇠�2 + q2 � i�!

critical order-parameter fluctuations

Theory does not account for experiment in systems like CeCu6-xAux or YbRh2Si2

H. v. Löhneysen J. Phys. CM 8 (1998); H. v. Löhneysen et al. JMMM 177-181 (1998)

A. Schröder et al. PRL 80 (1998)

( )( )( )αα ωωγ

χsign1

2 iqT ++∝+qQ

74.0≈α 7.2≈⇒ z

Fermions coupled to singular bosons

Y. B. Kim, A. Furusaki, X. G.Wen and P. Lee, Phys. Rev. B 50 17917 (1994), B. L. Altshuler, L. B. Ioffe & A. J. Millis, Phys. Rev. B 50, 14048 (1995), A.  Abanov, A. V. Chubukov & J. Schmalian, Adv. Phys. 52 119 (2003), S.-S. Lee, Phys. Rev. B 78, 085129 (2008), M. A. Metlitski & S. Sachdev, Phys. Rev. B 82, 075128 (2010), D. F. Mross, J. McGreevy, H. Liu & T. Senthil, Phys. Rev. B 82 045121 (2010), A.  L. Fitzpatrick, et al. Phys. Rev. B 88 125116 (2013), B. Meszena, P. S�aterskog, A. Bagrov & K. Schalm, Phys. Rev. B 94, 115134 (2016)

...

hot and cold regions of the Fermi surface, with singular behavior in hot parts

beyond the order-parameter fluctuation approach :

Good metals Landau quasi-particles

G (k,!) =Z

! + i�� "⇤k+Ginc (k,!)

L. D. Landau, Sov. Phys. JETP 3, 920 (1957); 5, 101 (1957).

S = �kBX

k,�

(nk� log nk� + (1� nk�) log (1� nk�))

quantum numbers map onto free Fermi gas

•  same “accounting”

•  quasi-particle response

�s,aqp,l = ⇢0F

m⇤/m

1 +F s,a

l2l+1

C =m⇤

m

⇡2k2B3

⇢0FTheat capacity:

susceptibilities: Pomeranchuk instabilities

I. Pomeranchuk, Sov. Phys. JETP 8, 361 (1958).

Critical quasi-particles

C.M. Varma et al, Phys. Rev. Lett. 63 (1989); T. Senthil, PRB 78 (2008); P. Wölfle and E. Abrahams, PRB 84 (2011).

quasi-particle concept beyond Fermi liquid theory

G (k,!) =Z(!)

! + i�! � "⇤k+Ginc (k,!)

scale-dependent weight:

Z (!) / |!|⌘

(Z (! ! 0) ! 0)0 ⌘ < 1

Critical quasi-particles (= unparticles)

spectral function

If : the width of the peak is smaller than its position

-1.0 -0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

w

AHwL

-1.0 -0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

w

AHwL

31

41

61

81 ,,,=η

ω ω

43=η

( )ωA ( )ωA21<η 12

1 <<η

⌃�! + i0+

�= �a |!|1�⌘

⇣sign (!) cot

⇣⇡⌘2

⌘+ i

⌘Z (!) / |!|⌘ ()

⌘ <1

2

Critical quasi-particles

C.M. Varma et al, Phys. Rev. Lett. 63 (1989); T. Senthil, PRB 78 (2008); P. Wölfle and E. Abrahams, PRB 84 (2011).

quasi-particle concept beyond Fermi liquid theory

G (k,!) =Z(!)

! + i�! � "⇤k+Ginc (k,!) Z (!) / |!|⌘

•  well defined Fermi surface:

•  dynamic scaling exponent:

•  marginal Fermi liquid:

nk � n(0)k / (vk · (k� kF ))

1�⌘⌘

zF =1

1� ⌘ C/T / T�⌘

C/T / log T⌘ ! 0

What can one do with such a theory?

naively: analyze Cooper instability

1 = gppTX

n

ZdkG (k,!)G (�k,�!)

A. Balatsky, Philos. Mag. Lett. 68, 251 1993. A. Sudbo, Phys. Rev. Lett. 74, 2575 1995. L. Yin and S. Chakravarty, Int. J. Mod. Phys. B 10, 805 1996.

G (k,!) = bG⇣bk, b

11�⌘ !

⌘

dgppd log(1/E)

= �⌘gpp + g2pp pairing instability is weakened

However: dynamic nature of the pairing interaction ignored

gpp ! gpp(!) / |!|�� A. V. Chubukov and J. Schmalian, PRB 72, 174520 (2005) J.-H. She and J. Zaanen, PRB 80, 184518 2009

Detour: Pomeranchuk instabiliy:

spontaneous deformation of the Fermi surface:

�pF,� (✓) = p(0)l,�Pl (cos ✓)

�E < 0 F s,al < � (2l + 1)

I. J. Pomeranchuk, On the stability of a Fermi liquid, Soviet Physics JETP 8,361(1959)

deformation is energetically favored if

�s,aqp,l = ⇢0F

m⇤/m

1 +F s,a

l2l+1

divergent quasi-particle susceptibility

l = 1spin instability: dynamic generation of spin-orbit coupling

C. Wu + S.-C. Zhang. Phys. Rev. Lett. 93, 036403 (2004) C. Wu, K. Sun, E. Fradkin, and S.-C. Zhang, Phys. Rev. B 75 115103 (2007)

E. I. Kiselev, M. S. Scheurer, P. Wölfle & J. Schmalian, PRB B 95, 125122 (2017)

coherent and incoherent response

G =Z

! � ✏k + i0!+Ginc =)

vertex correction due to coherent states

near the Fermi surface fully incoherent

response = �s,a

l (q = 0,! ! 0)

�s,al = (Z�s,a

l )2

m⇤

m ⇢0F

1 +F s,a

l2l+1

+ �s,ainc,l

Ward identity [Oq, Hint]� = 0 Oq =

Z

k †k+q,↵O

↵�k k�

Z

k(i!1 � (✏k+q1 � ✏k))O

↵�k G(4)

↵��� (k, q1, q2) = O��q2

⇣G(2)

q1+q2 �G(2)q2

⌘

coherent and incoherent response

G =Z

! � ✏k + i0!+Ginc =)

vertex correction due to coherent states

near the Fermi surface fully incoherent

response

1. conserved quantities: (charge, spin, momentum)

= �s,al (q = 0,! ! 0)

�s,al = Z�1

)

�s,al = (Z�s,a

l )2

m⇤

m ⇢0F

1 +F s,a

l2l+1

+ �s,ainc,l

�s,ainc,l = 0

�s,al = �s,a

qp,l =m⇤

m ⇢0F

1 +F s,a

l2l+1

pure quasi-particle response

coherent and incoherent response

G =Z

! � ✏k + i0!+Ginc =)

vertex correction due to coherent states

near the Fermi surface fully incoherent

response

e.g. Galilei invariant charge current:

= �s,al (q = 0,! ! 0)

)

�s,al = (Z�s,a

l )2

m⇤

m ⇢0F

1 +F s,a

l2l+1

+ �s,ainc,l

m⇤

m= 1 +

F s1

3

�sinc,l=1 = 0

�sl=1 = ⇢0F

pure quasi-particle response

coherent and incoherent response

G =Z

! � ✏k + i0!+Ginc =)

vertex correction due to coherent states

near the Fermi surface fully incoherent

response

2. current of a conserved quantities

= �s,al (q = 0,! ! 0)

)

�s,al = (Z�s,a

l )2

m⇤

m ⇢0F

1 +F s,a

l2l+1

+ �s,ainc,l

purely incoherent at the “PI”

�al=1 = 1 +

F a1

3vanishes at the

“Pomeranchuk instability”

�ainc,l=1 = ⇢0F

✓1� m

m⇤

✓1 +

F a1

3

◆◆

There is no instability as F a1 ! �3

coherent and incoherent response

G =Z

! � ✏k + i0!+Ginc =)

vertex correction due to coherent states

near the Fermi surface fully incoherent

response

3. generic non-conserved quantities

= �s,al (q = 0,! ! 0)

�s,al = (Z�s,a

l )2

m⇤

m ⇢0F

1 +F s,a

l2l+1

+ �s,ainc,l

Incoherent contribution to a susceptibility is generically large even for a Fermi liquid!

vertex of quasi-particles may be singular!

Back to critical quasi-particles

G (k,!) =Z(!)

! + i�! � "⇤k+Ginc (k,!) Z (!) / |!|⌘

0 ⌘ < 1

-1.0 -0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

w

AHwL

-1.0 -0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

w

AHwL

31

41

61

81 ,,,=η

ω ω

43=η

( )ωA ( )ωA

Hot spots (SDW-QCP)

Hot and cold regions of the Fermi surface, with singular behavior in hot parts

Ar. Abanov, AV Chubukov, J. S., Adv. in Phys 52 (2003); Ar. Abanov and A. V. Chubukov, PRL 93 (2004); M. A. Metlitski and S. Sachdev, PRB 82 (2010).

⌃ (khs,!) / i! |!|d�32

� (q,!)�1 / (q�Q)2 � i�! · · ·

Hot spots (SDW-QCP)

Cold regions: corrections due to composite energy-density fluctuations S. A. Hartnoll, D. M. Hofman, M. A. Metlitski, and S. Sachdev, PRB B 84 (2011)

φ φ φφ ⋅→

off-shell

→ ⌃ (kF ,!) / i! |!|d�32

sub-leading but non-analytic correction to Fermi liquid behavior (if d>3/2)

bootstrap approach

scattering process: = take your favored diagram

Z = 1 ! Z(!) / |!|⌘replace bare by critical quasi particles:

determine self energy ) Z(!)

demand self consistency

Same philosophy as Landau theory, but for singular quasi particles.

Scattering of singular quasiparticles

energy

microscopic model effective low-energy model ⌃>

k (!) �>k,q (!,⌦)

W⇤

matching at intermediate scales:

r = /Z1/2

�> ! g⇤

⌃qp (!) = Z (!)⌃> (!)

= take your favored diagram

E. Abrahams, J.S., and P. Wölfle, Phys. Rev B 90, 045105 (2014)

=Σq.p.hot =Σq.p.cold =Π

perturbation theory of renormalized quasi particles at low energies

P. Wölfle, J.S., and E. Abrahams, Reports on Progress in Physics (2017)

Scattering of singular quasiparticles

Γ= =

•  Singular low energy behavior (formally sub-leading) is boosted by high energy behavior •  There is always a weak coupling solutions:

•  strong coupling solution: Z(!) / |!|⌘Z = const.

⌘ =2d� 3

4d

dynamic scaling exponent

z =4d

3

Can one make this more systematic?

full solution in a “static” bosonic background J. Schmalian, D. Pines, and B. Stojkovic, Phys. Rev. Lett. 80, 3839 1998 .

a trick to sum all diagrams

recent development to “sum all diagrams” in the quantum regime

B. Meszena, P. Säterskog, A. Bagrov, and K. Schalm, PRB 94, 115134 (2016) P Säterskog, B. Meszena, and K. Schalm, PRB 96, 155125 (2017)

Nf ! 0, kF ! 1, Nkf = const

Scaling theory (collective modes)

E. Abrahams, J.S., and P. Wölfle, Phys. Rev B 90, 045105 (2014)

� (q,!)�1 / ⇠�2 + (q�Q)2 � i�!/Z(!)2

Z(!, T, x� xc) = b�⌘zZ⇣bz!, bzT, b1/⌫(x� xc)

⌘

⌫ =1

2 + ⌘z=

3

3 + 2d! 3

7 � = 1 + ⌫⌘z =4d

3 + 2d! 8

7 � = ⌫d/2 =! 3

7

“hyperscaling” for critical fermions:

Scaling theory (critical fermions)

f(T ) = b�(1+zF )f (bzF T )

there are two critical length scales:

Two dynamic scaling exponents z =4d

3zF =

4d

3 + 2d< zand

⇠ / T�1/z ⇠F / T�1/zF � ⇠

( )( )( )αωωγ

χsign1

2 iq +∝+qQ

( ) 75.02 ==dα 74.0exp ≈α

comparison with CeCu6-xAux (d=2)

8/1/ −∝TTC

E. Abrahams, J.S., and P. Wölfle, Phys. Rev B 90, 045105 (2014)

Renormalization group flow

�c(↵c)

↵c

weak coupling

strong coupling

↵⇤c

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

P. Wölfle, J.S., and E. Abrahams, Reports on Progress in Physics (2017)

the system flows to strong coupling à at lowest energies the power-law behavior may change

weak coupling à Hertz-Moria behavior

pole of the β-function Novikov-Shifman-Vainshtein-Zakharov function (n-extended SU(N) super symetric Y.-M. theories)

� (↵) =↵2

2⇡

(4� n)N

1� 2�n2⇡ N↵

•  Critical quasi-particles might be a powerful concept to combine Fermi liquid theory and quantum criticality.

•  Phenomenological approach: key assumption scale matching (high

energy dynamics boosts singularities at low temperatures) •  Coupling to composite modes (higher loop fluctuations) boosted in a

critical metal

•  Two non-trivial critical modes and two diverging length scales.

•  Results in good agreement with experiments on YbRh2Si2 (3-d fluctuations) and CeCu6-xAux (quasi-2-d fluctuations).

Conclusion

G (k,!) =Z(!)

! + i�! � "⇤k+Ginc (k,!) Z (!) / |!|⌘

⇠F / T�1/zF � ⇠

⌃ (kF ,!) / i! |!|d�32