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1 EPiQS Intertwined Orders 2016 Workshop Rigorous Bounds on Pomeranchuk Instabilities Egor Kiselev, Mathias Scheurer, Peter Wölfle, and Jörg Schmalian Karlsruhe Institute of Technology

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Page 1: Rigorous Bounds on Pomeranchuk Instabilitiesqpt.physics.harvard.edu/abs/Schmalian-poster.pdf1 EPiQS Intertwined Orders 2016 Workshop Rigorous Bounds on Pomeranchuk Instabilities Egor

1 EPiQS Intertwined Orders 2016 Workshop

Rigorous Bounds on Pomeranchuk Instabilities

Egor Kiselev, Mathias Scheurer, Peter Wölfle, and Jörg Schmalian

Karlsruhe Institute of Technology

Page 2: Rigorous Bounds on Pomeranchuk Instabilitiesqpt.physics.harvard.edu/abs/Schmalian-poster.pdf1 EPiQS Intertwined Orders 2016 Workshop Rigorous Bounds on Pomeranchuk Instabilities Egor

2 EPiQS Intertwined Orders 2016 Workshop

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

�E =

X

k�

✏(0)k �nk� +

X

kk0��0

f��0(cos ✓k,k0

) �nk��nk0�0

Landau expansion of the excitation energy

Page 3: Rigorous Bounds on Pomeranchuk Instabilitiesqpt.physics.harvard.edu/abs/Schmalian-poster.pdf1 EPiQS Intertwined Orders 2016 Workshop Rigorous Bounds on Pomeranchuk Instabilities Egor

3 EPiQS Intertwined Orders 2016 Workshop

Pomeranchuk instabiliy:

quasi-particle susceptibility:

Os,al =

X

k�

(�)1,2 Yl,0

⇣k̂⌘nk,� order parameter formed by quasi-particles

u no divergence for in Galilei invariant systems u 3He spin and charge response is very different (near instability for l=1 in the spin channel?)

�sl=1

m⇤

m= 1 +

1

3F s1

G. Baym and Ch. Pethick. Landau Fermi-liquid theory: concepts and applications J. Wiley & Sons (2008)

�s,aq.p.l =

@Os,al

@hs,al

= �(0)l

m⇤/m

1 +F s,a

l2l+1

Page 4: Rigorous Bounds on Pomeranchuk Instabilitiesqpt.physics.harvard.edu/abs/Schmalian-poster.pdf1 EPiQS Intertwined Orders 2016 Workshop Rigorous Bounds on Pomeranchuk Instabilities Egor

4 EPiQS Intertwined Orders 2016 Workshop

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)

J. E. Hirsch, Phys. Rev. B 41, 6820 (1990) J. E. Hirsch, Phys. Rev. B 41 6828 (1990)

u spin-split states in metals, a proposal for Cr

u dynamic generation of spin-orbit coupling

Proposals for spin instabilities l = 1

Page 5: Rigorous Bounds on Pomeranchuk Instabilitiesqpt.physics.harvard.edu/abs/Schmalian-poster.pdf1 EPiQS Intertwined Orders 2016 Workshop Rigorous Bounds on Pomeranchuk Instabilities Egor

5 EPiQS Intertwined Orders 2016 Workshop

C.M. Varma + L. Zhu., Phys. Rev. Lett. 96, 036405 (2006)

A. V. Chubukov and D. L Maslov Phys. Rev. Lett. 103, 216401 (2009).

u hidden order parameter in URu2Si2: Helicity order

u instabilities near a ferromagnetic quantum critical point

Proposals for spin instabilities l = 1

phase diagram: A. Villaume, et al., Phys. Rev. B 78, 012504 (2008)

FM

Page 6: Rigorous Bounds on Pomeranchuk Instabilitiesqpt.physics.harvard.edu/abs/Schmalian-poster.pdf1 EPiQS Intertwined Orders 2016 Workshop Rigorous Bounds on Pomeranchuk Instabilities Egor

6 EPiQS Intertwined Orders 2016 Workshop

Y. Yoshioka and K. Miyake, J. Phys. Soc. Jpn. 81 023707 (2012)

u  instability for Sr3Ru2O7

u persistent current states in bilayer graphene

J. Jung, M. Polini, and A. H. MacDonald, Phys. Rev. B 91 155423 (2015)

Proposals for spin instabilities l = 1

Page 7: Rigorous Bounds on Pomeranchuk Instabilitiesqpt.physics.harvard.edu/abs/Schmalian-poster.pdf1 EPiQS Intertwined Orders 2016 Workshop Rigorous Bounds on Pomeranchuk Instabilities Egor

7 EPiQS Intertwined Orders 2016 Workshop

Ward identity

single-particle operator with

H =

Z

x,↵

†↵

✓�~2r2

2m+ U (x)

◆ ↵

+

Z

x,x

0⇢ (x)V (x, x0) ⇢ (x0)consider arbitrary

non-relativistic Hamiltonian

[Oq, Hint]� = 0

Z

k(i!1 � (✏k+q1 � ✏k))O↵�

k G(4)↵��� (k, q1, q2) = O��

q2

⇣G(2)

q1+q2 �G(2)q2

No divergence of the spin-current and charge-current susceptibilities!

Qµ,j = �i †↵�

↵�µ rj � Jj = �i †

↵rj ↵

Oq =

Z

k †k+q,↵O↵�

k k�

�Q,Q (q ! 0,! = 0) = �J,J (q ! 0,! = 0) =n

m

Page 8: Rigorous Bounds on Pomeranchuk Instabilitiesqpt.physics.harvard.edu/abs/Schmalian-poster.pdf1 EPiQS Intertwined Orders 2016 Workshop Rigorous Bounds on Pomeranchuk Instabilities Egor

8 EPiQS Intertwined Orders 2016 Workshop

Bloch-Bohm argument for spin-currents assume a ground state with finite spin-current

| i

hQµ,ji = �i⌦ �� †

↵�↵�µ rj �

�� ↵6= 0

construct a trial state

|�i = ei�P·Rr

†↵�

µ↵�r � | i

h� |H|�i = h |H| i+ �P · hQµ (q = 0)i+O ��P2

For finite spin-current one can a lower the ground state energy.

=) hQµ,ji = 0(argument can be generalized to finite temperatures)

no spin-currents

charge currents: D. Bohm, Phys. Rev. 75, 502 (1949) spin currents, see also: N. Bray-Ali, Z. Nussinov, Phys. Rev. B 80, 012401 (2009)

Page 9: Rigorous Bounds on Pomeranchuk Instabilitiesqpt.physics.harvard.edu/abs/Schmalian-poster.pdf1 EPiQS Intertwined Orders 2016 Workshop Rigorous Bounds on Pomeranchuk Instabilities Egor

9 EPiQS Intertwined Orders 2016 Workshop

implications for Fermi-liquid theory

coherent and incoherent spectrum

G =Z

! � ✏k + i0!+Ginc

A. I. Larkin and A. B. Migdal, Sov. Phys. JETP 17, 1146 (1963), A. J. Leggett, Phys. Rev. A 140, 1869 (1965)

=)incoherent 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 = 0

)

�s,al = (�s,a

l )2 Z�(0)

l

1 +F s,a

l2l+1

+ �s,al

�s,al =

Z�1�(0)l

1 +F s,a

l2l+1

Page 10: Rigorous Bounds on Pomeranchuk Instabilitiesqpt.physics.harvard.edu/abs/Schmalian-poster.pdf1 EPiQS Intertwined Orders 2016 Workshop Rigorous Bounds on Pomeranchuk Instabilities Egor

10 EPiQS Intertwined Orders 2016 Workshop

implications for Fermi-liquid theory 2. charge-current: (with and without momentum conservation)

�sl=1 =

n

m

✓1� m

m⇤

✓1 +

F s1

3

◆◆�sl=1 = 1 +

F s1

3

vertex vanishes for over-compensates q.p.-divergence ! Pomeranchuk instability = complete break-down of the quasi-particle picture

often small (vanishes for Galilei invariance)

3. spin-current: (not conserved!)

�al=1 = 1 +

F a1

3

often large

F a1 ! �3

�al=1 =

n

m

✓1� m

m⇤

✓1 +

F a1

3

◆◆

response dominated by incoherent part

3. generic non-conserved quantities: incoherent part can be large vertex will likely not cancel divergent q.p. contribution

Page 11: Rigorous Bounds on Pomeranchuk Instabilitiesqpt.physics.harvard.edu/abs/Schmalian-poster.pdf1 EPiQS Intertwined Orders 2016 Workshop Rigorous Bounds on Pomeranchuk Instabilities Egor

11 EPiQS Intertwined Orders 2016 Workshop

conclusions u  there are no charge Pomeranchuk instabilities

(protected by charge conservation) u  there are no spin Pomeranchuk instabilities for

nonrelativistic systems (protected by spin conservation) u  the divergence of the quasi-particle susceptibility

is eliminated by vertex corrections: does not signal a conventional Pomeranchuk instability!

Q: what happens instead? see also C.M. Varma, Phil. Mag. 85, 1657 (2005)

u  finite incoherent contributions to the order-parameter

susceptibilities for all generic channels

F s,a1 ! �3

l = 1

l = 1

l = 1

l > 0