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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
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
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
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
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
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
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
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)
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
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
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