What is a Pomeranchuk instability?
Hiroyuki Yamase
National Institute for Materials Science (NIMS), Tsukuba, Japan
Ref. PRB 72, 035114 (2005)
Tsukuba
Tokyo
Narita
Airport
1
Outline of my talk
1. forward scattering model
2. typical phase diagram
3. analytical understanding
4. universal ratios (c.f. 2∆/kB T = 3.54 in BCS theory)
5. self-masking
6. toward real materials
collaboration
W. Metzner (MPI-FKF, Germany)
P. Jakubczyk (Warsaw University, Poland)
R. Zeyher (MPI-FKF, Germany)
A. A. Katanin ( Institute of Metal Physics, Russia)
V. Oganesyan (Princeton University, USA)
Fermi surface:
usually same symmetry as the point-group symmetry of the lattice
e.g., square lattice
2
What is Pomeranchuk instability?
Recently it was found that
symmetry of the FS can be broken spontaneously
2D t-J model
H.Y. & Kohno, JPSJ 69, 332 (2000)
H.Y. & Kohno, JPSJ 69, 2151 (2000)
2D Hubbard model
Halboth & Metzenr, PRL 85, 5162 (2000)
Valenzuela & Vozmediano, PRB 63, 153103
(2001)
✕ orientational symmetry
Pomeranchuk’s stability criterion for isotropic Fermi liquids
I. J. Pomeranchuk, JETP 8, 361 (1958)
dFSD: also strongly correlated systems
dFSD: instability also without breaking his criterion
electronic nematic state S.A. Kivelson et al., Nature 393, 550 (1998)
dFSD: q=0 instability and no underlying charge stripes
3
f-model
H.Y., V. Oganesyan, W. Metzner, PRB 72, 35114 (2005)
g>0 : attractive interaction of the dFSD
What is Pomeranchuk instability?
Recently it was found that
symmetry of the FS can be broken spontaneously
2D t-J model
H.Y. & Kohno, JPSJ 69, 332 (2000)
H.Y. & Kohno, JPSJ 69, 2151 (2000)
2D Hubbard model
Halboth & Metzenr, PRL 85, 5162 (2000)
Valenzuela & Vozmediano, PRB 63, 153103
(2001)
✕ orientational symmetry
4
f-model
H.Y., V. Oganesyan, W. Metzner, PRB 72, 35114 (2005)
g>0 : attractive interaction of the dFSD
mean-field analysis
order parameter of the dFSD
self-consistency equation
4
Typical phase diagram
t'/t=-1/6, t''/t=0, g/t=1.0
saddle points: (π, 0), (0, π)
0.3
-1
µ/t
1st order at low T
2nd order at high T
0
maximal Tc (µ) near µ= εvH
0.25
ky
1
0
1
-1
-2
-2
-3
-3
-2
-1
0
kx
1
2
3
0.3
0.05
0
0
T c2nd"
0
kx
1
2
3
open Fermi surface
in the symmetry broken phase
0.2
0.1
T c2nd
Ttri
T cPS
"
-1
µ/t
(g)
0.15
0.05
-2
-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3
0.25
0.1
-3
T=0.01t
T=0.15t
0
-1
(e)
0
-1
0.1
t'/t = -1/6
t"/t = 0
0.2 g/t = 1.0
u/t = 0
0.15
(c)
T= 0.01t
µ/t
0.15
0.05
3 µ= -0.4t
2 T= 0.01t
µ= -0.9t
-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3
0.2
η /t
(b)
η /t
0.7
-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3
(f)
0.25
g/t = 1.0
g/t = 0
"
-3
N0 (µ)
0.9
0.8
T c2nd"
-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3
µ/t
T/t
T/t
ky
n
T c2nd
Ttri
T c1st
0.05
2
0.4
-1
(d)
1 T=0.01t
0.1
3
0.6
0.5
t'/t = -1/6
t"/t = 0
0.2 g/t = 1.0
u/t = 0
0.15
0
-1
0.7
0
van Hove singularity: εvH
=4t' = -2t/3
log-divergence in DOS
(a)
0.25
t'/t = -1/6
t"/t = 0
0.8
0
0.7 0.75
0.8
0.85
n
0.9 0.95
µ= - 0.87t
µ= - 2t/3
µ= - 0.50t
0.05
0.1
T/t
0.15
0.2
0.25
1
phase separated region
5
Landau expansion
ω(η)-ω(0)= 12 a2 η2 +
N2 (µ)
2.5
N2 (µ)
1 a η4
4! 4
a2 = g -1 - N2 (µ)
a4 = - N''
4 (µ)
T=0
t'/t = -1/6
t"/t = 0
2
+
p
Np (µ) = - 2 Σ d k f ' (ε0k - µ)
L
1.5
a 2 =0, a 4 <0 : necessarily 1st order
a 2 =0, a 4 >0 : possible 2nd order
0
0.1
0.2
0
µ−εvH
0.8
0.3
N0 (µ)
20
T=0
t'/t = -1/6
t"/t = 0
0.7
0.6
15
N2' (µ)
-0.1
N0 (µ)
10
5
N2' (µ)
2000
T=0
t'/t = -1/6
t"/t = 0
0
T=0
t'/t = -1/6
t"/t = 0
1000
-10
0.5
N"4 (µ)
1500
-5
500
-15
0.8
-0.2
-0.1
0
µ−ε
0
vH
0.1
0.2
0.3
20
T = 0.01t
N0 (µ)
15
t'/t = -1/6
t"/t = 0
0.7
-20
-0.3
10
0.6
-0.2
-0.1
0
0.1
0
vH
µ−ε
N2' (µ)
0.2
2000
T = 0.01t
t'/t = -1/6
t"/t = 0
5
-0.1
0
0.1
0
µ−εvH
0.2
0.3
0
0
µ−εvH
0.1
0.2
0.3
N"4 (µ)
-4000
-10
-0.2
-0.1
-2000
-20
-0.3
T = 0.01t
-6000
-15
0.4
-0.3
0
-0.2
0
-5
0.5
0
-0.3
0.3
N"4 (µ)
0.4
-0.3
N2' (µ)
-0.2
N0 (µ)
-0.3
N"4 (µ)
1
-0.2
-0.1
0
0.1
0
µ−εvH
0.2
0.3
-8000
-0.3
t'/t = -1/6
t"/t = 0
-0.2
-0.1
0
0.1
0
µ−εvH
0.2
0.3
6
Universal properties
7
Layered systems: interlayer coupling
two possible stacking patterns
η<0
η<0
η>0
η<0
3
2
η<0
η>0
ferro-type
(F) stacking
ky
1
0
-1
-2
-3
-3 -2 -1
0
kx
1
2
3
antiferrotype (AF)
stacking
3
2
ky
Ising symmetry
1
0
-1
-2
-3
-3 -2 -1
0
kx 1
2
3
AF stacking
no macroscopic anisotropy!
spontaneous symmetry breaking is self-masked!!
What is the generic tendency of the dFSD stacking?
8
Self-masking of Pomeranchuk instability
H.Y., PRL 102, 116404 (2009)
order parameters
: AF stacking
: F stacking
calculate the coupling constant J
stacking does not depend on details of interlayer couplings
usually AF stacking as long as
no macroscopic anisotropy appears !!
spontaneous symmetry breaking is self-masked !!
9
Sr3Ru2O7
metamagnetism
QCEP? Non-FL
2.0
1.5
Perry et al.
PRL 86, 2661
(2001)
H-T phase diagram
1.0
Grigera et al.
Science 294, 329
(2001)
strong anisotropy of ρ, dFSD!?
2nd
2nd
dFSD
dFSD
Grigera et al.,
Science 306,
1154 (2004)
Borzi et al,
Science 315, 214
(2007)
10
Everything is OK for Sr3Ru2O7?
Pomeranchuk instability around the van Hove singularity
Kee & Kim, PRB 71, 184402 (2005)
Doh et al., PRL 98, 126407 (2007)
HY & Katanin, JPSJ 76, 073706 (2007)
HY, PRB 76, 155117 (2007)
Ho & Schofield, EPL 84, 127007 (2008)
HY, PRB 80, 115102 (2009)
Fischer & Sigrist, PRB 81, 064435 (2010)
Puetter et al., PRB 81, 081105 (2010)
presence of some QCP ?
2nd
dFSD
van Hove?
Raghu et al., PRB 79, 214402 (2009)
Lee & Wu, PRB 80, 104438 (2009)
more profound story! ?
multiple singularity at a continuous phase transition
jump of the longitudinal magnetic susceptibility!
its divergence in a critical region!
non-ordering and ordering susceptibilities
YbRh2Si2, CeRu2Si2, β-TbAlB4
HY&Jakubczyk, submitted
Misawa & Imada, JPSJ 78, 084707 (2009)
analysis of critical fluctuations
van Hove singularity: LGW theory is not applicable!!
functional RG framework Jakubczyk, Metzner, HY, PRL 103, 220602 (2009)
maybe more
11
Relevance to cuprates
1. La-based cuprates with LTT
different scenario from stripes
2. Y-based cuprates
YBa2Cu3O6.6
YBCO6.85, YBCO6.6
V.Hinkov et al., Nature 430, 650 (2004)
V.Hinkov et al., Nat. Phys. 3, 780 (2007)
microscopic theory from view of dFSD
H.Y. & W. Metzner, PRB 73, 214517 (2006)
YBCO6.45 (Tc=35K)
V.Hinkov et al., Science 319, 597 (2008)
phenomenological theory from view of dFSD
H.Y., PRB 79, 052501 (2009)
Kim et al., PRB 77, 184514 (2008)
Huh et al., PRB 78, 064512 (2008)
12
Other systems
N=2 Landau level
2DEG in a magnetic field
( )
Lilly et al, PRL 82, 394 (1999)
Du et al, Solid State Comm. 109, 389 (1999)
Review:
Wexler & Ciftja, Int. J. Mod. Phys. B 20, 747 (2006)
5/3
9/2
11/2
4/3
200
xx
higher Landau levels: ν = 9/2, 11/2
T=150mK
300
5/2
100
0
... 3
0
2
2
the same physics as the dFSD or something different?
=2
N=1
4
N=0
6
B (Tesla)
8
10
Lilly et al, (1999)
Graphene
honeycomb lattice
Valenzuela & Vozmediano, New J. Phys. 10, 113009 (2008)
ultracold dipolar neutral fermions
dipole interaction ≈ Σ<i,j> V ni nj
Lin, Zhao, Liu, PRB 81, 045115 (2010)
typical materials with high TcvH
may have AF stacking
spontaneous symmetry breaking is self-masked!!
13
Summary
forward scattering model
T0
2nd
!Λ e−1/2¯g
Ttri
dFSD (PI)
Ising symmetry
Ttri /T0 = 0.56
1st
µ1
0
two possible stacking patterns
η<0
η>0
η<0
3
2
ferro-type
(F) stacking
ky
1
0
-1
-2
-3
-3 -2 -1
0
kx
1
2
3
antiferrotype (AF)
stacking
self-masking
3
2
ky
η<0
η<0
η>0
1
0
-1
-2
-3
-3 -2 -1
0
kx 1
2
3
competing order
Sr3Ru2O7
high-Tc cuprates
2DEG in field
graphene
dipolar fermi gas
14
Ph.D. positions in NIMS in Tsukuba
nims yamase
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Narita
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15