1. Review of Fermi liquid theory - Harvard Universityqpt.physics.harvard.edu/talks/natal15c.pdf ·...
Transcript of 1. Review of Fermi liquid theory - Harvard Universityqpt.physics.harvard.edu/talks/natal15c.pdf ·...
1. Review of Fermi liquid theory Topological argument for the Luttinger theorem
2. Fractionalized Fermi liquid A Fermi liquid co-existing with topological order for the pseudogap metal
3. Quantum matter without quasiparticles(A) A mean-field model of a non-Fermi liquid, and
charged black holes(B) Field theory of a non-Fermi liquid (Ising-nematic
quantum critical point)(C) Theory of transport in strange metals: application to
the (less) strange metal in graphene
Quantum criticality of Ising-nematic ordering in a metal
x
yOccupied states
Empty states
A metal with a Fermi surfacewith full square lattice symmetry
x
y
Spontaneous elongation along y direction:Ising order parameter � < 0.
Quantum criticality of Ising-nematic ordering in a metal
Spontaneous elongation along x direction:Ising order parameter � > 0.
x
yQuantum criticality of Ising-nematic ordering in a metal
Ising-nematic order parameter
� ⇠Z
d2k (cos kx
� cos ky
) c†k�
ck�
Measures spontaneous breaking of square latticepoint-group symmetry of underlying Hamiltonian
Spontaneous elongation along x direction:Ising order parameter � > 0.
x
yQuantum criticality of Ising-nematic ordering in a metal
x
y
Spontaneous elongation along y direction:Ising order parameter � < 0.
Quantum criticality of Ising-nematic ordering in a metal
Pomeranchuk instability as a function of coupling �
��c
��⇥ = 0⇥�⇤ �= 0
or
Quantum criticality of Ising-nematic ordering in a metal
T
Phase diagram as a function of T and �
��⇥ = 0
Quantum critical
⇥�⇤ �= 0
TI-n
Quantum criticality of Ising-nematic ordering in a metal
��c
T
��⇥ = 0
Quantum critical
⇥�⇤ �= 0
TI-n Classicald=2 Isingcriticality
Quantum criticality of Ising-nematic ordering in a metal
Phase diagram as a function of T and �
��c
T
��⇥ = 0
Quantum critical
⇥�⇤ �= 0
rc
TI-n
Quantum criticality of Ising-nematic ordering in a metal
D=2+1 Ising
criticality ?
Phase diagram as a function of T and �
�
T
��⇥ = 0
Quantum critical
⇥�⇤ �= 0
rc
TI-n
D=2+1 Ising
criticality ?
Quantum criticality of Ising-nematic ordering in a metal
Phase diagram as a function of T and �
�
T
��⇥ = 0
Quantum critical
⇥�⇤ �= 0
rc
TI-n
Strongly-coupled“non-Fermi liquid”
metal with no quasiparticles
Quantum criticality of Ising-nematic ordering in a metal
Phase diagram as a function of T and �
�
T
��⇥ = 0⇥�⇤ �= 0
rc
TI-n
Strongly-coupled“non-Fermi liquid”
metal with no quasiparticles
Quantum criticality of Ising-nematic ordering in a metal
Fermiliquid
Fermiliquid
Quantum critical
Phase diagram as a function of T and �
�
T
��⇥ = 0⇥�⇤ �= 0
rc
TI-n
Strongly-coupled“non-Fermi liquid”
metal with no quasiparticles
StrangeMetal
Quantum criticality of Ising-nematic ordering in a metal
Fermiliquid
Fermiliquid
Phase diagram as a function of T and �
�
L = f†↵
✓@⌧ � r2
2m� µ
◆f↵
+ u f†↵f
†�f�f↵
The Fermi liquid
�� kF !
Occupied states
Empty states
The Fermi liquid: RG
FIG. 3: Alternative low energy formulation of Fermi liquid theory. We focus on an extended patch
of the Fermi surface, and expand in momenta about the point ~k0 on the Fermi surface. This yields
a theory of d-dimensional fermions in (7), with dispersion (14). The co-ordinate y represents the
d� 1 dimensions parallel to the Fermi surface.
(5) in one-dimensional. One benefit of (7) is now immediately evident: it has zero energy
excitations when
vFkx +
k
2y
2= 0, (8)
and so (8) defines the position of the Fermi surface, which is then part of the low energy
theory including its curvature. Note that (7) now includes an extended portion of the Fermi
surface; contrast that with (5), where the one-dimensional chiral fermion theory for each n̂
describes only a single point on the Fermi surface.
The gradient terms in (7) define a natural momentum space cuto↵, and associated scaling
limit. We will take such a limit at fixed ⇣, vF and . Notice that momenta in the x direction
scale as the square of the momenta in the y direction, and so we can choose v2Fk2x+
2k
4y < ⇤4.
Notice that as we reduce ⇤, we scale towards the single point ~k0 on the Fermi surface, as we
7
• Expand fermion kinetic energy at wavevectors about
~k0, bywriting f↵(~k0 + ~q) = ↵(~q)
L = f†↵
✓@⌧ � r2
2m� µ
◆f↵
+ u f†↵f
†�f�f↵
The Fermi liquid: RG
FIG. 3: Alternative low energy formulation of Fermi liquid theory. We focus on an extended patch
of the Fermi surface, and expand in momenta about the point ~k0 on the Fermi surface. This yields
a theory of d-dimensional fermions in (7), with dispersion (14). The co-ordinate y represents the
d� 1 dimensions parallel to the Fermi surface.
(5) in one-dimensional. One benefit of (7) is now immediately evident: it has zero energy
excitations when
vFkx +
k
2y
2= 0, (8)
and so (8) defines the position of the Fermi surface, which is then part of the low energy
theory including its curvature. Note that (7) now includes an extended portion of the Fermi
surface; contrast that with (5), where the one-dimensional chiral fermion theory for each n̂
describes only a single point on the Fermi surface.
The gradient terms in (7) define a natural momentum space cuto↵, and associated scaling
limit. We will take such a limit at fixed ⇣, vF and . Notice that momenta in the x direction
scale as the square of the momenta in the y direction, and so we can choose v2Fk2x+
2k
4y < ⇤4.
Notice that as we reduce ⇤, we scale towards the single point ~k0 on the Fermi surface, as we
7
L[ ↵
] = †↵
�@⌧
� i@x
� @2y
� ↵
+ u †↵
†�
�
↵
• Expand fermion kinetic energy at wavevectors about
~k0, bywriting f↵(~k0 + ~q) = ↵(~q)
L = f†↵
✓@⌧ � r2
2m� µ
◆f↵
+ u f†↵f
†�f�f↵
The Fermi liquid: RG
S[ ↵
] =
Zd
d�1y dx d⌧
h
†↵
�@
⌧
� i@
x
� @
2y
�
↵
+ u
†↵
†�
�
↵
i
The Fermi liquid: RG
S[ ↵
] =
Zd
d�1y dx d⌧
h
†↵
�@
⌧
� i@
x
� @
2y
�
↵
+ u
†↵
†�
�
↵
i
The kinetic energy is invariant under the rescaling x ! x/s,
y ! y/s
1/2, and ⌧ ! ⌧/s
z, provided z = 1 and
! s
(d+1)/4.
Then we find u ! us
(1�d)/2, and so we have the RG flow
du
d`
=
(1� d)
2
u
Interactions are irrelevant in d = 2 !
The Fermi liquid: RG
S[ ↵
] =
Zd
d�1y dx d⌧
h
†↵
�@
⌧
� i@
x
� @
2y
�
↵
+ u
†↵
†�
�
↵
i
The kinetic energy is invariant under the rescaling x ! x/s,
y ! y/s
1/2, and ⌧ ! ⌧/s
z, provided z = 1 and
! s
(d+1)/4.
Then we find u ! us
(1�d)/2, and so we have the RG flow
du
d`
=
(1� d)
2
u
Interactions are irrelevant in d = 2 !
The fermion Green’s function to order u2has the form (upto logs)
G(~q,!) =A
! � qx
� q2y
+ ic!2
So the quasiparticle pole is sharp. And fermion momentum distribution
function n(~k) =Df†↵
(
~k)f↵
(
~k)Ehad the following form:
The Fermi liquid: RG
S[ ↵
] =
Zd
d�1y dx d⌧
h
†↵
�@
⌧
� i@
x
� @
2y
�
↵
+ u
†↵
†�
�
↵
i
The Fermi liquid: RG
S[ ↵
] =
Zd
d�1y dx d⌧
h
†↵
�@
⌧
� i@
x
� @
2y
�
↵
+ u
†↵
†�
�
↵
i
n(k)
kkF
A
The fermion Green’s function to order u2has the form (upto logs)
G(~q,!) =A
! � qx
� q2y
+ ic!2
So the quasiparticle pole is sharp. And fermion momentum distribution
function n(~k) =Df†↵
(
~k)f↵
(
~k)Ehad the following form:
• Fermi wavevector obeys the Luttinger relation kdF ⇠ Q, the
fermion density
• Sharp particle and hole of excitations near the Fermi surface
with energy ! ⇠ |q|z, with dynamic exponent z = 1.
• The phase space density of fermions is e↵ectively one-dimensional,
so the entropy density S ⇠ T . It is useful to write this is as S ⇠T (d�✓)/z
, with violation of hyperscaling exponent ✓ = d� 1.
The Fermi liquid
L = f†✓@⌧ � r2
2m� µ
◆f
+ 4 Fermi terms�� kF !
Occupied states
Empty states
• Fermi wavevector obeys the Luttinger relation kdF ⇠ Q, the
fermion density
• Sharp particle and hole of excitations near the Fermi surface
with energy ! ⇠ |q|z, with dynamic exponent z = 1.
• The phase space density of fermions is e↵ectively one-dimensional,
so the entropy density S ⇠ T . It is useful to write this is as S ⇠T (d�✓)/z
, with violation of hyperscaling exponent ✓ = d� 1.
The Fermi liquid
L = f†✓@⌧ � r2
2m� µ
◆f
+ 4 Fermi terms
⇥| q|�
�� kF !
Occupied states
Empty states
L = f†✓@⌧ � r2
2m� µ
◆f
+ 4 Fermi terms
• Fermi wavevector obeys the Luttinger relation kdF ⇠ Q, the
fermion density
• Sharp particle and hole of excitations near the Fermi surface
with energy ! ⇠ |q|z, with dynamic exponent z = 1.
• The phase space density of fermions is e↵ectively one-dimensional,
so the entropy density S ⇠ T . It is useful to write this is as S ⇠T (d�✓)/z
, with violation of hyperscaling exponent ✓ = d� 1.
The Fermi liquid
⇥| q|�
�� kF !
Occupied states
Empty states
Pomeranchuk instability as a function of coupling �
��c
��⇥ = 0⇥�⇤ �= 0
or
Quantum criticality of Ising-nematic ordering in a metal
E�ective action for Ising order parameter
S⇥ =⇤
d2rd⇥�(⌅�⇤)2 + c2(⇤⇤)2 + (�� �c)⇤2 + u⇤4
⇥
Quantum criticality of Ising-nematic ordering in a metal
E�ective action for Ising order parameter
S⇥ =⇤
d2rd⇥�(⌅�⇤)2 + c2(⇤⇤)2 + (�� �c)⇤2 + u⇤4
⇥
E�ective action for electrons:
Sc =⌃
d�
Nf⇧
�=1
�
⇤⇧
i
c†i�⇤⇥ ci� �⇧
i<j
tijc†i�ci�
⇥
⌅
⇥Nf⇧
�=1
⇧
k
⌃d�c†k� (⇤⇥ + ⇥k) ck�
Quantum criticality of Ising-nematic ordering in a metal
��⇥ > 0 ��⇥ < 0
Coupling between Ising order and electrons
S�c
= � g
Zd⌧
NfX
↵=1
X
k,q
�q (cos kx� cos ky
)c†k+q/2,↵ck�q/2,↵
for spatially dependent �
Quantum criticality of Ising-nematic ordering in a metal
S�c
= � g
Zd⌧
NfX
↵=1
X
k,q
�q (cos kx� cos ky
)c†k+q/2,↵ck�q/2,↵
S⇥ =⇤
d2rd⇥�(⌅�⇤)2 + c2(⇤⇤)2 + (�� �c)⇤2 + u⇤4
⇥
Sc =Nf�
�=1
�
k
⇥d�c†k� (⇤⇥ + ⇥k) ck�
Quantum criticality of Ising-nematic ordering in a metal
• � fluctuation at wavevector ~q couples most e�ciently to fermions
near ±~k0.
• Expand fermion kinetic energy at wavevectors about ±~k0 andboson (�) kinetic energy about ~q = 0.
Quantum criticality of Ising-nematic ordering in a metal
• � fluctuation at wavevector ~q couples most e�ciently to fermions
near ±~k0.
• Expand fermion kinetic energy at wavevectors about ±~k0 andboson (�) kinetic energy about ~q = 0.
Quantum criticality of Ising-nematic ordering in a metal
L[ ±,�] =
†+
�@⌧
� i@x
� @2y
� + + †
��@⌧
+ i@x
� @2y
� �
��⇣ †+ + + †
� �
⌘+
1
2g2(@
y
�)2
M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075127 (2010)
Quantum criticality of Ising-nematic ordering in a metal
(a)
(b)Landau-damping
L = †+
�@⌧
� i@x
� @2y
� + + †
��@⌧
+ i@x
� @2y
� �
� �⇣ †+ + + †
� �
⌘+
1
2g2(@
y
�)2
One loop � self-energy with Nf
fermion flavors:
⌃�
(~q,!) = Nf
Zd2k
4⇡2
d⌦
2⇡
1
[�i(⌦+ !) + kx
+ qx
+ (ky
+ qy
)2]⇥�i⌦� k
x
+ k2y
⇤
=N
f
4⇡
|!||q
y
|
Quantum criticality of Ising-nematic ordering in a metal
(a)
(b)Electron self-energy at order 1/N
f
:
⌃(
~k,⌦) = � 1
Nf
Zd2q
4⇡2
d!
2⇡
1
[�i(! + ⌦) + kx
+ qx
+ (ky
+ qy
)
2]
"q2y
g2+
|!||q
y
|
#
= �i2p3N
f
✓g2
4⇡
◆2/3
sgn(⌦)|⌦|2/3
L = †+
�@⌧
� i@x
� @2y
� + + †
��@⌧
+ i@x
� @2y
� �
� �⇣ †+ + + †
� �
⌘+
1
2g2(@
y
�)2
Quantum criticality of Ising-nematic ordering in a metal
(a)
(b)Electron self-energy at order 1/N
f
:
⌃(
~k,⌦) = � 1
Nf
Zd2q
4⇡2
d!
2⇡
1
[�i(! + ⌦) + kx
+ qx
+ (ky
+ qy
)
2]
"q2y
g2+
|!||q
y
|
#
= �i2p3N
f
✓g2
4⇡
◆2/3
sgn(⌦)|⌦|2/3
L = †+
�@⌧
� i@x
� @2y
� + + †
��@⌧
+ i@x
� @2y
� �
� �⇣ †+ + + †
� �
⌘+
1
2g2(@
y
�)2
⇠ |⌦|d/3 in dimension d.
Quantum criticality of Ising-nematic ordering in a metal
L = †+
�@⌧
� i@x
� @2y
� + + †
��@⌧
+ i@x
� @2y
� �
� �⇣ †+ + + †
� �
⌘+
1
2g2(@
y
�)2
Schematic form of � and fermion Green’s functions in d dimensions
D(~q,!) =1/N
f
q2? +|!||q?|
, Gf
(~q,!) =1
qx
+ q2? � isgn(!)|!|d/3/Nf
In the boson case, q2? ⇠ !1/zb with zb
= 3/2.In the fermion case, q
x
⇠ q2? ⇠ !1/zf with zf
= 3/d.
Note zf
< zb
for d > 2 ) Fermions have higher energy thanbosons, and perturbation theory in g is OK.Strongly-coupled theory in d = 2.
Quantum criticality of Ising-nematic ordering in a metal
L = †+
�@⌧
� i@x
� @2y
� + + †
��@⌧
+ i@x
� @2y
� �
� �⇣ †+ + + †
� �
⌘+
1
2g2(@
y
�)2
Schematic form of � and fermion Green’s functions in d = 2
D(~q,!) =1/N
f
q2y
+
|!||q
y
|
, Gf
(~q,!) =1
qx
+ q2y
� isgn(!)|!|2/3/Nf
In both cases qx
⇠ q2y
⇠ !1/z, with z = 3/2. Note that the
bare term ⇠ ! in G�1f
is irrelevant.
Strongly-coupled theory without quasiparticles.
Quantum criticality of Ising-nematic ordering in a metal
Simple scaling argument for z = 3/2.
L = †+
�@⌧
� i@x
� @2y
� + + †
��@⌧
+ i@x
� @2y
� �
� �⇣ †+ + + †
� �
⌘+
1
2g2(@
y
�)2
Quantum criticality of Ising-nematic ordering in a metal
Simple scaling argument for z = 3/2.
L = †+
�@⌧
� i@x
� @2y
� + + †
��@⌧
+ i@x
� @2y
� �
� �⇣ †+ + + †
� �
⌘+
1
2g2(@
y
�)2
X X
Quantum criticality of Ising-nematic ordering in a metal
Simple scaling argument for z = 3/2.
L = †+
�@⌧
� i@x
� @2y
� + + †
��@⌧
+ i@x
� @2y
� �
� �⇣ †+ + + †
� �
⌘+
1
2g2(@
y
�)2
X X
Under the rescaling x ! x/s, y ! y/s
1/2, and ⌧ ! ⌧/s
z, we
find invariance provided
� ! � s
! s
(2z+1)/4
g ! g s
(3�2z)/4
So the action is invariant provided z = 3/2.
Quantum criticality of Ising-nematic ordering in a metal
⇥| q|�
�� kF !
FL Fermi liquid
• kdF ⇥ Q, the fermion density
• Sharp fermionic excitations
near Fermi surface with
⇥ ⇥ |q|z, and z = 1.
• Entropy density S ⇥ T (d��)/z
with violation of hyperscaling
exponent � = d� 1.
• Entanglement entropy
SE ⇥ kd�1F P lnP .
• Fermi surface
with kdF ⇠ Q.
• Di↵use fermionic
excitations with z = 3/2to three loops.
• S ⇠ T (d�✓)/z
with ✓ = d� 1.
• SE ⇠ kd�1F P lnP .
⇥| q|�
�� kF ! �� kF !
NFLNematic
QCP
• kdF ⇥ Q, the fermion density
• Sharp fermionic excitations
near Fermi surface with
⇥ ⇥ |q|z, and z = 1.
• Entropy density S ⇥ T (d��)/z
with violation of hyperscaling
exponent � = d� 1.
• Entanglement entropy
SE ⇥ kd�1F P lnP .
FL Fermi liquid
n(k)
kkF
• Fermi surface
with kdF ⇠ Q.
• Di↵use fermionic
excitations with z = 3/2to three loops.
• S ⇠ T (d�✓)/z
with ✓ = d� 1.
• SE ⇠ kd�1F P lnP .
M. A. Metlitski and S. Sachdev,Phys. Rev. B 82, 075127 (2010)
⇥| q|�
�� kF ! �� kF !⇥| q
|�
• kdF ⇥ Q, the fermion density
• Sharp fermionic excitations
near Fermi surface with
⇥ ⇥ |q|z, and z = 1.
• Entropy density S ⇥ T (d��)/z
with violation of hyperscaling
exponent � = d� 1.
• Entanglement entropy
SE ⇥ kd�1F P lnP .
FL Fermi liquid
NFLNematic
QCP
• Fermi surface
with kdF ⇠ Q.
• Di↵use fermionic
excitations with z = 3/2to three loops.
• S ⇠ T (d�✓)/z
with ✓ = d� 1.
• SE ⇠ kd�1F P lnP .
⇥| q|�
�� kF !⇥| q
|�
�� kF !
• kdF ⇥ Q, the fermion density
• Sharp fermionic excitations
near Fermi surface with
⇥ ⇥ |q|z, and z = 1.
• Entropy density S ⇥ T (d��)/z
with violation of hyperscaling
exponent � = d� 1.
• Entanglement entropy
SE ⇥ kd�1F P lnP .
FL Fermi liquid
NFLNematic
QCP
T
��⇥ = 0⇥�⇤ �= 0
rc
TI-n
Strongly-coupled“non-Fermi liquid”
metal with no quasiparticles
StrangeMetal
Quantum criticality of Ising-nematic ordering in a metal
Fermiliquid
Fermiliquid
Phase diagram as a function of T and �
�