Fermi surfaces and the AdS/CFT correspondence

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Fermi surfaces and the

AdS/CFT correspondence

Indian Institute of Science, Bangalore, Dec 8, 2010

Lecture notes

arXiv:1010.0682

arXiv: 1012.0299

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Science 317, 1356 (2007)

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Hubbard model on a bilayer triangular lattice

H = H_a + H_b + H_ab H_a = −t �

�ij�

c^†_aiαc_aj_α + H.c. + (�_a − µ) �

i

(n_ai_↑ + n_ai_↓)

+ U �

i

�

n_ai_↑ − 1 2

� �

n_ai_↓ − 1 2

�

H_b = −t �

�ij�

c^†_biαc_bj_α + H.c. + (�_b − µ) �

i

(n_bi_↑ + n_bi_↓)

+ U �

i

�

n_bi_↑ − 1 2

� �

n_bi_↓ − 1 2

�

_a

< �

_b

< �

_a

+ U a

b

(4)

Small w

Fermi liquid (FL) phase

Adiabatically connected to the state at U=0 Layer a

Area A

¹

Layer b

Area A

²

(5)

Fermi liquid (FL) phase

Adiabatically connected to the state at U=0

Area A

¹

A

¹

+ A

²

2π

²

= N

Large w

A

²

= 0

(6)

1. The Kondo lattice model

The fractionalized Fermi liquid (FL*) phase

2. Mean field theories of the FL* phase

Infinite-range Kondo lattice models and AdS

2

x R

²

3. Gauge theory of the FL and FL* phases

U(1)xU(1) theories and semi-holographic phases

Outline

(7)

2

x R

²

3. Gauge theory of the FL and FL* phases

U(1)xU(1) theories and semi-holographic phases

Outline

(8)

Kondo lattice model

Perform canonical transformation of

Hubbard model on layer a

to a Heisenberg antiferromagnet

(9)

Kondo lattice model

� J

_H

(i, j ) S �

_i

· S �

_j

Conduction electrons from layer b

�

k

ε

_k

c

^†_kα

c

_kα

(10)

Kondo lattice model

Kondo exchange J

_K

�

i

S �

_i

· c

^†_iα

�σ

_αβ

c

_iβ

J

_K

∼ w

²

U

� ε c

^†

c

(11)

Kondo lattice model

� J

_H

(i, j ) S �

_i

· S �

_j

Large Fermi surface Fermi liquid (FL)

Kondo exchange J

_K

�

i

S �

_i

· c

^†_iα

�σ

_αβ

c

_iβ

J

_K

∼ w

²

U

Obtained as J

_K

→ ∞ . Electron Fermi surfaces

have total area A

2π

²

= N

�

k

ε

_k

c

^†_kα

c

_kα

(12)

Kondo lattice model

(13)

Kondo lattice model

� J

_H

(i, j ) S �

_i

· S �

_j

(14)

Kondo lattice model

Spin liquid of electrons

on layer a

(15)

Kondo lattice model

� J

_H

(i, j ) S �

_i

· S �

_j

on layer a

(16)

Kondo lattice model

on layer a

(17)

Kondo lattice model

� J

_H

(i, j ) S �

_i

· S �

_j

�

k

ε

_k

c

^†_kα

c

_kα

(18)

Kondo lattice model

� ε c

^†

c

Kondo

exchange

perturbative

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Kondo lattice model

� J

_H

(i, j ) S �

_i

· S �

_j

Kondo exchange perturbative

�

k

ε

_k

c

^†_kα

c

_kα

Obtained when J

_K

is

“irrelevant”.

Electron Fermi surfaces have total area

A − 1

2π

²

= N

T. Senthil, S. Sachdev, and M. Vojta, Phys. Rev. Lett. 90, 216403 (2003).

T. Senthil, M. Vojta, and S. Sachdev, Phys. Rev. B 69, 035111 (2004).

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Infinite-range Kondo lattice models and AdS

2

x R

²

Outline

(21)

2

x R

²

3. Gauge theory of the FL and FL* phases

Outline

(22)

�

i<j

J

_H

(i, j ) S �

_i

· S �

_j

A mean-field theory of a spin liquid

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�

i<j

J

_H

(i, j ) S �

_i

· S �

_j

J

_H

(i, j ) Gaussian random variables.

A quantum Sherrington-Kirkpatrick model of SU(N ) spins.

Described by the quantum mechanics of a spin

fluctuating in a self-consistent

time-dependent magnetic field: a realization the finite

entropy density

AdS

₂

× R

^d

state

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AdS

2

realization in the quantum SK model

Focus on a single S� spin, and represent its imaginary time fluctuations by a unit vector S� = �n(τ)/2 which is controlled by the partition function

Z =

�

D�n(τ) δ(�n²(τ) − 1) exp (−S) S = i

2

� 1

0

du

� 1/T

0

dτ �n ·

� ∂�n

∂u × ∂�n

∂τ

�

−

� 1/T

0

dτ �h(τ) · �n(τ) The first term is a Wess-Zumino term, with the “extra dimension”

u defined so that �n(τ, u = 1) ≡ �n(τ) and �n(τ, u = 0) = (0, 0, 1).

The field �h(τ) represents the “environment”, which which we take to be a Gaussian random variable with the correlation

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AdS

2

realization in the quantum SK model

Solution of Z for such an � h(τ ) yields

� � n(τ ) · � n(0) � = B

� �

� � π T

sin(π T τ )

� �

h

with the exponent h = 2 − γ . The self-consistency condition for the infinite-range model requires that the two-point correlation of

� h is proporionaly to that of � n. This leads to h = γ , which implies

h = γ = 1.

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�

i<j

J

_H

(i, j ) S �

_i

· S �

_j

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�

i<j

J

_H

(i, j ) S �

_i

· S �

_j

J

_H

(i, j ) Gaussian random variables.

A quantum Sherrington-Kirkpatrick model of SU(N ) spins.

A mean-field theory of FL*

Kondo exchange perturbative

�

k

ε

_k

c

^†_kα

c

_kα

(28)

Effective low energy theory for conduction electrons

The operators acting on the low energy subspace are layer a electrons c

_i

and layer b spins S �

_i

.

For the c

_i

we have the eﬀective theory S

^c

=

� d

^d

k (2π )

^d

�

dτ

�

c

^†_kα

� ∂

∂τ − ε

_k

�

c

_kα

+ J

_K

�

i

S �

_i

· c

_iα

�σ

_αβ

c

_iβ

�

(29)

_i

and layer b spins S �

_i

.

For the c

_i

^c

=

� d

^d

k (2π )

^d

�

dτ

�

c

^†_kα

� ∂

∂τ − ε

_k

�

c

_kα

− J

_K

2 F

_kα^†

c

_kα

− J

_K

2 c

^†_kα

F

_kα

�

Here the F

_iα

are strongly renormalized operators on layer b, which project onto the low energy theory as

F

_iα

= �

�σ

_αβ

· S �

_i

�

c

_iβ

Effective low energy theory for conduction electrons

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Effective low energy theory for conduction electrons

From this we obtain the conduction electron self energy

_i

and layer b spins S �

_i

.

For the c

_i

^c

=

� d

^d

k (2π )

^d

�

dτ

�

c

^†_kα

� ∂

∂τ − ε

_k

�

c

_kα

− J

_K

2 F

_kα^†

c

_kα

− J

_K

2 c

^†_kα

F

_kα

�

Here the F

_iα

are strongly renormalized operators on layer b, which project onto the low energy theory as

F

_iα

= �

�σ

_αβ

· S �

_i

�

c

_iβ

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• The quantum SK model has z = ∞ conformal spin correlations and a finite ground state entropy density: similar to AdS₂×R^d.

• The conduction electrons are ‘probe fermions’ coupling to the SK model by

S^c =

� d^dk (2π)^d

�

dτ

�

c^†_kα

� ∂

∂τ − ε_k

�

c_kα−V F_kα^† c_kα−V c^†_kαF_kα

�

where F_iα are operators probing the z = ∞ correlations of AdS₂×R^d (T. Faulkner and J. Polchinski, arXiv:1001.5049.)

F_iα ∼ 1 U

��σ_αβ · S�_{f i}�

c_iβ

• This leads to a ‘probe fermion’ self energy which is identical to the AdS × R^d theory of the holographic metal (T. Faulkner,

Connection to semi-holographic metals

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S^c =

� d^dk (2π)^d

�

dτ

�

c^†_kα

� ∂

∂τ − ε_k

�

F_iα ∼ 1 U

��σ_αβ · S�_{f i}�

c_iβ

Connection to semi-holographic metals

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Connection to semi-holographic metals

S^c =

� d^dk (2π)^d

�

dτ

�

c^†_kα

� ∂

∂τ − ε_k

�

F_iα ∼ 1 U

��σ_αβ · S�_{f i}�

c_iβ

• This leads to a ‘probe fermion’ self energy which is identical to the AdS × R^d theory of the holographic metal (T. Faulkner,

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1. The Kondo lattice model

2

x R

²

Outline

(35)

The fractionalized Fermi liquid (FL*) phase

2

x R

²

Outline

(36)

Gauge theory of FL and FL* phases

We want to keep better track of the charge on layer a. For this we introduce a ‘fictitious’ quantum rotor on each a lattice site. Each rotor has a periodic angular co-ordinate ϑ_i with period 2π; hence the states of the rotors are eⁱⁿ^ri^ϑⁱ where n_ri is a rotor angular

momentum, which takes all positive and negative integer values.

We will use the state with all n_ri = 0 to represent the states with one electron each a lattice site. Then we write

c_aα = e⁻ⁱ^ϑf_α (1)

where we have dropped the implicit site index, and f_α are neutral fermions (‘spinons’) which keep track of the orientation of the

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We can now identify the 4 states on each a lattice site with states of the rotor and spinons:

|0� ⇔ e⁻ⁱ^ϑ|0� c_aα^† |0� ⇔ f_α^†|0�

c_a^†_↑c_a^†_↓|0� ⇔ eⁱ^ϑf_↑^†f_↓^†|0� (2) Note that these allowed states obey the constraint

f_α^†f_α − n_r = 1. (3)

Associated with this constraint is the U(1) gauge invariance

f_α → f_αeⁱ^ζ , ϑ → ϑ + ζ. (4)

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We can now write down an eﬀective continuum U(1) gauge theory which captures the low energy physics of the Hubbard model. The degrees of freedom are the b layer electrons c_α, the a layer spinons f_α, and the bosonic rotors

b ∼ e⁻ⁱ^ϑ. (5)

L = L^f + L^b + L^c L^f = f_α^†

� ∂

∂τ + �_f − iA_τ − 1

2m_f (∇ − iA)²

� f_α L^b = �

(∂_µ − (�_r − µ)δ_µτ − iA_µ + iA_ext,µ) b^†�

× �

(∂_µ + (�_r − µ)δ_µτ + iA_µ − iA_ext,µ) b�

+ s|b|² + u|b|⁴ L^c = c_α^†

� ∂

∂τ − µ − iA_ext,τ − 1

2m (∇ − iA_ext)²

�

c_α

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a source term which couples to the current of the globally conserved electromagnetic charge.

The continuum theory in Eq. (6) has a U(1)×U(1)_ext symmetry associated with the transformations

f_α → f_αeⁱ^ζ , b → b⁻ⁱ^ζ , c_bα → c_b_α

f_α → f_α , b → bⁱ^ζ^˜ , c_b_α → c_b_αeⁱ^ζ^˜ (7) There are Fermi surface area constraints associated with these two U(1) symmetries:

�

α

�f_α^†f_α�

−

�∂L^b

∂µ

�

= A¹

2π² = N^a. (8)

�

α

�c_b^†_αc_bα� +

�∂L^b

∂µ

�

= A²

2π² = N − N^a. (9) N^a is the density of electrons on layer a in the projected Hilbert

space: our present lattice derivation was for N^a = 1, but the

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These two area constraints apply only if the U(1)×U(1)_ext

symmetry is not spontaneously broken. This is the case only in the FL* phase:

�b� = 0 in the FL* phase. (10)

Layer a

Layer b Spinon

Fermi surface of area

A

¹

2π

²

= 1

Electron Fermi surface

of area A²

2π² = N − 1

(41)

In the FL phase the U(1)×U(1)_ext symmetry is broken down to a diagonal U(1) because

�b� �= 0 in the FL phase. (11) Only the sum of the constraints in Eqs. (8) and (9) applies, and

this leads to expected area constraint.

Electron

Fermi surface(s) of total area

A

¹

2π

²

= N

(42)

FL phase

In the FL phase, we condense the b boson, and focus on the

fluctuations of its phase b = e⁻ⁱ^ϑ. Then the eﬀective theory of the FL phase of Eq. (6) is

L^FL = K (∂_µϑ − A_µ + A_ext,µ)² + Π_f (A_µ) + L^c (12) where Π_f is the eﬀective action obtain after integrating out the f spinons

The structure of Eq. (12) is nearly identical to the

semi-holographic of metals by Nickel and Son. They argued that there is generically an emergent gauge field A_µ which links the UV fields (the ‘electrons’, c ) to the IR fields near the horizon (the

Connections to semi-holographic RG

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FL* phase

Now the b field is not condensed, and so we can integrate it out, and obtain an eﬀective theory for the electrons and the spinons

L^FL_∗ = L^f + J_K �

c_α^†σ_αβ^a c_β� �

f_γ^†σ_γδ^a f_δ�

+ L^c (13)

We can now rewrite this as

L^FL_∗ = L^f − J_K 2

�F_α^†c_α + c_α^†F_α�

+ L^c (14)

where F_α is a IR fermion invariant under the emergent U(1) F_α ≡ − �

σ_αβ^a f_γ^†σ_γδ^a f_δ�

c_δ (15)

This is precisely the semi-holographic thery of Faulkner and