Introduction in Bosonization I

Outline Motivation The Fermion-Boson Correspondence The Luttinger Model

Vadim Cheianov

Lancaster University

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model

1 Motivation

2 The Fermion-Boson Correspondence

3 The Luttinger Model

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Non-interacting or Interacting?

In every physical system there are correlations between constituent particles due to various interactions ”Non-interacting systems” - systems whose physics can be understood using the single particle picture and where correlations are an annoying ingredient reducing the predictive power of the theory ”Interacting systems” - systems whose essential physical properties would not exist without interactions

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Systems Where Interactions are Essential

Here are some textbook examples: Ferro and antiferromagnets Superconductors Fractional quantum Hall systems Kondo impurities Systems at Coulomb Blockade Luttinger liquids

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model The Aim of this Lecture Course

To give an elementary introduction into the physics of 1D Luttinger liquids, the basics of Bosonization technique and the scaling theory.

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Outline

Part I This is a formal part, where we will discuss the fermion-boson correspondence, the Luttinger model and the Bosonization.

Part II In this part a contact to reality will be made. The concept of Luttinger Liquid will be introduced and physical examples will be given.

Part III In this part applications of Luttinger Liquid theory to quantum impurity problems will be discussed.

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model

The Fermion-Boson Correspondence

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model The Harmonic Oscillator

Bosonic Oscillator Fermionic Oscillator The Bosonic algebra: The Fermionic algebra:

[b, b†] = 1 ψψ† + ψ†ψ = 1

[b, b] = [b†, b†] = 0 {ψ, ψ} = {ψ†, ψ†} = 0 The Fock space: The Fock space:

b|0i = 0, |ni = (b†)n|0i ψ|0i = 0, |1i = ψ†|0i

The Hamiltonian The Hamiltonian

† † HB = ~ωNB , NB = b b HF = ~ωNF , NF = ψ ψ

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Occupation Numbers

Bosons Fermions The extended Bosonic algebra: The extended Fermionic algebra:

† † bn, bn, n = 1,..., N ψn, ψn, n = 1,..., M

† † [bn, bm] = δnm {ψn, ψm} = δnm The Fock space is spanned by The Fock space is spanned by states of given occupation num- states of given occupation numbers 0 ≤ nk ≤ ∞ bers 0 ≤ nk ≤ 1

† n1 † nN † n1 † nM |n1 ... nN i = (b1) ... (bN ) |0i |n1 ... nN i = (ψ1) ... (ψM ) |0i

† † † † NB = b1b1 + ··· + bN bN NF = ψ1ψ1 + ··· + ψM ψM

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model

Fermion-Boson Correspondence: NF = 1

For extended fermionic oscillator † algebra ψk , ψk , 0 ≤ k ≤ ∞ and

∞ X † HF = kNk , Nk = ψk ψk k=0

The Correspondence of States:

The eigenstates of HF for NF = 1 are in one to one correspondence † to the eigenstates of HB = b b.

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model

Fermion-Boson Correspondence: NF = 1

For extended fermionic oscillator † algebra ψk , ψk , 0 ≤ k ≤ ∞ and

∞ X † HF = kNk , Nk = ψk ψk k=0

The Correspondence of States:

The eigenstates of HF for NF = 1 are in one to one correspondence † to the eigenstates of HB = b b.

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model

Fermion-Boson Correspondence: NF = 1

For extended fermionic oscillator † algebra ψk , ψk , 0 ≤ k ≤ ∞ and

∞ X † HF = kNk , Nk = ψk ψk k=0

The Correspondence of States:

The eigenstates of HF for NF = 1 are in one to one correspondence † to the eigenstates of HB = b b.

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model

Fermion-Boson Correspondence: NF = 1

For extended fermionic oscillator † algebra ψk , ψk , 0 ≤ k ≤ ∞ and

∞ X † HF = kNk , Nk = ψk ψk k=0

The Correspondence of States:

The eigenstates of HF for NF = 1 are in one to one correspondence † to the eigenstates of HB = b b.

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model

Fermion-Boson Correspondence: NF = 2

Fermion Hamiltonian Eigenstates of H and H ∞ F B X † HF = −1 + kψk ψk , Fermions Bosons E k=0 |110000 ... i |00i 0 |101000 ... i |10i 1 for NF = 2. |011000 ... i |01i 2 |100100 ... i |20i Boson Hamiltonian |010100 ... i |11i 3 |100010 ... i |30i † † HB = b1b1 + 2b2b2

Energy levels and their degeneracies coincide for HF and HB

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model

Fermion-Boson Correspondence: general NF

Consider a pair of Hamiltonians

∞ N X † X † HF = −EN + ~ω kψk ψk and HB = ~ω mbmbm. k=0 m=1

where HF constrained onto the subspace of ﬁxed particle number NF = N. Here EN = ~ωN(N − 1)/2. Theorem

The eigenvalues of HB and HF coincide and are given by En = ~ωn, where n is a non-negative integer. Each level En has the same degeneracy D(n) for HB and HF .

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Fermion-Boson Correspondence. Exercises.

Problem 1 Prove the theorem in the previous slide.

Problem 2

For the eigenvalue En of HB calculate the degeneracy D(n) of the corresponding eigenspace. Calculate large n asymptotics of D(n) for n N and for n N.

Problem 3

P −βEn Calculate the quantum partition function Z(β) = n D(n)e , free energy and the speciﬁc heat of the system described by the Hamiltonian HB (HF ) in the large N limit.

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Weyl Fermion in 1+1 Dimensions I

Consider the Hamiltonian: ∞ v X 1 H = m − : ψ† ψ : F r 2 m m m=−∞

The ground state is the Dirac vacuum:

† ψ−m+1|0i = ψm|0i = 0, m ≥ 1

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Weyl Fermion in 1+1 Dimensions II

Introduce a local ﬁeld: ∞ 1 X i mx ψ(x) = √ e r ψm 2πr m=−∞

{ψ(x), ψ†(x0)} = δ(x − x0) Time evolution is generated by Hamiltonian

Z 2πr † HF = v dx : ψ (x)(−i∂x )ψ(x): 0

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Singularities and Normal Ordering

Consider a point-split product ψ†(x)ψ(x + ). It is singular in the following sense: 1 hψ†(x)ψ(x + )i = + O(1), → 0 2πi This can be used for an alternative deﬁnition of normal ordering:

1 : ψ†(x)ψ(x) := lim ψ†(x)ψ(x + ) − →0 2πi

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Weyl Fermion: The Current Anomaly

The density operator is ρ(x) = ψ†(x)ψ(x). Normally, the density operator satisﬁes [ρ(x), ρ(x0)] = 0 For the Weyl fermion ψ†(x)ψ(x) is singular The non-singular (normal ordered) expression is ρ(x) =: ψ†(x)ψ(x): The Current Anomaly: 0 i 0 0 The normal ordered density satisﬁes [ρ(x), ρ(x )] = 2π δ (x − x )

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Weyl Fermion. Exercises.

Problem 1 Derive the formula for the current anomaly. (Hint: get rid of the singular operators in the result of the commutation before taking the → 0 limit)

Problem 2 Derive the equation of motion for the Weyl fermion ψ(x, t). Use the equation of motion to show that all local operators of the theory satisfy O(x, t) = O(x − vt)

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Chiral Boson in 1+1 Dimensions I

Consider a Hamiltonian v X v H = mb† b + N2 B r m m 2r m≥1

where N is the ”angular momentum” operator of a rotator algebra

[ϕ0, N] = i, ϕ0 + 2π = ϕ0.

The Fermion-Boson Correspondence in 1+1 D The energy levels and their degeneracies for the Weyl fermion and the chiral boson on a cylinder coincide. N for the chiral boson corresponds to the particle number of fermions.

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Chiral Boson in 1+1 Dimensions II

Introduce a local ﬁeld:

x X 1 † −i mx i mx ϕ(x) = ϕ + N + i b e r − b e r 0 r p m m m≥0 |m|

this ﬁeld satisﬁes ϕ(x, t) = ϕ(x, t) + 2π and can be considered as a map from a cylinder onto a unit circle. The commutation relations are [ϕ(x), ϕ(x0)] = −iπsgn(x − x0) The Hamiltonian becomes Z 2πr v 2 HB = dx(∂x ϕ) 4π 0

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Local Fermion-Boson Correspondence

By comparison of local operator algebras one can establish a local Fermion-Boson correspondence called Bosonization Example: Bosonization of density operator For the Weyl fermion For the chiral boson 0 i 0 0 0 0 [ρ(x), ρ(x )] = 2π δ (x − x ) [ϕ(x), ϕ(x )] = −iπsgn(x − x )

⇓ 1 ρ(x) = ∂ ϕ 2π x

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Bosonization of Fermion Field

Local Fermion Algebra

ψ(x)ψ(x0) + ψ(x0)ψ(x) = 0, ψ(x)ψ†(x0) + ψ†(x0)ψ(x) = δ(x − y)

[ρ(x), ψ(x0)] = −ψ(x)δ(x − x0)

Bosonization of the local fermion algebra 1 ρ(x) = ∂ ϕ, ψ(x) = c : eiϕ(x) : 2π x Here : eiϕ :≡ eiϕ+ eiϕ− , that is all annihilation operators in ϕ are put to the right of creation operators. The constant c depends on the ultraviolet regularization.

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Bosonization Exercies

Problem 1 1 iϕ(x) Prove that the operators 2π ∂x ϕ, : e : satisfy the same algebra as ρ(x), ψ(x). Hint: use the Campbell- Hausdorﬀ formula for two operators whose commutator is a c-number:

eAeB = eB eAe[A,B]

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Free Massless Dirac Fermions in 1+1 Dimensions

Free Massless Dirac Fermion = 2 Weyl Fermions

1 X ikx ψL,R (x) = √ e ψL,R (k) 2πr k Z h † † i HF = v dx ψR (−i∂x )ψR − ψL(−i∂x )ψL

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Bosonization of Free Massless Dirac Field

Each Weyl fermion is Bosonized by its chiral boson ϕL(R).

0 0 0 0 [ϕL(x), ϕL(x )] = iπsgn(x−x )[ϕR (x), ϕR (x )] = −iπsgn(x−x ) v Z H = dx (∂ ϕ )2 + (∂ ϕ )2 B 4π x R x L Bosonization Rules 1 ρ (x) = ∂ ϕ , ψ (x) = eiϕR (x) R 2π x R R 1 ρ (x) = ∂ ϕ , ψ (x) = e−iϕL(x) L 2π x L L

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Free Compact Boson Field in 1+1 Dimensions

Change of Variables

φ = ϕR + ϕL and θ = ϕR − ϕL v Z H = dx (∂ θ)2 + (∂ φ)2 B 2π x x

Fields φ and Π = ∂x θ/π are canonically conjugate. In Lagrangian formulation: 1 Z S = dxdt (∂ φ)2 − v 2(∂ φ)2 B 2πv t x

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Summary.

Weyl=Chiral Boson

Z v Z 1 v dxψ†(−i∂ )ψ = dx(∂ ϕ)2, ψ = eiϕ, ρ(x) = ∂ ϕ x 4π x 2π x

Massless Dirac=Free Boson

Z h i v Z v dx ψ† (−i∂ )ψ − ψ†(−i∂ )ψ = dx (∂ θ)2 + (∂ φ)2 R x R L x L 2π x x

i(θ−φ) i(θ+φ) 0 0 ψL = e , ψR = e , [φ(x ), ∂x θ(x)] = iπδ(x − x )

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model

The Luttinger Model

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model The Luttinger Model

The Luttinger model is probably the simplest model describing interacting relativistic fermions. The interaction is characterized by a dimensionless coupling γ.

The Luttinger Hamiltonian

Z h † † i HLUT = v dx : ψLi∂x ψL − ψR i∂x ψR + γρL(x)ρR (x) :

Note, that all terms are normal ordered!

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Perturbation Theory to Luttinger Model

The Schwinger-Dyson perturbation theory:

The perturbation theory contains divergent terms which need to be renormalized in the spirit of Gell-Mann and Low RG. The beta function of this theory vanishes to all orders in parameter γ.

Exact resummation of the perturbation series using the axial Word identities I. E. Dzyaloshinsky and A. I. Larkin, Sov. Phys. JETP 38, 202 (1974)

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Bosonization of the Luttinger Model

The kinetic term of the Luttinger Hamiltonian is the massless Dirac Hamiltonian. Its Bosonization is a Boson Hamiltonian. The bosonization of the interaction term is 1 1 : ρ (x)ρ (x) :=: (∂ φ + ∂ θ) (∂ φ − ∂ θ) : R L 2π x x 2π x x therefore Z h † † i HLUT = v dx ψLi∂x ψL − vψR i∂x ψR + γρL(x)ρR (x) ⇓ v Z n γ o HB = dx (∂ θ)2 + (∂ φ)2 + [(∂ φ)2 − (∂ θ)2] LUT 2π x x 2π x x

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Bosonization of the Luttinger Model

Luttinger Hamiltonian. Standard Notations. v Z 1 H = c dx (∂ φ)2 + K(∂ θ)2 LUT 2π K x x where 0 0 [∂x θ(x), φ(x )] = −iπδ(x − x )

r γ 2 v = v 1 − is the sound velocity c 2π s 1 − γ/2π K = is the Luttinger parameter 1 + γ/2π

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Luttinger Model. Exercises.

Problem 1

Using the Heisenberg equation i∂t A = [A, H] show that ﬁeld φ satisﬁes the wave equation

2 2 2 ∂t φ − vc ∂x φ = 0

Problem 2 By performing the Legendre transform of the bosonized Luttinger Hamiltonian ﬁnd the Lagrange density of the Bose ﬁeld φ (in this case π∂x θ should be treated as the momentum density).

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Luttinger Model. Lagrangian Formulation.

Imaginary time Luttinger Action Action for the φ ﬁeld: " # 1 Z β Z 1 ∂φ2 ∂φ2 Si = dτ dx + vc 2πK 0 vc ∂τ ∂x

Action for the θ ﬁeld: " # K Z β Z 1 ∂θ 2 ∂θ 2 Si = dτ dx + vc 2π 0 vc ∂τ ∂x

Note the duality θ → φ, K → K −1.

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Correlation Functions: Bosons I

We start with the imaginary time correlator

G(x, τ) = hT φ(x, τ)φ(x0)i

It is a Fourier transform of X Z dk G(x, τ) = β−1 eikx−iωnτ G(iω , k) 2π n ωn

where G(iωn, k) in a free Boson theory it is given by

πvc K 2πn G(iωn, k) = 2 2 2 , ωn = ωn + vc k β

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Correlation Functions: Bosons II

Temperature Correlator of Bose fields

2π 2π K h − (|x|+ivc τ)i K h − (|x|−ivc τ)i G(x, τ) = − ln 1 − e βvc − ln 1 − e βvc 4 4 In the limit of zero temperature T → 0 this becomes K G(x, τ) = − ln(x2 + v 2τ 2) + c 4 c In real time t = −iτ there are light cone singularities at

x = ±vc t

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Gaussian Integration

Write the quadratic action of boson in a symbolic form 1 S = φG−1φ i 2 Then for a source ﬁeld η there is a Gaussian Integration Formula Z iηφ 1 −S iηφ − 1 ηGη hTe i = Dφe i e = e 2 Z For example,

0 0 0 0 hTeiφ(x,τ)e−iφ(x ,τ )i = c eG(x−x ,τ−τ )

here c = e−hφ(0)2i is an (inﬁnite) constant.

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Correlation Function of Fermions. T = 0

The right-moving Fermion is given by

iφR (x) iθ(x)+iφ(x) ψR (x) = e ≡ e

Applying Gaussian Integration Formula to this expression we ﬁnd Gaussian Integration Formula c hT ψ (x, τ)ψ† (x0)i = R R ∆ ∆¯ (x + ivc τ) (x − ivc τ) where (1 + K)2 (1 − K)2 ∆ = and ∆¯ = 4K 4K

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Correlation Function of Fermions. T = 0

The structure of correlation function c hT ψ (x, τ)ψ† (x0)i = R R ∆ ∆¯ (x + ivc τ) (x − ivc τ) suggests that in interacting system the ”right” electron is no more a pure right-mover. It rather splits into two counterpropagating wave-packets. This is called charge fractionalization.

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Particle occupation numbers.

The particle occupation numbers are found as Z −ikx † 0 ∆+∆¯ −1 nR (k) = dxe hT ψR (x)ψR (x )i = n0 + csgn(k)|k|

(K − 1)2 ∆ + ∆¯ − 1 = > 0 2K Instead of the sharp Fermi step there is a continuous distribution with a power-law singularity at k = 0.

Vadim Cheianov Introduction in Bosonization I Outline Motivation The Fermion-Boson Correspondence The Luttinger Model Summary.

Using the Fermion-Boson correspondence we solved exactly a non-trivial interacting system of fermions. We found that the spectrum of the system is described by bosons (phonons) whose velocity is renormalized by interactions. The interaction eﬀects are encoded in the Luttinger parameter K. For K 6= 1 charge fractionalization and the disappearance of Fermi step are observed.

Vadim Cheianov Introduction in Bosonization I