Lecture I 1
Lecture I: Collective Excitations: From Particles to Fields Free Scalar Field Theory: Phonons
The aim of this course is to develop the machinery to explore the properties of quantum systems with very large or infinite numbers of degrees of freedom. To represent such systems it is convenient to abandon the language of individual elementary particles and speak about quantum fields. In this lecture, we will consider the simplest physical example of a free or non-interacting many-particle theory which will exemplify the language of classical and quantum fields. Our starting point is a toy model of a mechanical system describing a classical chain of atoms coupled by springs.
. Discrete elastic chain
RI-1 φ I M ks
(I-1)a Ia (I+1)a
Equilibrium positionx ¯n na; natural length a; spring constant ks ≡ Goal: to construct and quantise a classical field theory for the collective (longitundinal) vibrational modes of the chain
. Discrete Classical Lagrangian:
k.e. p.e. in spring N z }| { z }| { X m 2 ks 2 L = T V = x˙ (xn+1 xn a) − 2 n − 2 − − n=1
assume periodic boundary conditions (p.b.c.) xN+1 = Na + x1 (and setx ˙ n ∂txn) ≡
Using displacement from equilibrium φn = xn x¯n − N X m ˙2 ks 2 L = φ (φn+1 φn) , p.b.c : φN+1 φ1 2 n − 2 − ≡ n=1 In principle, one can obtain exact solution of discrete equation of motion — see PS I
However, typically, one is not concerned with behaviour on ‘atomic’ scales:
1. for such purposes, modelling is too primitive! viz. anharmonic contributions
2. such properties are in any case ‘non-universal’
Aim here is to describe low-energy collective behaviour — generic, i.e. universal
In this case, it is often permissible to neglect the discreteness of the microscopic entities of the system and to describe it in terms of effective continuum degrees of freedom.
Lecture Notes October 2006 Lecture I 2
. Continuum Lagrangian
Describe φn as a smooth function φ(x) of a continuous variable x; makes sense if φn+1 φn a (i.e. gradients small) − Z L=Na 1/2 3/2 X 1 φn a φ(x) , φn+1 φn a ∂xφ(x) , dx a → x=na − → x=na n −→ 0
N.B. [φ(x)] = L1/2
φ(x) φI
RI
Lagrangian functional Lagrangian density z }| { Z L z }| 2 { m 2 ksa 2 L[φ] = dx (φ, ∂xφ, φ˙), (φ, ∂xφ, φ˙) = φ˙ (∂xφ) 0 L L 2 − 2 . Classical action
Z Z Z L S[φ] = dt L[φ] = dt dx (φ, ∂xφ, φ˙) 0 L
N-point particle degrees of freedom continuous classical field φ(x) • 7→ Dynamics of φ(x) specified by functionals L[φ] and S[φ] • What are the corresponding equations of motion...? ——————————————–
. Hamilton’s Extremal Principle: (Revision) Z Suppose classical point particle x(t) described by action S[x] = dt L(x, x˙)
Configurations x(t) that are realised are those that extremise the action
i.e. for any smooth function η(t), the “variation”, 1 δS[x] lim→0 (S[x + η] S[x]) = 0 is stationary ≡ − ; Euler-Lagrange equations of motion Z t Z t 2 S[x + η] = dt L(x + η, x˙ + η˙) = dt (L(x, x˙) + η∂xL + η∂˙ x˙ L) + O( ) 0 0 = 0 z }| { Z by parts Z d δS[x] = dt (η∂xL +η∂ ˙ x˙ L) = dt ∂xL (∂x˙ L) η = 0 − dt
t Note: boundary term, η∂x˙ L vanishes by construction |0 Lecture Notes October 2006 Lecture I 3
"#(x,t) ! !(x,t) x
t
. Generalisation to continuum field x φ(x)? 7→ Apply same extremal principle: φ(x, t) φ(x, t) + η(x, t) with7→ both φ and η periodic in x, i.e. φ(x + L) = φ(x) Z t Z L 2 2 S[φ + η] = S[φ] + dt dx mφ˙η˙ ksa ∂xφ∂xη + O( ). 0 0 − t L Integrating by parts boundary terms vanish by construction: ηφ˙ = 0 = η∂xφ |0 |0 Z t Z L ¨ 2 2 δS = dt dx mφ ksa ∂xφ η = 0 − 0 0 − 2 2 2 Since η(x, t) is an arbitrary smooth function, m∂t ksa ∂x φ = 0, i.e.− φ(x, t) obeys classical wave equation
General solutions of the form: φ+(x + vt) + φ−(x vt) p − where v = a ks/m is sound wave velocity and φ± are arbitrary smooth functions
φ+ x=νt x=-νt φ–
. Comments Low-energy collective excitations – phonons – are lattice vibrations • propagating as sound waves at constant velocity v
Trivial behaviour of model is consequence of simplistic definition: • Lagrangian is quadratic in fields linear equation of motion 7→ Higher order gradients in expansion (i.e. (∂2φ)2) dispersion 7→ Higher order terms in potential (i.e. interactions) dissipation 7→
L is said to be a ‘free (i.e. non-interacting) scalar (i.e. one-component) field theory’ •
In higher dimensions, field has vector components transverse and longintudinal modes • 7→
Variational principle is example of functional analysis – useful (but not essential method for this course) – see lecture notes
Lecture Notes October 2006 Lecture II 4
Lecture II: Collective Excitations: From Particles to Fields
Quantising the Classical Field
Having established that the low energy properties of the atomic chain are represented by a free scalar classical field theory, we now turn to the formulation of the quantum system.
. Canonical Quantisation procedure
Recall point particle mechanics:
1. Define canonical momentum, p = ∂x˙ L 2. Construct Hamiltonian, H = px˙ L(p, x) − 3. Promote position and momentum to operators with canonical commutation relations
x x,ˆ p p,ˆ [ˆp, xˆ] = i~,H Hˆ 7→ 7→ − 7→ Natural generalisation to continuous field:
∂ 2 1. Canonical momentum, π(x) L , i.e. applied to chain, π = ∂ ˙ (mφ˙ /2) = mφ˙ ≡ ∂φ˙(x) φ 2. Classical Hamiltonian Hamiltonian density (φ, π) H Z z }| { 2 h i 1 2 ksa 2 H[φ, π] dx πφ˙ (∂xφ, φ˙) , i.e. (φ, π) = π + (∂xφ) ≡ − L H 2m 2
3. Canonical Quantisation
(a) promote φ(x) and π(x) to operators: φ φˆ, π πˆ 7→ 7→ 0 0 (b) generalise commutation relations, [ˆπ(x), φˆ(x )] = i~δ(x x ) −N.B. [δ(−x x0)] = [Length]−1 (Ex.) − Operator-valued functions φˆ andπ ˆ referred to as quantum fields
Hˆ represents a quantum field theoretical formulation of elastic chain, but not yet a solution.
As with any function, φˆ(x) andπ ˆ(x) can be expressed as Fourier expansion:
φˆ(x) 1 X ˆ ˆ 1 Z L=Na φˆ(x) = e±ikx φk , φk dx e∓ikx πˆ(x) L1/2 πˆ πˆ ≡ L1/2 πˆ(x) k k k 0 P ˆ runs over all discrete wavevectors k = 2πm/L, m , Ex: confirm [ˆπk, φk0 ] = i~δkk0 k ∈ Z − Advice: Maintain strict conventions(!) — we will pass freely between real and Fourier space.
Lecture Notes October 2006 Lecture II 5
ˆ† ˆ ˆ† ˆ Hermiticity: φ (x) = φ(x), implies φk = φ−k (similarlyπ ˆ). Using
δk+k0,0 z }| { Z L Z L 2 X 0 1 i(k+k0)x X 2 dx (∂φˆ) = (ikφˆ )(ik φˆ 0 ) dx e = k φˆ φˆ k k L k −k 0 k,k0 0 k
2 mωk/2 z }|2 { X h 1 ksa 2 ˆ ˆ i 1/2 Hˆ = πˆkπˆ−k+ k φkφ−k , ωk = v k , v = a(ks/m) 2m 2 | | k i.e. ‘modes k’ decoupled Comments:
Hˆ describes low-energy excitations of system (waves) • in terms of microscopic constituents (atoms)
However, it would be more desirable to develop picture where • relevant excitations appear as fundamental units:
. Quantum Harmonic Oscillator (Revisited)
pˆ2 1 Hˆ = + mω2qˆ2 2m 2 Defining ladder operators r r mω i † mω i † 1 aˆ xˆ + pˆ , aˆ xˆ pˆ ; Hˆ = ~ω aˆ aˆ + ≡ 2~ mω ≡ 2~ − mω 2
If we find state 0 s.t.a ˆ 0 = 0 ; Hˆ 0 = ~ω 0 , i.e. 0 is g.s. | i | i | i 2 | i | i Using commutation relations [ˆa, aˆ†] = 1, one may then show n aˆ†n 0 | i ≡ | i 1 is eigenstate with eigenvalue ~ω(n + 2 )
!
Comments: Although single-particle, a-representation suggests many-particle interpretation
0 represents ‘vacuum’, i.e. state with no particles •| i † aˆ 0 represents state with single particle of energy ~ω • | i aˆ†n 0 is n-body state, i.e. operatora ˆ† creates particles • | i Lecture Notes October 2006 Lecture II 6
† 1 † In ‘diagonal’ form Hˆ = ~ω(ˆa aˆ + ) simply counts particles (viz.a ˆ aˆ n = n n ) • 2 | i | i and assigns an energy ~ω to each
. Returning to harmonic chain, consider r r mωk ˆ i † mωk ˆ i ak φk + πˆ−k , ak φ−k πˆk ≡ 2~ mωk ≡ 2~ − mωk N.B. By convention, drop hat from operators a i~δkk0 z − }| { † i ˆ ˆ with [ak, ak0 ] = [ˆπ−k, φ−k0 ] [φk, πˆk0 ] = δkk0 2~ − . And obtain (Ex. - PS I)
X 1 Hˆ = ω a† a + ~ k k k 2 k
Elementary collective excitations of quantum chain (phonons) † created/annihilated by operators ak and ak
Spectrum of excitations is linear ωk = v k (cf. relativistic) | |
Comments:
Low-energy excitations of discrete model involve slowly varying collective modes; • i.e. each mode involves many atoms;
Low-energy (k 0) long-wavelength excitations, • → 7→ i.e. universal, insensitive to microscopic detail;
Allows many different systems to be mapped onto a few classical field theories; • Canonical quantisation procedure for point mechanics generalises to quantum field theory; • Simplest model actions (such as the one considered here) are quadratic in fields • – known as free field theory;
More generally, interactions ; non-linear equations of motion viz. interacting QFTs. • Lecture Notes October 2006 Lecture II 7
. Other examples? †Quantum Electrodynamics
EM field — specified by 4-vector potential A(x) = (φ(x), A(x)) (c = 1)
Z 4 1 µν Classical action : S[A] = d x (A), = FµνF L L −4
Fµν = ∂µAν ∂νAµ — EM field tensor − Classical equation of motion:
Euler Lagrange eqns. Maxwell0s eqns. z − }| { β ∂ z αβ}| { ∂Aα ∂ L = 0 ∂αF = 0 L − ∂(∂βAα) 7→ Quantisation of classical field theory identifies elementary excitations: photons
for more details, see handout, or go to QFT!
Lecture Notes October 2006 Lecture III 8
Lecture III: Second Quantisation
We have seen how the elementary excitations of the quantum chain can be presented in terms of new elementary quasi-particles by the ladder operator formalism. Can this approach be generalised to accommodate other many-body systems? The answer is provided by the method of second quantisation — an essential tool for the development of interacting many-body field theories. The first part of this section is devoted largely to formalism – the second part to applications aimed at developing fluency. Reference: see, e.g., Feynman’s book on “Statistical Mechanics”
. Notations and Definitions
Starting with single-particle Schrodinger equation,
Hˆ ψλ = λ ψλ | i | i how can one construct many-body wavefunction?
!λ nλ nλ 0 0 !4 1 1 !3 1 1 !2 3 1 !1 5 1 !0 bosons fermions
Particle indistinguishability demands symmetrisation: e.g. two-particle wavefunction for fermions i.e. particle 1 in state 1, particle 2... state 1, particle 1 1 z }| { ψF (x1, x2) (ψ1(x1) ψ2(x2) ψ2(x1)ψ1(x2)) ≡ √2 − 1 In Dirac notation: 1, 2 F ( ψ1 ψ2 ψ2 ψ1 ) | i ≡ √2 | i ⊗ | i − | i ⊗ | i N.B. denotes outer product of state vectors ⊗ . General normalised, symmetrised, N-particle wavefunction of bosons (ζ = +1) or fermions (ζ = 1) −
1 X P λ1, λ2, . . . λN ζ ψλ ψλ ... ψλ | i ≡ p Q∞ | P1 i ⊗ | P2 i ⊗ | PN i N! λ=0 nλ! P
nλ — no. of particles in state λ; (for fermions, Pauli exclusion: nλ = 0, 1) •
Lecture Notes October 2006 Lecture III 9
P P : Summation over N! permutations of λ1, . . . λN • {required by} particle indistinguishability Parity — no. of transpositions of two elements which brings permutation • P ( 1, 2, N ) back to ordered sequence (1, 2, N) P P ···P ··· In particular, for fermions, x1, . . . xN λ1, . . . λN is Slater determinant, det ψi(xj) h | i Evidently, “first quantised” representation looks clumsy! motivates alternative representation... . Second quantisation
† Define vacuum state: Ω , and set of field operators aλ and adjoints a — no hats! | i λ N 1 Y † aλ Ω = 0, a Ω = λ1, λ2, . . . λN pQ∞ λi | i λ=0 nλ! i=1 | i | i cf. ladder operators for phonons N.B. ambiguity of ordering? Field operators fulfil commutation relations for bosons (fermions)
† † † [aλ, aµ]−ζ = δλµ, [aλ, aµ]−ζ = [aλ, aµ]−ζ = 0
where [A,ˆ Bˆ]−ζ AˆBˆ ζBˆAˆ is the commutator (anti-commutator) ≡ − † Operator a creates particle in state λ, and aλ annihilates it • λ Commutation relations imply Pauli exclusion for fermions: a† a† = 0 • λ λ Any N-particle wavefunction can be generated by application of set of • N operators to a unique vacuum state e.g. 1, 2 = a†a† Ω | i 2 1| i Symmetry of wavefunction under particle interchange maintained by • commutation relations of field operators e.g. 1, 2 = a†a† Ω = ζa†a† Ω = ζ 2, 1 | i 2 1| i 1 2| i | i (Providing one maintains a consistent ordering convention, the nature of that convention doesn’t matter)
a a a ... 2 1 0 F F F 0 a+ a+
. Fock space: Defining N to be linear span of all N-particle states λ1, . . . λN , F | i ∞ Fock space is defined as ‘direct sum’ N F ⊕N=0F † operators a and a connect different subspaces N F Lecture Notes October 2006 Lecture III 10
General state φ of the Fock space is linear combination of states • | i with any no. of particles Note that vacuum Ω (sometimes written as 0 ) is distinct from zero! • | i | i . Change of basis: † † a˜ Ω λ| i aλ Ω z}|{ X z}|{| i Using resolution of identity 1 P λ λ , we have λ˜ = λ λ λ˜ ≡ λ | ih | | i | i h | i λ
† X † X i.e. a = λ λ˜ a , and a˜ = λ˜ λ aλ λ˜ h | i λ λ h | i λ λ
e.g. Fourier representation: aλ ak, a˜ a(x) ≡ λ ≡ eikx/√L Z L X z }| { 1 −ikx a(x) = x k ak, ak = dx e a(x) h | i √ k L 0
† . Occupation number operator:n ˆλ = aλaλ measures no. of particles in state λ e.g. (bosons)
† 1 + aλaλ † † n † z}|†{ † n−1 † n † 2 † n−1 † n a aλ(a ) Ω = a aλa (a ) Ω = (a ) Ω + (a ) aλ(a ) Ω = = n(a ) Ω λ λ | i λ λ λ | i λ | i λ λ | i ··· λ | i Ex: check for fermions So far we have developed an operator-based formulation of many-body states. However, for this representation to be useful, we have to understand how the action of first quantised operators on many-particle states can be formulated within the framework of the second quantisation. To do so, it is natural to look for a formulation in the diagonal basis and recall the action of the particle number operator. To begin, let us consider...
Second Quantised Representation of Operators
. One-body operators: i.e. operators which address only one particle at a time
N N 2 X X pˆn ˆ1 = oˆn, e.g. k.e. Tˆ = O 2m n=1 n=1 P∞ Supposeo ˆ diagonal in orthonormal basis λ , i.e.o ˆ = λ=0 λ oλ λ , oλ = λ oˆ λ | i | i h | h | | i 2 e.g. k.e., λ p and op = p /2m | i ≡ | i N ! 0 0 X 0 0 λ , λ ˆ1 λ1, λN = oλ λ , λ λ1, λN h 1 ··· N |O | ··· i i h 1 ··· N | ··· i i=1 ∞ 0 0 X = λ , λ oλnˆλ λ1, λN , h 1 ··· N | | ··· i λ=0
Lecture Notes October 2006 Lecture III 11
∞ ∞ X X † Since this holds for any basis state, ˆ1 = oλnˆλ = oλa aλ O λ λ=0 λ=0
i.e. in diagonal representation, simply count number of particles in state λ and multipy by corresponding eigenvalue of one-body operator P Transforming to general basis (recall aλ = λ ν aν) νh | i
X † X † ˆ1 = µ λ oλ λ ν a aν = µ oˆ ν a a O h | i h | i µ h | | i µ ν λµν µν
i.e. ˆ1 scatters particle from state ν to µ with probability amplitude µ oˆ ν O h | | i . Examples of one-body operators:
ˆ R † P † 1. Total number operator: N = dx a (x)a(x) = k akak ˆ P † ˆ 1 2. Electron spin operator: S = αβ Sαβaαaβ, Sαβ = α S β = 2 σαβ where α =h , | ,| andi σ are Pauli spin matrices ↑ ↓ 1 0 z 1 0 1 + † σz = Sˆ = (ˆn↑ nˆ↓), σ+ = σx + iσy = Sˆ = a a↓ 0 1 7→ 2 − 0 0 7→ ↑ − 3. Free particle Hamiltonian
2 Z L 2 2 X p † Ex. † ( ~ ∂x) a a = dx a (x) − a(x) 2m p p 2m p 0
Z L 2 ˆ ˆ ˆ † pˆ i.e. H = T + V = dx a (x) + V (x) a(x) wherep ˆ = i~∂x 0 2m −
. Two-body operators: i.e. operators which address two-particles
E.g. symmetric pairwise interaction: V (x, x0) V (x0, x) (such as Coulomb) acting between two-particle≡ states N.B. 1/2 for double counting
1 Z Z Vˆ = dx dx0 x, x0 V (x, x0) x, x0 2 | i h | When acting on N-body states,
N 1 X Vˆ x1, x2, xN = V (xn, xm) x1, x2, xN | ··· i 2 | ··· i n6=m
In second quantised form, it is straightforward to show that (Ex.)
1 Z Z Vˆ = dx dx0 a†(x)a†(x0)V (x, x0)a(x0)a(x) 2
Lecture Notes October 2006 Lecture III 12 i.e. annihilation operators check for presence of particles at x and x’ – if they exist, asign the potential energy and then recreate particles in correct order (viz. statistics)
1 Z Z N.B. dx dx0 V (x, x0)ˆn(x)ˆn(x0) does not reproduce the two-body operator 2
. In non-diagonal basis
ˆ X † † 0 ˆ 0 2 = µ,µ0,λ,λ0 a 0 a a a 0 , µ,µ0,λ,λ0 µ, µ 2 λ, λ O O µ µ λ λ O ≡ h |O | i λλ0µµ0
1 X e.g. in Fourier basis: a†(x) = eikxa† can show that (Ex.) L1/2 k k Z 1 0 † † 0 0 0 X † † dxdx a (x)a (x )V (x x )a(x )a(x) = V (q)a a ak +qak −q 2 − k1 k2 2 1 k1,k2,q
k’,σ’ k,σ V(q)
k’+q,σ’ k–q,σ Feynman diagram:
Lecture Notes October 2006 Lecture IV 13
Lecture IV: Applications of Second Quantisation
1. Phonons: oscillator states k form a Fock space: for each mode k, arbitrary| i state of excitation can be created from vacuum
† † k = a Ω , ak Ω = 0, [ak, a 0 ] = δkk0 | i k| i | i k ˆ P † Hamiltonian, H = k ~ωk(akak + 1/2) is diagonal: † † ˆ k1, k2, ... = ak ak Ω is eigenstate of H with energy ~ωk1 + ~ωk2 + | i 1 2 · · · | i ···
2. Interacting Electron Gas
(i) Free-electron Hamiltonian
Z 2 (0) X † pˆ † 0 0 Hˆ = dx c (x) + V (x) cσ(x), [cσ(x), c 0 (x )] = δ(x x )δσ,σ0 σ 2m σ − σ=↑,↓
(ii) Interacting electron gas:
Z Z 2 (0) 1 0 X † † 0 e 0 Hˆ = Hˆ + dx dx c (x)c 0 (x ) c 0 (x )c (x) 2 σ σ x x0 σ σ σσ0 | − | . Comments:
. Phonon Hamiltonian is example of ‘free field theory’: involves field operators at only quadratic order... . (whereas) electron Hamiltonian is typical of an interacting field theory and is infinitely harder to analyze...
To familiarise ourselves with second quantisation, in the remainder of this and the next lecture, we will explore several case studies: ‘Atomic limit’ of strongly interacting elec- tron gas: electron crystallisation and Mott transition; Quantum magnetism; and weakly interacting Bose gas
Tight-binding and the Mott transition
According to band picture of non-interacting electrons, a 1/2-filled band of states is metallic. But strong Coulomb interaction of electrons can effect a transition to a crystalline phase in which electrons condense into an insulating magnetic state – Mott transition. We will employ the second quantisation to explore the basis of this phenomenon.
. ‘Atomic Limit’ of crystal
How do atomic orbitals broaden into band states? Show transparencies
Lecture Notes October 2006 Lecture IV 14
E E " s=1 ! 1B 1 " "1 1A
! s=0 0 "0A "0 x "0B V(x)
!/ a E E
x 0 k a (n!1)a na (n+1)a
Weak overlap of tightly bound orbital states narrow band of Bloch states ψks , specified by band index7→ s, k [ π/a, π/a] in first Brillouin| i zone. ∈ − Bloch states can be used to define ‘Wannier basis’, cf. discrete Fourier decomposition
B.Z. N 1 X −ikna 1 X ikna 2π ψns e ψks , ψks e ψns , k = m | i ≡ √N | i | i ≡ √N | i Na k∈[−π/a,π/a] n=1
!n0(x)
x
(n!1)a na (n+1)a
In ‘atomic limit’, Wannier states ψns mirror atomic orbital s on site n | i | i † † cns Ω c (x) Ω ψns(x) z }|| {i Z z}|{| iz }| { Field operators associated with Wannier basis: ψns = dx x x ψns | i | i h | i Z † † c dx ψns(x)c (x) ns ≡
P ∗ 0 0 and using completeness ψ (x )ψns(x) = δ(x x ) ns ns − † X ∗ † † c (x) = ψns(x)cns, [cns, cn0s0 ]+ = δnn0 δss0 ns
Lecture Notes October 2006 Lecture IV 15
† i.e. (if we include spin index σ) operators cnsσ/cnsσ create/annihilate electrons at site n in band s with spin σ
. In atomic limit, bands are well-separated in energy. If electron densities are low, we may focus on lowest band s = 0.
Transforming to Wannier basis, X Z pˆ2 Hˆ = dx c† (x) + V (x) c (x) σ 2m σ σ=↑,↓ Z Z 1 0 X † † 0 0 0 + dx dx c (x)c 0 (x )V (x x )c 0 (x )c (x) 2 σ σ − σ σ σσ0 X † X † † = tmncmσcnσ + Umnrscmσcnσ0 crσ0 csσ0 mn,σ mnrs,σσ0
pˆ2 ∗ where “hopping” matrix elements tmn = ψm [ 2m + V (x)] ψn = tnm and “interaction pa- rameters” h | | i 1 Z Z e2 U = dx dx0ψ∗ (x)ψ∗(x0) ψ (x0)ψ (x) mnrs 2 m n x x0 r s | − | (For lowest band) representation is exact: but, in atomic limit, matrix elements decay exponentially with lattice separation
(i) “Tight-binding” approximation: ( m = n (0) X † X † tmn = t mn neighbours , Hˆ c cnσ t c cnσ + h.c. − ' nσ − n+1σ 0 otherwise nσ nσ B.Z. 1 X In discrete Fourier basis: c† = e−iknac† nσ √N kσ k∈[−π/a,π/a]
δkk0 N z }| { X † X 1 X −i(k−k0)na −ika † X † t c cnσ + h.c. = t e e c ck0σ + h.c. = 2t cos(ka) c ckσ − n+1σ − N kσ − kσ nσ kk0σ n kσ
(0) X † Hˆ = ( 2t cos ka)c ckσ − kσ kσ