Fermion Number Conservation

D FREQUENCYINTEGRALS AND MATSUBARA SUMS

and

I_2,σ(k) = 1 4Z_σ,k

htanh^Ωσ,k/Zσ,k−µ

2 β−tanh^Ωσ,k/Zσ,k+µ

2 βⁱ (D.8)

µ→0

−−−→0 (D.9)

β→∞

−−−−→ −sgn (µ) 1

2Z_σ,k Θ |µ| −^Ωσ,k/Zσ,k

(D.10)

µ→0, β→∞

−−−−−−−→0, (D.11)

withq⁰ →iω_n and ω_n= ^(2n+1)π_β for fermionic boundary conditions. Regarding the second integral, it can be helpful to introduce a regularization term which is equal to the case ofµ= 0, in order to apply the residue theorem

I_2,σ(k) = Z dq⁰

2π

i (q⁰+µ)Z_σ,k (q⁰+µ)²Z_σ,k² −Ω²_σ,k −

Z dq⁰ 2π

iq⁰Z_σ,k (q⁰Z_σ,k)²−Ω²_σ,k

| {z }

=0

. (D.12)

From the evaluation of Eq. (D.3) one can additionally extract X

n

1 n+¹/2+ ix

1

n+¹/2+ iy = π

x−y[tanh(πx)−tanh(πy)] (D.13) which is a usefule relation in order to evaluate Matsubara sums and is also exploited to determine the frequency integrals that occur in the vacuum polarization function in Sec. 5.2.

D.2 FERMION NUMBERCONSERVATION

where the Matsubara sum of the first term has already been calculated in Eq. (D.4), the Matsubara sum within the free ernergy can be evaluated. The second term has been dropped, due to the fact that it neither depends on the chemical potential nor on the temperature (with fermionic frequencies ωn= (2n+ 1)π T), what effectively regularizes the integral and determines the vacuum energy to be zero. So we finally find

Ω_f(T, µ)/N²=− T N²

X

k,σ

ln cosh(β(ε_k,σ−µ_σ)/2) + ln cosh(β(ε_k,σ+µ_σ)/2) (D.18)

T→∞= − T N²

X

k,σ

(ε_k,σΘ(ε_k,σ− |µ_σ|) +|µ_σ|Θ(|µ_σ| −ε_k,σ)), (D.19) from what the fermion number per unit cell can be simply extracted and is given by

n_σ = 1 N²

dΩ_f dµσ

= sign(µ_σ) ^X

k,σ

1

N²Θ(|µ_σ| −ε_k,σ) (D.20)

= sign(µσ) Z _|µ|

0 dε ρ(ε), (D.21)

with ρ() being the density of states [16], originally calulated in Ref. [12] in 1953.

Consequently forµ→ ±∞one finds plus or minus one quasiparticle per unit cell and per spin degree of freedom, here negative values refer to hole states of the energy band. However, important to note here is that the neutrality point for electron states and holes, so for graphene at half-filling, is found for µ_↑ = µ_↓ = 0 for two fermion flavors.

E

D YSON -S CHWINGER E QUATIONS

E.1 Path Integral Formalism

In the following the formalism of second quantization is translated to the path inte-gral formalism of a quantum field theory by the introduction of coherent fermionic states via

|ψi=e^{−Piψic†i} |0i and hψ|=h0|e^{−Piciψ∗i} , (E.1) with the fermionic creation and annihilation operators c^†_i and ci for the quasi-particles in coordinate space as already introduced in Sec. 2.1. These states com-posed by Grassmann numbers and the corresponding creation/annihilation operator build a basis of Eigenstates of the respective operator

ci |ψi=−ψ_i |ψi and hψ|c^†_i =hψ|(−ψ_i^∗). (E.2) For a detailed description and derivation of the basic principles we recommend Ref. [116] or Ref. [123]. Note that in the considered theory the vacuum state in-dicated by |0i or h0| refers to the many-particle state of the half-filled hexagonal lattice.

Starting from the Hamiltonian in second quantization, the partition function that leads to the generating functional for correlation functions in Euclidean space-time is given by

Z= tre^−βH = Z

Y

x

dξ_x^∗dξ_xdη^∗_xdη_x e^{−Px(ξx∗ξx+ηx∗ηx)} h−ξ, −η|e^−βH|ξ, ηi , (E.3) for convenience we drop the subscript E (as introduces in App. B) for Euclidean space-time variables throughout the whole chapter. The fermionic many-particle

E DYSON-SCHWINGER EQUATIONS

state|ξ, ηiis characterized by two complex Grassmann valued fieldsηandξ. In order to distinguish between spin-up and spin-down fermions we introduce the notation

ξ_x =ψ_x,↑ and η_x=ψ_x,↓. (E.4)

For any normal ordered function F(c^†_x,↑, cx,↑, c^†_x,↓, cx,↓) the matrix element is then given by [116]

hξ, η| F(c^†_x,↑, c_x,↑, c^†_x,↓, c_x,↓)|ξ,e ηi_e =F(ξ_x^∗,ξ^e_x, η_x^∗,η_ex) e^{Pxξx∗eξx+η∗xeηx}. (E.5) With the Trotter-Theorem [70] the partition function from Eq. (E.3) is factorized via

e^−βH ≈e^−δHe^−δH. . . e^−δH, (E.6) withδ= _N^β

t and a complete set of states building the unity Z Y

x

dξ_x^∗dξxdη_x^∗dηx e^{−Px(ξ∗xξx+η∗xηx)} |ξ, ηi hξ, η|=1, (E.7) is inserted between all exponentials. With antiperiodic boundary conditions

h−ξ, −η|=h−ξ_x,0,−η_x,0|=hξ_x,Nt, ηx,Nt|, (E.8) and a new integration variable for each insertion which is labeled by a second index t, one obtains

Z=

Z ^Nt−1 Y

t=0

Y

x

dξ_x,t^∗ dξx,tdη_x,t^∗ dηx,t e⁻ P

x,t(ξ^∗_x,t+1ξx,t+1+η^∗_x,t+1ηx,t+1) hξ_x,Nt, ηx,Nt|e^−δH|ξ_x,Nt−1, ηx,Nt−1i

× hξ_x,Nt−1, ηx,Nt−1|e^−δH|ξ_x,Nt−2, ηx,Nt−2i. . .hξ_x,1, η_x,1|e^−δH|ξ_x,0, η_x,0i. In the next step the expectation values have to be evaluated for the Hamiltonian of interest, here given by the free Hamiltonian from Eq. (2.25)

H₀=−κ ^X

hi,ji,σ

(c^†_i,σc_j,σ+c^†_j,σc_i,σ) + ^X

is,s,σ

(−1)^smσc^†_is,σc_is,σ−^X

i,σ

µσc^†_i,σc_i,σ, (E.9) plus the two-particle interaction, as already introduced in Eq. (5.2),

H_tp = 1 2

X

i,j,σ,σ⁰

:c^†_i,σc_i,σ(λ1+Ve_ij⁺) c^†_j,σ0c_j,σ⁰:

| {z }

≥0

−1 2

X

i,j,σ,σ⁰

:c^†_i,σc_i,σ (λ1−Ve_ij⁻) c^†_j,σ0c_j,σ⁰:

| {z }

≥0

+H_os.

(E.10) Here the on-site interactionH_osis additionally separated from the interaction matrix with Ve_ii⁺ = Ve_ii⁻ = 0 and we formally introduce an arbitrary term proportional to λ ∈ R, to ensure the invertibility of (λ1+Ve_xy⁺) and (λ1−Ve_xy⁻). Note that contributions proportional to the particle number operator has been absorbed in the chemical potential (see Eq. (3.7)).

In order to take a closer look on the predominant on-site interaction, we use a kind of complete Fierz transformation

(c^†_ic_i−1)(c^†_ic_i−1) =−1

3(c^†_i~σ c_i)(c^†_i~σ c_i), (E.11)

E.1 PATHINTEGRALFORMALISM

to gain another representation of the on-site contribution which directly couples to the order parameter of an SDW state.

The spin index notation here is replaced by a two dimensional vector notation with c_i = (c_i,↑, c_i,↓) and analogically for the creation operator, ~σ is refering to the Pauli matrices σ^l in spin space were the matrix product should be understood as follows: ~σ_αβ~σ_γδ =^P3_m=1σ_αβ^m ·σ^m_γδ. In order to keep track of both possibilities in the derivation of DSE’s an artificial parameterα was inserted which equals toα= 1 for the theory investigated in the main part of this work,

H_os=αU 2

X

i

: (c^†_ic_i)² :−(1−α)U 6

X

i

: (c^†_i~σ c_i) (c^†_i~σ c_i) :. (E.12) Next, the required expectation values within the partition function are evaluated with the use of Eq. (E.5) and contributions which are quadratic in the density oper-ator were linearized by a Hubbard-Stratonovich transformation (HST) (App. A.2).

As an example, the HST for the long-range contribution is considered in the follow-ing, where the bosonic fieldsσ⁺andσ⁻have been introduced. These fields therefore couple to the positive or negative semidefinite part of the interaction,

exp



−δ 2

X

x,y,t,t⁰

(ξ_x,t+1^∗ ξ_x,t+η^∗_x,t+1η_x,t) (λ1+Ve_xy⁺) (ξ_y,t^{∗ 0}₊₁ξ_y,t⁰+η^∗_y,t⁰₊₁η_y,t⁰) (E.13)

+δ 2

X

x,y,t,t⁰

(ξ_x,t+1^∗ ξx,t+η^∗_x,t+1ηx,t) (λ1−Ve_xy⁻) (ξ_y,t^{∗ 0}₊₁ξ_y,t⁰+η^∗_y,t⁰₊₁η_y,t⁰)



=

× q

det [(λ1+δV^exy⁺)⁻¹] q

det [(λ1−δV^exy⁻)⁻¹]

Z Dσ⁺ (2π)^D/2

Z Dσ⁻

(2π)^D/2 (E.14)

×exp



−δ 2

X

x,y,t,t⁰

σ_x,t⁺ (λ1+Ve⁺)⁻¹_xyσ_y,t⁺0 −iδ^X

x,t

σ⁺_x,t(ξ_x,t+1^∗ ξx,t+η_x,t+1^∗ ηx,t)





×exp



−δ 2

X

x,y,t,t⁰

σ_x,t⁻ (λ1−Ve⁻)⁻¹_xyσ_y,t⁻0 +δ^X

x,t

σ_x,t⁻ (ξ^∗_x,t+1ξx,t+η_x,t+1^∗ ηx,t)



. Here the product of integrals over all fields in Euclidean space-time is abbreviated by

Dξ =

Nt−1

Y

t=0

Y

x

dξx,t. (E.15)

With the view of a compact notation we additionally absorb the prefactors in the in-tegral measures (D=N²·N_t). Furthermore we analogically introduce the Hubbard-fields Φ and ρ~ which couple to the on-site interaction and its Fierz transformed version (Eq. (E.12)).

In the next step we introduce the fermionic fields in sublattice space, this repre-sentation simply divides the summation from above over all lattice points x (with all fermionic fields being zero on unoccupied lattice points due to Eq. (2.15)) into a summation over all lattice points of the corresponding sublattice (either AorB), labeled by an index i,

ξ¯_i = ¯ξ_i(t) = ξ_x^∗_a(t) ξ_x^∗_b(t)

!T

·γ⁰ and ξ_i=ξ_i(t) = ξ_xa(t) ξx_b(t)

!

, (E.16)

E DYSON-SCHWINGER EQUATIONS

for the bosonic fields we use Φ^T_i = Φ^T_i (t) = Φ_xa(t)

Φ_xb(t)

!T

, and Φ_i = Φ_i(t) = Φ_xa(t) Φ_xb(t)

!

, (E.17) with the gamma matrices in sublattice space, given in App. A.2. With a transition to a continuous time integral with δ →0 and Nt → ∞for a fixed value of β (note also the continuous time argument in Eq. (E.17) and Eq. (E.16)) we obtain the partition function of the form

Z= Z

DξDξ¯ D¯ηDηDφD~ρ Dσ⁺Dσ⁻ exp





−

β=δ·Nt

Z

τ=0

dτ ^X

i

L_i[ ¯ξ, ξ,η, η, φ, ~¯ ρ, σ⁺, σ⁻]







= Z

DξDξ¯ D¯ηDηDφDρDσ⁺Dσ⁻ e^{−SE[¯ξ,ξ,¯η,η,φ,~ρ,σ+,σ−]}. (E.18) In order to cover the sum over A and B sublattices within a matrix notation, the sublattice projectors Γ^Aand Γ^B already defined in Eq. (3.15) are used to obtain the following Lagrangian,

L_i=^hξ¯_iγ⁰∂_tξ_i+ ¯η_iγ⁰∂_tη_i−κ

2

X

l=0

ξ¯_iγ¹ξ_i+∆l+m_↑ ξ¯_iξ_i−µ_↑ ξ¯_iγ⁰ξ_i−κ

2

X

l=0

η¯_iγ¹η_i+∆l +m_↓ η¯iηi−µ_↓ η¯iγ⁰ηi+ 1

2α U Φ²_i + i^X

s

(Φ^s_i +σ^+s_i + iσ_i^−s) ( ¯ξiγ⁰Γ^sξi+ ¯ηiγ⁰Γ^sηi)

+ 3

2(1−α)U ~ρi2−^X

s

ρ^s_1,i (¯ηiγ⁰Γ^sξi+ ¯ξiγ⁰Γ^sηi)−^X

s

iρ^s_2,i (¯ηiγ⁰Γ^sξi−ξ¯iγ⁰Γ^sηi)

−^X

s

ρ^s_3,i ( ¯ξ_iγ⁰Γ^sξ_i−η¯_iγ⁰Γ^sη_i) +1

2σ^+T_i [(λ1+V⁺)⁻_ij¹]σ_j⁺+1

2σ_i^−T[(λ1−V⁻)⁻_ij¹]σ⁻_j

# , (E.19)

where s = 0,1 denotes the sublattice index, corresponding to the A or B lattice, respectively, and ~ρ = (ρ₁, ρ₂, ρ₃). The first terms can exactly be identified by the Euclidean propagator from Eq. (B.15) and the Euclidean action is eventually given by

SE[ ¯ξ, ξ,η, η,¯ Φ, ~ρ, σ⁺, σ⁻] = Z P

˜ x,˜y

ξ¯x_˜G⁰_E,↑(˜x,y)˜ ⁻¹ξy_˜+ Z P

˜ x,˜y

η¯x_˜G⁰_E,↓(˜x,y)˜ ⁻¹ηy_˜ (E.20) +

Z P

˜ x

1

2α U Φ²_x˜+ i ^X

s

Z P

˜ x

(Φ^s_x˜+σ_x^+s_˜ + iσ_x⁻_˜^s) ( ¯ξ_x˜γ⁰Γ^sξ_˜x+ ¯η_x˜γ⁰Γ^sη_˜x) +

Z P

˜ x

3

2(1−α)U ~ρ_˜x²−^X

s

Z P

˜ x

ρ^s_1,˜x (¯η_i˜xγ⁰Γ^sξ_x˜+ ¯ξ_x˜γ⁰Γ^sη_x˜)

−i^X

s

Z P

˜ x

ρ^s_2,˜x (¯η_˜xγ⁰Γ^sξ_˜x−ξ¯_˜xγ⁰Γ^sη_x˜)−^X

s

Z P

˜ x

ρ^s_3,˜x( ¯ξ_x˜γ⁰Γ^sξ_˜x−η¯_˜xγ⁰Γ^sη_˜x) +1

2 Z P

˜ x,˜y

σ^+T_˜x [(λ1+V⁺)⁻¹_x˜˜y]σ⁺_y˜ +1 2

Z P

˜ x,˜y

σ_x⁻_˜^T[(λ1−V⁻)⁻¹_x˜˜y]σ⁻_y˜

# , (E.21)

E.1 PATHINTEGRALFORMALISM

with the generating functional Z[j] =

Z

Dφexp

−S_E[φ] + Z

d³x j_α(x)φ_α(x)

, (E.22)

φ ={ξ,ξ, η,¯ η,¯ Φ, ~ρ, σ⁺, σ⁻} and j ={ψ¯^↑, ψ^↑,ψ¯^↓, ψ^↓, h₁, ~h₂, j⁺, j⁻} should abbrevi-ate the entirety of all fields and sources. The generating functional for connected n-point function is then given by (see Ref. [85, 124, 125] for a general introduction)

W[j] = lnZ[j], (E.23) which accounts for the correctly normalized expectation values in presence of all sources within the resultingn-point functions. If all sources are set to zero,W[0] = 0 is retained from Z[0] = 1.

The full propagator hφ₁(x)φ₁(y)i resulting from the functional derivative ofZ[j]

is easily connected to the associated counterpart of Z[j], the so called connected Green function G(x−y) ,

G_αβ(x−y) =hφ_α(x)φ_β(y)i_conn.= δ²W[j]

δjα(x)δjβ(y)

_j=0=hφ_α(x)φ_β(y)i − hφ_α(x)ihφ_β(y)i

= δ²Z[j]

δjα(x)δjβ(y)

j=0−φ^c_α(x)φ^c_β(y). (E.24)

Here the derivatives are defined as right- and left-derivatives, respectively, δ

δξ(x)¯ =: left derivative, and δ

δξ(x) =: right derivative. (E.25) The conventions in Euclidean space-time follow them from Ref. [110].

With a Legendre transformation with respect to the sourcesjα(x) =δΓ[φ^c]/δφ^c_α(x) , Γ[φ^c] =−W[j] +

Z

d³x φ^c_α(x)jα(x), (E.26) the generating functional for 1PI n-point functions Γ[φ^c] depending on the corre-sponding field expectation values with nonzero sourcesφ^c_α(x) =δW[j]/δj_α(x) =hφ_αi_j is obtained. The complete set of classical field variables is given by

φ^c={ξ^c,ξ¯^c, η^c,η¯^c,Φ^c, ~ρ^c, σ^+c, σ^−c}.

The DSE’s result from performing a total derivative on an integral equation which surely vanishes [110], from Eq. (E.22) one immediately finds

0 =− δ δφ_β(y)

Z

Dφe^{−S[φ]+Rdxjα(x)φα(x)}= Z

Dφ e^{−S[φ]+Rdxjα(x)φα(x)} δ

δφ_β(y)S[φ]−j_β(y)

=:^D δ

δφ_β(y)S[φ]−j_β(y)^E

j. (E.27)

By replacing all field dependencies within the expectation value by the derivative with respect to the corresponding source,Z[j] can be separated as a factor,

− δS δφβ(y)

δ δj

+j_β(y)Z[j] = 0, (E.28)

E DYSON-SCHWINGER EQUATIONS

in terms of the generating functional for connected Green functions this can be rewritten as

− δS δφ_β(y)

δW δj + δ

δj

+j_β(y) = 0, (E.29)

where the additional derivative with respect to j in the replacement for the field dependency take account for the product rule. Finally, for the 1PI generating func-tional we obtain

δΓ[φ]

δφ^c_β(y)− δS δφ^c_β(y)

"

φ^c+δ²W δjδj

δ δφ^c

#

= 0, (E.30)

where the field componentφ_α(x) more precisely has to be replaced by φ_α(x)−→φ^c_α(x) +

Z

z

X

γ

δ²W δj_γ(z)δj_α(x)

δ

δφ^c_γ(z), (E.31) for instance. These equations build the starting point for the derivation of DSE’s for full, connected or propern-point functions, respectively.

In the following we will take the Ansatz from Eq. (E.28) as a starting point and use the relations of App. E.4 to obtain the DSE’s for 1PI Green functions. Although starting from Eq. (E.30) and use Eq. (E.98) would probably be less laborious, follow-ing Eq. (E.28) and elaborate the 1PI part afterwards clearly contribute to a better understanding.

Performing an additional derivative on Eq. (E.28) with respect to a source entails δ

δjγ(z) Z

Dφ e^{−S[φ]+Rdxjα(x)φα(x)} δ

δφ_β(y)S[φ]−jβ(y) (E.32)

= Z

Dφ e^{−S[φ]+Rdxjα(x)φα(x)} φ_γ(z) δ

δφ_β(y)S[φ]−φ_γ(z)j_β(y)−δ_βγδ(y−x), (E.33) from that the DSE directly follows,

D δS[φ]

δφ_β(y)φ_γ(z)^E

j=0 =δ_βγδ(y−x), (E.34) with all sources set to zero.

Im Dokument Dyson-Schwinger Equations for strongly interacting fermions on the hexagonal graphene lattice (Seite 106-114)