Action expansion

According to this, n^A_↑ −n^B_↓ = 0 except for D^>U = 1. Although the equation suggests the inexistence of a broken-symmetry phase, the singular behaviour for a fixed value ofUsuggests otherwise. This is an indicator that the extremum ofSdoes not correspond to a minimum in the broken-symmetry phase and that we must approach the problem from a different standpoint.

We will, however, make use of the solution arrived at here, which amounts to the saddle-point solution of the action, valid for the normal phase. But before proceeding, we derive an additional result, which relates the Fermi levelµforU6= 0and the Fermi level of the non-interactive system µ0, in virtue of charge conservation. This reads

n = n^A_↑ +n^B_↑ +n^A_↓ +n^B_↓

= 2D^>(µ0−ε^>_Λ) + 2D^<(−∆^<−ε^<_Λ)

(5.9)

= 2D^> µ−U 2

X

τ,σ

n^τ_σ−ε^>_Λ

!

+ 2D^<(−∆^<−ε^<_Λ), (5.13)

where, in the second equality, the electron densities in the non-interactive system were used, ε^>_Λ, ε^<_Λ < 0are cone-dependent cutoffs and nis the total electron density. Noting that, in the normal phase, the following relations hold

X

σ

n^A_σ =X

σ

n^B_σ =X

τ

n^τ_↑ =X

τ

n^τ_↓ = n

2 , (5.14)

we arrive at

µ=µ0+U n

2 . (5.15)

This relation holds for the normal phase as well as for the broken symmetry phase in the low doping regime, but not for any other, as will be shown further on. Note that due to the constancy of the DOS and, therefore, simplicity of the integrals forT →0, from now on most computations involving electron densities will be written directly, as in Eq. (5.13).

¯

n^τ_σ holds. Absorbing the saddle-point contribution into the inverse of the propagator Gˆ_∗, we arrive at the bare propagator in the non-interactive systemGˆ0,

~

Gˆ₀^−1τ

k,σ =i~ωn−ε^τ_k,σ+µ−UX

τ⁰

n^τ_σ¯⁰ =i~ωn−ε^τ_k,σ+µ0,

where Eqs. (5.14) and (5.15) were used in the second equality. Integrating out the fermionic fields,

S[{δφ^τ_σ}]/~=N βUX

q

X

τ,τ⁰

i¯n^τ_↑δq,0+δφ^τ_q,↑

i¯n^τ_↓⁰δq,0+δφ^τ_−q,↓⁰

−Tr ln

"

β~ Gˆ₀⁻¹+iU

~ X

τ

δφˆ^τ

!#

, (5.17) factoring out the bare propagator and simplifying the quadratic term in virtue of Eq. (5.14), we obtain the grand-partition function

Z=Z0

ˆ

D{δφ^τ_σ}e^{−S0[{δφτσ}]/~}, (5.18) whereZ0is the grand-partition function of the non-interactive system and the newφ-field action S⁰reads

S⁰[{δφ^τ_σ}]/~=−N βUn 2

2

+N βUin 2

X

q

X

τ

δφ^τ_q,↑+δφ^τ_−q,↓

+N βUX

q

X

τ,τ⁰

δφ^τ_q,↑δφ^τ_−q,↓⁰ −Trln

"

1 + iU

~ Gˆ0

X

τ

δφˆ^τ

# .

Now, due to the magnitude of the perturbations{δφ^τ_q,σ}relatively to the electron densities{n^τ_σ}, hypothesized previously, it is possible to obtain the action up to quadratic order inδφby Taylor expanding the fermionic contribution of the action which, in virtue of the trace, reads

F[{δφ^τ_σ}] = Tr ln

"

1 +iU

~ Gˆ0

X

τ⁰

δφˆ^τ0

#

=

=iU

~ X

σ,τ

X

k,k⁰

Gˆ0

^τ,σ

k,k⁰

X

τ⁰

δφˆ^τ0

!^σ¯

k⁰,k

+1 2

U

~ 2

X

σ,τ

X

k,k⁰,q,q⁰

Gˆ0

^τ,σ

k,k⁰

X

τ⁰

δφˆ^τ0

!^σ¯

k⁰,q⁰

Gˆ0

^τ,σ

q⁰,q

X

τ⁰

δφˆ^τ0

!^¯σ

q,k

.

(5.19)

Note that Gˆ₀τ,σ

k⁰,k

= Gˆ₀τ

k,σ

δ_k⁰_,k and P

τ⁰δφˆ^τ0σ¯ k⁰,k

= P

τ⁰δφ^τ_k−k⁰ 0,¯σ . Note also that, within the validity of the parabolic approximation, this expansion is exact, as shown in Sec. D.2 of Appendix D. We now make an ansatz for the decoupling fieldsφ, imposing space- and time- homogeneity

δφ^τ_q,σ ≡iη_σ^τδq,0,∀τ, σ . (5.20) This ansatz is variational, in the sense that the solution will be determined in virtue of minimiza- tion of the action: equivalently to the motivation of Eq. (5.8), the structure of the path integral Eq. (5.18) is such that the solutions which minimize the action constitute the dominant contribu- tion to the grand-partition function.

Why is this approach necessary, considering it is founded essentially on the same principle as the previous one? Eq. (5.12) suggests that the saddle-point solution fails to account for a minimum in the broken symmetry phase, so that we must take a slightly wider look at the action in the vicinity of the transition. Indeed, this approach will allow for a more intricate manipulation of the action, essentially by including constraints due to conservation laws unaccounted for so far.

Eq. (5.19) then becomes

F({η^τ_σ})/~=−U

~ X

σ,τ

X

k

P

τ⁰η_σ^τ_¯⁰ iωn−ε^τ_k,σ+µ0

−1 2

U

~ ²

X

σ,τ

X

k

P

τ⁰η^τ_¯σ⁰ iωn−ε^τ_k,σ+µ0

!2

. (5.21)

We can perform the Matsubara frequency integrations, yielding

F({η^τ_σ})/~=−βUX

σ

X

τ⁰

η_σ^τ_¯⁰X

k,τ

nF(ε^τ_k,σ−µ0)−βU² 2

X

σ

X

τ⁰

η^τ_¯σ⁰

!² X

k,τ

∂nF(ε^τ_k,σ−µ0)

∂ε^τ_k,σ , (5.22) where the derivative of the Fermi distribution in the second term appears in virtue of taking the residue of a second-order pole [118]. Replacing summations we have, for the 1st order and 2nd order terms, respectively,

X

σ

X

τ⁰

η_σ^τ_¯⁰X

k,τ

nF(ε^τ_k,σ−µ0) =

T→0

= N

D^> µ0−ε^>_Λ

−D^<(∆^<+ε^<_Λ) X

τ

η_σ^τ(5.13)= Nn 2

X

τ,σ

η^τ_σ,

(5.23)

X

σ

X

τ⁰

η^τ_σ¯⁰

!² X

k,τ

∂nF(ε^τ_k,σ−µ0)

∂ε^τ_k,σ =

T→0

= −N

"

D^>

ˆ −∆^>

−ε^>_Λ

dε δ(µ₀−ε) +D^<

ˆ −∆^<

−ε^<_Λ

dε δ(µ₀−ε)

# X

σ

X

τ⁰

η_σ^τ0

!²

=−N D^>X

σ

X

τ⁰

η^τ_σ⁰

!2

,

(5.24)

We can now use charge conservation to impose a constraint on the variational parameters{η^τ_σ}.

Retrieving again the fermionic propagator from the effective action of Eq. (5.17), applying the

ansatz of Eq. (5.20) yields

~

Gˆ⁻¹τ k,σ

=i~ω_n−ε^τ_k,σ+µ₀−UX

τ⁰

η_σ^τ_¯⁰.

This is, in fact, the propagator in the broken symmetry phase, which implies that the non- interactive electronic bands become modified by a spin-dependent shift, reading

E_k,σ^τ =ε^τ_k,σ+UX

τ⁰

η_σ^τ_¯⁰=ε^τ_k,σ−σU 2

X

τ⁰,σ⁰

σ⁰η_σ^τ00+U 2

X

τ⁰,σ⁰

η^τ_σ⁰0, (5.25)

which shall be referred to as magnetic bands (or cones). The second equality is obtained in virtue of

X

τ⁰

η^τ_σ¯⁰ = 1 2

X

τ⁰,σ⁰

η_σ^τ00−σ 2

X

τ⁰,σ⁰

σ⁰η_σ^τ00.

Using this result to compute electron densities in the interactive system, we can now compare the cone fillings between the non-interactive and the interactive system, setting a constraint to the parameters{η_σ^τ}in virtue of charge conservation. This reads

n= 2D^>(µ0−ε^>_Λ) + 2D^<(−∆^<−ε^<_Λ)

=D^> µ0−UX

τ

η_↓^τ−ε^>_Λ

!

+D^> µ0−UX

τ

η^τ_↑−ε^>_Λ

!

+ 2D^<(−∆^<−ε^<_Λ),

⇒ X

τ,σ

η^τ_σ= 0 , (5.26)

rendering the 1st order term of the fermionic contribution of the expanded action, Eq. (5.22), vanishing. Moreover, note that

X

σ

X

τ⁰

η_σ^τ0

!²

= X

τ,σ

η^τ_σ

!²

−2X

τ,τ⁰

η_↑^τη^τ_↓⁰, (5.27)

holds for the 2nd order term and, in turn

X

τ,τ⁰

η_↑^τη^τ_↓⁰ = 1 4

X

τ,σ

η^τ_σ

!²

−1 4

X

τ,σ

ση_σ^τ

!²

(5.28)

⇒X

σ

X

τ⁰

η^τ_σ⁰

!2

= 1 2

X

τ,σ

η^τ_σ

!2

+1 2

X

τ,σ

ση^τ_σ

!2

,

so that the fermionic contribution reads

F({η^τ_σ})/~=N βU D^>U X

τ,σ

ση^τ_σ 2

!² ,

yielding, for the the total action,

S⁰({η^τ_σ})/~=−N βUn 2

2

−N βUn 2

X

τ,σ

η_σ^τ−N βUX

τ,τ⁰

η_↑^τη^τ_↓⁰−N βD^>U² X

τ,σ

ση^τ_σ 2

!2

=N βU(1−D^>U) X

τ,σ

ση_σ^τ 2

!²

−N βUn 2

² ,

where, in the second equality, Eqs. (5.26) and (5.28) were used. Finally, recovering thatη_↓^A = η^B_↑ = 0holds - accounting for the fact that, in the low doping regime, the↓-Aand↑-B are not affected by charge imbalance - yields

S⁰(ξ)

N β~U =S⁰({η_σ^τ}) N β~U +n

2 2

= 1−D^>U ξ

2 ²

, (5.29)

where ξ = P

τ,σση_σ^τ = η^A_↑ −η^B_↓. Note that, for D^>U > 1, the action is minimized by maxi- mizing the variational parameterξand, thus, charge imbalance becomes non-vanishing. This implies the existence of a broken symmetry phase, wherein the hole-doped system becomes spontaneously spin and valley polarized, provided that the on-site Coulomb interaction energy is greater than the critical value

Uc= 1 D^> ,

with the DOSD^>given by, according to the fourth equality of Eq. (5.11), D^>= 2π

ABZ

m^>

~²

=

√3 4π

a

~ 2

m^>, (5.30)

whereABZ = (2/√

3)(2π/a)² is the area of the Brillouin Zone andm^> is the effective mass of charge carriers in the high-lying cones of the spin-split valence band, as given by Eq. (3.26). As expected, the onset of the phase transition is defined by a Stoner criterion. Values ofD^> and Uc for the different TMDCs are shown in Table 5.1.

Table 5.1: DOS of the high-lying conesD^> and critical valueUc for the various TMDCs - low doping. (Units: eV)

M X₂ MoS2 WS2 MoSe2 WSe2

D^{> −1}

0.0745 0.0503 0.0876 0.0565

U_c 13.4 19.9 11.4 17.7

These appear to be higher than current estimates for the interaction strength, ranging in 2÷10 eV[119], especially forWcompounds. There are reasons to believe, however, that the strength of the on-site Coulomb interaction (or, put simply, of the Hubbard coupling) might cover these critical values, based on studies carried out for graphene [114], where the energy of the partially screened on-site Coulomb interaction is found to be9.3 eV. TMDCs, in turn, due to their richer electronic structure, are expected to have significantly larger values for the Coulomb interaction, so that these values do not stand to compromise the realization of this phase.

Note, also, that the values used for the effective mass are an estimate, since they are ob- tained from the non-interacting, undoped electronic structure and, as such, are not accurate for the system studied here.

Failure of the saddle-point solution: charge conservation

This result accounts for the findings from the saddle-point solution: indeed, forU = U_c the action is flat and, trivially, the order parameter is arbitrary, and forU > Ucthe action inverts its concavity, so that its extremum is actually a maximum in the broken symmetry phase.

In this sense, the action derived so far is not physical, since it implies that charge imbalance would increase unboundedly to minimize the action. The minimum of the action must then be fixed by charge conservation. In turn, charge conservation imposes that the cone↑-Acan only be filled up to its capacity, which reads, for the spin densityn^A_↑ −n^B_↓,

n^A_↑ −n^B_↓ =D^>

ˆ −∆^>

ε^>_Λ

dε−

ˆ µ₀−^U₂ξ ε^>_Λ

!

=D^>

ˆ −∆^>

µ0−^U₂ξ

dε

=D^>U

2 ξ−D^>(∆^>+µ0), On the other hand, charge imbalance in the magnetic cones is given by

n^A_↑ −n^B_↓ =D^>

ˆ dε

nF

ε−µ0−U ξ 2

−nF

ε−µ0+U ξ 2

=

ˆ µ₀+^U₂ξ µ0−^U₂ξ

dε

=D^>U ξ .

Equating, we arrive at a constraint for the parameterξ, reading D^>U

2 ξ+D^>(∆^>+µ0) = 0. (5.31) Note that the filling constraint of Eq. (5.31) is equivalent to imposing directly that the the top of

the↑-Acone coincides with the Fermi level, that is,

−∆^>=µ0+U ξ 2 ,

and, as will be shown, directly yields the mean-field solution in the broken symmetry phase, which we callξ_U. Now, we introduce this constraint in the unphysical action of Eq. (5.29) using a Lagrange multiplierλ, which is promptly eliminated by variatingξ,

δ

S⁰(ξ)−λ D^>U

2 ξ+D^>(∆^>+µ0)

= 0⇒λ=N βU1−D^>U D^>U ξ . Substituting back, we arrive at the physical action,

S(ξ) N β~U =−

U Uc

−1 ξ_U −ξ 2

ξ

2, U > U_c, ξ >0 . (5.32) This is the most general form of the action, up to second order of the variational order parameter ξ, valid for all doping regimes. The extremum ofSis then given by

∂S

∂ξ(ξ_U) = 0⇔ ξ_U = 2|µ₀| −∆^>

U , U > U_c , (5.33)

which, clearly, is a minimum in the broken symmetry phaseU > Uc:

∂²S

∂ξ² = N βU 2

U Uc

−1

>0.

Figure 5.1 depicts the physical picture for this transition, in terms of the cones in the magnetic bands. Additionally, it is possible to determineµA, the Fermi level down to which the↓-Acone remains completely filled, according to the conditionn^B_↑ =n^A_↓ forξ=ξ_U. This condition reads

n^B_↑ −n^A_↓ =D^<

ˆ −∆^<

µ_A−U ξU/2

dε= 0 ⇔ |µA|= ∆^>+ ∆^<

2 = ∆. Furthermore, the magnetization density (i.e. spin density) yields

m=n^A_↑ −n^B_↓ =D^>U ξU = 2D^>(|µ0| −∆^>), (5.34) which, as expected, is equal to the hole densityp= 2D^>(|µ0| −∆^>), meaning that all charge transferred to the↑-Acone originates from the↓-B cone, as required by charge conservation.

Indeed, considering the conservation law Eq. (5.26) and thatη_↓^A=η^B_↑ = 0holds, in this regime,

Figure 5.1: Magnetic bands, according to Eq. 5.25, for (a) the normal phase,D^>U <1, and (b) the magnetic phase,D^>U >1. In (b) we have assumedξ >0.

we can determine the solutionsη^A_↑,Uandη^B_↓,U in the broken symmetry phase, yielding

η_↑,U^A =−η^B_↓,U =|µ0| −∆^>

U . (5.35)

These solutions confirm our ansatz: they are proportional (by a factor ofD^>U) to the variation of charge density in each cone and, indeed, such a variation is smaller in magnitude that the total charge density of a cone.

Finally, note that the same results obtained with the formalism used can be obtained us- ing a variational mean-field approach based on the Gibbs-Bogoliubov-Feynman inequality, as discussed in Sec. D.1 of Appendix D.

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