5.2 Low-energy theory
5.2.1 Saddle-point solution
Our interest is determining the critical value of the intra-orbital interaction energyUwhich onsets a charge imbalance between spin populations of the system and, thus, breaking of time-reversal symmetry. If the system is to have a ground-state which breaks this symmetry, there must exist at least one configuration of the fields{φτσ}which is an extremum of the action. Put simply, we will now find
∀τ, σ , δ
δφτ−q,σS[{φτσ}] = 0 (5.8)
which, in turn, shall yield the critical valueUc, provided the extremum of the action is, in fact, a minimum. In turn, the configuration which minimizes the action is expected to be the dominant contribution to the grand-partition function, in virtue of the structure of the path integral. Carrying out the functional derivatives, noting particularly that
δ δφτ−q,σ
X
σ0
Tr lnβ~ Gˆ−1
σ0
= X
σ0,τ0
X
k0,k
Gˆτ0
k0,k,σ0
δ δφτ−q,σ
Gˆ−1 τ0
k,k0,σ0
,
a general result, valid in virtue of taking the functional derivative of the trace of an operator [117].
We find
φτq,σ = i N β~
X
k
"
Gˆ∗−1τ k,σ
δq,0+iU
~ X
τ0
φτq,¯0σ
#−1
, ∀τ, σ .
The solution of this set of equations requires making a physically motivated ansatz. In fact, we can expect a space- and time-independent configuration to minimize the action. As such, we consider the ansatz
φτq,σ=inτσδq,0,∀τ, σ .
The resulting self-consistent equations are simplified by performing the Matsubara frequency summations, yielding
nτσ = 1 N
X
k
nF ετk,σ−µ+UX
τ0
nτσ¯0
!
, ∀τ, σ . (5.9)
As expected, the action is stationary for homogeneous H-S fields which solution is the electron densities at each cone. As we will be shown, however, they do not yield a minimum of the action.
Before proceeding, it is imperative to introduce some assumptions. The first is regarding the position of the Fermi level, lowered by hole-doping, and the subsequent shift of the spin-split cones. Namely, we have to consider that the low-lying bands (↑in valleyB and↓ in valleyA) will only be affected by the charge imbalances for some intervals of the Fermi levelµ0, that is, the Fermi level set atU = 0. In fact, we can break this picture down to three regimes:
1. Low doping: down to some level ofµ0, sayµA, only the high-lying bands (↑in valleyAand
↓in valleyB) suffer an imbalance.
2. Intermediate doping: fromµA down to another levelµB, the↓-Avalley will contribute as well, since that valley rises in energy enough to be energetically favourable to lose some of its electrons to↑-A.
3. High doping: ↑-B will only come into play for a Fermi level below the maximum of this cone, since its energy is lowered.
For now, our aim is to determine the existence of a spin and valley polarized material, thus it is sufficient to consider a Fermi level for which the low-lying cones remain completely filled. That is, we shall work assuming thatnA↓ = nB↑ and that charge imbalance happens solely between the cones↑-Aand↓-B, thus takingnA↑ −nB↓ as an order parameter.
The second is regarding the description of the energy spectrum of the low-energy system.
More precisely, we consider to be within the range of validity of the parabolic approximation for the valence band. Futhermore, due to the spin-splitting at the edge of the valence band, we distinguish the highest-lying and lowest-lying branches of the spin-split valence band according to the different spin-valley couplings, so that we have, respectively,
εAk,↑=εBk,↓=ε>k =−∆>− ~2k2
2m>, and εBk,↑=εAk,↓ =ε<k =−∆<− ~2k2
2m<, (5.10) where the parameters∆ν,ν =>, <, are related to the spin-split half-gap by
∆ν = ∆ν−νλ
2 = ∆−νλ ,
where the binary indexν is assigned to positive and negative signs as{>, <} 7→ {+,−}. Note that these spectra yield constant DOS in 2D,
Dστ(ε) =Dν(ε)≡Dν.
Also, in further calculations we will take the zero of energy to beεF (the midpoint between the unsplit valence band and the conduction band, as in Eq. (3.8)) so thatµ <0.
Considering the foregoing argument, we use Eqs. (5.9), for the electron densities, and (5.10), for the DOS, to compute the order parameternA↑ −nB↓, yielding
nA↑ −nB↓ = 1 N
X
k
"
nF εAk,↑−µ+UX
τ0
nτ↓0
!
−nF εBk,↓−µ+UX
τ0
nτ↑0
!#
= 1
ABZ
ˆ d2k
"
nF εAk,↑−µ+UX
τ0
nτ↓0
!
−nF εBk,↓−µ+UX
τ0
nτ↑0
!#
= 2π
ABZ
m>
~2 ˆ Λ
0
~2kdk m>
"
nF εAk,↑−µ+UX
τ0
nτ↓0
!
−nF εBk,↓−µ+UX
τ0
nτ↑0
!#
= D>
ˆ −∆>
ε>Λ
dε
"
nF ε−µ+UX
τ0
nτ↓0
!
−nF ε−µ+UX
τ0
nτ↑0
!#
T→0
= D>
ˆ µ−UP
τ0nτ↓0 µ−UP
τ0nτ↑0
dε=D>U X
τ0
nτ↑0−X
τ0
nτ↓0
!
= D>U nA↑ −nB↓
(5.11)
⇒ 1−D>U
nA↑ −nB↓
= 0. (5.12)
According to this, nA↑ −nB↓ = 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 = nA↑ +nB↑ +nA↓ +nB↓
= 2D>(µ0−ε>Λ) + 2D<(−∆<−ε<Λ)
(5.9)
= 2D> µ−U 2
X
τ,σ
nτσ−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
nAσ =X
σ
nBσ =X
τ
nτ↑ =X
τ
nτ↓ = 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).