Evaluación Interna. (Matriz EFI)

7.4.1 Solitons

The Su-Schrie↵er-Heeger (SSH) model[123, 124] may provide an explanation for the observed structures. The SSH model describes a one-dimensional chain, characterised by two tunnelling amplitudes t₁ and t₂ between two sublattices A and B. It provides a model for studying fractionalized charges, if interfaces exist in the lattice potential to separate topologically distinct chains into multiple domains in the real space [125,126].

To model the SSH a double-well optical lattice has been shown to be a powerful tool [127,128]. A possible explanation for the emergence of the 0.25 G₀ structure could be that holes in Ge forms pairs with neighbouring holes, forming a symmetric ground and an anti-symmetric first excited state. In the ground state the two holes are coupled so move together or else some complex motion which involves pairs constantly reforming and breaking. Considering on-site repulsive interaction, the potential,

V = Uˆ X

j

(ˆn_j,a"nˆ_j,a#+ ˆn_j,b"nˆ_j,b#) (7.2) where U> 0 is the on-site interaction strength, j is the site index of the double-well lattice,", # are the spin up and spin down, respectively. Flatbands arise in the extreme case when t₂ = t₁ = 0. In this case localised orbitals ˆc^†_j"|0i = (ˆa^†_j"+ ˆb^†_j")|0i/p

2 and ˆ

c^†_j#|0i = (ˆa^†_j# + ˆb^†_j#)|0i/p

2 are degenerate eigenstates of this flatband with energy t1. Where ˆa and ˆb are the creation operators for spin-up or spin-down atoms at the left and right wells on site j. In this case a ferromagnet emerges at half filling and all electrons/holes fill up either spin up or spin-down chain. This is illustrated in fig. 7.5.

Figure 7.5: Charge fractionalization. top)At half-filling, a repulsive interaction leads to spontaneous symmetry breaking and a ferromagnet emerges. The red and green clouds represent the Wannier wave functions of the spin-up and spin-down particles, respectively. middle Doping an extra particle forms two domain walls (blue ovals). If t₂ = 0, the two domain walls are de-confined because of interchain tunneling t (black wiggle). bottom) If t = 0, the two domain walls are confined for any finite interchain tunneling t₂ (purple arrow). Figure adapted form ref. [124].

This state saves energy and does not cost extra kinetic energy. This means a repulsive interaction lifts the single-particle degeneracy, resulting in a ferromagnet with a twofold degeneracy, and the ground state can be |Gi1 =Q

jcˆ^†_j"|0i or |Gi2 =Q

jˆc^†_j#|0i. If this is the case then in a degenerate ferromagnet state when we apply an in-plane magnetic field this degeneracy lifts and hence we get the 0.125 G₀ structure. This is a highly speculative explanation and further measurements along with theoretical models would help with the explanation. For more details on the SSH model please see references [123,124,125,126].

When the source-drain voltage is increased this has the e↵ect of breaking the pairing so there is a transition to single electron behaviour which means that the conductances are increased by a factor of 2 as pairing is eliminated.

We can obtain the energy loss due to the pairing, when we plot the transconductance

Figure 7.6: Transconductance dG/dVsg plotted against d.c bias Vdc at B=0. Regions of light orange correspond to a large amplitude of dG/dV_g, indicating the locations of subband transitions.

dG/dVsg against source drain bias at B=0 T. The regions of light orange correspond to a large amplitude of dG/dV_sg, and indicate the locations of the subband transitions.

This is shown in figure 7.6. From the figure we can determine the energy loss due to the pairing, which is⇠ 0.25 meV.

Temperature dependence measurements have shown that both 0.25 G₀ and 0.125 G₀ are robust up to ⇠ 1 K. Figure 7.7 shows di↵erential conductance as a function of split gate voltage at 20 mK and 1 K. From the figure we can see that the 0.25 G₀ structure is stronger at 20 mK and we can still see it at 1 K. Similar to 0.25 G₀structure the 0.125 G₀ structure is robust to temperatures up to ⇠ 1 K. Figure 7.8 shows the di↵erential conductance as a function of split gate voltage at 20 mK and 900 mK at B

= 6 T.

a)

b)

20 mK

1 K

V

= 0.18 V

V

= 0.18 V

Figure 7.7: Di↵erential conductance as a function of split gate voltage at 20 mK (a) and 1 K (b) at B = 0 T, and V_g3 = 0.18 V.

a)

b)

B = 6 T

B = 6 T V

= 0.18 V

V

= 0.18 V 20 mK

900 mK

Figure 7.8: Di↵erential conductance as a function of split gate voltage at 20 mK (a) and 900 mK (b) at B = 6 T, and V_g3 = 0.18 V.

7.4.2 Fractional Wigner Crystal

The Luttinger liquid, describing low energy gapless one dimensional systems can provide another explanation [129,130]. In reference [131] it was proved that the Fermi operator

s(x), where s =± is the spin projection, of a generic one dimensional electron system can be very di↵erent with respect to the Luttinger model [131, 132]. In the standard relation

s⇠ e^i✓(x) X

p=±1

e ^ipkFxe^{ip (x)} (7.3)

where k_F the Fermi momentum and (x) and ✓(x) the usual bosonic fields, is replaced by a more general expression.

s⇠ e^i✓(x) X1 p= 1

c^(s)_p e ^ipkFxe^{ip (x)} (7.4)

where c^(s)_p is the model dependent coefficients. A model has been developed in ref-erence [133] for a strongly interacting Helical Luttinger liquid [134, 135], when the spin-momentum locking breaks the symmetry c^(s)_p = c^(s)p . Reference [133] develops a strongly interacting Luttinger liquid quantum spin Hall system in the presence of two-particle backscattering extending over the full helical edge. This system is charac-terised by charge oscillations which are half the wavelength of the usual Wigner crystal, which suggests the formation of a correlated state of fermions with charge e/2, with e the electron charge. This can be visualised by two-particle backscattering bringing two-particles from one branch of the dispersion relation to the other one, and there is a momentum transfer of 4k_F. From Landau’s theory of the Fermi liquid, the helical liq-uid in the presence of two-particle backscattering can be reformulated in terms of new quasi-particles. In the presence of strong interactions (Luttinger parameter K_L< 1/2, where K_L = ⌫_F/⌫), two-particle backscattering becomes relevant, and determines the nature of the quasi-particles. The two-particle backscattering can be viewed as a single quasi-particle backscattering term, thus the Femi momentum k_F⁰ of the quasi-particles becomes 2k⁰_F. As there are two quasi-particles, their charge becomes e/2 because of charge conservation. For a more thorough explanation please see reference [133].