Dynamic response functions and helical gaps in interacting Rashba nanowires with and without magnetic fields

A partially gapped spectrum due to the application of a magnetic field is one of the main probes of Rashba spin-orbit coupling in nanowires. Such a “helical gap” manifests itself in the linear conductance, as well as in dynamic response functions such as the spectral function, the structure factor, or the tunneling density of states. In this paper we investigate theoretically the signature of the helical gap in these observables with a particular focus on the interplay between Rashba spin-orbit coupling and electron-electron interactions. We show that in a quasi-one-dimensional wire, interactions can open a helical gap even without magnetic field. We calculate the dynamic response functions using bosonization, a renormalization group analysis, and the exact form factors of the emerging sine-Gordon model. For special interaction strengths, we verify our results by re-fermionization. We show how the two types of helical gaps, caused by magnetic fields or interactions, can be distinguished in experiments.

Figure 1

Single-particle spectrum of a Rashba wire with two transverse channels in the absence of magnetic field. The spin texture is indicated by the color coding. Virtual transitions between subbands conserve momentum and are associated with a spin flip. In combination with electron-electron interaction, this can produce spin-umklapp scattering as indicated by the Feynman diagram on top.

Figure 2

A schematic representation of the various response functions we consider. (a) The insertion of a single electron into a system with a $B$-field gap, resulting in a single quasiparticle of mass $M_1$ in the “conduction” band. This will result in threshold behavior for the spectral function at $\omega =M_1$. (b) The same process, but in the case of an umklapp gap, where in order to obey charge conservation we must create two $e/2$ quasiparticles of mass $M_2$, so the threshold value shifts to $\omega =2M_2$. (c) and (d) The process interrogated by the density structure factor, which will have the same threshold behavior, regardless of the underlying gap-generating mechanism.

Figure 3

Effective interactions in the lowest subband due to virtual transitions to higher subbands: spin-flip terms ${\widehat{V}}_{\mathrm{sf}}$ (denoted by solid arrows) and spin-exchange terms ${\widehat{V}}_{\mathrm{sx}}$ (denoted by dashed arrows).

Figure 4

This figure shows the position of the chemical potential and the four low-energy linearized modes used to bosonize the system. When bosonized, the inner modes near $k=0$ are described by the operators $\left({\varphi }_{-},{\theta }_{-}\right)$, the outer modes near $k=\pm k_F$ by $({\varphi }_{+},{\theta }_{+})$.

Figure 5

(a) Density plot of the structure factor for the system with either a magnetic field or an umklapp-generated gap, encoded in the soliton mass $M_{\gamma }$. Regions with darker shading have a higher value of $S(k,\omega )$. The line through the origin is a smeared $\delta$-function peak arising from the Luttinger liquid modes, whereas the region of high $S(k,\omega )$ emanating from $\omega =2M_{\gamma }$ at $k=0$ corresponds to the contribution from the gapped modes. The momentum is measured in multiples of $k_0=M_{\gamma }/v_{-}$. The vertical dashed lines correspond to the cuts for fixed momentum of $S(k,\omega )$. (b) Plots of $S(q,\omega )$ for fixed momentum $q$. The size of the Luttinger liquid peak grows linearly as a function of $k$, whereas the contribution from the gapped modes grows with $k^2$.

Figure 6

(a) Density plot of the spectral function with a magnetic field induced gap, whose size is encoded in the soliton mass $M_1$. Regions with darker shading have a higher value of $A(k,\omega )$. The fan of spectral weight spreading from $k_F=2m{\alpha }_R$ comes from the Luttinger liquid modes, whereas the smeared $\delta$-function peak emanating from $\omega =M_1$ at $k=0$ corresponds to the contribution from the gapped modes. The momentum is measured in multiples of $k_0=M_1/v_{-}$. The vertical dashed lines correspond to the cuts for fixed momentum of $A(k,\omega )$. (b) Plots of $A(q,\omega )$ for fixed momenta $q$. The Luttinger liquid contribution is broad and featureless for momenta $q<k_F$, whereas the contribution from the gapped modes gives a sharp peak.

Figure 7

(a) Density plot of the spectral function with an umklapp gap, whose size is encoded in the soliton mass $M_2$. Regions with darker shading have a higher value of $A(k,\omega )$. The fan of spectral weight spreading from $k_F=2m{\alpha }_R$ comes from the Luttinger liquid modes, whereas the square-root function which onsets near to $\omega =2M_2$ is the contribution from the gapped modes. The momentum is measured in multiples of $k_0=M_2/v_{-}$. The vertical dashed lines correspond to the cuts for fixed momentum of $A(k,\omega )$. (b) Plots of $A(q,\omega )$ for fixed momenta $q$. The Luttinger liquid contribution is broad and featureless for momenta $q<k_F$, whereas the contribution from the gapped modes gives a square-root onset.