Universal spin gap in interacting nanowires with spin-orbit coupling and inversion symmetry

Universal spin gap in interacting nanowires with spin-orbit coupling and inversion symmetry

Tommy Li Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark

This work shows that in nanowires of arbitrary spin with time-reversal and inversion symmetry, the Luttinger liquid phase is always destroyed by Coulomb repulsion and an arbitrarily weak spin-orbit coupling, regardless of the details of the one-dimensional confinement or spin-orbit interaction, whenever one pair of spin-degenerate bands is occupied. The interacting phase is universally characterized by gapless charge excitations and gapped spin excitations. It is proposed that a spin gap of order 10 µeV can be experimentally realized in free-standing hexagonal nanowires in narrow-gap semiconductors.

Interacting fermionic systems in one-dimension (1D) are paradigmatic examples of strongly correlated systems, displaying striking properties such as interaction-dependent critical exponents and spin-charge separation. The celebrated Luttinger liquid, which first appeared in models without backscattering, possesses a gapless spectrum of bosonic excitations Tomonaga (1950); Luttinger (1963); Mattis and Lieb (1965). For spinful systems with backscattering, however, the nature of the ground state was shown by Luther and Emery to depend on the nature of the interaction Luther and Emery (1974). When spin rotational symmetry is present, Luttinger liquid properties are maintained for repulsive interactions, while for attractive interactions, a new correlated state arises which is gapped to spin excitations but gapless to excitations of the total charge. The spin gap manifests as a vanishing of the single particle density of states at low energies Voit (1996), a flux periodicity of indicative of fermionic pairing Seidel and Lee (2005), and vanishing backscattering from impurities at the Fermi level Kainaris et al. (2018). These characteristics, which bear remarkable similarities with the superconducting state exist nevertheless in the absence of superconducting order.

A number of previous studies have demonstrated that the spin-gapped Luther-Emery state may be realized despite the absence of an attractive interaction in certain multiband systems Finkel’stein and Larkin (1993); Cheng and Tu (2011); Kraus et al. (2013); Lang and Büchler (2015); Guther et al. (2017); Li et al. () as well as Dirac semimental nanowires with an external magnetic field Zhang and Liu (2018). At the same time, previous studies on interacting nanowires with Rashba spin-orbit coupling (describing the common scenario in semiconductor nanowires) have exclusively focused on the case where either time-reversal or inversion symmetry are broken Yu et al. (2004); Moroz et al. (2000); Cheng and Zhou (2007); Sun et al. (2007). If two subbands are occupied, spin-charge separation is destroyed in these systems and the Luther-Emery state cannot be realized. In this work, the opposite case is considered, where inversion symmetry and time-reversal symmetry are both maintained, and the lowest pair of bands remain degenerate. A Renormalization Group (RG) and bosonization analysis reveals that an arbitrarily weak spin-orbit interaction is sufficient to open a spin gap, regardless of the geometry of the wire or screening surfaces, the total spin of the system, or the nature of the spin-orbit interaction as long as these two symmetries are preserved. This result depends on the explicit consideration of the three-dimensional structure of the single-particle states as well as the two-particle scattering amplitudes for a Coulomb interaction which is screened by symmetric surfaces of arbitrary geometry, and yet applies generally to all systems with the required symmetries. In particular, this result implies that the ground state of any clean semiconductor nanowire possessing intrinsic spin-orbit coupling is always the Luther-Emery phase, as long as the engineering of the system does not introduce inversion asymmetry.

Figure 1: Examples of possible geometries for screening of the nanowire: (left) two planes symmetrically placed around the wire, and (right) a single conductor enclosing the wire.

General symmetry analysis. Consider a system of fermions confined to 1D which we may generally describe by a long-wavelength effective Hamiltonian where the interaction term is


where , are three dimensional coordinates, and are the spin operators, the fermionic creation operators are -component spinors, and is the Coulomb interaction which is screened by conducting surfaces external to the wire. For semiconductor devices populated by pure or hole states we have respectively, while for narrow gap semiconductors we may account for mixing between electron and hole states via spin-operators which act on the direct sum of the electron and hole sectors. In the absence of external magnetic fields, we may assume that is symmetric under time reversal (). We shall study the case where the Hamiltonian has an additional symmetry under combined spatial and spin rotation by an angle about the wire axis as well as inversion along the axis of the wire (and thus the Hamiltonian is also symmetric under inversion of three spatial coordinates). In the case of and type semiconductors, this situation includes hexagonal or rectangular wires with Rashba interaction in which anisotropic spin-momentum couplings due to the lattice are either negligible at the relevant energy scales or the wire axis coincides with a symmetry axis of the crystal. We may then classify degenerate time-reversed pairs of single-particle states via their eigenvalues under the combined spin and spatial -rotation. Switching to 2D coordinates we may express the single particle states in a basis of spin states with polarization along the wire axis,


and the functions satisfy


The phases in (2) are chosen so that . The infrared properties of the system are determined by interactions involving particles close to the Fermi level. We will consider the situation where two bands are occupied, so that there are four Fermi points corresponding to the states .

In the following analysis the interaction term in the Hamiltonian is required to satisfy both translational invariance along the wire axis and -rotational symmetry in the cross section. Since the Coulomb interaction is spatially isotropic these symmetries can only be violated by the geometry of screening planes near the wire. The analysis incorporates several physical scenarios, some of which are illustrated in Fig.  1: (left) two conducting planes placed symmetrically around the wire and (right) a single conductor enclosing the wire. We may also consider the situation where the system is screened only by a remote conducting surface at distance larger than both the dimensions of the cross section and the Fermi wavelength (not shown). In all cases the conducting surfaces are required to be homogeneous along the wire direction. For a remote screening plane parallel to the wire, the Coulomb interaction consists simply of the unscreened interaction in addition to the potential due to an image charge at distance from the wire and is axially isotropic. In the remaining cases the Coulomb interaction may be expanded in a basis of solutions of the Helmholtz equation which vanish on the screening surfaces,


where are the eigenvalues of the Laplace operator. The interaction Hamiltonian then involves processes of the form , , and corresponding terms generated by reversing and indices.

We may express the effective low-energy Hamiltonian density in terms of slowly varying chiral field operators , with associated with right and left movers, and the density operators and (where ) corresponding to the sum and difference of densities in opposite spin bands,


and the factor of containing the Fermi velocity on the left hand side is present for convenience to ensure that the interaction constants are dimensionless.

Figure 2: The RG flow of the couplings , .

After a bosonization mapping Giamarchi (2003) via operators satisfying associated with the densities , , we obtain a Hamiltonian density which is a sum with the general forms




where ()


The exact decomposition of the Hamiltonian into separate spin and charge sectors is a consequence of the symmetry between the and bands. In the absence of both time-reversal and inversion symmetry, the charge and spin fields become mixed Moroz et al. (2000); Yu et al. (2004); Cheng and Zhou (2007), preventing the realization of the Luther-Emery state. The Hamiltonian density is harmonic and describes gapless excitations of the total charge, while , which describes spin excitations, contains self-interactions which in general generate several possible interacting phases depending on the values of the parameters , and . The nature of the Coulomb interaction in combination with time-reversal and inversion symmetry, however, will enforce strict relations between the values of these parameters and the system will universally exhibit one phase of the Sine-Gordon model, in which excitations of the bosonic field are gapped and the ground state contains a nonvanishing expectation value . In order to demonstrate this, we must evaluate the interaction matrix elements between initial and final two-particle states , which are given by


where is the transferred momentum and is the Fourier transform of the interaction with respect to the -axis and the inner products are given in terms of the spin components (2) via


and is symmetric under exchanges of and indices. In the Sine-Gordon Hamiltonian (7), the parameter is proportional to the matrix element corresponding to backscattering,


where for convenience the inner products are denoted


and choose to be real. Since , we have , and to first order the scaling dimensions of the operators and are and respectively, so to first order we might expect that the term proportional to in (7) becomes relevant at low energies while the term proportional to is irrelevant. It is well known, however, in the theory of Sine-Gordon models that the true infrared properties of the system emerge only when interactions are accounted for to second order Giamarchi (2003). The scaling relations for the couplings to second order are given by


with where is the UV cutoff and is an energy scale which runs toward the infrared limit. If either the interactions are initially zero, they remain zero under the RG flow. This situation occurs, for example when the nanowire possesses an additional rotational symmetry whose selection rules forbid interactions of the form and therefore . In this case the RG equations may be integrated easily, and the system is gapped when the bare couplings satisfy . The symmetries of our system imply that this condition is always satisfied, since represents a sum of interfering forward and backward scattering processes , and


and the inequality is rooted in the fact that is odd under both inversions of the spatial coordinate and longitudinal momentum, (Eq. 10). Solution of (13) then shows that the system flows to strong coupling at an energy scale , which indicates the presence of a spin gap given by


where represent the bare value of the couplings.

Figure 3: Calculated values of the Fermi energy at which (15) predicts a gap (panels a,b,e,f) and the corresponding ratios (panels c,d,g,h), as a function of . The left (right) set of panels shows (). The values are: (panels a,c) and (dashed), (panels b,d) , (e,g) and (dashed), (f,h) . In panels a,c,d,e,g,h the curves increase uniformly with increasing magnitude of , while the curves decrease uniformly with increasing magnitude of in panels b,f. The range of used in the plots are chosen to ensure that only one pair of spin degenerate bands intersects the Fermi level. The topmost panels illustrate the dispersion of the lowest two pairs of degenerate bands for the values of used in the panels below. These results are independent of material parameters.

For the case when , the RG equations exhibit a duality relation , so that the flow to strong coupling of one of either couplings implies the vanishing of the other. The RG flows of the couplings ,   are plotted in Fig.  2. The phase diagram consists of four regions separated by the dashed lines, with a fixed point at . The value of is given in terms of the Coulomb interaction by


which satisfies , implying that the couplings initially lie above the lower separatrix (dashed line in Fig. 2), and flow in the portion of the phase diagram indicated by the blue lines towards strong coupling, , , . We therefore find that the spin sector is gapped for both and and the ground state develops a nonvanishing expectation value .

Results for nanowires with Rashba interaction. Consider now a specific model for a cylindrical -type semiconductor wire, in which the wavefunction is strongly peaked at the radius , a situation which may arise due to surface effects. The single particle Hamiltonian and dispersion are given by


with being the Pauli matrices acting on spin, the Rashba coefficient, the transverse confining potential, and the ratio of the spin-orbit interaction strength to the splitting between the lowest angular momentum modes. The wavefunctions have the generic form


where depend on the strength of the spin-orbit interaction as well as the density. The interaction due to axial symmetry, and may be obtained analytically in the form of the modified Bessel functions. The gap (15) depends on , as well as the strength of the interaction, which we may parametrize via the effective Bohr radius and Rydberg energy . The perturbative expression (15) is quantitatively accurate for , and for we may crudely estimate that the gap reaches a maximum value . Fig.  3 shows the Fermi energy for which , and the corresponding values of the ratio , as a function of for various values of , in the range of values of for which only the lowest pair of degenerate spin bands is occupied.

The interaction is strongly affected by modification of the single-particle dispersion due to the spin-orbit interaction, shown in the topmost panels in Fig.  3. At special values the lowest band becomes flat at (17), leading to a strong enhancement of the interaction at low densities, . The ratio plotted in Fig.  3 diverges logarithmically for , and therefore when the spin-orbit interaction is tuned to these values the spin gap may in principle become arbitrarily large. The value of is given by


where is the Euler-Mascheroni constant. At the same time, the required value of also vanishes as


The most promising experimental candidates for the observation of the spin gap are free standing hexagonal nanowires in narrow gap semiconductors, where both spin-orbit coupling and surface confining effects are strong. In InAs, the parameters and in InSb, , thus the typical nanowire radius is comparable to . Approximating these wires via the hollow cylinder model discussed, we obtain maximum gaps 15µeV and 19µeV for values respectively for InAs, and 8, 10µeV in InSb.

We may alternatively study the case of a hollow rectangular wire with dimensions . Assuming the corners are slightly rounded over a region much smaller than the side lengths so that no reflection occurs, the transverse orbital wavefunctions in the absence of spin-orbit coupling are given by


for within a small distance of the edge of the rectangle, is a real wavefunction confined within an asymmetric potential well, is the outward pointing normal vector to the edge of the rectangle, and is a path length. Accounting for spin-orbit interaction in first-order perturbation theory, the spin components of the wavefunction are


where parametrizes the strength of the spin-orbit interaction and are dimensionless constants which decrease with . When , and interactions of the form are forbidden and the results are similar to that for the cylindrical case. For or , and becomes purely imaginary. As a result, the matrix elements and become equal in magnitude, which implies that the initial couplings move onto the separatrix in the region , of the RG flow (Fig. 2), and the system appears to be gapless. Accounting for higher orders in the spin-orbit interaction however, we find that is no longer purely imaginary and the system remains gapped.

Summary. This work has shown that a class of 1D interacting fermionic systems with spin-orbit coupling exhibits a spin gap, namely those possessing time reversal and spatial inversion symmetries. This result may be contrasted with previous studies on interacting nanowires with Rashba spin-orbit coupling Yu et al. (2004); Moroz et al. (2000); Cheng and Zhou (2007); Sun et al. (2007) where inversion asymmetry and magnetic fields play a crucial role and spin excitations are ungapped. The size of the spin gap has been calculated for a hollow cylindrical wire, and we have seen how in a rectangular geometry it may be controlled by tuning of the aspect ratio. As previous studies have shown, the spin gap is manifested in several observable phenomena, including an energy gap in the tunneling conductance Voit (1996); Li et al. (). This gap can be of the order of 10 µeV in narrow-gap semiconductors, with appropriate tuning of the density, and thus observable in conventional experiments in sufficiently clean and long nanowires.

Acknowledgments. The author acknowledges M. Burrello and K. Flensberg for important discussions. This work was supported by the Danish National Research Foundation.


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