A Second order perturbative RG calculation for the case with U(1) symmetry

Signatures of Majorana Kramers Pairs in superconductor-Luttinger liquid and superconductor-quantum dot-normal lead junctions

Abstract

Time-reversal invariant topological superconductors are characterized by the presence of Majorana Kramers pairs localized at defects. One of the transport signatures of Majorana Kramers pairs is the quantized differential conductance of when such a one-dimensional superconductor is coupled to a normal-metal lead. The resonant Andreev reflection, responsible for this phenomenon, can be understood as the boundary condition change for lead electrons at low energies. In this paper, we study the stability of the Andreev reflection fixed point with respect to electron-electron interactions in the Luttinger liquid. We first calculate the phase diagram for the Luttinger liquid-Majorana Kramers pair junction and show that its low-energy properties are determined by Andreev reflection scattering processes in the spin-triplet channel, i.e. the corresponding Andreev boundary conditions are similar to that in a spin-triplet superconductor - normal lead junction. We also study here a quantum dot coupled to a normal lead and a Majorana Kramers pair and investigate the effect of local repulsive interactions leading to an interplay between Kondo and Majorana correlations. Using a combination of renormalization group analysis and slave-boson mean-field theory, we show that the system flows to a new fixed point which is controlled by the Majorana interaction rather than the Kondo coupling. This Majorana fixed point is characterized by correlations between the localized spin and the fermion parity of each spin sector of the topological superconductor. We investigate the stability of the Majorana phase with respect to Gaussian fluctuations.

pacs:

I Introduction

The search for topological superconductors, which host Majorana zero modes (MZMs), has becomes an active pursuit in condensed matter physics Reich (2012); Brouwer (2012); Wilczek (2012); Alicea (2012). Such exotic modes are predicted to obey non-Abelian braiding statisticsMoore and Read (1991); Nayak and Wilczek (1996); Read and Green (2000), and have potential application in topological quantum computations Kitaev (2002); Nayak et al. (2008). Many theoretical proposals for realizing topological superconductors in the laboratory have been put forward recently Fu and Kane (2008, 2009); Sau et al. (2010); Alicea (2010); Lutchyn et al. (2010); Oreg et al. (2010); Cook and Franz (2011); Sau and Sarma (2012); Nadj-Perge et al. (2013), and, more excitingly, devices for detecting MZMs were successfully fabricated in the laboratory and the preliminary signatures of MZMs were observed Mourik et al. (2012); Das et al. (2012); Deng et al. (2012); Finck et al. (2013); Churchill et al. (2013); Nadj-Perge et al. (2014); Deng et al. (2014); Higginbotham et al. (2015); Albrecht et al. (2016); Zhang et al. (2016). Most research activity has focused on the topological superconductors belonging to class D ( i.e., SCs with broken time-reversal-symmetry) and supporting an odd number of MZMs at a topological defect Altland and Zirnbauer (1997); Schnyder et al. (2008); Kitaev (2009). However, Majorana zero modes can also appear in pairs in time-reversal invariant topological superconductors (TRITOPS) belonging to class Schnyder et al. (2008); Kitaev (2009); Teo and Kane (2010). Those MZM pairs are referred to as “Majorana Kramers pairs” (MKPs), and their stability is protected by the time-reversal (TR) symmetry and the quasiparticle excitation gap. Recently, several theoretical proposals were put forward to realize TRITOPS Wong and Law (2012); Deng et al. (2012); Zhang et al. (2013); Nakosai et al. (2013); Keselman et al. (2013); Gaidamauskas et al. (2014); Klinovaja et al. (2014); Schrade et al. (2015). Transport signatures of MKPs and their detection schemes using a QPC were also recently investigated in a quantum spin Hall system Li et al. (2016).

Most previous works on MKPs considered non-interacting (or effectively non-interacting) models. It is well-known, however, that interactions in one-dimensional systems are very important Gangadharaiah et al. (2011); Lobos et al. (2012); Fidkowski et al. (2012); Affleck and Giuliano (2013) and in some cases may even modify the classification of non-interacting systems Fidkowski and Kitaev (2011). For non-interacting systems, the presence of a MKP leads to a quantized conductance of due to perfect Andreev reflection at the junction. This quantization of the conductance is due to the constraints imposed by TR symmetry which leads to complete decoupling of MKP in the non-interacting models. The situation is different, however, in the presence of interparticle interactions, and the fate of the perfect Andreev reflection fixed point is unclear. In this paper, we study the stability of MKPs with respect to electron-electron interactions and consider two generic systems - a) MKP coupled to an interacting Luttinger liquid (see Fig. 1 a)); b) MKP coupled to an interacting quantum dot (see Fig. 1 b)).

Figure 1: (Color online) a) Schematic setup consisting of a) a junction between the Luttinger liquid and TRITOPS, and b) a quantum dot coupled to a normal lead and a TRITOPS. Here, denotes the point in the lead which couples to the MKPs or quantum dots.

We first consider a spinful Luttinger liquid lead with spin symmetry coupled to a TRITOPS with a single MKP per end. In this case the boundary problem has an additional symmetry. We find that for weak repulsive interactions, with being the Luttinger parameter, the Andreev reflection fixed point () is stable and the normal reflection fixed point () is unstable. For intermediate interaction strength , the phase diagram depends on microscopic details, i.e. on the strength of four-fermion interactions allowed by TR symmetry, which causes a Berezinsky-Kosterlitz-Thouless (BKT) transition between and . Finally, for sufficiently strong repulsive interactions , the two electron backscattering term becomes relevant, and drives the system to a stable normal reflection fixed point.

In the presence of spin-orbit coupling, the corresponding boundary problem may break symmetry. In this case, allowed processes involve spin-preserving and spin-flip Andreev scattering which drive the system to different boundary conditions for lead electrons: spin-preserving Andreev boundary () condition corresponds to and spin-flip Andreev boundary () condition corresponds to . Thus, the corresponding phase diagram depends on the relative strength of the corresponding Andreev scattering amplitudes. We find that the boundary conditions in this case are similar to those in a spin-triplet superconductor-Luttinger liquid junction and are stable with respect to weak repulsive interactions. In this sense, the physics is fundamentally different from an s-wave superconductor-Luttinger junction where weak repulsive interactions destabilize Andreev reflection fixed point Fidkowski et al. (2012).

In this paper, we also study the effect of local repulsive interactions by considering a MKP coupled to a quantum dot (QD) and an -invariant normal lead (NL). In the limit of a large Coulomb interaction in the QD and single-electron occupancy, we investigate the competition between Kondo and Majorana correlations. When the coupling to the MKP is absent (), the system flows to the Kondo fixed point with the corresponding boundary conditions for NL electrons where denote right and left movers. As we increase the coupling constant , the system exhibits a crossover from the Kondo dominated regime to a Majorana dominated regime where the QD spin builds up a strong correlation with the MKP. The latter is characterized by boundary conditions . Thus, the problem at hand represents a new class of boundary impurity problems where spin in the dot is coupled to the fermion parity of a topological superconductor.

In order to understand thermodynamic and transport properties of this Majorana fixed point, we have developed a slave-boson mean-field theory (please refer to Refs. Coleman (1984); Bickers (1987) for Anderson impurity models) for this system. We show that the Majorana dominated regime corresponds to a new (i.e. different from Kondo) saddle-point solution. We have analyzed the stability of this mean-field solution with respect to Gaussian fluctuations (in the spirit of Refs. Read and Newns (1983); Coleman (1987)) finding that the mean field theory is stable (in the quasi-long range order sense) and can be used to calculate different observable quantities. We use this approach to calculate differential tunneling conductance as a function of applied voltage bias.

The paper is organized as follows. In Secs. II.1 and II.2, we introduce the model of a MKP - Luttinger liquid junction, and consider the boundary problem with and without (e.g., due to Rashba spin-orbit coupling in the lead) symmetry. In Sec. III, we study the signatures of a MKP in a QD-NL junction using both the renormalization group (RG) analysis and the slave-boson mean-field theory. We also consider the Gaussian fluctuations around the mean-field solution, and analyze the stability of the slave-boson mean-field solution. Finally, we conclude in Sec. IV.

Ii Majorana Kramers pair - Luttinger liquid junction

In this section we consider the setup shown in Fig. 1 a) consisting of a semi-infinite spinful Luttinger liquid coupled weakly to a TRITOPS. We assume that the topological gap of the superconductor is much larger than the other relevant energy scales (i.e. tunneling amplitudes , and see the text below Eq. (4) for definitions). Thus, in the low-energy approximation the superconductor Hamiltonian consists of only the MKPs localized at its opposite ends. In this section, we will use to describe the operators at the boundary , and use (similarly for , and ) as the initial value in RG flow with the initial length cutoff .

ii.1 Majorana Kramers pair coupled to -invariant Luttinger liquid

Theoretical Model

We first consider an -invariant interacting nanowire coupled to a MKP. The Hamiltonian for the 1D lead can be written as the spinful Luttinger model

(1)

where and are velocity and Luttinger parameter for charge and spin modes, respectively. The bosonic fields satisfy the commutation relation . We use here the following convention for the Abelian bosonization procedure Giamarchi (2003):

(2)

where represents right/left moving modes, is an ultraviolet cutoff length scale, denotes fermion spin, and is the Klein factor.

The total Hamiltonian is given as

(3)

where the coupling between the Luttinger liquid lead and the MKP. We neglect here the ground-state degeneracy splitting energy. The most general form of the TR invariant boundary Hamiltonian describing the coupling between the MKP and Luttinger liquid and including only two and four-fermion operators reads

(4)

where are Majorana operators with . Here , and are set to be real. The first two terms represent tunneling between the lead and the MKP with the amplitudes . Under TR symmetry , field operators transform as

(5)
(6)

(i.e. ) and the coupling constants in the Hamiltonian need to complex conjugated. TR symmetry requires that with being real. Assuming the spin-quantization axis is fixed in the whole system, the overall Hamiltonian has spin-rotation symmetry, leaving it invariant under the unitary transformation:

(7)
(8)

Here represents a spin-rotation matrix by an angle . Thus, electron tunneling between Luttinger liquid and topological superconductor preserves the spin. The last two terms and represent normal, and anomalous backscattering terms, which, in fact, will also be generated by the tunneling terms in the RG flow in the presence of interactions in the Luttinger liquid.

Weak coupling RG analysis near normal reflection fixed point

We now study the stability of the weak coupling normal reflection fixed point using perturbative RG analysis. In the ultraviolet, the boundary conditions for lead electrons at are given by (i.e. perfect normal reflection). In terms of bosonization language, this boundary condition corresponds to and pinning . Once we turn on the boundary couplings , and , boundary conditions for lead electrons may change depending on the strength of interaction in the lead. Let us study now the stability of this normal reflection fixed point. After integrating out the fields away from , the corresponding imaginary-time partition function becomes

(9)

with

(10)

and the boundary coupling term reads

(11)

where is the ultraviolet cutoff. Here we used short-hand notation denoting the fields at .

We now perform a perturbative RG procedure by separating the bosonic fields into slow, and fast modes and integrating out the fast modes. After some manipulations, the new effective action can be calculated using the cumulant expansion:

(12)

where the average describes an integration over the fast modes. The details of this calculation are presented in the Appendix A, and we simply summarize the RG equations here

(13)
(14)
(15)

Here where is the ratio of the cutoff change from to with . One can notice that is a relevant perturbation and grows under RG. Therefore, in the non-interacting case when , the system will flow to the perfect Andreev reflection fixed point () corresponding to the boundary condition  Fidkowski et al. (2012) and quantized differential conductance at zero temperature.

Figure 2: Illustration of the RG flow diagram near the normal reflection fixed point for the case with spin symmetry. The red line indicates the regime where is stable. Here, we set

Let us now try to understand the effects of interactions. In this section, we will focus on an spin-invariant lead () and repulsive interactions in the nanowire . In this case, the coupling is irrelevant and can be neglected, and RG equations simplify to

(16)
(17)

The coupling is relevant for not too strong repulsive interactions. It becomes marginal, however, if initial value of is equal to the special value . Indeed, then above RG equations (after a slight redefinition of variables) are identical to the anisotropic Kondo model Giamarchi (2003), the solution of which is well-known. If the initial value of is zero, and the parameter (i.e. ), the system will flow to strong coupling fixed point whereas for , the system will flow to fixed point for small and flow to strong coupling for larger . The perturbative RG flow is summarized in Fig. 2.

Weak coupling RG analysis near perfect Andreev reflection fixed point

As shown in the previous section, the normal reflection fixed point is unstable for weak repulsive interactions and the system flows to the perfect Andreev fixed point corresponding to the boundary conditions which, in bosonic variables corresponds to pinning and fields at . Thus, the fluctuating degrees of freedom are the fields and and the corresponding boundary action reads

(18)

We now consider perturbations near the Andreev fixed point which are consistent with time-reversal and the spin- symmetry of the Luttinger liquid lead. The only fermion bilinear boundary perturbation preserving aforementioned symmetries is

(19)

In addition, one has to also consider the following four-fermion perturbation consistent with the above symmetries:

(20)

which corresponds to two-electron backscattering. The perturbative RG equations for and are given by

(21)
(22)

One can see that the first term is irrelevant since whereas the second coupling becomes relevant for indicating that fixed point becomes unstable for strong repulsive interactions. Taking into account the perturbative RG analysis near both and fixed points, we conjecture the qualitative phase diagrams shown in Fig. 3.

Figure 3: Illustration of the flow between the normal reflection fixed point and the Andreev reflection fixed point for the case with spin symmetry. For the analysis at , the boundary perturbation bends the transition line, i.e. the brown dashed line connecting at and at . We also assume the initial value in RG is zero.

Differential tunneling conductance

We now discuss transport signatures of MKPs. The simplest experiment to detect the presence of a MKP is the differential conductance measurement. We focus on the case with an symmetric wire and calculate at zero voltage bias as a function of temperature. The RG flow between the normal and Andreev reflection defines a crossover temperature , which roughly corresponds to the width of the zero bias peak. Although the conductance for the whole crossover regime requires involved calculations, the conductance around and fixed points can be obtained using perturbation theory, see, e.g., Ref. [Lutchyn and Skrabacz, 2013].

First of all, we consider the case , where fixed point is stable. In the ultraviolet (i.e. near the unstable normal reflection fixed point), the leading relevant perturbation is the coupling to the MKP, , which has scaling dimension . Near the stable Andreev reflection fixed point (i.e. in the infrared), the deviation from the quantized value comes from the leading irrelevant operators which cause backscatterings, i.e. single-electron backscattering shown in Eq. (19) with scaling dimension and two-electron backscattering shown in Eq. (20) with scaling dimension .

Here, for , the single-electron backscattering shown in Eq. (19) is the leading irrelevant operator. We can now obtain scaling of the conductance with temperature at zero bias (assuming the initial value of coupling is zero, i.e. ):

(23)

where are numerical coefficients of the order one. Similarly, one can obtain voltage corrections to the conductance at zero temperature. Interestingly, the analogous coefficient vanishes in the non-interacting limit and, therefore, the scaling of the conductance with voltage and temperature is different at , see Ref. [Lutchyn and Skrabacz, 2013] for details.

Next, we consider , where is stable in the infrared. In this case, we start near the high energy unstable fixed point and calculate the conductance by perturbing with the two-electron backscattering operator which is the leading relevant operator in this regime. Thus, we obtain

(24)

where are numerical coefficients.

The calculation of the conductance in the regime depends on microscopic details (i.e. strength of ), and is outside the scope of the paper.

ii.2 The effect of the Rashba spin-orbit coupling in the lead

Theoretical Model

In this section, we consider the effect of Rashba spin-orbit coupling (SOC) in the nanowire. When coupling to MKP, the spin eigenstates of the MKP do not have to be the same as the the spin eigenstates of the nanowire. Therefore, tunneling between the lead and the TRITOPS will have both spin-preserving and spin-flip components. In order to see how the spin flip tunneling is generated, we consider the direction of the Rashba coupling which has an angle rotation compared to that of the MKP. The corresponding tight binding model can written as

(25)
(26)

One can see that the above Hamiltonian respects TR symmetry. We apply the following unitary transformation

(27)

and then the bulk and boundary Hamiltonians become

(28)

and

(29)

where and , and for spin-. Therefore, the spin-flip tunneling is non-zero for any , i.e. due to the presence of SOC. One can simply check that, in the presence of both and , the symmetry shown in Eq. (8) is broken. In this case, the boundary condition at the Andreev reflection fixed point is determined by the relative magnitude of and . For the discussion of boundary condition and bosonization procedure in the normal reflection fixed point, please refer to Appendix B.

It is instructive to analyze the boundary conditions in the non-interacting case using the scattering matrix approach. The unitary scattering matrix is defined as (see, e.g., Ref. Nilsson et al. (2008))

(30)

where is the Hamiltonian for the MKP (2 by 2 matrix) which vanishes in the limit with and being respectively the length and coherence length of the superconductor. Note that the local term is not allowed by TR symmetry. The matrix describes the coupling between the MKP and the lead degrees of freedom in the basis :

(31)

Note that we assume that lead Hamiltonian is diagonal here. Therefore, represent helicity eigenstates in the case of a Rashba model. Using Eq. (30), we can represent the scattering matrix at as

(32)

The components and describe normal and Andreev reflection, respectively. As pointed out in Ref. Li et al. (2016), the normal part is zero so we focus on the non-diagonal components:

(33)

where the diagonal term is the coefficient of the same-spin Andreev reflection , and the off-diagonal term is the coefficient of the spin-flip Andreev reflection . As we change the angle of SOC, , from to , the Andreev reflection boundary condition changes continuously from with (i.e. and ) to and . We denote this boundary condition for as spin flip Andreev reflection boundary condition , which describes an Andreev reflection with spin-flip processes. Upon increasing to , the boundary condition becomes (i.e. and ).

Figure 4: RG flow diagram for the junction without symmetry: (a) for and , (b) for and , and (c) for and , and the green line indicates the conjectured BKT phase transition. The inset table summarizes the important time-reversal invariant boundary perturbations near , , and fixed points. For , along each line of the RG flow, the phase diagram as a function of is similar to the that shown in Fig. 2.

Here we would like to emphasize that the boundary condition is different from the Andreev boundary condition in s-wave spin-singlet superconducting junction where and (see, e.g., Ref.Maslov et al., 1996). Notice different signs in this case for spin-up and spin-down components. The boundary condition in our case corresponds to spin-triplet Andreev reflection which typically is realized at junctions between a normal lead and a spin-triplet p-wave superconductor. Indeed, if we denote spin-triplet pair potential as , then different orientations of the -vector correspond to () and () boundary conditions. This difference between conventional (s-wave) spin-singlet Andreev boundary conditions and boundary conditions considered here becomes very important later when we consider allowed boundary perturbations.

RG analysis near normal reflection fixed point

Let’s now analyze the interaction effects in the lead. In the absence of U(1) spin-rotation symmetry, we can have additional terms in the boundary action:

(34)

We have omitted here the irrelevant terms, e.g. , analogous to those considered in Sec. II.1. After the bosonization, the boundary action reads

(35)

Note the appearance of the new marginal term described by coupling constant .

We now perform a perturbative RG analysis up to the second-order in coupling coefficients. The details of the calculations are presented in Appendix C. Here we summarize our results for :

(36)
(37)
(38)
(39)

The generation of the term (proportional to ) originates from the processes involving two different spin channels of the lead whereas the generation of the term (proportional to ) comes from processes within the same spin channel. Both of these terms can be generated only in the presence of the interaction in the lead. This fact follows from the definition of the function

(40)

Here the function is defined as

(41)

and originates from the integration over relative coordinate, during the RG procedure, see Appendix C. In the non-interacting limit, , , and thus, the RG equation for becomes

(42)

where numerical constant . As mentioned, both and cannot be generated in the RG in the absence of interactions in the lead (i.e. ).

Using Eqs.(36) it is instructive to analyze first the flow in the non-interacting limit, in which case . Both and are relevant and flow to strong coupling. As follows from the discussion in the previous section, the exact boundary condition at the Andreev reflection fixed point is determined by the initial values of and and we can identify the corresponding limits by looking at the scattering matrix, i.e. corresponds to , corresponds to and finally corresponds to , see Fig. 4a.

We now analyze the RG flow for not-too-strong repulsive interactions . First of all, one can notice that even if we start with initial conditions , , the corresponding four-fermion terms are going to be generated by the RG procedure. Here is initial length cutoff. Since the couplings and affect the RG flow differently, we now have 4-parameter phase diagram. Based on the perturbative RG equations, one can see that both and will grow under RG, see Fig.  4. Thus, normal reflection fixed point is unstable in this parameter regime.

RG analysis near spin-flip Andreev reflection fixed point

We now analyze the stability of the spin-flip Andreev reflection fixed point which corresponds to the following boundary conditions:

(43)
(44)

In the bosonization language, the bosonic fields and are pinned, and the Klein factors have the relation and . Now, let us study all the fermion bilinear perturbations at the boundary allowed by TR symmetry. First of all, one can show that the normal backscattering is vanishing in this case, in agreement with the scattering calculation in Sec. II.2.1. Indeed, using the boundary conditions (43) one can show that

(45)

Note that for s-wave spin-singlet superconductor the boundary conditions are different: and , and the backscattering term does not vanish. Since this term is relevant for , the Andreev reflection fixed point is unstable in an s-wave superconductor-LL junction.

Let’s now consider allowed Andreev reflection bilinear processes. Among those, the only allowed bilinear term is spin-conserving Andreev reflection:

(46)

Additionally, we also consider the following four-fermion term

(47)

which corresponds to two-electron backscattering. The leading order perturbative RG equations for and are give by

(48)
(49)

One can see that the first term is marginal for symmetric Luttinger liquid lead , whereas the second coupling becomes relevant for indicating that the fixed point becomes unstable for strong repulsive interactions. If the spin symmetry is broken in the lead, the fixed point becomes unstable for , and the system will flow towards the fixed point. On the other hand, the is stable for .

RG analysis near spin-conserving Andreev fixed point

As shown in Sec. II.2.1, the boundary conditions near fixed point are with or . Thus, the boson fields are and are pinned at the boundary, and the Klein factors satisfy the relations . In the -conserving case, there are only irrelevant perturbations for such as two-electron backscattering

(50)

Additionally, if symmetry is broken, the spin-flip Andreev reflection processes are allowed

(51)

The leading order perturbative RG equations for and are give by

(52)
(53)

One can see that the first term is marginal for symmetric Luttinger liquid lead , whereas the second coupling becomes relevant for indicating that fixed point becomes unstable for strong repulsive interactions. If the spin symmetry is broken in the lead, the fixed point becomes unstable for , and the system will flow towards the fixed point. On the other hand, the is stable for . Exactly at , both and terms are marginal and compete with each other. Thus, generically both spin-conserving and spin-flip Andreev reflection processes will be present and their relative strength depends on microscopic details. This conclusion is consistent with the non-interacting results (