Kondo physics from quasiparticle poisoning in Majorana devices
We present a theoretical analysis of quasiparticle poisoning in Coulomb-blockaded Majorana fermion systems tunnel-coupled to normal-conducting leads. Taking into account finite-energy quasiparticles, we derive the effective low-energy theory and present a renormalization group analysis. We find qualitatively new effects when a quasiparticle state with very low energy is localized near a tunnel contact. For attached leads, such “dangerous” quasiparticle poisoning processes cause a spin single-channel Kondo effect, which can be detected through a characteristic zero-bias anomaly conductance peak in all Coulomb blockade valleys. For more than two attached leads, the topological Kondo effect of the unpoisoned system becomes unstable. A strong-coupling bosonization analysis indicates that at low energy the poisoned lead is effectively decoupled and hence, for , the topological Kondo fixed point re-emerges, though now it involves only leads. As a consequence, for , the low-energy fixed point becomes trivial corresponding to decoupled leads.
pacs:71.10.Pm, 73.23.-b, 74.50.+r
Majorana bound states (MBSs) in topological superconductors are presently attracting a lot of attention (1); (2); (3); (4); (5). Recent progress suggests that they can be experimentally realized as end states of topological superconductor (TS) nanowires. Such TS wires effectively implement the well-known Kitaev chain by contacting helical nanowires (i.e., nanowires with strong spin-orbit coupling in a properly oriented Zeeman field) with bulk -wave superconductors. Shortly after the first report of MBS signatures via zero-bias anomalies in the tunneling spectroscopy (6), a second generation of topological nanowires has emerged. These are based on InAs with high-quality proximity coupling to superconducting Al (7), which allows one to achieve the hard superconducting proximity gap (8); (9) needed for the unambigous observation of Majorana fermions. Evidence for MBSs in such second-generation wires has recently been observed in Coulomb blockade spectroscopy experiments (10). Intense experimental efforts are now devoted to elucidating the nonabelian braiding statistics expected for MBSs. Devices with strong Coulomb effects may be very useful in this regard (11). A possible complication in Majorana devices can arise from the presence of low-lying fermionic quasiparticle states. Many works have studied such “quasiparticle poisoning” effects in the absence of topologically protected modes, for instance, see Refs. (12); (13); (14); (15). Given the crucial role of parity conservation for detecting MBS signatures (1); (2); (3), even a single quasiparticle may drastically affect experimental results for Majorana devices. Indeed, quasiparticle poisoning has already been analyzed in this context, but only for noninteracting Majorana systems (16); (17); (18).
In the present work, we instead study the effects of low-lying quasiparticle states in the context of Coulomb-blockaded Majorana devices. The setup is sketched in Fig. 1. We consider a floating mesoscopic superconductor with charging energy , onto which helical nanowires have been deposited. Due to the proximity effect, each TS wire hosts a MBS pair. Below, the island together with the wires is referred to as “Majorana-Cooper box”, which is tunnel-coupled to normal-conducting leads. The leads could, e.g., be due to non-superconducting “overhanging” nanowire parts, see Fig. 1, where we assume that each TS wire end is contacted by at most one lead, i.e., . For nontrivial quantum transport behavior, the minimal case of interest is . Importantly, fermion parity on the box is conserved as long as charge quantization is enforced by a sufficiently large charging energy.
For temperatures well below the proximity gap, one may argue that quasiparticle states are not occupied with significant thermodynamic weight. Even for a sub-gap bound state, as long as it is not located near a MBS, the poisoning timescale (on which the occupation of this state will change) should be very long because all matrix elements connecting this quasiparticle state to other low-energy electronic levels, such as MBSs or lead electrons, are small. A more “dangerous” situation arises for sub-gap states located near the TS wire ends, which may occur in practice because the proximity-induced pairing gap also closes there (19). When the TS wires are tunnel-contacted by leads or quantum dots, tunneling processes via the quasiparticle state will then compete with those involving topologically protected MBSs. In order to identify such “dangerous” quasiparticle states, it is important to understand this competition and the resulting physical consequences.
Previous work on the setup in Fig. 1 has ignored all quasiparticle states apart from the MBSs. In that case, for attached leads, one arrives at the “Majorana single-charge transistor” (20); (21); (22), where the non-locality of the fermion mode built from the two Majorana operators allows for electron teleportation (20); (23) and for long-distance entanglement generation between a pair of quantum dots (24); (25). For leads, one instead encounters the so-called “topological Kondo effect” (TKE) (26); (27); (28); (29); (30); (31); (32); (33); (34); (35). In spite of charge quantization due to Coulomb blockade, the box ground state is -fold degenerate. This fact can be understood by noting that the Majorana space is a priori -fold degenerate, but the parity constraint due to charge quantization now removes half of the states. For , the remaining degree of freedom can be viewed as a quantum impurity “spin”, where the “real-valuedness” condition of the Majorana operators implies the symmetry group SO( instead of SU(2) (26); (29). This spin is effectively exchange-coupled to the lead electrons, and the corresponding screening processes culminate in the TKE, which is of overscreened multi-channel type, represents a non-Fermi liquid fixed point, and is detectable through the temperature dependence of the linear conductance tensor (26); (27); (28); (29); (30).
Let us now briefly motivate why quasiparticle poisoning is expected to be important for the setup of Fig. 1. In the absence of poisoning, the in-tunneling of a lead electron into the Majorana-Cooper box has to be followed after a short time by the out-tunneling of an electron from the box to some other lead (27). However, with an additional low-energy quasiparticle state present near the TS wire end, the system has a new option: The in-tunneling process can be compensated for by the out-tunneling of a quasiparticle. Such effects could significantly modify the TKE for , as well as the teleportation or long-range entanglement phenomena for . This question is also important in view of the fact that the Majorana-Cooper box is a basic building block in Majorana surface code proposals (36); (37); (38). In these proposals, the effective quantum impurity spins (each of which is encoded by one Majorana-Cooper box) are arranged on a two-dimensional lattice, which then is employed for quantum information processing.
The structure of the remainder of this paper is as follows. In Sec. II, we model the setup in Fig. 1, with an emphasis on new aspects introduced by quasiparticle poisoning. In the absence of poisoning, our Hamiltonian below reduces to previously studied models. For clarity, we mainly focus on the case of a single relevant quasiparticle state of energy . We derive the effective low-energy theory, , by a Schrieffer-Wolff transformation in Sec. III.1. For , we predict an enhancement of the Kondo temperature for the TKE, see Sec. III.2. Quasiparticle poisoning thus is not necessarily detrimental to the observation of this non-Fermi liquid state: it may actually help to access the regime. In Sec. IV, we turn to the simplest case, where despite of the effectively spinless nature of the system, is equivalent to the anisotropic (XYZ) spin single-channel Kondo model (39), which flows to an isotropic Fermi liquid strong-coupling fixed point on energy scales below . We determine the respective Kondo temperature, , and discuss the zero-bias anomaly conductance peak caused by the many-body Kondo resonance. Next, in Sec. V, we turn to the case of arbitrary . In Sec. V.1, we apply Abelian bosonization (39) to study the most challenging case , see also App. A. In Sec. V.2, we determine the perturbative renormalization group (RG) equations, cf. App. B, and we show that the TKE is destabilized by dangerous quasiparticle poisoning processes. However, the strong-coupling analysis presented in Sec. VI shows that for , a TKE with symmetry group SO() re-emerges at low temperatures. The effective change is rationalized by noting that only leads (those not attached to the TS wire end that hosts the poisoning quasiparticle) will contribute to the low-energy sector. For , the RG flow instead proceeds to a fixed point corresponding to effectively decoupled leads. We finally present our conclusions in Sec. VII. Throughout the paper, we employ units where .
In this paper, we present a theoretical analysis for the low-energy transport properties of the generic setup in Fig. 1. The central element of the setup is the Majorana-Cooper box, where nanowires are in proximity to the same floating mesoscopic superconductor. When driven into the topologically nontrivial phase, each of these TS wires hosts a pair of zero-energy Majorana end states (1); (40). We shall assume sufficiently long TS wires such that the hybridization between different MBSs can be neglected; for a discussion of these effects, see, e.g., Ref. (29). Recent experiments have shown that this requirement can be fulfilled for available InAs/Al nanowires (10). The box is then connected to (with ) normal-conducting leads by tunnel couplings, see Fig. 1. The Hamiltonian is thereby written as
where captures Coulomb charging effects, models finite-energy quasiparticles in the TS, describes the normal-conducting leads, and is a tunneling Hamiltonian connecting the box to the leads. We next describe these contributions.
In concrete realizations, the leads may be defined by the “overhanging” non-superconducting wire parts, see Fig. 1. We model them as semi-infinite one-dimensional (1D) channels of noninteracting spinless (helical) fermions. For the case of point-like tunneling studied below, this model also describes transport for bulk (2D or 3D) electrodes (21); (41). With the coordinate for a given lead (), we have a pair of right- and left-movers in each wire, , and the electron field operator is , where is the Fermi momentum. On low energy scales, the generic lead Hamiltonian now takes the form
where denotes the Fermi velocity. At , the boundary conditions are enforced, and point-like tunneling processes involve the lead operators and .
Turning to the Hamiltonian of the Majorana-Cooper box, , we first note that the proximity-induced pairing gap in the TS wires is typically well below the bulk superconducting gap (1). We therefore take into account quasiparticles only in the TS wires, which are either continuum (above-gap) states (cf. also Sec. VII) or localized (sub-gap) bound states. Each TS wire corresponds to a -wave superconductor, where the proximity-induced gap profile can be chosen real-valued in a suitable gauge (42). With Fermi velocity , the single-particle Bogoliubov-de Gennes (BdG) Hamiltonian for a given wire reads (1); (2)
where the Pauli matrices act in particle-hole (Nambu) space. Particle-hole and conjugation symmetry properties are expressed by
For eigenstates of the BdG equation, , Eq. (4) implies the symmetry relations
The last relation in Eq. (5) can be rationalized by noticing that in a rotated basis, , the BdG equation admits purely real solutions.
We now switch to a second-quantized formulation and introduce the Nambu field operator for a given TS wire, where refer to left/right-moving field operators in the TS. With the BdG single-particle Hamiltonian (3), the quasiparticle Hamiltonian for a single wire follows in the form
For , let us now define conventional fermion operators for particle-like (hole-like) excitations of energy . Taking into account Eq. (5), then has the mode expansion
where the real-valued phase follows by solving the BdG equation and the energy summation extends over positive BdG eigenvalues. The MBS contribution will be taken into account in the tunneling Hamiltonian below. However, zero-energy modes do not contribute to the quasiparticle Hamiltonian (6). Inserting Eq. (7) into Eq. (6), and subsequently summing over all TS wires (), we obtain
In addition to the finite-energy quasiparticles just discussed, we also have zero-energy MBSs on the box, which are localized near the TS wire ends. They correspond to a set of self-adjoint operators, , subject to the anticommutator algebra . A pair of Majorana operators defines a fermion (1); (2); (3), e.g., for each TS wire, where the Majorana algebra implies standard fermion anticommutation rules for the and operators.
Turning to the interaction contribution, with the single-electron charging energy and a dimensionless backgate parameter , capacitive Coulomb charging effects on the box are contained in (20); (21); (42)
where is the Cooper pair number operator and counts both the occupation of states in the zero-energy Majorana sector and of finite-energy quasiparticle states,
The respective eigenvalues and take integer values. We note that the condensate phase is conjugate to , with the canonical commutator . As a consequence, the operator annihilates one Cooper pair.
From now on, we shall assume that only a single quasiparticle state with energy is relevant, plus the hole state required by particle-hole symmetry. Apart from its simplicity, this minimal case is also of considerable practical interest: Noting that the gap function vanishes near the TS wire ends, an important example for such a low-energy quasiparticle comes from sub-gap bound states that may be formed near a tunnel contact (19). While the model and the techniques used here can be directly extended to the case of many low-energy quasiparticles, the effects of different quasiparticles are not simply additive in this interacting system, cf. the discussion before Eq. (46) in Sec. V.2. Taking into account only one quasiparticle state, say, near the left end of TS wire , cf. Fig. 1, Eq. (8) simplifies to
The quasiparticle state also couples to electrons in normal lead by tunneling processes encoded in the tunneling Hamiltonian . We here assume pointlike tunneling, , and for the moment ignore charge conservation issues. Employing Eq. (7) and the definition of the Nambu spinor, we obtain
with the generally complex-valued tunnel amplitude . The Majorana operators are built from quasiparticle fermion operators,
As a consequence of particle-hole symmetry, see Eqs. (4) and (5), and the assumption of pointlike tunneling, the lead fermion therefore couples to the “hybrid” fermion (and its conjugate). Taking into account also the tunneling processes involving the topologically protected Majorana fermions , cf. Refs. (20); (21), we arrive at the tunneling Hamiltonian
Without loss of generality, the tunnel amplitudes connecting MBSs to the respective leads can be taken real-valued and positive (42).
At this stage, we pause to incorporate the charge conservation condition for our floating (not grounded) device. As has been shown in Ref. (21), this condition can be taken into account by the following steps. First, for in Eq. (12), we put
such that “anomalous” processes will be accompanied by the splitting of a Cooper pair, which in turn is implemented by the operator . Second, after rewriting the Majorana operators in terms of and fermions, a similar replacement is performed in the Majorana part of Eq. (14). With these changes, is explicitly charge conserving.
Finally, in order to arrive at maximally transparent expressions, we remove the term in the charging contribution , see Eq. (9), by a gauge transformation, Using , and similarly for the fermions, we find that is still given by Eq. (2) and by Eq. (11). The charging energy term reads
where the charge operator has integer eigenvalues , with canonical commutator . This implies that the operator () adds (removes) charge to (from) the box. The tunneling Hamiltonian now takes the form
with the quasiparticle tunneling contribution
Iii Effective low-energy Hamiltonian
iii.1 Schrieffer-Wolff transformation
In this section, we derive an effective low-energy description for the general model discussed in Sec. II, which holds under the following conditions. First, we take into account only one quasiparticle state at energy , localized near a TS wire end. (We briefly discuss the case of delocalized above-gap quasiparticles in Sec. VII.) Second, the charging energy should be the dominant energy scale,
Third, we assume that is close to an integer. In the regime defined by Eq. (19), the system then exhibits charge quantization, .
According to Eq. (18), the lead fermion is tunnel-coupled to three Majorana fermions, namely to the topologically protected Majorana operator and to the Majorana fermions describing the real and imaginary parts of the quasiparticle operators and , resp., see Eq. (13). Through the quasiparticle Hamiltonian in Eq. (11), and also couple with strength to the two additional Majorana fermions representing the imaginary and real part of and , respectively. However, even though this -coupling constitutes a relevant perturbation in the RG sense, it does not affect the scaling properties of the system for temperatures within the window
Next, it is beneficial to switch to new Majorana operators and , representing linear combinations of , and . This step allows us to decouple one of these three Majorana fermions from the problem. To that end, using Eq. (17) and writing , we define
where and are new (real-valued positive) tunnel couplings. Equation (21) is evidently consistent with the Majorana operator algebra, in particular . In order to simplify the notation, we finally rename , as well as the respective MBS coupling to the lead, . The tunneling Hamiltonian for the contact to lead is therefore given by
where it is worth stressing that, in effect, only a single “poisoning” Majorana fermion () remains in the problem.
We now employ a standard Schrieffer-Wolff transformation (43) to project the system to the low-energy Hilbert space spanned by states with quantized box charge . This projection takes into account virtual excitations of higher-order charge states and has been described for the same system in the absence of poisoning in Ref. (26). Including poisoning effects, we now arrive at the effective low-energy Hamiltonian , where
with real-valued non-negative “exchange couplings”
For , the first term in Eq. (23) reduces to the TKE model (26). Indeed, in the absence of , the low-energy box degrees of freedom correspond to a “spin” operator of symmetry group SO, which has the components (29) and is exchange-coupled to a lead electron “spin” density at , see Eq. (23). The exchange couplings are marginally relevant under RG transformations and scale towards an isotropic strong-coupling fixed point describing the TKE. The second term in Eq. (23) is new and describes additional exchange-type couplings involving the poisoning Majorana fermion .
Together with in Eq. (2), the Hamiltonian (23) defines our low-energy model for quasiparticle poisoning in a Majorana device operating under strong Coulomb blockade conditions. In this model, we consider a quasiparticle state localized near one tunnel contact, such that effectively several MBSs will be tunnel-coupled to the same lead. Similar but different models have also been studied recently by others (33); (34). As discussed below, this modification of the clean TKE has interesting consequences that may be observable in Coulomb spectroscopy experiments.
iii.2 Intermediate quasiparticle energy
Before studying through a bosonization analysis, let us briefly turn to the regime of intermediate quasiparticle energy, , where an effective low-energy theory only involving the topologically protected Majorana fermions is applicable. Indeed, in this regime, since occupation of the quasiparticle states now comes with the large energy cost , see in Eq. (11), we can project also to the ground-state sector of .
Using the tunneling Hamiltonian in Eqs. (17) and (18), we first perform a Schrieffer-Wolff transformation in order to project away the higher-order charge states. Subsequently, since we are interested in energy scales well below , we also project to the ground-state sector of by a second Schrieffer-Wolff transformation. The resulting low-energy Hamiltonian is given by , with
plus an RG-irrelevant potential scattering term . (We note that here refers to the “original” Majorana operator, without the transformation in Eq. (21).) When compared to the small- exchange term [ in Eq. (23)], we observe that all terms related to the quasiparticle state have disappeared, except for a renormalization of the couplings . Instead of Eq. (24), which gives already in the absence of poisoning, we now find with
We mention in passing that for several quasiparticles with energy above , Eq. (26) simply acquires independent corrections of the form quoted here. Tunneling processes via the quasiparticle state () therefore increase the couplings, which can be rationalized by noting that an additional channel for cotunneling processes through the box has now become available. This channel is due to the high-energy quasiparticle state. The increase then implies an upward renormalization of the Kondo temperature characterizing the TKE for . For isotropic couplings, one finds , with the lead density of states (26). We conclude that quasiparticle poisoning will not necessarily destroy the TKE. To the contrary, when a quasiparticle state is localized near a tunnel contact and has energy , access to the regime becomes easier through the described enhancement mechanism.
Iv Conventional Kondo physics:
From now on, we shall discuss the more challenging case of a low-energy quasiparticle, where we can effectively put . The simplest scenario considers attached leads, which we discuss in this section. For , there are only three independent exchange couplings,
Their bare (initial) values follow from Eq. (24). The effective low-energy Hamiltonian, with in Eq. (23), is then equivalent to the fully anisotropic single-channel Kondo model. To establish this correspondence, we introduce a “quantum impurity spin” operator with components
Noting that the coupling of this spin operator to the identity operator, , does not generate RG-relevant scaling operators, and taking into account the exchange couplings defined in Eq. (27), we find that in Eq. (23) is equivalent to the fully anisotropic (XYZ) exchange term
with Pauli matrices in lead space.
The anisotropic single-channel Kondo model can be solved by the Bethe ansatz (44). The model scales towards a strong-coupling Fermi liquid fixed point (39), where the exchange couplings become more and more isotropic. In order to obtain the Kondo temperature, , determining the crossover scale from weak to strong coupling, we consider the standard RG equations for this problem. With the flow parameter , with for running short-time cutoff (45), and the couplings in Eq. (27), we arrive at the symmetric RG equations
It is straightforward to show from Eq. (30) that two invariants during the RG flow are given by
Under the assumption , Eq. (30) yields
By integration of Eq. (32), we then extract the Kondo temperature as the scale at which diverges,
where is the elliptic integral of the first kind (46), denotes the bandwidth, and is the lead density of states. For almost isotropic initial conditions, Eq. (33) can be simplified and reduces to the more familiar expression .
For , the spin single-channel Kondo fixed point will be approached, where deviations from isotropy are dynamically suppressed. The low-temperature behavior thus corresponds to conventional Kondo physics, where the formation of a many-body Kondo resonance allows for resonant tunneling through the Majorana-Cooper box. The predicted Kondo physics should be experimentally observable in setups similar to the one of Ref. (10) through a narrow conductance peak of width around zero bias voltage (“Kondo ridge”). The linear conductance between leads 1 and 2 then approaches the quantized value for , see Ref. (39),
with a coefficient of order unity. The temperature dependence of the conductance here follows from Fermi liquid theory. Importantly, these Kondo ridges are predicted to appear in all Coulomb valleys, in contrast to conventional quantum dots where they are found in “odd” valleys only (39).
We conclude that for , “dangerous” quasiparticle poisoning processes are responsible for a single-channel Kondo effect. The resulting conductance peak structure can easily be distinguished from standard Kondo features due to the electronic spin in quantum dots, as well as from the resonant Andreev reflection peaks found in noninteracting (grounded) Majorana devices (1); (2); (3) which are independent of the backgate parameter .
V Topological Kondo effect and quasiparticle poisoning
v.1 Abelian bosonization
In order to discuss the general case of leads, it is convenient to employ Abelian bosonization for the lead fermions (41); (39). Within this approach, the 1D fermion operators have the equivalent bosonized form
where is a microscopic short-distance lengthscale. The dual pairs of boson fields ( have the commutator algebra . Equation (35) also makes use of auxiliary Majorana operators with the anticommutator algebra , which represent the Klein factors needed to ensure anticommutation relations for fermions on different wires. This Klein factor representation allows for significant technical advantages in Majorana devices (27); (28). The lead Hamiltonian (2) has the bosonized form
where yields the boundary conditions .
In order to bosonize in Eq. (23), it is convenient to employ the shorthand notation
As we show in App. A, by combining the physical Majorana fermions ( and ) with the Majorana fermions representing the Klein factors, parity conservation allows one to efficiently capture the dynamics of all these Majorana fermions, for arbitrary number of leads , by just a single “pseudospin” operator with components (where ). Following the steps in App. A, the bosonized form of is given by
where . We note that a factor has been absorbed in the exchange couplings and , see Eq. (24). Instead of and with , we employ the linear combinations
The second term in Eq. (38) contains contributions that are initially absent, for , see Eq. (24). However, we shall see in Sec. V.2 that such contributions are dynamically generated during the RG flow.
We conclude that describes a pseudospin coupled to bosonic modes, where the pseudospin dynamics encodes quasiparticle poisoning effects in this strongly blockaded Majorana device. Finally, we note that the bosonized description also allows one to incorporate weak electron-electron interactions in the leads in an exact manner (27); (28). However, we do not consider such effects below.
v.2 RG equations
We now discuss the RG equations for arbitrary . We have derived them for the bosonized Hamiltonian , with in Eq. (38) and , by using the operator product expansion technique (45). We have also confirmed the correctness of the RG equations by an independent derivation using the fermionic representation of in Sec. III.1. For arbitrary , the closed set of one-loop RG equations, with the indices and , is then given by
Note that the flow of the couplings and is now contained in the couplings , see Eq. (39). The bare (initial) couplings and have been specified in Eq. (24). This equation also determines the from Eq. (39). Moreover, we always have . In the above RG equations, we have dropped all RG-irrelevant couplings that are generated during the RG flow but have vanishing initial value. We mention in passing that for , these RG equations become equivalent Eq. (30) in Sec. IV.
Equation (43) implies that , which is dictated by current conservation and stems from the gauge invariance of the system, namely the invariance of the effective action [see Eq. (61) in App. A] with respect to the simultaneous shift of all (where denotes imaginary time) by an arbitrary constant. Noting that the current through the respective tunnel contact is determined by , current conservation implies . Using bosonization identities (39), this relation is equivalent to , and therefore has to be invariant under a uniform shift of all .
Let us now briefly check that known results for the clean TKE are recovered. In the absence of poisoning, which corresponds to removing the Majorana fermion by setting , i.e., and , Eqs. (40) and (41) reproduce the RG equations for the TKE (26),
Interestingly, one can also arrive at the TKE by trading the “true” Majorana fermion for the “poisoning” Majorana fermion . To illustrate this point, let us consider the case , such that the tunnel coupling between and the attached lead vanishes. For , we then have and . Renaming , the general RG equations [Eqs. (40)–(43)] again reduce to the TKE equations (44).
The presence of a poisoning Majorana fermion () in the effective low-energy Hamiltonian has several consequences. First, it implies the opening of additional “forward scattering” channels, in Eq. (23), where an electron is transferred from lead to some other lead (), and likewise for the conjugate processes. Second, a new “backscattering” channel will open for the poisoned lead, . Under the RG flow, this effect also generates additional terms , i.e., backscattering will appear in the other leads as well. (Of course, all these processes are subject to the current conservation constraint in Eq. (43).) One can therefore expect a rich interplay between nonlocal features similar to teleportation (20), due to forward scattering between different leads, and local backscattering effects within each lead.
As a minimal description capturing the above physics, we now simplify the full set of RG equations [Eqs. (40)–(43)] by approximating the couplings as follows. In our simplified version of the RG equations, we assume that there are only four independent couplings, denoted by , and below. With indices and , the exchange couplings are expressed as
The initial values for and obtained from Eq. (24) are positive, with . We here assume the same coupling for each “unpoisoned” () lead, with initial value . Writing as in Eq. (46), we observe that does not scale under the RG because of current conservation, see Eq. (43), and therefore plays no role in what follows. Inserting Eq. (46) into Eqs. (40)–(42), we arrive at the simplified RG equations,
As a first check, let us briefly verify that Eq. (47) correctly captures the expected TKE in the clean limit, see Eqs. (44) and (45). In the absence of poisoning, we have , resulting in and . We then readily obtain from Eq. (47), in accordance with Eq. (45). A second check comes from comparing the results of a numerical integration of the full RG equations [Eqs. (40)-(43)] to the corresponding predictions obtained from the simplified RG equations. We present this comparison in App. B, which shows that for , the simplified description is justified. For , the RG flow towards isotropic couplings as expressed by Eq. (46) is not yet established.
We next observe that Eq. (47) implies a constant growth of the ratio during the RG flow, . Noting also that the couplings and grow, we see that is dynamically suppressed against these couplings. Neglecting in Eq. (47) and considering the case , the RG equations (47) simplify to
These equations predict , i.e., , which in turn implies that . We conclude that all three couplings scale uniformly towards a strong-coupling regime, , where one eventually leaves the validity regime of the perturbative RG approach. The analysis in App. B shows that the above suppression of couplings also takes place for . This suppression is then followed by a flow towards isotropic couplings (this happens only for ).
To conclude this section, the perturbative RG equations indicate that the clean TKE found for will be destroyed by “dangerous” quasiparticle poisoning. Nevertheless, see App. B, a simplified four-parameter description is sufficient for , where one coupling turns out to be irrelevant. This dynamical suppression of couplings also holds for , while then isotropy is not reached. We exploit these insights now when turning to the strong-coupling regime.
Vi Strong coupling regime
Let us now discuss the physics for encountered at very low temperatures, where the strong-coupling regime is approached. We begin our analysis with the case , where it is justified to employ the isotropic couplings in Eq. (46), and later return to the case .
For , the coupling decouples and can be neglected, while the remaining couplings [ and in Eq. (46)] become isotropic and simultaneously approach the strong-coupling regime, see Sec. V.2 and App. B. Using the shorthand notation (37), the exchange interaction in Eq. (38) then takes the form (47)
We now perform a unitary transformation,
in order to gauge away the term. As a result, the exchange term in takes the form
The strong-coupling regime is now accessible by (i) diagonalizing for static field configurations , (ii) minimizing the corresponding ground-state energy , and (iii) subsequently taking into account quantum fluctuations (caused by