Superconductivity of Composite Particles in Two-Channel Kondo Lattice
Emergence of odd-frequency -wave superconductivity is demonstrated in the two-channel Kondo lattice by means of the dynamical mean-field theory combined with the continuous-time quantum Monte Carlo method. Around half filling of the conduction bands, divergence of an odd-frequency pairing susceptibility is found, which signals instability toward the superconductivity. The corresponding order parameter is equivalent to a staggered composite-pair amplitude with even frequencies, which involves both localized spins and conduction electrons. A model wave function is constructed for the composite order with use of symmetry operations such as charge conjugation and channel rotations. Given a certain asymmetry of the conduction bands, another -wave superconductivity is found that has a uniform order parameter. The Kondo effect in the presence of two channels is essential for both types of unconventional superconductivity.
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Unconventional superconductivity refers to such pairing states that have non-trivial symmetry in spin and/or space-time structures. Among those states, we address the odd-frequency (OF) pairing state berezinskii74 (), which breaks the gauge symmetry, but has a zero pairing amplitude at equal time. Possible relevance of the OF pairing to real materials was first pointed out by Berezinskii for He berezinskii74 (). After the discovery of high-temperature superconductivity in cuprates, the OF pairing has aroused broad interest emery92 (); balatsky92 (); abrahams93 (); coleman94 (); zachar96 (); coleman97 (); vojta99 (); fuseya03 (); yada08 (); shigeta09 (); hotta09 (); shigeta11 (); kusunose11-2 (); yanagi12 (); heid95-2 (); martisovits98 (); martisovits00 (); tanaka12 (); emery93 (); balatsky93 (); coleman93 (); schrieffer94 (); coleman95 (); abrahams95 (); heid95 (); jarrell97 (); anders02 (); anders02-2 (); hoshino11 (); belitz99 (); solenov09 (); kusunose11 (); sakai04 (); coleman99 (); flint11 (); shigeta13 () as one of candidate mechanisms for unconventional superconductivity.
The OF pairing state can be viewed from a different perspective; it has been recognized that the OF superconductivity is alternatively regarded as a composite pairing state with even frequencies (EF) emery92 (); balatsky93 (). From another argument, it has also been suggested that OF superconductivity has a tendency to favor a spatial inhomogeneity coleman93 (); heid95 (), and that finite density of states remains at the chemical potential balatsky92 (); coleman93 ().
One of possible realizations of OF superconductivity is proposed in the two-channel Kondo systems. Emery and Kivelson have shown for the two-channel Kondo impurity that the OF pairing susceptibility is enhanced at the impurity site emery92 (). They have further elaborated on a variant of the two-channel Kondo lattice (TCKL) in one dimension emery93 (), and have demonstrated the divergence of the OF pairing susceptibility at zero temperature. For the TCKL in higher dimensions, microscopic calculations have been performed, however, without finding divergent susceptibility jarrell97 (). Another calculation for the corresponding Anderson lattice anders02 (); anders02-2 () did find the divergent susceptibility, which vanished as the system goes to the TCKL limit. So far no microscopic theory has established the OF pairing in the TCKL at finite temperature.
In this paper, we present highly accurate numerical results for the pairing susceptibility in high dimensional TCKL, and demonstrate that the TCKL realizes the -wave OF superconductivity with the staggered order parameter. We also derive the corresponding composite order parameter and model wave functions for the composite orders explicitly, using symmetry arguments such as charge conjugation and channel rotations. It is further shown, with a certain channel asymmetry, that another -wave superconductivity occurs with EF in the TCKL. Our key strategy is to exploit the charge conjugation that relates the diagonal long-range order to the off-diagonal one.
The TCKL Hamiltonian jarrell96 () is given by with
The operator () annihilates the conduction electron with energy , channel and spin at wave vector (site ). The spin for conduction electrons couple with the local spin antiferromagnetically (i.e. ). The chemical potential controls the total number of conduction electrons. Note that this model has double SU(2) symmetries; SU(2) for spin and SU(2) for channel degrees of freedom. We assume the bipartite lattice with , where corresponds to a vector at edges of the Brillouin zone. This assumption makes the model invariant at half filling () under particle-hole transformations, i.e., charge conjugation, as will be discussed later.
Physically, the degree of freedom described by is interpreted as orbital for the non-Kramers doublet system with configuration in Pr and U cox98 (). The labels and are then interpreted as orbital and real spin, respectively. On the other hand, in Kramers systems as in Ce with configuration, is regarded as a real spin of electrons. The labels and are then regarded as real spin and orbital, respectively. The non-Kramers doublet system has the channel (real spin) symmetry protected by the time-reversal symmetry, while in the Kramers doublet case the channel (orbital) symmetry is only approximate. In the following, we call and simply as ‘spin’ and ‘channel’, respectively, unless otherwise stated explicitly.
We use the dynamical mean-field theory (DMFT) kuramoto85 (); georges96 () for analysis of the TCKL, and the continuous-time quantum Monte Carlo method rubtsov05 (); gull11 () for the numerical impurity solver. We take the semi-circular density of states with being the unit of energy. We have confirmed that the behaviors are qualitatively the same if we take the Gaussian density of state for example.
We begin with the pairing susceptibilities in the TCKL. Following the literature jarrell97 (); anders02 () we use for each pairing type the labels C and S indicating ‘channel’ and ‘spin’, and s and t indicating ‘singlet’ and ‘triplet’. We introduce the following operators dependent on imaginary time to describe possible pairings:
where is the antisymmetric unit tensor, and . Note that due to the Pauli principle, meaning that these cannot form the equal-time pairings.
In order to calculate susceptibilities, we introduce the two-particle Green function by where is the time-ordering operator, and represents one of the labels in Eqs. (3–6). The Fourier transform is defined by
with . Using this quantity, we define the EF and OF pairing susceptibilities with by
where denotes CsSt or CtSs, while denotes CsSs or CtSt. The form factor is defined by jarrell97 (); anders02-2 (); sakai04 (). Namely we extract the EF part for CsSt and CtSs, and the OF part for CsSs and CtSt. Equation (8) is the usual susceptibility and must be positive freericks93 (). On the other hand, the OF susceptibility given by Eq. (9) is no longer positive definite due to the presence of , but still signals the instability toward the pairing state by its divergence hoshino11 (). The critical temperature thus obtained is insensitive to the choice of the form factor provided is odd against .
Figure 1(a) shows the temperature dependence of at and with being the average number of conduction electrons per site. Here we consider the two ordering vectors and , which we call ferro (F) and antiferro (AF), respectively. Among the eight susceptibilities, only the one with AF-CsSs diverges at from the negative side, signaling the onset of the OF superconductivity with the ordering vector . We have confirmed that similar behaviors are obtained when we take the other parameters such as and . Since the normal state of the TCKL is a non-Fermi liquid state as seen in the electrical resistivity jarrell96 (), the present system becomes superconducting directly from the non-Fermi liquid.
Together with the diagonal orders that have been obtained in our previous study hoshino13 (), the phase diagram of the TCKL is completed as shown in Fig. 1(b). Here the diagonal orders are characterized by the vector operators
which describe spin (), channel (), and composite () orders, respectively.
The transition temperature to the superconducting AF-CsSs order is lower than the AF spin order at (half filling) as shown in Fig. 1(b). As decreases, however, the AF-CsSs dominates the AF spin order. Since the AF-CsSs order can be best visualized at half filling, we consider the AF-CsSs state mainly at half filling by neglecting the AF spin order.
We derive the composite order parameter corresponding to the (odd-frequency) AF-CsSs phase by combining the particle-hole and channel-rotation symmetries. At half filling, the transition temperatures for the AF-CsSs and F channel orders are the same within the numerical accuracy as seen in Fig. 1(b), which indicates a degeneracy between these two orders. In fact, these two orders are obtained from each other by symmetry operations at half filling. To demonstrate this, we introduce a particle-hole transformation that acts only on channel as
On the other hand, and are not affected by . The half-filled Hamiltonian and composite quantity for the F-channel order are invariant under this transformation. Here the phase factor in Eq.(13) is necessary to make the kinetic energy invariant under .
By contrast, the transverse components in Eq.(12) are affected by . The explicit form of is given by
This composite quantity gives the EF order parameter corresponding to the AF-CsSs phase. Thus the AF-CsSs and F-channel orders are exactly degenerate at half filling. This is also interpreted as a reflection of the SO(5) symmetry at affleck92 (); hattori12 (). The degeneracy is lifted by the chemical potential as shown in Fig. 1(b).
Another form of the order parameter can be constructed that involves conduction electrons only. The simplest derivation is to start again from the F-channel order, and apply . In the F-channel phase with , the conduction electrons with are nearly free, while the ones with form the Kondo insulator at half filling hoshino11 (). Hence the difference of the kinetic energies between channels arises, which can be regarded as a secondary order parameter nourafkan08 (); hoshino13 (). We write the quantity in the SU(2) symmetric form as
Performing the particle-hole transformation, we obtain as
Note that this expression is similar to the so-called pairing yang89 (). However, a difference lies in the form factor present in Eq. (16). Direct calculation shows that and are related to the time-derivative of the OF order given by Eq. (3). If one would investigate the instability toward superconductivity using the EF susceptibility for or , explicit calculation of the correlation function would be much more tedious. Hence the OF susceptibility defined in Eq. (9) provides a convenient tool serving for the same purpose.
Let us proceed to construct wave functions for the ordered phases. A simple form, though crude, should be useful to visualize a composite order. Starting from the F-channel state at half filling, we obtain a superconducting state by symmetry operations on the wave function. According to Ref. hoshino11 (), the F channel state described by consists of itinerant electrons for and the Kondo singlets for . We introduce a simplified wave function describing this F channel state as
where ‘HBZ’ means a half Brillouin zone. The part labeled by represents free conduction electrons at half filling. The local Kondo singlet state with channel is written as by with being the localized-spin state at site . The local picture of Eq. (17) is illustrated in the upper-right part of Fig. 2.
To obtain the off-diagonal order, we combine with a channel mixing unitary transformation defined by
which rotates from to axis in channel space. In view of and , it is reasonable to postulate a model wave function for the AF-CsSs state as
Note that transforms the vacant state into the states occupied doubly by electrons with channel at each site. As seen in the second line of Eq. (19), the states with one and three local conduction electrons per site are superposed, indicating the broken gauge symmetry. As is clear from this expression, both channels participate to make the local spin-singlet state. The lower-right part of Fig. 2 illustrates this local state. The first line of Eq. (19) includes the Bogoliubov quasi-particles composed of particle at and hole at , which in general have a finite density of states at the Fermi level.
We now consider the region around in Fig.1(b) where the AF-channel order is dominant. By the particle-hole transformation , we can relate the electronic state at to the half-filled case under asymmetric channel potential. We introduce the new Hamiltonian
Starting from for , we end up in with 1/2 electron per site for , while 3/2 electrons for . Then we obtain the sum . Namely the hole-doped TCKL described by is transformed into the TCKL at half filling under the channel field. Physically, simulates such systems that have two conduction electrons per site but with inequivalent conduction bands. This situation may arise in a Kondo lattice with Kramers degeneracy, since the channels in this case correspond to spatial orbitals that have no degeneracy in general.
On the other hand, the AF-channel order at are transformed as
Namely, the AF-channel order in corresponds to the charge density wave given by Eq. (21), as well as the -wave superconductivity given by Eq. (22) in the new Hamiltonian . These two orders are degenerate at by symmetry. We have numerically confirmed that this degeneracy is lifted with , and the -wave superconductivity is more stable. This -wave pairing can be understood from the strong coupling limit, following interpretation of the AF channel order in cox98 (); schauerte05 (). Then, the local image of the state is given by the lower-right panel of Fig. 2 without the Bogoliubov quasi-particle part. In contrast to the AF-CsSs state, the superconductivity with has a full gap in the density of states.
In a similar manner, we can show that the off-diagonal AF-CsSs order in is transformed into the diagonal composite order in . The spin order remains the same after the transformation. Thus the phase diagram for is obtained without further calculation if we replace by in Fig. 1(b).
Finally let us briefly discuss possible relevance of our results to real physical systems. The primary candidate for the non-Kramers doublet system with the order is UBe, which has first been proposed as the two-channel Kondo system by Cox cox87 (). The superconductivity in UBe appears directly from the non-Fermi liquid state ott83 (), which is consistent with what we have found in the TCKL. For Kramers-doublet systems, on the other hand, a channel (orbital) asymmetry is inevitable. Then the modified Hamiltonian seems relevant to describe the two-orbital Kondo lattice. Recent specific-heat measurement in CeCuSi kittaka13 () has reported a full-gap superconductivity reminiscent of multiband superconductivity. Hence we suggest possible relevance of the state to CeCuSi.
In conclusion, taking the TCKL, we have demonstrated instability of the non-Fermi liquid state toward the OF -wave superconductivity with finite center-of-mass momentum, and with channel-singlet and spin-singlet pairing. The OF pairing state is alternatively characterized by an EF composite order. We have further derived another EF -wave superconductivity with the uniform order parameter by considering asymmetry in the channels. Further studies inside the superconducting phases will provide a better understanding of the unconventional superconductivity.
We are grateful to Yusuke Kato for stimulating discussions and valuable comments on our paper. We also appreciate Kazumasa Hattori, Hiroaki Kusunose and Youichi Yanase for fruitful discussions. S.H. acknowledges the financial support from Japan Society for Promotion of Science.
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