Generalized Spin Fluctuation Feedback in Heavy Fermion Superconductors

Experiments reveal that the superconductors UPt3, PrOs4Sb12 and U1−xThxBe13 undergo two superconducting transitions in the absence of an applied magnetic field. The prevalence of these multiple transitions suggests a common underlying mechanism. A natural candidate theory which accounts for these two transitions is the existence of a small symmetry breaking field, however such a field has not been observed in PrOs4Sb12 or U1−xThxBe13 and has been called into question for UPt3. Motivated by arguments originally developed for superfluid He we propose that a generalized spin fluctuation feedback effect is responsible for these two transitions. We first develop a phenomenological theory for He that couples spin fluctuations to superfluidity, which correctly predicts that a high temperature broken time-reversal superfluid He phase can emerge as a consequence. The transition at lower temperatures into a time-reversal invariant superfluid phase must then be first order by symmetry arguments. We then apply this phenomenological approach to the three superconductors UPt3, PrOs4Sb12 and U1−xThxBe13 revealing that this naturally leads to a high-temperature time-reversal invariant nematic superconducting phase, which can be followed by a second order phase transition into a broken time-reversal symmetry phase, as observed.

summation convention. The order parameter d iα is a 3 × 3 matrix with complex entries, where i is the spin index and α is the orbital index and both run over x, y and z. By comparing to experiments, the 3 He-A phase was identified with the Anderson-Brinkman-Morel (ABM) state with d xx = ∆ √ 2 , d xy = i ∆ √ 2 and all other d ij = 0, while B state was associated with Balian-Wethamer (BW) which has d ij = ∆ √ 3 δ i,j [27,28]. Weak coupling theory shows that the BW state is stable for all temperatures [29], implying that a strong coupling approach is needed to explain the existence of the high temperature high pressure A phase. Anderson-Brinkman [30] used SFFE to stabilize the A phase, which relies on the pairing glue in 3 He being paramagnetic fluctuations. This implies that the formation of the superfluid alters the pairing interaction, where the type of modification depends on which state is formed [27]. Thus the A state can be stabilized despite being unstable under weak coupling theory. The A-B transition in this case is first order as the B state is not a subgroup of the A state.
Here SFFE shall be recaptured in a phenomenological manner, by coupling the superfluid order parameter to paramagnetic fluctuations, and calculating the change to the bare free energy. The coupling is constructed to be invariant under independent rotations in orbital and spin space, and is where m i is the magnetic order parameter. We assume A 1 is parametrically small and positive (i.e. A 1 → 0) to indicate that we have large fluctuations close to a magnetic transition. The magnetic partition function is given as Z m = Dm i e − d 3 xf sf −m = Dm i e − d 3 xAij mimj , where A ij contains couplings between magnetic and superconducting orders, and give corrections the bare superconducting free energy density. Integrating out the quadratic magnetic fluctuations, gives an effective free energy density where the β i are associated with the free energy density without coupling to fluctuations, here assumed to be derived from weak-coupling theory and the terms with the K's originate from SFFE. Close to the paramagnetic instability, for which A 1 is small, the terms that dominate are those that are proportional to A −2 1 and we ignore terms of order A −1 1 . The K 2 term in (1) shows that paramagnetic fluctuations can favor non-unitary states [33][34][35][36], but are neglected here due to their weaker A −1 1 dependence. Weak coupling theory gives β 2 = β 3 = β 4 = −β 5 − 2β 1 = 6 5 s, where s is a positive valued constant [27,28]. When the SFFE is turned off (i.e. K i = 0), we see that the BW state is energetically favorable with f ef f = 5 3 s, while the A state has a slightly larger free energy density of f ef f = 2s. The SFFE coupling lowers the energy of the A state by K 2 1 /3A 2 1 compared to the B state and thus for large fluctuations (i.e. K 2 1 /3A 2 1 > 1/3s) can stabilize the A state. The second A-B transition stems from the different temperature dependence of the weak coupling terms versus the SFFE terms. It can be shown using a microscopic theory that the terms originating from the SFFE have a 1 T dependence while weak coupling terms scale as 1 T 2 [27,31,37]. This implies that at high temperatures, strong fluctuations may stabilize the A phase, while at lower temperature the weak coupling terms will dominate and the system will undergo a first order transition into the state preferred by weak coupling theory i.e. the B phase. Analogous arguments will be applied to the heavy fermions, where as the temperature is lowered there will be a sign change the coefficients in the free energy density, though the A-B transition will be second order.
UPt 3 .-UPt 3 is a hexagonal crystal with D 6h point group symmetry and has two distinct phases under zero field, a high temperature A phase and a low temperature B phase [15,38]. However, unlike 3 He, the A phase is TRS while the B phase is TRSB as seen in muon spin relaxation (µSR) and polar Kerr measurements [5,14]. UPt 3 has four 2D (reps) labeled E 1u/g and E 2u/g , where the order parameter transforms like η 1 ∼ k x k z , η 2 ∼ k y k z and η 1 ∼ k 2 x −k 2 y , η 2 ∼ 2k x k y respectively. The free energy density is the same for all the E reps and is given as 2 determines the behavior below T c , with β 2 > 0 favoring the (1, i) state, and β 2 < 0 stabilizing the (1, 0) state. To explain the existence of multiple phases, the currently accepted model involves coupling superconductivity to AFM order [20,21]. However experiments raise questions over the existence of true AFM order. The Bragg peaks in INS are not resolution limited near the superconducting transition of 550mK and order truly appears around 20mK [23] far away from T c , which is further confirmed by SQUID measurements [40]. We interpret these experiments to imply the presence of AFM fluctuations, instead of AFM order, where a generalized SFFE may stabilize a TRS A state. We shall proceed analogously to 3 He, and assume that the B TRSB broken state is the weak coupling state, while strong fluctuations can stabilize a TRS A phase. The fluctuations are characterized by wave vectors 22,23] which are associated with the magnetic order parameters m 1 , m 2 and m 3 respectively. The coupling of superconductivity to the magnetic fluctuations is constructed to be invariant under D 6 × U (1) × T , where T is time-reversal symmetry and is expressed as [21] Integrating out the fluctuations as before gives the following free energy density: Importantly, the generalized SFFE changes the η 2 i 2 coefficient, and for large fluctuations can stabilize the A state, instead of the TRSB B by making this coefficient negative. This also implies that two transitions will occur only if the B state is a TRSB state. In particular, if the B state was TRS then the SFFE terms would simply further stabilize the TRS nematic state. A formal phenomenological description of the second transition requires a free energy density that is eighth order in the order parameter [41,42]. Here we use the arguments discussed earlier for the microscopic theory of SFFE, which show that as the temperature is lowered the β 2 term grows faster the K 2 2 /A 2 1 term, and results in a sign change of η 2 i 2 , thereby allowing a second transition to a TRSB state.
To gain further insight, we model the A-B transition as an effective phenomenological theory, by assuming that SFFE stabilizes the TRS A (1, 0) state, and the TRSB B continuously state grows out of this i.e. (1 +η 1i , 0 +η 2i ), whereη i is small near the transition. Time-reversal symmetry allows us to classify the order parameterη i for the A-B transition into a real partη R which is invariant under T , and an imaginary partη I which changes sign under T . The condition that the second transition break TRS, allows us to consider only the imaginary order parameter. The (1, 0) state has D 2 (C 2 ) × T and D 2 × T [33] symmetry for the E 1u/g and E 2u/g reps respectively. The order parameterη 1I belongs to the A 1 rep of D 2 , whileη 2I belongs to the B 1 rep.
The observation of a polar Kerr signal for the A-B transitions [5] further constrains the possible order parameters, as only those order parameters which belong to the same representation as the magnetic moments (m x , m y , m z ) will show a Kerr effect and shall be referred to as Kerr active (also labeled as belonging to a ferromagnetic class [43]). This rules out the A 1 order parameter as it is Kerr inactive. Thus the A-B transition can be modeled by the effective order parameter η 2I with the following free energy f A→B = α 1Iη 2 2I + β 1Iη 4 2I . We shall use this approach later to shed insight into the possible symmetries of the order parameters.
PrOs 4 Sb 12 .-PrOs 4 Sb 12 (POS) is a Pr based tetrahedral heavy fermion skutterudite superconductor with a T h point group, and like UPt 3 has two distinct phases [44]. Polar Kerr and µSR measurements show a TRSB B phase [6,18], while the A phase is TRS. POS has been studied by phenomenological methods [45], however there is no satisfactory mechanism for the double transition. INS experiments indicate the presence of AFQ fluctuations with a Q = (1, 0, 0) [25,46], which is a single Q order, invariant under the point group operations. The order parameter of these AFQ fluctuations is 3D with components that transform as m 1 ∼ k y k z , m 2 ∼ k x k z and m 3 ∼ k x k y [47,48]. The AFQ fluctuations can stabilize a time-reversal symmetric A phase for both the E and T reps as shown below.
For the E reps, where the order parameter transforms as Integrating out the AFQ fluctuations, we obtain the effective free energy density This generalized SFFE may again stabilize a TRS A state (φ 1 , φ 2 ) with D 2 × T symmetry [45,49], instead of the TRSB B phase (1, i) with T (D 2 ) symmetry by changing the sign of the (η 1 η * 2 − η 2 η * 1 ) 2 term, from positive to negative. The A phase has the two components with the same magnitude but an arbitrary phase [45]. Similar to UPt 3 two transitions are possible only when the B state is TRSB. The A-B transition is modeled similar to UPt 3 , however both η 1I/2I have A 1 Kerr inactive symmetry and are ruled out due to presence of Kerr effect, thereby eliminating the 2D E reps scenario for PrOs 4 Sb 12 .
For the 3D T reps, with order parameter which transforms for example as η 1 ∼ k y k z , η 2 ∼ k x k z and η 3 ∼ k x k y , the coupling is Integrating out the Gaussian AFQ fluctuations gives the following effective free energy density: Again the SFFE has changed the coefficient of the bare free energy density and hence allows for the possibility of a TRS A state. Interestingly, here we may have two transition even if the A state is TRSB, due the indeterminant sign of the correction to the β 3 coefficient. However since this does not agree with the experimental identification of the B state being TRSB, so we don't consider this possibility. This rep has two states which are TRS, the (1, 0, 0) state with D 2 (C 2 ) × T symmetry and the (1, 1, 1) state with C 3 × T symmetry. Both of these allow for a transition to a TRSB B state which is Kerr active and hence provide two viable channels for the transition. The physics of this is similar to the 2D case for UPt 3 and PrOs 4 Sb 12 , and is worked out in the supplementary material [50], the results of which are collected in Table I. This model assumes that AFQ fluctuations act as the pairing glue, and we suggest INS scattering and tunneling spectra as done in UPd 2 Al 3 to confirm this unconventional glue [51,52].
U 1−x Th x Be 13 .-U 1−x Th x Be 13 is a cubic material with O h point group, which also has two transitions [9,17], but only for a doping range of 2 % < x < 4 %. The B phase is again a TRSB state [9]. AFM fluctuations are seen in INS with a wave vector of Q 3 = (1/2, 1/2, 0) [26]. We consider both the E and T reps and model the system with O h symmetry, having AFM fluctuations with wave vector Q 3 . The star of Q 3 gives two additional wave vectors Q 2 = (1/2, 0, 1/2) and Q 1 = (0, 1/2, 1/2), each of which correspond to a 1D order parameter m 1 , m 2 , m 3 . Here η 1 and η 2 transform exactly as E reps of PrOs 4 Sb 12 is The K 3 term present in Eq. 4 is absent above, due to additional symmetry elements present in the O h as compared to T h point group. The correction to the free energy density from these fluctuations is There are two possible TRS A states. The details of the A-B effective theory follows that of UPt 3 and is in the supplementary material [50], with the possible A state being (1,0) with D 4 ×T symmetry and (0,1) with D 4 (D 2 )×T symmetry [43]. These states are Kerr inactive, and the 2D order parameter can be ruled out if a polar Kerr signal is seen.
For the 3D order parameter case, we note that, U 1−x Th x Be 13 has four T reps, one which transforms exactly like T reps of PrOs 4 Sb 12 which we call T 2g/u and the other T 1g/u which transforms as . The coupling for these T reps is The K 3 term found in Eq. 6 is absent because of higher O h symmetry compared to the T h symmetry, while the K 4 term is forbidden as m i m j is not translationally invariant for this Q vector. The effective free energy density obtained is Interestingly, here unlike PrOs 4 Sb 12 there is no change to β 2 , which is result of the K 4 coupling being absent for Eq. 11 in contrast to Eq. 6. There are four possible TRS A state, two for each T reps, i.e. (1,0,0) and (1,1,1). However due to there being no change to coefficient of the η 2 i 2 term, (1,0,0) is the only TRS A state which allows for a viable transition to a TRSB B state [33]. This state has D 4 (C 4 ) × T and D (2) 4 (D 2 ) × T symmetry for the T 1 and T 2 reps respectively [43]. The A-B transition, follows similar to UPt 3 , and is modeled in the supplementary information [50]. Polar Kerr measurements may be useful as they can rule out the E reps scenario, and can eliminate other 3D T order parameters.
Conclusions.-In summary, we have argued that the mechanism for two transitions in heavy fermions needs to be revisited. In analogy to superfluid 3 He we have argued that a generalized SFFE may stabilize a TRS high temperature A phase. We provide a simple phenomenological model to capture this effect, and show that generalized SFFE provides a unifying mechanism for two transitions in UPt 3 , PrOs 4 Sb 12 and U 1−x Th x Be 13 . D 4 (C 4 ) × T and D (2) 4 (D 2 ) × T symmetry for the T 1 and T 2 reps respectively [1]. The A-B transition is modeled similar to PrOs 4 Sb 12 i.e. (1 +η 1I , 0 +η 2I , 0 +η 3I ), except hereη 1I belongs to the A 1 rep of D 4 and would again have the s + is physics with the standard free energy for 1D reps while (η 2I ,η 3I ) belong to the 2D Kerr active E reps of D 4 . The free energy for the E rep is For β 2 > 0 will pick the η I (1,0) and for β 2 < 0 the η I (1,1) state, both of which break time-reversal symmetry.