Calorimetric evidence for two phase transitions in Ba1−xKxFe2As2 with fermion pairing and quadrupling states

Materials that break multiple symmetries allow the formation of four-fermion condensates above the superconducting critical temperature (Tc). Such states can be stabilized by phase fluctuations. Recently, a fermionic quadrupling condensate that breaks the Z2 time-reversal symmetry was reported in Ba1−xKxFe2As2. A phase transition to the new state of matter should be accompanied by a specific heat anomaly at the critical temperature where Z2 time-reversal symmetry is broken (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T}_{{{{{{{{\rm{c}}}}}}}}}^{{{{{{{{\rm{Z2}}}}}}}}} \, > \, {T}_{{{{{{{{\rm{c}}}}}}}}}$$\end{document}TcZ2>Tc). Here, we report on detecting two anomalies in the specific heat of Ba1−xKxFe2As2 at zero magnetic field. The anomaly at the higher temperature is accompanied by the appearance of a spontaneous Nernst effect, indicating the breakdown of Z2 symmetry. The second anomaly at the lower temperature coincides with the transition to a zero-resistance state, indicating the onset of superconductivity. Our data provide the first example of the appearance of a specific heat anomaly above the superconducting phase transition associated with the broken time-reversal symmetry due to the formation of the novel fermion order.


INTRODUCTION
The formation of bound states of fermions can lead to new states of matter: symmetry-breaking condensates. The most investigated case is the condensate of paired electrons (Cooper pairs). It results in a new state of matter: superconductivity, characterized by a spontaneously broken U(1) gauge symmetry [1][2][3]. A complex order parameter ∆ describing the simplest single-band spin-singlet superconducting state is associated with non-vanishing averages of two-fermion annihilation operators < c ↑ c ↓ >. It describes the flow of Cooper pairs, each carrying twice the electron charge "e". Bound states of 4, 6 etc. electrons would also be bosons, but within the standard Bardeen-Cooper-Schrieffer (BCS) theory, such condensates do not form. Four-fermion electronic condensates can appear via a fluctuations-driven mechanism if the system breaks multiple symmetries [4][5][6][7][8][9]. Among four-fermion orders, there is a counterpart of Cooper pair superconductivity: a charge-4e superconducting order parameter can be constructed involving nonzero averages of the kind < c ↑i c ↓i c ↑j c ↓j >, where i and j are component indices. Although that state also breaks U (1) local symmetry and it is a superconductor, it is different in several important aspects from charge-2e superconductors: the coupling constant of the order parameter to the electromagnetic field is proportional to 4e leading to several different effects such as half-quanta vortex excitations.
These properties are used in current attempts to find such states in experiments [10]. A bosonic counterpart of such states was also discussed in the context of an ultracold atomic mixture close to the Mott insulating state [11].
Besides the superconducting order, one can construct a great diversity of order parameters out of four fermionic operators. The variety of the potential new states described by such order parameters is much greater than possible orders arising from fermionic pairs. An especially interesting possibility is associated with fermion quadrupling condensates forming above the superconducting phase transition, leading to condensates with principally different properties than superconductors.
One such possible state appears when a fermion quadrupling condensate results in a Broken Time-Reversal Symmetry (BTRS) [6,[12][13][14][15][16]. At low temperatures, such a system is a multicomponent superconductor that breaks time-reversal symmetry and can be described by several complex fields ∆ i , where i is a component or band index [17][18][19][20]. Since applying the time-reversal operation twice returns the system to its original state, such a condensate breaks an additional two-fold (i.e. Z 2 ) symmetry. A new state of matter forms under temperature increase [13]. In this state, the averages of the pairing order parameters in each band are zero ∆ i = 0, but a nonvanishing composite order parameter ∆ 4f ∝ ∆ * i ∆ j = 0 appears [6,12,13,15]. This order parameter is of fourth order in fermionic fields < c ↑i c ↓i c † ↑j c † ↓j >. The state preserves the local U (1) symmetry and hence it is resistive to dc current. Instead, it breaks Z 2 time-reversal symmetry resulting in dissipationless local counterflows of charges between i and j components. These currents produce spontaneous magnetic fields around certain kinds of inhomogeneities and topological defects [13,14]. Many novel properties of this state can be described by an effective model, which is different from the Ginzburg-Landau effective models of superconductors and the Gross-Pitaevskii effective models of superfluids, but is rather related to the Skyrme model [14]. The weak spontaneous magnetic fields appearing at T Z2 c , above the superconducting critical temperature T c , and detected by µSR in Ba 1-x K x Fe 2 As 2 are consistent with the appearance of a spontaneous Nernst effect at the same temperature [13]. This, and other experimental data in ref. [13] provides the evidence for a 4-fermionic (quartic) state, which exists in a range of temperatures above T c .
A transition from a normal state to a new state, such as the quartic state, should result in a specific heat anomaly at T Z2 c . A second anomaly at T c should follow at a lower temperature. An example of these anomalies from Monte-Carlo simulations of a multiband model is shown in the theoretical analysis section below. However, the predicted two anomalies are expected to be difficult to detect experimentally since the quartic phase is a fluctuation-induced effect, in which phase fluctuations cause a tiny contribution on top of a background due to other degrees of freedom.
The phase-fluctuations contribution, in addition, can be washed out by inhomogeneities since T Z2 c strongly depends on doping [21]. Such double anomalies in zero external fields were not observed in experiments so far. Here, we investigate new samples of Ba 1-x K x Fe 2 As 2 with slightly different doping compared to the one studied in ref. [13] and report the observation of two anomalies in the zero-field specific heat. Importantly, these anomalies are consistent with spontaneous Nernst and electrical resistivity data, signalling two separate Z 2 and U (1) phase transitions. These data provide calorimetric evidence for a phase transition above T c associated with the formation of the quartic state.

EXPERIMENTAL RESULTS
The temperature dependence of the specific heat and the magnetic susceptibility measured on the sample (S NP ) from Ref. [13] is shown in panel (a) of Fig. 1. There is a significant splitting between the onset temperatures for the specific-heat anomaly and the diamagnetic susceptibility.
As discussed in Ref. [13], this splitting is related to the precursor formation of electronic bound states and eventually to the Z 2 phase transition above the T c . However, the expected distinct The second anomaly, highlighted by black arrows, appears at the Z 2 transition (see Fig. 2).
anomaly in the specific heat at T Z2 c was not resolved in samples investigated in ref. [13]. In this work, we performed systematic specific heat measurements of several new samples (S2 − S4) with a doping level of x ≈ 0.75, close to x = 0.77 for S NP sample. According to the previous studies [21,22], this doping level is within the range where the superconducting state breaks time-reversal symmetry. In addition, we had two reference samples (S1, and S5) that do not break time-reversal symmetry. The specific-heat data ∆C el /T are summarized in Fig. 1. The raw data are given in extended data Fig. ED1 for samples S1, S3-S5, and in Fig. 2b for sample S2.   The reference samples (panels (b) and (f) in Fig. 1) show conventional behaviour with a single phase transition at T c expected for standard superconductors: the appearance of non-dissipative currents observed in the susceptibility data match with the onset of the specific-heat anomaly.
The samples with the BTRS superconducting state (panels (c-e)) show a qualitatively different behaviour, which is similar to the previously investigated S NP sample shown on panel (a). In contrast to sample S NP , the samples investigated in this work show a step-like anomaly above T c . These anomalies cannot be attributed to a superconducting phase with higher T c since no superconducting signal is observed in the magnetic susceptibility at the corresponding temperature (right axis in Fig. 1). For a detailed analysis of a possible inhomogeneity effect on the susceptibility and specific heat, see [13].
To investigate whether this anomaly can be associated with the Z 2 time-reversal-symmetrybreaking phase transition, we performed more detailed investigations on sample S2, which has the most pronounced double anomalies in the specific heat. The data are summarized in Fig. 2 and Fig. ED2. Fig. 2a shows the electrical resistivity plotted versus squared temperature. The resistivity follows a conventional Fermi liquid behaviour in the normal state with the residual resistivity ratio RRR ≈ 70. This indicates high sample quality and the absence of proximity to magnetic critical points. The inset in panel (a) shows the difference between the experimental data and the T 2 fitting curve. It is seen that resistivity deviates from T 2 -behavior below T SCF ≈ 18 K.
T SCF is associated with the onset of detectable effects of superconducting fluctuations [13]. Note that, compared to about 10 K for a sample S NP , here the temperature difference between T SCF and T c ≈ 12.6 K is smaller.
The specific heat data measured in zero and applied magnetic field B c = 16 T are shown in panel (b). The magnetic field doesn't suppress the specific heat anomaly completely. Therefore, to subtract the phonon contribution from the zero-field specific heat, we fit the in-field specific heat above 10 K and use the fitting curve to represent the values from the phonon background.
Details of the fitting procedure can be found in Refs. [21,22]. The result is shown in panel (c), left axis, and compared with the temperature dependence of the zero-field spontaneous Nernst effect, right axis. The onset of the spontaneous Nernst signal gives the critical temperature of the BTRS transition at T Z2 c ≈13.25 K, indicating the formation of the 4-fermion ordered state characterized by intercomponent phase locking arising at a significantly lower temperature than the crossover associated with local superconducting fluctuations T SCF . That is consistent with the theoretical expectations that the relative phase ordering of the four-fermion order parameter ∆ * i ∆ j = 0 should occur below the onset of incoherent fluctuations. Remarkably, as shown in panel (c), the T Z2 c transition temperature coincides with the high-temperature anomaly in the specific heat.
Next, our data shows that T Z2 c splits from the superconducting critical temperature T c . For this purpose, the temperature dependence of the zero-field electrical resistivity (left axis) is compared with the Seebeck effect (right axis) in panel (d) of Fig. 2. Both quantities are signalling supercon-ducting phase transitions at T c ≈12.6 K defined by the temperature at which the resistivity and the Seebeck effect are zero. This temperature is lower than T Z2 c ≈13.25 K. Notably, T c coincides with the lower anomaly (sharp maximum) in the specific heat, indicating that the appearance of zero resistance is caused by the appearance of a finite superconducting order parameter [13]. Note that the double-step anomaly in the zero-field specific heat is well visible in the raw data shown in panel (b). These observations allow us to conclude that our data provide calorimetric evidence for a Z 2 phase transition above T c .

THEORETICAL ANALYSIS
To demonstrate the two singularities in the specific heat we use the simplest phenomenological two-component free-energy functional that yields a similar phase diagram (for detail, see [14,15,23]) We use a simple London model with two phases φ 1,2 originating from two complex fields Ψ α = |Ψ α |e iφα . The existence of a fermion quadrupling phase in that model was previously reported in [13,15] without discussing the specific heat.
The model where the Z 2 time-reversal symmetry is associated with the two equivalent minima for the phasedifference mode, i.e. φ 1,2 = φ 1 − φ 2 = ±π/2, spontaneously broken at low temperatures. This represents one of the simplest models with a BTRS fermion quadrupling phase in zero external magnetic field [15]. Here we consider the model situation where we neglect the coupling to an electromagnetic field. Finite values of the magnetic field penetration length increase the size of the fermion-quadrupling phase [15]. Here, we discuss the specific heat of the system in the presence of a finite splitting between the two critical temperatures of the order of 10 % according to experimental observations. We note that in multicomponent models, when transition splitting is small, a transition could become weakly first order [24,25], which would alter the specific heat picture. As shown in Fig.3, our Monte-Carlo (MC) simulations reveal that the specific heat C v develops two anomalies in the proximity of the two critical temperatures T c and T Z2 c , respectively. We note that the model we consider is a phase-only model that only accounts for the phase-  [26][27][28]. This is an especially challenging experiment in our case since the splitting between the Z 2 and U (1) phase transitions and the strength of superconducting fluctuations are very sensitive to small doping variation, as observed in this study and consistent with the discussion in [6,12,13]. Here, we found that a few per cent changes in doping level narrow the splitting by about 50%. The strongest splitting of about 1.8 K was observed for S NP sample from Ref. [13] with x = 0.77, and it reduced to 1 K and lower for The main result of this study is that in high-quality Ba 1-x K x Fe 2 As 2 samples, we observed two specific-heat anomalies. One correlates with the onset of superconductivity while the other coincides with the spontaneous breaking of time-reversal symmetry, detected by the appearance of a spontaneous Nernst effect. The breaking of the Z 2 symmetry above the superconducting transition temperature and its dependence on doping and external magnetic field allowed earlier to establish that it is associated with the formation of a fermionic quadrupling order [13]. The current data provide calorimetric evidence for the existence of this novel phase in the Ba 1-x K x Fe 2 As 2 system at zero magnetic field. The second result of this study is the verification of the fermion quadrupling order at a different doping from the one reported in [13].

Samples
Phase purity and crystalline quality of the plate-like Ba 1−x K x Fe 2 As 2 single crystals were examined by X-ray diffraction (XRD) and transmission electron microscopy (TEM). The K doping level x of the single crystals was determined using the relation between the c-axis lattice parameter and the K doping obtained in previous studies [29]. The selected single-phase samples had a mass to generate it. The Nernst coefficient (S xy ) is related to the transverse electric field produced by a longitudinal thermal gradient [30]. In order to create an in-plane thermal gradient on the bar-shaped samples, a resistive heater (R = 2.7 kΩ) was connected on one side of the sample, while the other side was attached to a thermal mass. The temperature gradient was measured using a Chromel-Au-Chromel differential thermocouple, calibrated in magnetic field, attached to the sample with a thermal epoxy (Wakefield-Vette Delta Bond 152-KA). The Nernst and Seebeck signals were collected using two couples of electrodes (made of silver wires bonded to the sample with silver paint), aligned perpendicular to or along the thermal gradient direction, respectively.
The magnetic field B was applied in the out-of-plane direction. In order to separate the standard Nernst effect S xy from the spurious Seebeck component (caused by the eventual misalignment of the transverse contacts), the Nernst signal has been antisymmetrized by inverting the B direction.
The spontaneous Nernst signal, which is finite only in proximity to the superconducting transition, has been obtained by subtracting the Seebeck (S xx ) component from the B-symmetric part of the Nernst signal as described in Ref. [13]. In the thermoelectric measurements, the temperature difference ∆T sample across the sample (measured by the thermocouple) did not exceed 3% of the measurement temperature T fixed by the thermal mass.

Details of Monte Carlo simulations
The Monte Carlo (MC) simulations are performed by considering a three-dimensional cubic lattice of linear dimension L and lattice spacing h = 1. For the numerical calculations, we implement the Villain approximation [31] scheme, where the compactness of the phase difference is ensured by writing: where ∆ µ φ r i = φ r i+µ − φ r i is the discrete phase difference between two nearest neighbours lattice sites r i+µ and r i along µ =x,ŷ,ẑ, and β = 1/T is the inverse temperature. The discrete Villain Hamiltonian for the model (1) reads: where S µ = 1 2 [u 2 r,µ,1 + u 2 r,µ,2 ] − ν(u r,µ,1 · u r,µ,2 ) + η 2 cos[2(φ r,1 − φ r,2 )], and u r i ,µ,α = ∆ µ φ r,j − 2πn r,µ,α , with α = 1, 2 label the two components. We performed MC simulations of the Villain Hamiltonian Eq. (S1), locally updating the two phase fields φ 1 , φ 2 ∈ [0, 2π) by means of the Metropolis-Hastings algorithm. A single MC step here consists of the Metropolis sweeps of the whole lattice fields while, to speed-up the thermalization at lower temperatures, we implemented a parallel tempering algorithm. Typically, we propose one set of swap after 32 MC steps. For the numerical simulations presented in this work, we performed a total of 2 × 10 5 Monte Carlo steps, discarding the transient time occurring within the first 50000 steps.
As discussed in [13,15], we assessed the SC critical temperature T c by computing the helicitymodulus sum Υ + . That is defined as the linear response of the system with respect to a twist of the two-component phases along a given direction µ: with: In our MC simulations, the helcity-modulus sum has been computed along µ =x, so in what follows we mean Υ + ≡ Υ x + .
The critical temperature T c is then extracted by taking the thermodynamic limit of the finitesize crossings of the quantity LΥ + , as shown in Fig.3(a) for ν = 0.6 and η 2 = 0.1.
On the other hand, we extracted the critical temperature T Z2 c by introducing a Z 2 Ising order parameter m, equal to +1 or −1 according with the two possible sign of the ground-state phase difference φ 1,2 = ±π/2. The Z 2 critical temperature is then determined by means of finite-size where . . . stays for the thermal average over the MC steps, is indeed expected to be a universal quantity at the critical point. The finite-size crossing points of U for ν = 0.6 and η 2 = 0.1 are shown in Fig.3(b).
Finally, the specific heat C v shown in Fig.3(c)-(d) is defined as: where E is the total energy of the system at a given temperature T .
The error bars of all the observables are estimated via a bootstrap resampling method. In the figures shown, when not visible, the estimated error bars are smaller than the symbol sizes.