Phase-fluctuation Induced Time-Reversal Symmetry Breaking Normal State
Meng Zeng, Lun-Hui Hu, Hong-Ye Hu, Yi-Zhuang You, Congjun Wu
PPhase-fluctuation Induced Time-Reversal Symmetry Breaking Normal State
Meng Zeng, Lun-Hui Hu, Hong-Ye Hu, Yi-Zhuang You, and Congjun Wu Department of Physics, University of California, San Diego, California 92093, USA Department of Physics, Pennsylvania State University, University Park, State College, PA 16802, USA
Spontaneous time-reversal symmetry (TRS) breaking plays an important role in studying stronglycorrelated unconventional superconductors. When the superconducting gap functions with differentpairing symmetries compete, an Ising ( Z ) type symmetry breaking occurs due to the locking ofthe relative phase ∆ θ via a second order Josephson coupling. The phase locking can take placeeven in the normal state in the phase fluctuation regime before the onset of superconductivity. If∆ θ = ± π , then TRS is broken, otherwise, if ∆ θ = 0, or, π , rotational symmetry is brokenleading to a nematic state. In both cases, the order parameters possess a 4-fermion structurebeyond the scope of mean-field theory. We employ an effective two-component XY -model assistedby a renormalization group analysis to address this problem. In addition, a quartetting, or, charge-“4e”, superconductivity can also occur above T c . Monte-Carlo simulations are performed and theresults are in a good agreement with the renormalization group analysis. Our results provide usefulguidance for studying novel symmetry breakings in strongly correlated superconductors. Introduction–.
Unconventional superconductors (e.g.high- T c cuprates [1], heavy-fermion systems [2], and iron-based superconductors [3]) have aroused considerable at-tention for their novel symmetry structures in addition tothe U(1) gauge symmetry breaking. Time-reversal sym-metry (TRS) as well as parity and charge conjugationare fundamental discrete symmetries, hence, spontaneousTRS-breaking superconductivity is of particular impor-tance [4–13]. Various TRS-breaking pairing structuresare theoretically proposed, including d ± id [14, 15], p ± ip [16, 17], s ± id [4, 18], p ± is [ ? ], and s + is [19], and exper-imental evidence has been reported in various systems,such as Re Zr [20, 21], UPt [22, 23], PrOs Sb [24],URu Si [25, 26], SrPtAs [27], LaNiC [28], LaNiGa [29, 30], Bi/Ni bilayers [31], and CaPtAs [32] (For detailsrefer to a recent review [33].). They are often probed bythe zero-field µ -spin relaxation, or, rotation [34–36], andthe polar Kerr effect [37, 38]. TRS breaking signatureshave also been reported in iron-based superconductors[39, 40].If the TRS breaking arises from a complex pairingstructure, it is often presumed that it develops after theonset of superconductivity. However, these two transi-tions are of different natures: Superconductivity is U (1)symmetry breaking and TRS is Z , hence, they cantake place at different temperatures. It is interesting tofurther check whether TRS breaking can occur beforethe superconducting transition. In fact, phase fluctua-tions are prominent in strongly correlated superconduc-tors above but close to T c , such as high T c cuprates [41]and iron-based superconductors [42]. We emphasize thatin a two-gap superconductor, the TRS breaking can besolely determined by the relative phase ∆ θ between twogap functions. The phases of two channels may fluctuatein a coordinated way such that ∆ θ is locked leading toTRS breaking, while the total phase is disordered, hence,the system remains normal.In this article, we show that there exists an Ising sym- metry breaking normal phase in a generic 2D two-gapsuperconductors when the gap functions belong to differ-ent symmetries and are near degeneracy. The key ingre-dient here, as mentioned above, is the superconductingphase fluctuation. Hence, it is a phase fluctuation in-duced TRS-breaking, or, a nematic normal state. By thesymmetry principle, the two gap functions couple via asecond order Josephson term. In the phase fluctuationregime, the low energy physics is described by a coupledtwo-component XY -model, which is mapped to a cou-pled sine-Gordon model and analyzed by the renormal-ization group (RG) method. The phase-locking, or, the Z symmetry breaking temperature can be considerablylarger than the superconducting T c . A quartetting [43],or, charge-“4 e ” phase [44], can also appear above T c . Allthese phases exhibit the 4-fermion type order parameters,and thus are difficult to analyze in mean-field theories.Monte-Carlo (MC) simulations are performed to studythe coupled XY -model at different coupling strengths,and the results agree with the RG analysis.We start with the Ginzberg-Landau (GL) free-energyof superconductivity with two gap functions. Eachone by itself is time-reversal invariant. These two gapfunctions belong to two different representations of thesymmetry group, say, the s -wave and d -wave symme-tries of a tetragonal system, or, different componentsof a two-dimensional representation, say, the p x and p y -symmetries. They cannot couple at the quadratic levelsince no invariants can mix them at this level. Bear-ing this in mind, the GL free-energy is constructed as F = F + F with F = γ | (cid:126) ∇ ∆ | + γ | (cid:126) ∇ ∆ | + α ( T ) | ∆ | + α ( T ) | ∆ | + β | ∆ | + β | ∆ | + κ | ∆ | | ∆ | , (1) F = λ (cid:0) ∆ ∆ ∗ + ∆ ∗ ∆ (cid:1) , (2)where α , ( T ) are functions of temperatures, and theirzeros determine their superconducting transition temper- a r X i v : . [ c ond - m a t . s up r- c on ] M a r atures when the two gap functions decouple. γ , and β , are all positive to maintain the thermodynamic stability.If the gap functions form a two-dimensional representa-tion of the symmetry group, then α = α , β = β ,and γ = γ , otherwise, they are generally independent.Nevertheless, we consider the case that they are nearlydegenerate, i.e., α ≈ α , when they belong to two dif-ferent representations, such that they can coexist.The F -term only depends on the magnitude of ∆ , ,hence, is phase insensitive. We assume that the two gapfunctions can form a quartic invariant as the F -term, asin the cases of s and d -waves, and p x and p y -waves. The F -term does depend on the relative phase between ∆ , ,which can be viewed as a 2nd order Josephson coupling.To minimize the free energy, the relative phase betweentwo gap functions ∆ θ = θ − θ = ± π at λ >
0, i.e.,they form ∆ ± i ∆ , breaking TRS spontaneously. Onthe other hand, when λ <
0, ∆ θ = 0, or, π . Theyform the nematic superconductivity ∆ ± ∆ , breakingthe rotational symmetry. The magnitude of the mixedgap function remains isotropic in momentum space in theformer case, while that in the latter case is anisotropic.The value of λ depends on the energetic details of a con-crete system. At the mean-field level, the free energy isa convex functional of the gap function distribution inthe absence of spin-orbit coupling, [5, 45, 46]. This fa-vors a relatively uniform distribution of gap function inmomentum space, corresponding to the complex mixing∆ ± i ∆ , i.e., λ >
0. Nevertheless, the possibility of λ < ± ∆ , which breaks the rotational symmetryleading to nematic superconductivity.The above GL analysis only works in the supercon-ducting phases in which both ∆ , develop non-zero ex-pectation values. However, it does not apply to the phasefluctuation regime above T c . Let us parameterize the gapfunctions as ∆ , = | ∆ , | e iθ , . In the phase fluctuationregime, the order magnitudes | ∆ , | are already signif-icant, and their fluctuations can be neglected. On thecontrary, the soft phase fluctuations dominate the lowenergy physics, and the system remains in the normalstate before the long-range phase coherence develops.New states can arise in the phase fluctuation regime inwhich neither of ∆ , is ordered. A possibility is that thesystem remains in the normal state but ∆ θ is pinned:If ∆ θ = ± π , then Im∆ ∗ ∆ is ordered, which breaksTRS; if ∆ θ = 0 , π , then Re∆ ∗ ∆ is ordered, whichbreaks rotation symmetry. Similar physics occurs in the p -orbital band Bose-Hubbard model, where the boson op-erators in the p x,y -bands play the role of ∆ , , respec-tively. The transitions of superfluidity and TRS dividethe phase diagram into four phases of superfluidity stateswith and without TRS breaking, and the Mott insulat-ing state with and without TRS breaking, where TRS here corresponds to the development of the onsite orbitalangular momentum by occupying the complex orbitals p x ± ip y [47, 48]. The TRS-breaking normal states werealso studied in the context of competing orders in super-conductors [49, 50]. Another possibility is that the totalphase θ c = θ + θ is pinned, i.e., ∆ ∆ is ordered. Thiscorresponds to the quartetting instability , i.e., a four-fermion clustering instability analogous to the α -particlein nuclear physics. The competition between the pairingand quartetting instabilities in one dimension has beeninvestigated by one of the authors [43]. Later it was alsostudied in the context of high-T c cuprates as the charge-“4 e ” superconductivity [44].However, all the above states involve order parame-ters consisting of 4-fermion operators. Hence, they arebeyond the ordinary mean-field theory based on fermionbilinear order parameters. To address these novel states,we map the above GL free-energy to the XY -model ona bilayer lattice, and employ the renormalization group(RG) analysis combined with the Monte-Carlo simula-tions. Since there should be no true long-range or-der of the U(1) symmetry at finite temperatures, wemean the quasi-long-ranged ordering of the Kosterlitz–Thouless (KT) transition. The model is expressed as H = − J (cid:88) (cid:104) i,j (cid:105) cos( θ i − θ j ) − J (cid:88) (cid:104) i,j (cid:105) cos( θ i − θ j )+ λ (cid:48) (cid:88) i cos2( θ i − θ i ) , (3)where θ , are compact U(1) phases with the modulus2 π . J , are the intra-layer couplings estimated as J , ≈ γ , | ∆ , | , and λ (cid:48) is the inter-layer coupling estimatedas λ (cid:48) ≈ λ | ∆ | | ∆ | .Following the dual representation of the 2D classic XY -model as detailed in Supplemental Material (SM)Sect. I, the above model Eq. (3) can be mapped to thefollowing multi-component sine-Gordon model, which isoften employed for studying coupled Luttinger liquids[51, 52]. Its Euclidean Lagrangian in the continuum isdefined as L = (cid:82) d x L ( x ) [53], where L ( x ) = 12 K ( ∂ µ φ ) + 12 K ( ∂ µ φ ) + g θ cos2( θ − θ ) − g φ cos2 πφ − g φ cos2 πφ , (4)where φ , are the dual fields to the superconductingphase fields of θ , , and the Luttinger parameters K , = J , /T . g θ is proportional to λ (cid:48) in Eq. (3); g φ ,φ are pro-portional to the vortex fugacities of the phase fields θ , ,respectively. For simplicity, all of these g -eology couplingconstants have absorbed the short-distance cutoff of thelattice.We first consider the case that the superconductingchannels are non-degenerate. Without loss of generality, J > J > J = 1 is set as the en- FIG. 1. Phase diagram from the tree-level scaling analysis.0 < J < J is assumed and J = 1 is set as the energy scale. ergy scale. All the coupling constants g , , g θ are smallsuch that the RG analysis can be justified. The cou-pling constants g φ , and g θ are renormalized at the treelevel, whereas K , only become renormalized at the oneloop level. The scaling dimensions of the g -eology cou-plings are calculated based on the spin-wave approxi-mation [54]: ∆ φ , = T , /T where T , = π J , . and∆ θ = T /T ∗ with T ∗ = 2 π ( J − + J − ) − . T or T marksthe KT temperature in each channel, respectively, whentwo superconducting channels decouple, while T ∗ marksthe temperature scale below which the two channels cou-ple together.The tree-level RG equations can be represented in aunified way as d g/ d ln l = 2 (1 − ∆ g ) g , where g repre-sents g φ , and g θ , and ∆ g is the scaling dimension ofthe corresponding operator. Then the resulting phasediagram is sketched in Fig. 1 based on the tree-levelanalysis. As shown in Fig. 1, T > T and T ∗ > T .The Ising type symmetry breaking takes place below T ∗ – the TRS breaking at λ >
0, and the rotation symmetrybreaking at λ <
0. Before crossing T , the two channelsare already locked, hence, the superconducting transitionis determined by T , and no phase transition occurs at T . The lines of T and T ∗ cross at J = , and thendivide the area into four phases. At J /J < , as low-ering the temperature, the system from the normal state(Phase I), first undergoes the superconducting transitionin channel 1 but channel 2 remains normal (Phase II). Asfurther lowering the temperature blow T ∗ , two channelsare locked developing the Ising-type symmetry breaking(Phase III). In contrast, at J /J > , the system main-tains normal phase as crossing T ∗ and develops the Ising-type symmetry breaking (Phase IV). In this regime, θ , fluctuate together while maintaining a fixed ∆ θ . Af-ter further lowering the temperature below T , θ and θ as a whole develop the quasi-long-range superconductingorder.Next we focus on the case that the two channels aredegenerate, i.e., J = J ≡ J , g φ = g φ ≡ g φ and K = K = J/T ≡ K . Due to the permutation sym- metry between these two channels, the coupled theorycan be rewritten in terms of the θ ± , φ ± channels de-fined as θ ± ≡ √ ( θ ± θ ) , φ ± ≡ √ ( φ ± φ ). Since thequadratic Josephson coupling term tends to fix the phasedifferences between two channels, this term also tends toalign the vortices in two channels at the same location.Hence, we also add the couplings between vortices in twochannels as cos2 πφ cos2 πφ and sin2 πφ sin2 πφ in theLagrangian in Eq. (4). Since such terms are less relevantcompared to the original vortex terms in each channel,they do not change the previous tree-level analysis. Thenthe Lagrangian in terms of θ ± , φ ± becomes L ( x ) = 12 K + ( ∂ µ φ + ) + 12 K − ( ∂ µ φ − ) + g θ cos2 √ θ − − g φ cos √ πφ + cos √ πφ − − g φ + cos2 √ πφ + − g φ − cos2 √ πφ − . (5)The RG equations at the one-loop level can be derivedas d g φ d ln l = (cid:16) − π K + + K − ) (cid:17) g φ , d g θ d ln l = (cid:18) − πK − (cid:19) g θ , d g φ ± d ln l = (2 − πK ± ) g φ ± , d K + d ln l = − π (cid:16) g φ + g φ + (cid:17) K , d K − d ln l = − π (cid:16) π ( g φ + g φ − ) K − − g θ (cid:17) , (6)where both of the initial values of K ± equals J/T . How-ever, they receive different renormalizations under theRG process.Fixed points g φ + g φ − g θ g φ K + K − I ∞ ∞ ∞ ∞ ∞ + ∞ IV ∞ ∞ ∞ V 0 ∞ ∞ TABLE I. Fixed points of couplings under RG, each of whichcorresponds to a phase.
The RG equations Eq.(6) possess four fixed points assummarized in Tab.I, which correspond to phases I, III,IV, and V. The first three also appear in Fig.1, whilePhase II therein is a consequence of two unbalanced chan-nels, hence, it does not appear here. If g φ ± → ∞ and g θ →
0, then K ± → g φ → ∞ . This is the normal statephase I since vortices proliferate in both channels. As forthe case of g φ ± → g θ → ∞ , we have K ± → + ∞
0 0.01 0.02 0.03 0.04 0.05 0.06 g
1 0.90.80.60.5 T / J Phase IPhase V Phase III
0 0.03 0.05 0.08 0.1 0.13 0.15 0.18 g
1 0.80.60.40.2 T / J Phase I Phase III Phase IV (a)(b) (Normal State) (Z -breaking Normal State)(Z -breaking SC)(Normal State)(Quartetting State) (Z -breaking SC) FIG. 2. Phase diagram v.s. temperature and g θ by numeri-cally integrating the RG Eqs. (6). The initial values of cou-pling constants are g φ + = g φ − = 0 . g φ = 0 . a ), and g φ + = 0 . g φ − = 0 .
15, and g φ = 0 .
01 for ( b ). Phase IV isthe TRS breaking normal state, or, the nematic normal satebreaking the Ising symmetry, and Phase V is the quartteting,or, the charge-“4e” phase. and g φ →
0. Hence, this is phase III in which both chan-nels are superfluid and phase-locked. If g φ + → ∞ and g φ − →
0, then K + → K − → ∞ and g φ →
0. In thiscase, θ − is pinned while the vortex in the channel of θ + proliferates, i.e., this is phase IV. This is the interest-ing Z -broken normal state with the relative phase lock-ing. Instead, If g φ − → ∞ and g φ + →
0, then K − → K + → ∞ and g φ →
0. In this case, the θ + becomespower-law superfluidity, while the relative phase channelbecomes normal. In other words, phase V corresponds tothe quartetting state, or, the charge-“4e” state.The phase diagrams at different values of vortex fugac-ities g φ ± are presented in Fig. 2 ( a ) and ( b ) by numer-ically integrating the RG equations. In Fig. 2( a ), theIsing symmetry breaking normal state phase IV, appearsin the intermediate temperatures. It can be the TRSbreaking state, or, the nematic state depending on the∆ θ is pinned at ± π , or, 0 or π , respectively. Since itis a consequence of the relative phase locking, the phase (a) (b) Phase I (Normal State)
Phase IV
Phase III (Normal State with Z ₂ broken) (SC with Z ₂ broken) (a) (b) Phase I (Normal State) (Z -breaking Normal State) Phase IVPhase III (Z -breaking SC) Is θ T Is T SC FIG. 3. Phase diagram of Eq. 3 at J = J = 1 by MCsimulations. ( a ) The data collapse analysis for the Ising tran-sition based on S (0) L η − v.s. ( T /T Is − L /ν , with η = 1 / ν = 1, and T Is = 1 .
17, where L is the linear size of the sys-tem. The error bars are smaller than the marker size for allthe figures. ( b ) The phase diagram with two critical tempera-tures T Is and T SC extracted by using the scaling analysis for g θ = 0 . , . , , region expands as enlarging g θ . When g φ + (cid:28) g φ − , Fig.2(b) shows the quartteting phase at the intermediate tem-perature scale at small values of g θ .Now we present the MC simulations for the micro-scopic coupled XY -model Eq. (3) in the case of bal-anced channels, i.e., J = J (See SM Section II fordetailed information.). The superfluidity transition ischaracterized by the jump of the spin stiffness corre-sponding to the overall phase angle θ c = θ + θ . Wefocus on the Ising-type phase transition by using thefollowing structure factor to characterize it, S (0) = L (cid:80) ij sin ∆ θ ( i ) sin ∆ θ ( j ). Fig. 3( a ) shows the datacollapsing fitted by using the critical exponents of the2D Ising model. Fig. 3( b ) shows two phase transitions:The Ising-type transition occurs at a higher temperatureabove the superfluidity transition, which is consistentwith the RG analysis.The above RG and MC studies are based on generalsymmetry properties. Iron-based superconductors aremulti-band gap superconductors, thus are promising sys-tem to test our mechanism [55]. Recently, a gap openingwas observed at the Dirac point of the topological surfacestates in FeTe − x Se x [40], which signatures the Kramerstype TRS breaking to Dirac fermions. The gap magni-tude is already finite at T c , showing that TRS breakingalready occurs above T c . Furthermore, the gap increasebelow T c roughly scales with the growth of the supercon-ducting gap, exhibiting its intimate relation to supercon-ductivity. On the other hand, evidence of strong phasefluctuations has been reported in FeSe superconductorsabove T c [42]. Hence, our theory of phase fluctuation in-duced TRS-breaking normal state is a possible explana-tion for the TRS breaking above T c as shown in SM Sec-tion III. More experimental measurements such as Kerrrotations are desired to further test this scenario.In summary, we have analyzed the Ising-type symme-try breaking in the phase fluctuation regime in two-gapsuperconductors in 2D. It applies to both the phase fluc-tuation induced TRS breaking normal state and nematicstate. The only difference is that the relative phase ∆ θ is locked at ± π , or, at 0 or π . The key point is thatthe two gap functions of different symmetries can phase-couple via a 2nd order Josephson term, such that, theirrelative phase can be locked while the total phase re-mains disordered. The RG analysis shows that the Isingsymmetry breaking temperature can be higher than T c , and the phase coherence of a quartetting, or, charge-“4 e ”phase, can also exist above T c . All these three phasesabove T c are beyond the mean-field theories since theirorder parameters are of 4-fermion type. The above anal-yses are also confirmed by Monte-Carlo simulations. Ourresults apply to the novel symmetry breaking in the nor-mal states in strongly correlated unconventional super-conductors. Note added:
Upon the completion of this manuscript,we became aware of two manuscripts on related top-ics [56, 57]. Nevertheless, the emphases are different:their main point is the competition between charge-“4e”physics and the nematic state, and ours is the TRS-breaking normal state.
Supplementary Material for “Phase-fluctuation Induced Time-Reversal Symmetry Breaking Normal State”
This supplementary material contains,I. A review of the 2D classical XY -modelII. Monte-Carlo simulations for coupled XY -modelIII. Spontaneous time-reversal symmetry breaking states in FeSe − x Te x I. A review of the 2D classical XY -model In this section, we review the duality transformation from the XY -model to the sine-Gordon model, the operatorproduct expansions, scaling dimensions, and the one-loop calculation in the renormalization group (RG) analysis. A. Map the XY -model to the sine-Gordon model We follow Ref. [58] to review the duality between the XY -model and the sine-Gordon model. The Hamiltonian ofa single-component XY -model with the coupling constant J is given by, H XY = − J (cid:88) (cid:104) i,j (cid:105) cos( θ i − θ j ) . (7)To map the XY -model to the sine-Gordon model, we start with the Villain approximation, e − K (1 − cos θ ) ≈ ∞ (cid:88) n = −∞ e − K ( θ − nπ ) , (8)which is valid when K is large. In this case, the dominant contribution comes from the regime that cos θ ≈
1, i.e. θ ≈ nπ . Performing Taylor expansion around each of these values, we have e − K (1 − cos θ ) ≈ (cid:80) n e − K ( θ − nπ ) .Using the Villain approximation, the Partition function of the XY -model in Eq. (7) is given by Z XY = (cid:90) π (cid:89) i dθ i π e − βH XY = (cid:90) π (cid:89) i dθ i π e βJ (cid:80) (cid:104) i,j (cid:105) cos( θ i − θ j ) = (cid:90) π (cid:89) i dθ i π (cid:89) (cid:104) i,j (cid:105) (cid:88) m ij e − K/ θ i − θ j − m ij π ) , (9)where K = βJ = J/T and the Boltzmann constant is set to be 1 for simplicity; m ij are integers defined on each linkof the 2D lattice. Now we perform the Hubbard-Stratonovich transformation by introducing the continuous variables x ij defined on each link of the lattice. The Partition function becomes, Z XY = (cid:90) π (cid:89) i dθ i π (cid:90) ∞−∞ (cid:89)
In this part we use the operator product expansion (OPE) to calculate the scaling dimensions of the coupling termsconsisting of vertex operators of the form cos βφ in the free bosonic field φ and the vertex operators cos βθ in the dualfield θ , based on the free Lagrangian L = K ( ∂ µ φ ) .We start with the correlation functions of the following vertex operators. Following the notation in Ref. [59], thecorrelation function is given by, G β ( x − y ) ≡ (cid:104) e iβφ ( x ) e − iβφ ( y ) (cid:105) . (22)By using the operator identity: e A e B :=: e A + B : e (cid:104) AB + A B (cid:105) , where : ˆ O : means normal ordering, we have G β ( x − y ) = (cid:104) : e iβ ( φ ( x ) − φ ( y )) : (cid:105) e − β (cid:104) ( φ ( x ) − φ ( y )) (cid:105) = e β (cid:104) φ ( x ) φ ( y ) − φ ( x ) (cid:105) = lim a → (cid:18) a a + ( x − y ) (cid:19) β K π , (23)where a here is the short distance cutoff. The following fact is used to derive the above equation, (cid:104) φ ( x ) φ ( y ) − φ ( x ) (cid:105) = − K π ln a a + ( x − y ) . (24)Similarly, we have for the dual field θ : (cid:104) θ ( x ) θ ( y ) − θ ( x ) (cid:105) = − πK ln a a + ( x − y ) . (25)Therefore, we are able to obtain the following correlation functions for two different types of vertex operators: (cid:104) e iβφ ( x ) e − iβφ ( y ) (cid:105) ∼ | x − y | − β K π , (cid:104) e iβθ ( x ) e − iβθ ( y ) (cid:105) ∼ | x − y | − β πK , (26)based on which the scaling dimensions of the vertex operators can be calculated.By taking cos βφ = (e iβφ + e − iβφ ), then (cid:104) cos βφ ( x )cos βφ ( y ) (cid:105) = 14 (cid:16) (cid:104) e iβφ ( x ) e iβφ ( y ) (cid:105) + (cid:104) e iβφ ( x ) e − iβφ ( y ) (cid:105) + (cid:104) e − iβφ ( x ) e iβφ ( y ) (cid:105) + (cid:104) e − iβφ ( x ) e − iβφ ( y ) (cid:105) (cid:17) ∼ | x − y | − β K π where we have used the fact that (cid:104) e iβ φ ( x ) ... e iβ N φ ( x N ) (cid:105) = 0 in the thermodynamic limit when (cid:80) Nn =1 β n (cid:54) = 0 [60].From this we conclude that the scaling dimension of the cos βφ term is β K π . Similarly the cos βθ term has scalingdimension β πK . Using these results, the composite operators consisting of this two types of basic vertex operators,like the ones in the main text, can be readily calculated. C. The One-loop Correction
We consider the Lagrangian in terms of θ ± and φ ± with θ ± ≡ √ ( θ ± θ ) and φ ± ≡ √ ( φ ± φ ). A few cosineterms are also included as L ( x ) = 12 K ( ∂ µ φ + ) + 12 K ( ∂ µ φ − ) + g θ l D − ∆ θ cos(2 √ θ − ) − g φ l D − ∆ φ cos √ πφ + cos √ πφ − − g φ + l D − ∆ φ + cos2 √ πφ + − g φ − l D − ∆ φ − cos2 √ πφ − , (27)where the short-distance cutoff l is restored to make the couplings dimensionless [61].For the one-loop corrections for the RG equations, we consider first the simple case where the free bosonic Lagrangian L = K ( ∂ µ φ ) is perturbed by a generic vortex term L (cid:48) = g φ l D − ∆ φ cos βφ + g θ l D − ∆ θ cos αθ , then the partition functioncan be expanded as the following: Z = (cid:90) D [ φ ] e − S = Z ∗ (cid:16) (cid:90) dx g φ l D − ∆ φ (cid:104) cos βφ (cid:105) + (cid:90) dx g θ l D − ∆ θ (cid:104) cos αθ (cid:105) + 12 (cid:90) dxdy g φ g θ l D − ∆ φ − ∆ θ (cid:104) cos βφ ( x )cos αθ ( y ) (cid:105) + 12 (cid:90) dxdy g φ l D − φ (cid:104) cos βφ ( x )cos βφ ( y ) (cid:105) + 12 (cid:90) dxdy g θ l D − θ (cid:104) cos αθ ( x )cos αθ ( y ) (cid:105) + O ( g ) (cid:17) , (28)where Z ∗ represents the free theory partition function. As we know, the conformal invariance of the free theoryrequires that the cross term corresponding to g φ g θ vanishes at the one-loop level because the g φ and the g θ terms ingeneral have different scaling dimensions. So we only need to consider the g φ and g θ terms.Firstly, consider the g φ term. The OPE in terms of e iβφ is given by,: e iβφ ( x ) :: e − iβφ ( y ) : =: e iβ ( φ ( x ) − φ ( y )) : e − β (cid:104) ( φ ( x ) − φ ( y )) (cid:105) =: e iβ ( − φ (cid:48) ( x )( y − x ) − / φ (cid:48)(cid:48) ( y − x ) )+ O (( y − x ) ) : e − β (cid:104) ( φ ( x ) − φ ( y )) (cid:105) =: 1 − iβφ (cid:48) ( x )( y − x ) − i/ βφ (cid:48)(cid:48) ( x )( y − x ) − β / φ (cid:48) ( x )( y − x )) + O (( y − x ) : e − β (cid:104) ( φ ( x ) − φ ( y )) (cid:105) , (29)where the Taylor expansion is done based on the fact that | y − x | →
0. Therefore,: cos βφ ( x ) :: cos βφ ( y ) : = 12 Re (cid:16) : e iβφ ( x ) :: e − iβφ ( y ) : + : e − iβφ ( x ) :: e iβφ ( y ) : (cid:17) , =: 1 − β φ (cid:48) ( x )( y − x )) + O ( y − x ) : e − β (cid:104) ( φ ( x ) − φ ( y )) (cid:105) , ≈ : 1 − β / ∂ µ φ ) ( y − x ) : e − β (cid:104) ( φ ( x ) − φ ( y )) (cid:105) , ≈ | x − y | − β K π − β / ∂ µ φ ) : | x − y | − β K π +2 . (30)Similarly, the correlation function for cos θ operator is given by,: cos αθ ( x ) :: cos αθ ( y ) : ≈ | x − y | − α πK − α ∂ µ θ ) : | x − y | − α πK +2 . (31)For the g φ term in Eq. (28), (cid:82) dxdy g φ l D − φ (cid:104) cos βφ ( x )cos βφ ( y ) (cid:105) , which gives rise to the one-loop correction to the: ( ∂ µ φ ) : term, becomes − β / (cid:90) dxdy g φ l D − φ | x − y | − β K π +2 (cid:104) : ( ∂ µ φ ) : (cid:105) = − β / (cid:90) dx g φ l D − φ (cid:104) : ( ∂ µ φ ) : (cid:105) (cid:90) dy | x − y | − β K π +2 . (32)Now we do a change of scale by changing the cutoff l → l + δl = (1 + δ ln l ) l . This means the domain of the aboveintegration is changed from | x − y | > l to | x − y | > (1 + δ ln l ) l . Therefore, the corresponding change in the aboveintegration becomes, β / (cid:90) dx g φ l D − φ (cid:104) : ( ∂ µ φ ) : (cid:105) (cid:90) l< | x − y | < (1+ δ ln l ) l dy | x − y | − β K π +2 , (33)which in the case of D = 2 is, 12 2 πβ g φ δ ln l (cid:90) dx (cid:104) : ( ∂ µ φ ) : (cid:105) . (34)Comparing with the kinetic term K (cid:82) dx ( ∂ µ φ ) , we obtain the correction of K due to the g φ term, d (1 /K ) d ln l = 2 πβ g φ . (35)Similarly, the contribution from the g θ term is given by, dKd ln l = 2 πα g θ , (36)because the corresponding kinetic term for θ is K (cid:82) dx ( ∂ µ θ ) . In total we have the following one-loop flow equationfor Luttinger parameter K , dKd ln l = 2 π (cid:0) − β g φ K + α g θ (cid:1) . (37)With this result, the RG equations can be readily obtained. We summarize here the loop-level flow equations for thecoupled Luttinger theory defined in Eq. (27) dK + d ln l = − π (cid:16) g φ + g φ + (cid:17) K ,dK − d ln l = − π (cid:16) π ( g φ + g φ − ) K − − g θ (cid:17) , (38)which have been presented in the main text. II. Monte-Carlo simulations for coupled XY -model In this section, we perform the classical Monte-Caro simulation and discuss the derivation of the spin stiffness γ and the scaling of KT transition temperature. A. Derivation of the spin stiffness γ The Hamiltonian of the coupled XY -model is given by H = − J (cid:88)
Fig.4 shows one typical behavior (at g = 1) of the spin stiffness γ as a function of temperature T . The KT transitiontemperature for each system size is determined by intersecting the 2 T /π line with the stiffness curves. We can clearlysee that as system size goes large, the stiffness jump shows up. The temperatures at the crossings for different systemsizes can then be used to extrapolate the critical temperature at L → ∞ . T T / FIG. 4. Spin stiffness γ as a function of temperature T for different system sizes at g = 1. The purple line is 2 T / π . Theintersection point between γ ( T ) and 2 T / π indicates the KT transition temperature T c . Fig.5 shows the extrapolation of the KT transition temperature in the thermodynamic limit at different couplingstrengths g between the two XY -models. The extrapolated critical temperatures determine the phase boundarybetween the Z -breaking normal state and the Z -breaking SC phase in the main text.1 L T g=0.3g=1g=2g=3 FIG. 5. Extrapolation of the KT transition temperatures in the thermodynamic limit for different coupling strengths betweenthe two XY -models. III. Spontaneous time-reversal symmetry breaking normal states in the FeSe − x Te x superconductor In this section, we briefly discuss the application of our theory to the FeSe − x Te x superconductor, in which strongevidence to spontaneously time-reversal-symmetry breaking states has been observed by using the high-resolutionlaser-based photoemission method both in the superconducting and the normal states [40].Following Ref. [55], we consider two superconducting gap functions ∆ and ∆ , which possess different pairingsymmetries and each of them maintains time-reversal symmetry. The Ginzburg-Landau free energy is given by, F = α | ∆ | + β | ∆ | + α | ∆ | + β | ∆ | + κ | ∆ | | ∆ | + λ (cid:0) (∆ ∗ ∆ ) + c.c. (cid:1) , (44)where we assume α ≈ α so that the two pairing channels are nearly degenerate, discussed in the main text. And wefocus on λ > and ∆ as ∆ θ = ± π . Hence, the complex gap function∆ ± i ∆ spontaneously breaks time-reversal symmetry.Since the FeSe − x Te x superconductor has strong atomic spin-orbital coupling, as allowed by symmetry, the complexgap function can directly couple to the spin magnetization m z via a cubic coupling term as, F M = α m | m z | + iγm z (∆ ∆ ∗ − ∆ ∗ ∆ ) , (45)where α m > γ is proportional to the spin-orbit coupling strength [55]. This term satisfies both the U (1)symmetry and time-reversal symmetry. Because of α m >
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