depending on the d-vector symmetry in the vortex state of chiral and helical p-wave superconductors

Local NMR relaxation rates T

and T

depending on the d-vector symmetry in the vortex state of chiral and helical p-wave superconductors

Kenta K. Tanaka,^1,*Masanori Ichioka,^1,2,†and Seiichiro Onari^1,2

1Department of Physics, Okayama University, Okayama 700-8530, Japan

2Research Institute for Interdisciplinary Science, Okayama University, Okayama 700-8530, Japan

(Received 26 January 2018; published 10 April 2018)

Local NMR relaxation rates in the vortex state of chiral and helicalp-wave superconductors are investigated by the quasiclassical Eilenberger theory. We calculate the spatial and resonance frequency dependences of the local NMR spin-lattice relaxation rateT₁⁻¹and spin-spin relaxation rateT₂⁻¹. Depending on the relation between the NMR relaxation direction and thed-vector symmetry, the localT₁⁻¹ and T₂⁻¹ in the vortex core region show different behaviors. When the NMR relaxation direction is parallel to thed-vector component, the local NMR relaxation rate is anomalously suppressed by the negative coherence effect due to the spin dependence of the odd-frequencys-wave spin-triplet Cooper pairs. The difference between the localT₁⁻¹ andT₂⁻¹ in the site-selective NMR measurement is expected to be a method to examine thed-vector symmetry of candidate materials for spin-triplet superconductors.

DOI:10.1103/PhysRevB.97.134507

I. INTRODUCTION

The spin-triplet superconductors have attracted much at- tention since exotic states such as odd-frequency Cooper pairs and Majorana states are expected to be induced at the vortex core and surface regions. Although ruthenate superconductor Sr2RuO4, heavy fermion superconductor UPt3, and other ma- terials have been suggested as spin-triplet superconductors by many experimental and theoretical studies [1–6], thed-vector symmetries have not been identified. Thed-vector symmetry was discussed to explain experimental observations such as magnetic field orientation dependences of the Knight shift [4,5] and the Pauli-limit behavior ofH_c2 [6]. In addition to these approaches, we need new methods to clarify thed-vector symmetry in the spin-triplet superconductors.

In the spin-triplet chiralp-wave superconductors, the the- oretical studies for the local NMR spin-lattice relaxation rate T₁⁻¹ revealed that the local T₁⁻¹ in the vortex core region is anomalously suppressed [7–9]. The previous studies based on the Eilenberger theory found the site and resonance frequency dependences of T₁⁻¹, and the anomalous suppression of the localT₁⁻¹is derived from the negative coherence effect related to the odd-frequency s-wave spin-triplet Cooper pairs [10].

Experimentally, the local T₁⁻¹ were detected by the site- selective NMR measurements for high-Tc superconductors [11–13] and the conventional superconductor [14]. As shown in Fig.1(a), the localT₁⁻¹(r) as a function of internal fieldB(r) at the same positionrcan be observed by tuning the resonance frequency among the resonance line shape, since the internal field is in proportion to the resonance frequency. The spectrum ofB(r) in Sr2RuO4is observed byμSR measurement [15].

*[email protected]

†[email protected]

In the uniform state of three-dimensional chiral supercon- ductors with strong spin-orbit coupling and odd-parity pairing, significant suppression of the NMR relaxation rate is suggested for nuclear spins polarized along the nodal direction as a consequence of the spin-selective Majorana nature of nodal quasiparticles [16].

At the surface of the superfluid ³He B phase, the odd- frequency s-wave spin-triplet Cooper pairs were studied in relation to the static spin susceptibility [17], and Ising-type spin relaxation was discussed in relation to the Majorana state [18]. In addition, a strong relation between the Majorana zero-energy mode and the odd-frequency Cooper pair has been revealed [19–22]. These spin-dependent surface states have information of the pairing symmetry, and are expected to be studied also in the vortex core states in spin-triplet superconductors. Since the NMR relaxation rate can prove the direction of the conduction electrons’ spin, we expect that thed-vector structure in the vortex state is detected by site-selective NMR measurement with orientation control of the NMR relaxation direction. As shown in Fig. 1(b), for relaxation directionδM parallel (perpendicular) to the static applied fieldH, we observe the NMR spin-lattice relaxation rate T₁⁻¹ (the NMR spin-spin relaxation rate T₂⁻¹) [23–25].

When applied fieldsHare along thezdirection,T₁⁻¹byδMz comes from thexycomponent of dynamical spin susceptibility χ_xx+χ_yy. AndT₂⁻¹is fromχ_zz+χ_yyifδMx. Therefore, the difference betweenT₁⁻¹andT₂⁻¹may reflect the orientation of thedvector in spin-triplet superconductors.

In this paper, we study the local NMR relaxation rates T₁⁻¹ and T₂⁻¹ in the vortex state of chiral and helical p- wave superconductors. For chiral p-wave superconductors, we consider the two types of chiralp-wave states, dz and dx, where the direction of thed vector indicates thezandx axis, respectively. In particular, we discuss how the relaxation rates depend on the direction of the NMR relaxation directions, calculating the siterand internal fieldBdependences ofT₁⁻¹

f

FIG. 1. (a) The Redfield pattern of the resonance line shape of the NMR,P(B), for chiralp-wave pairing atH /Hc20.023 and T /Tc0=0.5. The resonance line shape is derived from the distribution of the internal magnetic fieldB(r)/Hpresented in the inset.B(r)/H is in proportion to the resonance frequencyf. In the inset, an arrow indicates radius r from the vortex center along the next-nearest- neighbor vortex direction. For example, the intensity at the maximum (minimum)B(r)/Hcomes from the vortex center (midpoint) at the r=0.0ax (r=0.5ax) region, and the peak intensity corresponds to the signal from the saddle point of the internal fields at the radius r=0.3ax far from the vortex center. (b) Schematic picture of the relation between the NMR relaxation directionδM and the static fieldHin the cases ofT₁⁻¹andT₂⁻¹.

and T₂⁻¹. These results help us to investigate the d-vector symmetry of chiral and helical p-wave superconductors by site-selective NMR measurement.

This paper is organized as follows. After the introduction, we describe our formulation of the quasiclassical Eilenberger equation in the vortex lattice state and the calculation method for the local NMR relaxation rates T₁⁻¹ andT₂⁻¹ in Sec. II.

The derivations ofT₁⁻¹andT₂⁻¹are explained in AppendixA based on the Eilenberger theory. In Sec.III, we investigate the site andB dependences of local T₁⁻¹ andT₂⁻¹ in the chiral and helicalp-wave superconductors to find the relation to the d-vector symmetry. In Sec.IVand AppendixB, to understand the d-vector dependence, we discuss the site dependence of the coherence terms and the odd-frequencys-wave spin-triplet Cooper pairs around a vortex. The last section is devoted to the summary.

II. FORMULATION

We calculate the spatial structure of vortices in the vortex lattice state by quasiclassical Eilenberger theory. The quasi- classical theory is valid when the atomic scale is small enough compared to the superconducting coherence length. For many

superconductors including Sr2RuO4, the quasiclassical condi- tion is well satisfied.

For simplicity, we consider the chiral or the helicalp-wave pairings on the two-dimensional cylindrical Fermi surface, k=(kx,k_y)=k_F(cosθ_k,sinθ_k), and the Fermi velocityvF= v_F0k/k_F. In the following, the hat symbol indicates the 2×2 matrix in spin space and the check symbol indicates the 4×4 matrix in particle-hole and spin spaces.

To obtain quasiclassical Green’s functions ˇg(iωn,r,k) in the vortex lattice state, we solve the Riccati equation derived from the Eilenberger equation [26,27]

−iv·∇g(iωˇ n,r,k)= ¹₂[iω˜_nτˇ₃−(r,k),ˇ g(iωˇ n,r,k)] (1) in the clean limit, where r is the center-of-mass coordinate of the pair, v=vF/v_F0, ˇτ₃ is the Pauli matrix defined in Eq. (A3), andiω˜n=iωn−v·Awith Matsubara frequencyωn. The quasiclassical Green’s functions and order parameter are described by

ˇ

g(iωn,r,k)= −iπ

g(iωˆ n,r,k) ifˆ(iωn,r,k)

−ifˆ¯(iωn,r,k) −g(iωˆ n,r,k)

, (2) (r,k)ˇ =

0 (r,k)ˆ

−ˆ^†(r,k) 0

, (3)

where ˇg²= −π²ˇ1. The spin spaces of ˆgand ˆare, respec- tively, defined by the matrix elements

g_{σ σ}(iωn,r,k)

=

g₀(iωn,r,k)ˆ1+

μ=x,y,z

gμ(iωn,r,k) ˆσμ

σ σ

, (4)

σ σ(r,k)=

i

μ=x,y,z

[dμ(r,k)·σˆμ] ˆσy

σ σ

, (5)

where σ,σ= ↑(up-spin) or ↓(down-spin), and d_μ is the μ component of thedvector. In addition, the matrix elements of the order parameter are defined by

_{σ σ}(r,k)=_{+,σ σ}(r)φp+(k)+_{−,σ σ}(r)φp−(k) (6) with the order parameter_{±,σ σ}(r) and the pairing function φ_p±(k)=k_x±ikyfor thep_±state.

Length, temperature, and magnetic field are, respectively, measured in units ofξ₀,T_c0, andB₀. Here,ξ₀=hv¯ _F0/2π kBT_c0 and B₀=φ₀/2π ξ₀² with the flux quantum φ₀. T_c0 is the superconducting transition temperature at a zero magnetic field. The energyE, pair potential, andω_nare in units of π k_BT_c0. In the following, we set ¯h=k_B=1. In this paper, our calculations are performed atH /B₀=0.02 andT =0.5Tc0. In the chiralp-wave and helicalp-wave states atT =0.5Tc0, the upper critical fieldsH_c2/B₀0.85.

We set the magnetic field along thezaxis, where the vector potential A(r)= ¹₂H×r+a(r) in the symmetric gauge.

H=(0,0,H) is a uniform flux density, and a(r) is related to the internal fieldB(r)=(0,0,B(r))=H+ ∇ ×a(r). The unit cell of the vortex lattice is set as a square lattice [1]. From the distribution ofB(r), we calculate the resonance line shape called the Redfield pattern given byP(B)=

δ[B−B(r)]dr in Fig.1.

To determine the pair potential ˆ(r) and the quasiclassical Green’s functions self-consistently, we calculate the order parameter ˆ_±(r) by the gap equation

ˆ_±(r)=gN₀T

|ω_n|ω_cut

φ_p^∗_±(k) ˆf(iωn,r,k)k, (7)

where . . .k indicates Fermi-surface average, (gN₀)⁻¹= lnT +2T

0<ωnωcutω⁻¹_n , and we use ω_cut=20k_BT_c0. In Eq. (7),p-wave pairing interaction is isotropic in spin space.

For the self-consistent calculation of the vector potential for the internal fieldB(r), we use the current equation

∇ ×(∇ ×A)= −2T κ²

0<ωn

vIm{g₀}k (8)

with the Ginzburg-Landau parameterκ=B₀/π k_BT_c√ 8π N₀. In our calculations, we useκ =2.7 appropriate to Sr2RuO4

as a candidate material for the chiral or helical p-wave superconductor. The results of this paper are not changed qualitatively by the choice of the Ginzburg-Landau parameter as long as a type-II superconducting vortex state is maintained.

We iterate calculations of Eqs. (1), (7), and (8) forωnuntil we obtain the self-consistent results ofA(r), ˆ(r,k), and the quasiclassical Green’s functions in the vortex lattice state.

In the chiralp-wave superconductors, we only consider the p₋ state, i.e., antiparallel vortex state, since the antiparallel vortex state is stable compared to the parallel vortex state [28,29]. For variety of d-vector orientation, we calculate two types of chiralp-wave states, dz anddx, which are, respectively, defined by d(k)∝(kx−ik_y)ˆzand (kx−ik_y) ˆx.

In the helical p-wave superconductors, we set thed vector asd(k)∝kxxˆ+kyyˆ=φp₊(k)d₋+φp₋(k)d₊in the uniform state at zero field, with d_±(k)=¹₂(1,±i,0). Thus, when we iterate calculations of Eqs. (1), (7), and (8), as an initial value, thedvector is set to bed(r,k)=(r)d(k) where(r) is the Abrikosov vortex lattice solution.

Next, using the self-consistently obtained A(r) and ˆ(r), we calculate ˇg(iωn,r,k)|iωn→E±iηfor real energyEby solving Eilenberger Eq. (1) withiωn→E±iη. In this paper, we de- fine the retarded and advanced Green’s functions ˇg^R(E,r,k)=

ˇ

g(iωn,r,k)|iωn→E+iη and ˇg^A(E,r,k)=g(iωˇ n,r,k)|iωn→E−iη, respectively. η is a small parameter, and we use η=0.01.

We use the components of ˇg^R(E,r,k) and ˇg^A(E,r,k) to numerically calculate local (T₁T)⁻¹and (T₂T)⁻¹in the vortex lattice state.

A. NMR spin-lattice relaxation rateT₁⁻¹

We consider the conventional form of the local hyperfine fields from the conduction electrons [24]. Therefore, relaxation ratesT₁⁻¹ andT₂⁻¹ of the nuclear magnetization are affected by the dynamical spin susceptibility of the local electronic state, which reflects spin states of the odd-frequencys-wave spin-triplet Cooper pairs at the atomic site in the vortex core region, as discussed later.

The NMR spin-lattice relaxation rate T₁⁻¹ by δMz is calculated fromxycomponents of dynamical spin susceptibil- ity,χxx+χyy∝χ₋₊, Therefore, as described in AppendixA

following Refs. [8,10],T₁⁻¹is given by [T1(T)T]⁻¹

[T1(Tc)Tc]⁻¹ = T₁^gg(T)T₋1

+ T₁^ff(T)T₋1

[T1(Tc)Tc]⁻¹

= _∞

−∞

W_sl^gg(E,r)+W_sl^ff(E,r)

4Tcosh²(E/2T) dE, (9) where

W_sl^gg(E,r)= a²²_↓↓(E,r,k)k a¹¹_↑↑(−E,r,k)k, (10) W_sl^ff(E,r)= − a_↓↑²¹(E,r,k)k a_↑↓¹²(−E,r,k)k (11) with

a_{σ σ}¹¹(E,r,k)= 1

2 g^R_{σ σ}(E,r,k)−g^A_{σ σ}(E,r,k) , a_{σ σ}²²(E,r,k)= 1

2 g¯^R_{σ σ}(E,r,k)−g¯^A_{σ σ}(E,r,k) , (12) a_{σ σ}¹²(E,r,k)= i

2 f_{σ σ}^R(E,r,k)−f_{σ σ}^A(E,r,k) , a_{σ σ}²¹(E,r,k)= i

2 f¯_{σ σ}^R(E,r,k)−f¯_{σ σ}^A(E,r,k) , whereσ,σ= ↑(up-spin) or↓(down-spin).T_c(<T_c0) is super- conducting transition temperature at a finite magnetic field, and

ˆ¯

g^R(E,r,k)=gˆ^R(−E,r,k). In Eq. (9), (T₁T)⁻¹is divided into two contributions, the density of state (DOS) term (T₁^ggT)⁻¹ fromW_sl^ggand the coherence term (T₁^ffT)⁻¹fromW_sl^ff.

B. NMR spin-spin relaxation rateT₂⁻¹

From the field theory of NMR relaxation rate [24], we derive the NMR spin-spin relaxation rate T₂⁻¹ by δMx from the dynamical spin susceptibilityχyy+χzz, based on the Eilenberger theory. Sinceχyygives¹₂T₁⁻¹,T₂⁻¹is given in the form

T₂⁻¹= ¹₂T₁⁻¹+¹₂T_2zz⁻¹, (13) where¹₂T_2zz⁻¹is the contribution fromχzz. In the following, we focus onT_2zz⁻¹instead ofT₂⁻¹.

As described in AppendixA,T_2zz⁻¹is given by 1

2

[T2zz(T)T]⁻¹ [T2zz(Tc)Tc]⁻¹ = 1

2

T_2zz^gg(T)T₋1

+ T_2zz^ff(T)T₋1

[T2zz(Tc)Tc]⁻¹

=

σ σ

Sσ σ

_∞

−∞

W_{σ σ}^gg(E,r)+W_{σ σ}^ff(E,r) 16Tcosh²(E/2T) dE,

(14) where

W_{σ σ}^gg(E,r)=

a_{σ σ}²²(E,r,k)

k

a_{σ σ}¹¹(−E,r,k)

k, (15) W_{σ σ}^ff(E,r)= −

a_{σ σ}²¹(E,r,k)

k

a¹²_{σ σ}(−E,r,k)

k, (16) Sσ σ =1 when σ =σ, and Sσ σ = −1 when σ=σ. Also in Eq. (14), (T_2zzT)⁻¹ is divided into two contributions of the DOS term (T_2zz^ggT)⁻¹ from W_{σ σ}^gg and the coherence term (T_2zz^ffT)⁻¹fromW_{σ σ}^ff.

2

2zz

2zz 2

2

2 2

2

2zz

FIG. 2. rdependence of (T₁T)⁻¹and (T_2zzT)⁻¹in (a) the chiralp- wavedz, (b) the chiralp-wavedx, and (c) the helicalp-wave states.

The dashed lines are for (T2T)⁻¹=(T1T)⁻¹/2+(T2zzT)⁻¹/2. The insets show the spatial distribution of (T1T)⁻¹in each state.H /Hc2 0.023 andT /Tc0=0.5.

In the spin-singlet pairing state and the normal state,T₁⁻¹= T₂⁻¹=T_2zz⁻¹, since

σ σW_{σ σ}^gg=2W_sl^gg and

σ σW_{σ σ}^ff = 2W_sl^ff.

III. SITE AND INTERNAL FIELD DEPENDENCES OF LOCAL NMR RELAXATION RATES

First, we study the local NMR relaxation rates [T1(r)T]⁻¹, [T2(r)T]⁻¹, and [T2zz(r)T]⁻¹ as a function of radius r on a line between next-nearest-neighbor (NNN) vortices in the vortex state of chiral and helicalp-wave superconductors. The results are presented in Fig. 2. At the outside of the vortex core regionr/ax >0.25, the difference between (T1T)⁻¹and (T2zzT)⁻¹is very small in all pairing cases in Fig.2. Therefore, we cannot obtain information about thed-vector symmetry from the observation of (T₁T)⁻¹ and (T₂T)⁻¹ in the uniform superconductivity outside of vortex cores.

On the other hand, in the vortex core regionr/ax <0.25, we can see the obvious difference between (T1T)⁻¹and (T2zzT)⁻¹. In Fig.2(a) for the chiralp-wave dzstate, (T₁T)⁻¹ shows the anomalous suppression of the relaxation rate around the vortex core [10], but (T2zzT)⁻¹ shows enhancement at the

vortex core. The enhancement reflects the accumulation of low-energy quasiparticles around the vortex core. In Fig.2(b) for the chiralp-wavedx state, at the vortex core (T2zzT)⁻¹ shows the anomalous suppression, and (T1T)⁻¹ shows the enhancement. From the differences between Figs. 2(a) and2(b), we realize that the suppression of the relaxation rate occurs when thed vector is parallel to the NMR relaxation direction δM. Therefore, we can extract information about thed-vector orientation from comparative observation ofT₁⁻¹ andT₂⁻¹ at the vortex core region in the site-selective NMR measurement.

In Fig.2(c)for the helicalp-wave state, therdependences of (T1T)⁻¹ and (T2zzT)⁻¹ show similar enhancement at the vortex core, but (T2zzT)⁻¹is smaller than (T1T)⁻¹in the vortex core. This is because thedvector in the helicalp-wave pairing is within thexyplane, and some components of thed vector parallel toδM(x) partially contribute to the suppression of (T_2zzT)⁻¹.

Next, we study the internal fieldB dependence of local (T1T)⁻¹and (T2zzT)⁻¹in the vortex state of chiral and helical p-wave superconductors to discuss how the difference between T₁⁻¹andT₂⁻¹for eachd-vector symmetry is detected in the site- selective NMR measurement. As presented in Fig.3, the local [T1(r)T]⁻¹is plotted as a function ofB(r) at the same position r. The signal from the higher (lower) field comes from inside (outside) of the vortex core. In Fig.3(a)for the chiralp-wave dzstate, (T₁T)⁻¹shows monotonically decreasing behavior as a function ofB. (T_2zzT)⁻¹shows also decreasing behavior in the rangeB1, but in high-field ranges (T2zzT)⁻¹ shows increasing behavior towards a large value at the vortex center.

In Fig.3(b)for the chiralp-wavedx state, (T2zzT)⁻¹shows similar monotonic decreasing behavior to those of (T1T)⁻¹in Fig.3(a). On the other hand, (T₁T)⁻¹in Fig.3(b)shows similar enhancement at the vortex core but the magnitude is smaller, compared to that of (T2zzT)⁻¹in Fig.3(a). In Fig.3(c)for the helicalp-wave state, (T1T)⁻¹and (T2zzT)⁻¹show decreasing behavior as a function ofB in the rangeB 1.5, but these relaxation rates show increasing behavior in high-field ranges.

The magnitude of (T_2zzT)⁻¹is about half of (T₁T)⁻¹. Therefore, the NMR relaxation rates show different behav- ior between (T1T)⁻¹ and (T2zzT)⁻¹at the vortex core region in chiralp-wavedz, chiralp-wavedx, and helicalp-wave states. The reason for this difference is related to the negative coherence effect and the odd-frequency spin-triplet Cooper pairs around the vortex center, as discussed in the next section.

IV. NEGATIVE COHERENCE EFFECT AND ODD-FREQUENCYs-WAVE SPIN-TRIPLET

COOPER PAIRS

To discuss the reason for the differences in anomalous suppressions of the NMR relaxation rates between cases presented in Figs.2 and3, we show the site dependence of the coherence terms (T₁^ffT)⁻¹ and (T_2zz^ffT)⁻¹ with the DOS term (T₁^ggT)⁻¹, and the amplitudes of odd-frequencys-wave spin-triplet Cooper pairs around the vortex core in the chiral and helicalp-wave superconductors. The results are presented in Fig.4along the NNN direction, and summarized in TableI.

We note that the local NMR relaxation rate is divided into two

2zz

2zz

2zz

2zz 2zz 2zz

FIG. 3. Points are for theBdependence of (T1T)⁻¹and (T2zzT)⁻¹ in (a) the chiralp-wavedz, (b) the chiralp-wavedx, and (c) the helicalp-wave states, corresponding to the cases in Fig.2. Only data points for (T1T)⁻¹1.2 are presented.H /Hc20.023 andT /Tc0= 0.5. In (a), (b), and (c), solid line indicates the Redfield pattern of the resonance line shapeP(B) in Fig.1(a)for the chiralp-wavedzand dxstates and helicalp-wave states, respectively. These are almost on the same line.

contributions of the DOS term and the coherence term so that 1/T_1(2zz) =1/T_1(2zz)^gg +1/T_1(2zz)^ff

As shown by a black solid line in Fig.4(a)for the chiral p-wavedzstate, the DOS term (T₁^ggT)⁻¹ is enhanced with approaching the vortex center, because (T₁^ggT)⁻¹ reflects the low-energy local DOS of bound states around the vortex. We note that (T₁^ggT)⁻¹is a small but finite value even at the outside, r/ax ∼0.2, since the bound states have small tails extending

toward the outside of the vortex core. Ther dependences of (T₁^ggT)⁻¹ and (T_2zz^ggT)⁻¹ show similar behavior also for the chiralp-wavedxstate and the helicalp-wave state.

Compared to (T₁^ggT)⁻¹and (T_2zz^ggT)⁻¹, contributions of the coherence terms (T₁^ffT)⁻¹ and (T_2zz^ffT)⁻¹ in Fig. 4(a) are negligible in the outside region of the vortex core, but become comparative contributions with approaching the vortex center.

For the chiralp-wave dz state, (T₁^ffT)⁻¹ is negative and (T_2zz^ffT)⁻¹is positive. The former negative contributions are the origin of the anomalous suppression of (T₁T)⁻¹at the vortex core [10], and the latter further enhances (T_2zzT)⁻¹at the vortex core. Their magnitudes satisfy|(T₁^ffT)⁻¹| ∼ |(T_2zz^ffT)⁻¹| in the whole range of r in Fig. 4(a). At the vortex center, (T₁^ffT)⁻¹ ∼ −(T₁^ggT)⁻¹∼ −24.7.

For the chiral p-wave dx state, the r dependence of (T_2zz^ffT)⁻¹ is the same as (T₁^ffT)⁻¹ in the chiralp-wavedz state. At the vortex center, (T_2zz^ffT)⁻¹∼ −(T_2zz^ggT)⁻¹∼ −24.7.

Therefore, (T2zzT)⁻¹in thedxstate shows similar anomalous suppression at the vortex core to those of (T1T)⁻¹in thedz state. The anomalous suppression at the vortex core occurs when the NMR relaxation directionδMd. On the other hand, (T₁^ffT)⁻¹ =0 in the dx state. Since the enhancement by (T₁^ffT)⁻¹does not work, (T₁T)⁻¹in Fig.3(b)is smaller than (T2zzT)⁻¹in Fig.3(a).

For the helical p-wave state with dxy, (T₁^ffT)⁻¹=0, and (T_2zz^ffT)⁻¹∼¹₂(T_2zz^ggT)⁻¹. This is because of the similar situation as in the chiralp-wave dx state. However, since only part of the d vector is parallel toδM, near the vortex center−(T_2zz^ffT)⁻¹is smaller than that of thedxstate. At the vortex center,|(T_2zz^ffT)⁻¹| ∼12.3 in the helicalp-wave state.

In the relations (T₁^ffT)⁻¹∼ −(T₁^ggT)⁻¹ for the chiral p-wave dz state, there are small deviations between them in our numerical calculation at finite temperature. However, as discussed in AppendixB, in the limit T →0 we expect (T₁^ffT)⁻¹ → −(T₁^ggT)⁻¹ so that (T₁T)⁻¹→0. This is also expected for the relations (T_2zz^ffT)⁻¹∼ −(T_2zz^ggT)⁻¹ for the chiralp-wavedxstate.

In the previous study for the localT₁⁻¹in the chiralp-wave dzstate [10], it is revealed that the negative coherence term (T₁^ffT)⁻¹, inducing the anomalous suppression of (T₁T)⁻¹ around the vortex core, is related to the odd-frequency s- wave spin-triplet Cooper pairFs(E=0,r). In addition, we found that [T₁^ff(r)T]⁻¹= −|Fs(E=0,r)|²in the low-energy limit at low T and the limit of the isolated vortex at low fieldsH, and the negative coherence term [T₁^ff(r)T]⁻¹tends to cancel the local DOS term [T₁^gg(r)T]⁻¹=N(E=0,r)², whereN(E=0,r) is the local DOS. A previous study using the Bogoliubov–de Gennes theory revealed the relationN(E= 0,r)∝ |Fs,↑↓(E=0,r)| in the chiral p-wave dz state for the vortex core quasiparticle states with Majorana zero-energy mode [30].

The spin-resolved local DOSNσ(E,r) is given by Nσ(E,r)=

Reg_{σ σ}^R (E,r,k)

k. (17)

where σ = ↑ or ↓ [27]. The local DOS is defined as 2N(E,r)=N_↓(E,r)+N_↑(E,r). And the spin-dependent

2zz 2zz

2zz

s 2zz

d||z d||x d||xy

2zz 2zz 2zz

s

FIG. 4. (a)rdependence of (T₁^ffT)⁻¹and (T_2zz^ffT)⁻¹in the chiral and helicalp-wave states. We plot−(T₁^ffT)⁻¹and (T_2zz^ffT)⁻¹for the chiral p-wavedzstate, and−(T2zz^ffT)⁻¹for the chiralp-wavedxstate. These are on the same line. For the helicalp-wave state, we plot−(T2zz^ffT)⁻¹. (b)rdependence of amplitude of odd-frequencys-wave spin-triplet Cooper pairs|Fs,σ σ(E=0,r)|.|Fs,↑↓(E=0,r)| = |Fs,↓↑(E=0,r)|for the chiral dzstate,|Fs,↑↑(E=0,r)| = |Fs,↓↓(E=0,r)|for the chiral dxstate, and|Fs,↑↑(E=0,r)|for the helical state show similar r dependence. In the inset,Fs,↓↓(E=0,r) for the helical state is suppressed in the vortex core region, compared toFs,↑↑(E=0,r). For comparison, (T₁^ggT)⁻¹and (T_2zz^ggT)⁻¹, andNσ(E=0,r), for the each state are presented by a black solid line in (a) and (b), respectively.

s-wave Cooper pair is given by Fs,σ σ(E,r)=

φ_s^∗(k)f_{σ σ}^R(E,r,k)

k (18)

with thes-wave pairing functionφ_s(k)=1 [10,31]. In the chi- ralp₋-wave superconductor considered in this paper, induced Cooper pair components around a vortex should satisfy the conditionLz+W =0, whereLzis an angular momentum for the induced components of Cooper pairs, andWis a winding number of the component around the vortex. The s-wave componentFs,σ σ(E=0,r) withL_z=0 has finite amplitude at the vortex center without the phase winding around the vortex (W =0).

In Fig. 4(b), we present the r dependence of the odd- frequencys-wave spin-triplet Cooper pairs around the vortex core. In the chiral p-wavedz state, (↑,↓) and (↓,↑) com- ponents are dominant as explained above. On the other hand, in the chiralp-wavedxstate, the dominant components are

|Fs,↑↑(E=0,r)| = |Fs,↓↓(E=0,r)|. In the helical p-wave state, the dominant component isFs,↑↑(E=0,r). These domi- nant components in the three states have the samerdependence as shown in Fig.4(b). The amplitude ofFs,↓↓(E=0,r) in the

helicalp-wave state is very small at the vortex core compared to|Fs,↑↑(E=0,r)|, since induced Cooper pair components in _↓↓satisfy the different conditionLz+W=2, and the ampli- tude of the induceds-wave component withLz=0 vanishes at the vortex center due to the phase windingW =2. Therefore, at the vortex center,|(T_2zz^ffT)⁻¹| in the helical p-wave state indicates the half value (12.3) to|(T₁^ffT)⁻¹|(24.7) in the dzstate and|(T₁^ffT)⁻¹|in thedxstate.

The finite odd-frequencys-wave spin-triplet Cooper pairs around a vortex core induce the coherence terms. The equal spin components (↑,↑) and (↓,↓) contribute only to T_2zz⁻¹. Therefore, (T₁^ffT)⁻¹=0 in the two cases of thedx state and the helical state. The spin components (↑,↓) and (↓,↑) contribute to bothT₁⁻¹andT_2zz⁻¹. In the low-temperature limit, from Eq. (B4) in AppendixB, the coherence terms of (T₁^ffT)⁻¹ and (T_2zz^ffT)⁻¹ are described by the zero-energy amplitude

|Fs,σ σ(E=0,r)| of the odd-frequency s-wave spin-triplet Cooper pair, as [T₁^ff(r)T]⁻¹= −|Fs,↑↓(E=0,r)|², and 2[T_2zz^ff(r)T]⁻¹ = −|Fs,↑↑(E=0,r)|²− |Fs,↓↓(E=0,r)|²+ TABLE I. Relation of the coherence terms with the DOS terms, and existence of the zero-energy odd-frequencys-wave spin-triplet Cooper pairs at the vortex center in the chiralp-wavedzanddxstates, and the helicalp-wave state. The value of|(T1^ggT)⁻¹|in the chiralp-wave dzstate is defined asC.

Main component of Coherence term Odd-frequencys-wave spin-triplet Cooper pair

order parameter

T₁^ffT₋₁

[δMz]

T_2zz^ffT₋₁

[δMx] Fs,σ σ(E=0,r=0)

Chiraldz −

T₁^ggT₋1

≡ −C + T_2zz^ggT₋1

= +C |Fs,↑↓| = |Fs,↓↑| =0

Chiraldx =0 −

T_2zz^ggT₋₁

= −C |Fs,↑↑| = |Fs,↓↓| =0

Helical (dxy) =0 −C/2 |Fs,↑↑| =0,|Fs,↓↓| =0

|Fs,↑↓(E=0,r)|²+ |Fs,↓↑(E=0,r)|². Therefore, (T_2zz^ffT)⁻¹ is positive fordzwith finite (↑,↓) and (↓,↑). And (T_2zz^ffT)⁻¹ is negative in the other two states, since finite components are (↑,↑) and (↓,↓). In the cases of thedzstate,Fs,↑↑(E= 0,r)=Fs,↓↓(E=0,r)=0. In the two cases of thedxstate and the helical state,Fs,↑↓(E=0,r)=Fs,↓↑(E=0,r)=0.

Since our calculations are performed at the finite tem- peratureT /T_c0=0.5, the results in Fig. 4 deviate from the relations in the low-T limit in Eq. (B4) by the contribu- tion from finite-energy states, but they satisfy the propor- tional relations [T₁^gg(r)T]⁻¹∝N(E=0,r)², [T₁^ff(r)T]⁻¹∝

−|Fs,↑↓(E=0,r)|², and also equivalent equations for the (T2zzT)⁻¹case. Therefore, the relation between the DOS term and the coherence term in TableIwithCis satisfied. The details of these relations are discussed later in AppendixB.

Lastly, we give some related discussions. In realistic ma- terials such as Sr2RuO4, the Fermi surface has multiband nature [1–3]. When multiband superconductivity is realized, the physical quantities are given by the summation of the contributions on the Fermi surfaces of the multibands. There- fore, if the dominant Fermi surfaces have the pair potential with chirality −1 of the chiral p-wave superconductivity, similar anomalous suppression of the local NMR relaxation rates by the negative coherence effects is expected to be observed in the vortex core region. The mechanism that the vortex state of the pair potential with chirality −1 induces the odd-frequency s-wave spin-triplet Cooper pairs and the negative coherence terms is universal, and can be applied to the multiband superconductivity.

Among the possible pairing states of Sr₂RuO₄, there re- mains a scenario of even-parity spin-singlet pairing [3,32,33].

As for the case of the spin-singlets-wave ord-wave super- conductors, since the odd-frequencys-wave Cooper pairs are not induced at the vortex center, the anomalous suppression of the NMR relaxation rates does not occur [10,34]. Therefore, we can examine the spin components of the pairing, singlet, or triplet, by the site-selective NMR measurements.

In this paper, we do not consider the Zeeman effect. The Zeeman magnetic field will quantitatively affect the NMR relaxation rates in the contribution of the relaxation process between up- and down-spin electrons. And detailed study belongs to future studies. However, when the orbital pair breaking due to the vortex is dominant, the scenario for the anomalous suppression of the NMR relaxation rates due to the odd-frequency Cooper pairs will survive.

V. SUMMARY

We studied the siterand the internal fieldBdependences of the local NMR relaxation ratesT₁⁻¹ andT₂⁻¹ in the vortex lattice state of chiralp-wave (dzordx) and helicalp-wave superconductors, based on the Eilenberger theory. We focused on how the anomalous suppression of the local T₁⁻¹ and T₂⁻¹ around the vortex core reflects the d-vector symmetry of the pair potential. The anomalous suppression occurs by the negative coherence term coming from the odd-frequency s-wave spin-triplet Cooper pairsFs,σ σ. The finite spin (σ σ) components of Fs,σ σ reflect the d-vector orientation, and determine in which ofT₁⁻¹andT₂⁻¹the anomalous suppression

occurs. Since the anomalous suppression can be observed when the NMR relaxation direction δM is parallel to the d-vector component, we may obtain the information of the d-vector symmetry by comparative observation of the local T₁⁻¹andT₂⁻¹at the vortex core region in the site-selective NMR measurement. In the chiralp-wavedz(dx) state, since the odd-frequencys-wave spin-triplet Cooper pairsFs,↑↓(↑↑) and Fs,↓↑(↓↓) are induced around the vortex core, the anomalous suppression in the localT₁⁻¹ (T₂⁻¹) occurs. In the helical p- wave state, since the odd-frequencys-wave spin-triplet Cooper pairsFs,↑↑are only induced, the difference between the local T₁⁻¹andT₂⁻¹is small. We hope that these theoretical results of the local NMR relaxation rates will be examined by experiment in spin-triplet superconductors. This observation can be also a method to detect the spin dependence of the odd-frequency s-wave spin-triplet Cooper pairs.

ACKNOWLEDGMENT

This work was supported by Japan Society for the Promo- tion of Science KAKENHI Grant No. JP16J05824.

APPENDIX A: NMR RELAXATION RATEST₁⁻¹ANDT₂⁻¹ We explain derivations of the NMR relaxation ratesT₁⁻¹ andT₂⁻¹in the quasiclassical Eilenberger theory. In the 4×4 matrix form for the Green’s functions

G(x,xˇ )=

G(x,xˆ ) Fˆ(x,x) Fˆ¯(x,x) G(x,xˆ¯ )

(A1) in particle-hole and spin spaces, the spin components of ˆG, ˆF, Fˆ¯, and ˆ¯Gare, respectively, defined as

Gσ σ(x,x)= − Tτ[ψσ(x)ψ_σ^†(x)], G¯σ σ(x,x)= − Tτ[ψ_σ^†(x)ψσ(x)], F_{σ σ}(x,x)= − T_τ[ψσ(x)ψσ(x)],

F¯_{σ σ}(x,x)= − T_τ[ψ_σ^†(x)ψ_σ^†(x)], (A2) where x =(r,τ) with the coordinate r and imaginary time τ. The brackets · · · denote the thermal average, and T_τ is the time-ordering operator. In the Eilenberger theory, the quasiclassical Green’s function is defined as ˇg=τˇ₃

dξkG,ˇ whereξk=(k)−μis the energy variable in thekspace.(k) is the dispersion relation of electrons, andμis the chemical potential. ˇτ₃is the Pauli matrix defined as

ˇ τ₃=

σˆ₀ ˆ0 ˆ0 −ˆσ₀

, (A3)

where ˆσ₀is the unit matrix. In AppendixA, we write the matrix components of ˇgas

ˇ g=

gˆ¹¹ gˆ¹² ˆ g²¹ gˆ²²

, (A4)

instead of the expression in Eq. (3).