Chiral Anomaly Phenomena in Weyl Superconductors Satoshi Fujimoto Department of Materials Engineering Science, Osaka University
Y. Ishihara (Osaka Univ.) T. Mizushima (Osaka Univ.) J. de Lisle (Osaka Univ. ) T. Kobayashi (Osaka Univ. ) Collaborators
Outline • Introduc-on ¡ • Torsional ¡chiral ¡magne-c ¡effect ¡in ¡Weyl ¡superconductors ¡ • Thermal ¡analogue ¡of ¡nega-ve ¡magnetoresis-vity ¡in ¡Weyl ¡ superconductors ¡ • Chiral ¡spin-‑polariza-on ¡effect ¡in ¡a ¡ferromagne-c ¡Weyl ¡ superconductor ¡UCoGe ¡ 3
Weyl superconductor due to broken TRS Weyl fermions = Bogoliubov quasiparticles from point-nodes characterized by monopole charge of point nodes of SC gap q m = ± 1 e.g. chiral p x + i p y wave SC � · Ω kk = q m δ ( k � b ) k z Berry curvature q m = +1 ∆ = 0 τ z = +1 b Weyl SC: H = τ z σ · k − σ · b ∆ � = 0 k y TRSB TRSB k x for b y � = 0 τ z = − 1 − b q m = − 1 ∆ = 0 σ : particle-hole space chirality τ z = ± 1 Time-reversal symmetry breaking is (sign of ) necessary for Weyl SC q m
Examples of Weyl superconductor/superfluid ∆ ↑↑ k = ∆ ↓↓ example ABM phase of He3 k = ∆ ( k x + ik y ) (Volovik) Weyl points (monopole charge -1) (thermal) Anomalous surface Hall effect BZ c.f. H. Ikegami et al, Science (2013) surface Majorana non-zero Chern number Weyl points(monopole charge +1) N.B. Weyl points have spin-degeneracy
Examples of Weyl superconductor example 2: URu 2 Si 2 Chiral d zx +id yz SC ∆ k = ∆ k z ( k x + ik y ) Thermal conductivity, specific heat, … Line nodes (Kasahara et al.; point nodes Yano et al.) (linear dispersion) • Kerr effect (Kapitulnik’s group, 2015) • Giant Nernst effect due to chiral SC fluctuation ( exp. : Matsuda’s group, 2016 ; N.B. Weyl points have spin-degeneracy theory : H. Sumiyoshi and S. F. , 2015)
Examples of Weyl superconductor non-unitary spin-triplet SC example 3: UCoGe Ferromagnetic SC d � d ∗ � = 0 d = ( a 1 k a + ia 2 k b , a 3 k b + ia 4 k a , 0) (Mineev, PRB66, 134504; ∆ ↑↑ � = ∆ ↓↓ Hattori, Tada et al., PRL108, 066403) point node Weyl point Fermi surface for majority spin (Samsel-Czekala et al.) N.B. Weyl points have no spin-degeneracy !! Genuine Weyl superconductor !!
Examples of Weyl superconductor example 4: B phase of UPt 3 Chiral f-wave SC ? (controversial) ∆ k = ∆ k z ( k x + ik y ) 2 (Schemm et al.) Point node (k-quadratic) double-Weyl (b) k y points k z (monopole Majorana arc C charge +2) anomalous thermal line node B Hall effect Point node A (k-quadratic) (Goswami-Nevidomskyy) double-Weyl points (monopole (Huxley et al.) charge -2) N.B. Weyl points have spin-degeneracy
PHYSICAL REVIEW LETTERS 26 NovEMaER 1990 VoLUME65, NUMBER 22 with x =0. 033. ' Rauchschwalbe, who studied Ref. 17. a sample We now discuss our conclusions these data, ad- from We now discuss the of the nature phase below T, , ~. T-x phase dressing the nature of the overall first dia- The fact that within errors the transitions at T, . ~ begin There is no evidence of a second phase transition gram. and terminate on the line of superconducting phase tran- at low fields below T, . i in pure UBet3 from either the sitions at T, ~ means that the order parameters for the pSR data or the H, i(T) data. This contradicts the data two phases must be strongly coupled. This could denote of Ref. 14, where from the T a small deviation depen- (AFM) phase coexisting a purely antiferromagnetic with in pure UBei3 below 0. 56 K was claimed dence as evi- and coupled to superconductivity I), or a sin- (hypothesis dence for a second phase, but supports previous specific- gle complex superconducting order parameter with * see an increase ' heat results. The fact that wt. in the difI'erent symmetry-group representations and a magnet- pSR linewidth below T, . ~ and two diA'erent quadratic ic (time-reversal-violating) II). ground state (hypothesis in H, i(T) only for x =0. 0193, temperature dependences Ultrasonic-attenuation data are consistent with hy- for x =0. 000, 0. 0100, or 0. 0245, and 0. 0355, but pothesis I. We note that local moments not on the Th sites' 0. 0600, is a clear indication that there are o, (0) proportional magnetic give rise to a dipolar would linewidth correlations only in regions of the phase diagram to x, is not seen (Table II). The fact that the ' where a which (0. 019 (x second specific-heat at x =0. 019, and peak has been observed magnetism appears abruptly not con- (0. 043). ' The previous suggestion' that with x &0. 019, is further the super- tinuously evidence lo- against at T, i=0. 86 K in UBetq Examples of Weyl superconductor conducting transition is to be cal Th moments; this is consistent with either hypothesis. at T, . 2 for 0. 019 ~ x . associated with the lower transitions II is supported Hypothesis by the fact that T, ~ & T, . i. (0. 043 is also ruled out by our data, because the latter That is, the Fermi surface is largely consumed by the su- phase transitions exhibit magnetic correlations while the perconducting transition at T, i. Thus ob- the large former does not. Within uncertainties in the Th concen- served specific-heat AC~ at T, ~ (comparable example 5: U 1-x Th x Be 13 jump odd parity pairing state ? to of about 0. 005, the T-x phase tration is con- that at T, i) would diagram be very surprising for a purely AFM structed as shown in Fig. 3, augmented approximately by phase, and would require an exceptional enhancement of specific-heat data for other Th concentrations. The the density of states near the zeros of the superconduct- possible d-vector (Shimizu et al.(2017)) dashed lines may not be absolutely vertical as drawn, but ing gap to account for the large Hence the con- AC&. near x =0. 019 and 0. 043 are sug- steep phase boundaries jump at T, 2 ( T, of the phase nectedness diagram and the large specific gested by the present data and the specific-heat data i to an AFM ground in state are proper- (Mizushima and Nitta (2017)) ties unlike those observed in other small-moment heavy- electron Rather magnets thi. s feature of (U, Th)Bei3 is 0. 9— similar to the two superconducting specific-heat anomalies in UPt~, and is also consistent E u representation (cubic symmetry) with hypothesis I I. T c for 0. 019 ~ x ( 0. 043. The increase of H, 'i (0) (Table I) both H, i(0) and a, (0) increase We note that with x } d l 2 ( k ) = 2ˆ z k z − ˆ x k x − ˆ with x must be due to an increase y k y , a in n, (0) or a decrease in m*. As argued an AFM transition at T, ~ previously 2 degenerate to decrease m* because magnetic be expected V would or- √ ~ 0. 5— I- der tends to suppress the f-moment fluctuations, spin l 1 ( k ) = 3(ˆ x k x − ˆ y k y ) to m*. Unfortunately, which contribute greatly it is not possible to predict how much m with x should change below T, . 2 without a detailed and believable microscopic I If n, (0) increases 1 theory. with x the correlation I between I 1 I H;i(0) and a, (0) could − − be explained under MAGNETlC hypothesis I II, because some models for time-reversal-violating y k y + � 2 ˆ I TRSB I te, d ( k ) = l 1 + i l 2 = ˆ x k x + � ˆ z k z I I superconducting states predict orbital currents I generated I I of the order parameter by inhomogeneities I produced by I I I I electron scattering I from nonmagnetic impurities. This non-unitary state 0. 0 j I 0. 0 2. 0 4. 0 X ('/0) ~-' at T=O in TABLE II. The x dependence of o, and [H, 'i] x (%) FIG. 3. Phase diagram for Ul —, Th, Bei&. Open symbols are U I —, Weyl SC !! Th, Be I ~. I frOm g. , ; CirCleS, T, frOm thiS wOrk. SquareS, 1 frOm mag- T, I, ' (. )/, ' ( . 3)l"- in H, I(T-). x ( k) x/1. 93 o;. (x)/o, (1. 93) M(H); inverted netization T, ~ from kink triangles, (Heffner et al.(1990)) The solid T, ~ from specific heat upright triangles are T, 1 and 1. 93 1. 00 1. 00 1. 00 in Ref. 17. The symbol (&) at x =0. 043 indicates of a merging 1. 11 ~ 0. 06 1. 14+ 0. 07 2. 45 1. 27 1 =0. 39 and as described Ref. 17. T, T, , T, K for 1. 31 ~ 0. 07 in 1. 21 + 0. 07 1 3. 55 1. 84 x =0. 0600 was determined resistively. 2818
Chiral Anomaly of Weyl semimetal violation of conservation law of axial current E ¼ q 3 L e 3 j µ 5 = j µ L − j µ 4 π 2 ~ E · ~ ∂ μ j μ B: 5 ¼ R µ = 0 , 1 , 2 , 3 k ( t, x, y, z ) and 2 b 0 is R L e ff = θ e 2 θ ( r ,t ) = 2 b · r − 2 b 0 t, where 2 b 2 π h E · B momentum J = e 2 Anomalous Hall effect π h b × E J = e 2 b 0 Chiral magnetic effect π h B negative magnetoresistance σ zz ∝ B 2 τ (Nielsen,Ninomiya, small B Burkov, Son, large B E � B ∝ B τ Spivak)
Chiral Anomaly of Weyl superconductor Weyl fermions in p-h space Axial current does not couple to E and B Chiral anomaly due to geometrical distortion (gravitational fields) 1 1 q q 192 π 2 ϵ μνρσ 1 e e ∂ μ j μ 5 ¼ 32 π 2 l 2 ϵ μνρσ ð η ab T a μν T b ρσ − 2 R ab ; μν e a ρ e b 4 R ab μν R cd ¼ σ Þ ρσ þ ð ρσ η ad η bc (Parrikar, Hughes, Leigh,Shopurian, Ryu) j µ 5 = v L n L − v R n R � : non-universal cutoff R ab : Riemann curvature µ ν e a : torsion : vielbein µ T 0 e.g. : temperature gradient (Shitade,Bradlyn,Read,Gromov,Abanov) 0 j T i : spatial rotation 0 j T i : vortex, topological texture of SC order parameter jk (l-vector, d-vector) i, j, k = x, y, z
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