Superconductivities of doped Weyl semimetals

Transcription

1 UIUC, Feb 5 th (2013) Superconductivities of doped Weyl semimetals Phys. Rev. B. 86, (2012) - Editors suggestion Gil Young Cho UC Berkeley Jens H Bardarson Yuan-Ming Lu Joel E Moore

2 Plan. Part 1 : superconductivities of doped WSM Motivations, Weyl semimetals? BCS or FFLO superconductivity? Topological Defects in FFLO SC-ivity UIUC, Feb 5 th (2013) Part 2 : Proximate phases of Z2 spin liquid on Kagome lattice (GYC, YML and AV, in preparation) Z2 spin liquid in Kagome lattice Proximate phases of Z2 spin liquid Ashvin Vishwanath

3 Part 1. I. Superconductivity of doped Weyl semimetals Hedgehogs in momentum space

4 Weyl semimetals = 3D graphene? graphene? Dirac fermion in 2D Honeycomb lattice of graphene H = v D (σ x k x + σ y k y ) μ Here σ a acts on sublattice basis (A-B sublattice)

5 Weyl semimetals = 3D graphene? graphene: H = v D (σ x k x + σ y k y ) μ Weyl SM: H = v D (σ x k x + σ y k y + σ z k z ) μ ~ two-component Weyl fermion in HEP In general, WSM is described by H = v a q σ a a=1..3 with the chirality c = sgn ( v 1 v 2 v 3 ) Material: iridates, layered top. Ins., TlBi Se 1 x S x 2, HgCr2Se3

6 What is interesting in WSM? Ref. Wan et. al. (2011), Hosur et. al. (2012), Burkov et.al. (2011), Yang et.al. (2011) A. robust-ness of Weyl nodes: - no mass term for a single Weyl node: e.g H = v σ k - pair annihilation is the only way to gap out! B. flat fermi arc surface state C. three-dimensional anomalous Hall effect: σ xy ~ Δθ e2 D. interesting transport phenomena E. Need to break time-reversal or/and inversion symmetry h Review article: Ari Turner, and Ashvin Vishwanath, arxiv: (2013)

7 Where can we find Weyl semimetal phase? Topological insulator + large enough Time-reversal symmetry breaking Material: Fe/Cr doped TlBi Se 1 x S x 2 Ref. Burkov et.al. (2011), Gil Young Cho (2011)

8 Top. Ins. -based Weyl semimetals Minimal model for Topological Insulators H = v D τ z σ k + τ x M Ref. Gil Young Cho (2011) e.g. near the Γ-point in Bi 2 Se 3 τ a : orbital degrees of freedom σ a : spin degrees of freedom We add a time-reversal breaking term coming from magnetic impurities (e.g. Fe or Cr) in bulk H zeeman = mτ 0 σ z

9 Top. Ins. -based Weyl semimetals H = v D τ z σ k + τ x M + mσ z Spectrum for m > M E K Z = ± m 2 M 2 away from the high sym. point Winding # is -1 Γ Winding # is +1 Kz Ref. Gil Young Cho (2011)

10 Top. Ins. -based Weyl semimetals Insulator Insulator M m a. Top. Ins. Triv. Ins. transition a fourcomponent Dirac fermion. b. A four-component Dirac fermion can be splitted into a pair of twocomponent Weyl fermions Material: Fe- or Cr- doped TlBi Se 1 x S x 2 Ref. Gil Young Cho (2011)

11 Superconductivities of doped Weyl semimetal? Fermi surfaces due to doping + Attractive interactions between the electrons

12 Why superconductivities? GY Cho, J Bardarson, YM Lu, and JE Moore, Phys. Rev. B. 86, (2012) I. Strong spin-orbit coupling e.g. A doped topological Insulator can become a topological superconductor II. Topological Winding # around Weyl nodes III. Disconnected Fermi pockets for small doping

13 GY Cho, J Bardarson, YM Lu, and JE Moore, Phys. Rev. B. 86, (2012) In this talk: Study possible superconducting states emerging from a doped Weyl semimetal

14 Our approach is Following the spirit of Fu and Berg (2010) A. Consider a minimal model for a Weyl semimetal & dope the model slightly B. Add a phenomenological attractive interaction to the model C. Classify possible superconductivities D. Compare energies of SC states

15 Minimal Lattice model for Weyl semimetals Ref. Yang et.al (2011) A. The minimal model has two bands touching each other at two Weyl points E It can be modeled with spin-ful electrons c, with spin-orbit coupling in a cubic lattice : Γ Kz c i, c j, site i site j H = iλ SOI { c i,α σ x αβ c i+x,β + c i,α σ y αβ c i+y,β } + h. c +M c i,α (σ z )c i,β t {c i, c j, c i, c j, } - Pauli matrices σ a acts on the spin index, of the electron c /

16 Minimal Lattice model for Weyl semimetals Ref. Yang et.al (2011) Minimal model on cubic lattice (broken time-reversal symmetry) H = H 1 k x, k y, M + H 2 (k z, m) H 1 k x, k y, M = σ x sin k x + σ y sin k y + Mσ z (2 cos k x cos k y ) H 2 k z, m = mσ z (cos k z cos Q) Important properties of the Hamiltonian I. Two Weyl nodes at k z = ±Q and k x = k y = 0 II. no spin rotational symmetry ( spin~ σ a ) (~ strong spin-orbit coupling) III. symmetry = lattice symmetry C 4h (~ pairing should be classified by C 4h )

17 Doping the minimal model A. There are two fermi pockets around the nodes with spin-momentum locking e.g. near the upper node H + = σ x k x + σ y k y + σ z (k z Q) μ Note: winding around the node is ±1 B. To study a superconducting instabilities, we add a simplest on-site attractive interaction δh = V 0 r n (r)n (r) e.g. phonon-mediated attractive interaction

18 Possible BCS superconducting states A. The interaction is completely local and thus the electrons are paired in singlet channel (due to Pauli s exclusion principle). Pairing interaction is local in real space = pairing gap is constant in momentum space B. So the mean-field state should be singlet BCS pairing δh = Δ ψ u k ψ d k + h. c. note: the total momentum carried by the pairing is zero; so it should be inter -nodal pairing

19 Possible BCS superconducting states BCS state: inter-nodal + singlet (trivial rep of the lattice sym.) Note: To be paired in singlet channel, spins need to be anti-parallel The nodes are Weyl nodes!

20 Possible FFLO superconducting states A. There is another competing superconducting state for the doped Weyl semimetal ~ FFLO pairing (FFLO state: intra-nodal + singlet ) δh = Δ ± ψ u ±Q + k ψ d ±Q k + h. c. note: the pairing Δ ± carry crystal momenta ±2Q note: this FFLO is fully gapped B. Why this state can be better than BCS state? - the effective DOS participating to the BCS pairing is reduced because the spin states at k and k are not anti-parallel. - However, FFLO state connects anti-parallel spins

21 Mean-field Energy of superconducting states

22 So.. FFLO state wins against BCS state in the doped Weyl semimetal

23 Physical properties of FFLO superconductivity in WSM??

24 FFLO superconducting states A. We have two independent pairings Δ ± for each node δh = Δ ± ψ u ±Q + k ψ d ±Q k + h. c. which can be compactly written as a wave in real space Δ r = Δ cos(2q r + δφ) a. so there should be density modulation in FFLO state b. center of momentum is fixed by the positions of Weyl nodes (in the iridates, we have 24 Weyl nodes) B. FFLO state has a half-quantum vortex and a usual full quantum vortex half-quantum vortex : a unit winding in only one of the two pairings full quantum vortex: a composite of two half-quantum vortices Any exotic interesting bound state to these vortices?

25 Lesson From superconducting doped Top. Ins. Any interesting bound mode to vortex? Ref. Fu and Berg (2010) Example. Topological SC from a doped Topological Insulator CuBiSe E i) Fermi surface is two-fold degenerate, related by T- symmetry kx ii) Each fermi surface encloses a non-trivial winding # +1 or -1 iii) Vortex realizes two zero modes related by T-symmetry, or a helical Majorana mode γ,k = γ, k, E = +vk γ,k = γ, k, E = vk Vortex

26 Similarty & Difference from doped Top. Ins. Similarities i) Fermi surface encloses a winding # ii) Fully gapped (robust topological feature) Differences i) No time-reversal symmetry = no helical state; at best chiral or gapped states at vortex ii) Order parameter space = S 1 S 1 in FFLO, instead of S 1 in usual SC = more topological defect types in FFLO = half-quantum vortex in FFLO e.g. spinful p+ip superconductor; we have a direction d S 1 and a phase e iθ S 1 see Ivanov (2000)

27 Half-quantum vortex in FFLO superconducting states A. Half-quantum vortex corresponds to arg(δ + ) arg(δ + ) + 2π arg Δ arg(δ ) i.e., only the phase of Δ + winds once B. The Fermi surface around the node carries a unit topological winding number C. Hence there is a zero mode or a chiral Majorana mode in the half-quantum vortex. (i.e., γ k = γ, E = +vk) k A quick way to see a chiral zero mode in the vortex: One node of the doped Weyl semimetal is half of the topological superconductor

28 Possible Majorana mode at a full vortex A. Bring two half-quantum vortices with opposite chiral Majorana modes Half-quantum vortices B. In the vortex, we have a helical Majorana mode which is not protected C. The helical Majorana mode will be gapped D. There are two possibilities for the gapped phase; weak pairing and strong pairing phases ref. Kitaev (2000) E. In the weak pairing phase, we have a bound Majorana zero mode at the end of the vortex A dangling Majorana mode Full quantum vortex

29 Conclusion I. We found FFLO state is competing with BCS state (FFLO is energetically better than BCS in our model) II. FFLO state I. Density modulation with the momenta fixed by position of Weyl nodes II. Chiral Majorana mode at half-quantum vortex III. Possible Majorana zero mode at full vortex

30 UIUC, Feb 5 th (2013) Part 2. II. Proximate phases of Z 2 spin liquid on Kagome lattice (Gil Young Cho, Yuan-Ming Lu and Ashvin Vishwanath, in preparation) Yuan-Ming Lu Ashvin Vishwanath

31 Z 2 spin liquid on Kagome lattice Heisenberg interaction on Kagome lattice : Materials: dmit, Herbertsmithite DMRG studies found a Z2 spin liquid! A. Gapped B. Topological Entanglement Entropy Ref. Yan Huse and White (2011) Jiang Wang and Balents (2012)

32 Which Spin liquid? Ref. Hastings (2000) Ran, Hermele, Lee, and Wen (2007) Hermele, Ran, Lee, and Wen (2008) Many different spin liquids from the fermionic rep. of spin-1/2 The (relatively) low energy state is, U(1) Dirac spin liquid (among the fermionic SL ansatz) This state features: A. Dirac spectrum for fermions B. U(1) Gauge theory Not consistent with DMRG result

33 Is there a Z2 spin liquid near the U(1) Dirac spin liquid? Ref. Lu, Ran, and Lee (2011) Requirement 1: gauge theory should be broken from U(1) to Z2 Pairing of Dirac fermions! Requirement 2: Invariant under the lattice symmetry operation + Invariant under the spin rotation operation Requirement 3: Pairing should gap out the Dirac fermion There is one and only one such pairing satisfying the requirements! s-wave pairing of Dirac fermion

34 What can we tell about this Z2 state? (Gil Young Cho, Yuan-Ming Lu and Ashvin Vishwanath, in preparation) Our claim is: If this Z2 spin liquid is the spin liquid found in DMRG study, it should have very specific proximate phases separated by a continuous transition from the spin liquid. A. Q=0 non-collinear magnetic ordered state B. VBS phase with the very specific bond-bond correlation

35 Proximate phases Special 5-tuplets of masses of Dirac fermions = WZW term for 5-tuplet masses = Unconventional second order transitions Ref. Wiegmann and Abanov (2000), Senthil and Fisher (2006), Grover and Senthil (2008), Ryu, Mudry, Hou, and Chamon (2009), Herbut (2010) etc. Underlying physics of WZW term: topological defect in one phase carries the quantum numbers related to the other phase Condensation of the defect = destroying one ordering + inducing the other order

36 Q=0 non-collinear magnetic order and Z2 spin liquid Monopole quantum #s of U(1) SL Vison spin-1/2 Q = 0 state

37 Q=0 non-collinear magnetic orders n r = V=

38 Result : Sachdev (1992) = Continuous transition?!

39 Nature of the VBS phase Order parameters for VBS pattern N are at M-points in BZ At least two of M-points should participate All the lattice symmetries (except translational symmetries) are broken 12-site unit cell ( quadrupled unit cell) e.g. Diamond pattern found in DMRG has 12-site unit cell patterns Two Dirac nodes BZ ky M- points kx

40 Ex. Bond ordering patterns at M-points BZ

41 Ex. Bond ordering patterns at M-points BZ

42 Conclusion I. We have studied the proximate symmetry broken phases of a particular Z2 spin liquid II. Q=0 non-collinear magnetic ordered state - This allows us to identify the fermionic Z2 spin liquid as the bosonic Q1=Q2 spin liquid III. VBS state - superposition of VBS patterns at M-points - 12-site unit cell with broken lattice symmetries IV. More direct probe?

43 Thanks! My previous research: (1) Topological BF theory description of topological Insulator (2011) (2) Quantum Phase transition and fractionalization in a topological insulator thin film with Zeeman and excitonic masses (2011) (3) Weyl semimetal in magnetically doped topological insulator (2011) (4) Dyon condensation in topological Mott insulator (2012) (5) Gapless edge state of BF field theory and Z2 spin liquids, (2012) (6) Superconductivities of doped Weyl semimetals, (2012) (7) Two dimensional symmetry protected phases with PSU(N) and time reversal symmetry, (2012) (8) Proximate phases of Z2 spin liquid on Kagome lattice, in preparetion My collaborators: C. Xu (UCSB) J. E. Moore (UCB) Y.B. Kim (U Toronto) Y.-M. Lu (UCB) A. Vishwanath (UCB) J.H Bardarson (UCB)