Magnetic Ordering on Topological Crystalline Insulator Surfaces Part 1. 3D topological insulators and their surface states Part 2. Ordering on the surface of magnetically doped TCIs Part 3. Domain walls and speculations Sahinur Reja - Indiana Luis Brey CSIC, Madrid Shixiong Zhang - Indiana 3D Topological Insulators I. Surface states protected by time-reversal symmetry Surface Brillouin zone: Paradigm: Bi1-xSbx Projection of time-reversal invariant points onto surface guaranteed to be doubly degenerate (Kramers doublet) Away from these points states repel surface Dirac cone surface Dirac cone

Gapless nature of surface states robust: protected by time-reversal symmetry From Hassan and Kane, RMP (2010) Model Hamiltonian for Bi2Se3 (Zhang et al. 2009): (111) surface (Figs from Zhang et al., Nat. Phys. 2009) Four relevant orbitals at point: To consider a surface to - to -z, start with kz =0. In vicinity of point: [ 2 2

] Hybridized pz orbitals of basis atoms ( , , 0 )= 0 2 ( + ) 0 + 2 ( + ) ( , , 0 )= 0 2 ( 2 + 2 ) 0 + 2 ( + ) [ ] Mutually anticommuting, like Pauli matrices Coefficients may represent 3D vector components of k dependent Zeeman Hamiltonian For M0B2 > 0, the origin is enclosed surface Dirac cone Topological insulator So gapless surface states can be constructed. Can do so explicitly with this form of H. [Liu et al. (2010); Silvestrov et al. (2012), Brey and HAF (2014)]

Example: Rectangular slab in a magnetic field Perpendicular surface Lateral surface EF Observations: 1. EF For narrow slab, can get = 1 QHE from two surfaces with Landau levels xy= (1/2)e2/h per surface 2. For thicker slabs, M left-movers, M+1 right movers

Need disorder or dephasing to get correct Hall quantization (Brey and HAF, PRB 2014) II. Surface states protected by crystalline symmetry: Topological crystalline insulators (TCI) (Liang Fu, MIT) Paradigm: Sn1-xPbxTe Surface states protected by mirror symmetry Semiconductors, standard model has 12 orbitals Direct gaps at L points (4 of these) L points are special: 1. Hnn(k=L) = 0 Sublattice a good quantum number 2. L direction hosts 3-fold rotational symmetry 3. L direction hosts 3 mirror planes Pb/Sn sites have effectively adjustable on-site energies controlled by x: virtual crystal approximation

System can be adjusted through topological transition! Continuum bulk Hamiltonian near L point with L perpendicular to (111) surface: q2 q3 out q1 BZ face perpendicular to k1 acts on sublattice; acts on C3 eigenstates

carries out a mirror reflection across x = y plane Set q3 = 0 and note mutually anticommute H1(q1, q2,0) encloses origin in this spinor space if m possible to construct gapless surface states Surface states at points as well, energies < 4 Surface Dirac cones! (Hsieh et al, 2012) Projection of H1(q) onto these states to linear order: Confirmation in ARPES and tight binding calculations. E/t (Polley et al., 2014)

Magnetic Impurities and Magnetism in TCI Systems (Sn,Pb)Te doped with sufficient magnetic impurity (Mn,Eu) concentration long-known to be ferromagnetic Impurities substitute for (Sn,Pb) Hole-doped semiconductors: impurity spins interact with bulk carriers holes via ferromagnetic exchange Story et al, 1990 If conduction electrons removed (e.g., compensation doping), impurity spins become decoupled and no magnetism. But if the system is a TCI, what happens at the surface? Levels repel if mirror symmetry breaks EF Mean-field approach: suppose impurity spins are uniformly polarized: constant

h a1, a2, proportional to components of perpendicular to L direction b3 proportional to component of along L direction A,B have expressions in terms of tight-binding parameters = 2 anisotropy factor for surface Dirac points Gaps of size 2b3 open at Dirac points: Mirror symmetry broken by Lowering of total electronic energy by a surface magnetization Favored direction is a competition among different surface Dirac points. Depends on chemical potential Numerical Simulations: Total energy calculations

(111) face Tight-binding model of (Sn,Pb)Te slab with (111) faces Superlattice of localized field only on a sites, chosen near surface so that coupling to two surface states is the same No field in bulk E along

along Total (free) energy computed for different orientations Free difference between and orientations h=

Fixed N Fixed EF ~ E F~ All same within a layer. So direction of magnetization can be manipulated using an electric gate! on 2/9 of sites - + + +

+ + + Checking ferromagnetism: Does energy increase when spins not aligned? 1. Numerical (spin stiffness at short distance) 2. Perturbation theory (long distance) Consider a generic gapped Dirac surface Hamiltonian (ignore anisotropy) Components of proportional to components of magnetization Consider a spatial variation: , Compute change in electronic energy E to second order in h

What is the Q dependence for small Q (quadratic order)? 2 1 Two situations: (a) Fermi surface present, result is independent of Q! Impurity spins are floppy EF (b) When EF is in the gap, (0) , 1 / (0)

EF As (0) 0, spin-spin interaction becomes long-ranged . Compare to graphene spin susceptibility (Q) Q) ) (Brey, HAF, Das Sarma, PRL 2007)) (1) For in-band, (Q) Q) ) independent of Q) for small Q) (2) For Q at Dirac point, (Q) Q) ) ~ Q) 1/r3 RKKY interaction Because (0) ~ along L direction, impurity spins become stiff as system thermally disorders at high T! III. Domain Walls and Thermal Disordering Simple model of surface magnetization : (appropriate spin stiffness) Number of energy minima e.g., two-fold symmetry (0) ~ b3 (0) ~ -b3

C1 = 1/2 C1 = -1/2 C1 = 1: Jackiw-Rebbi mid-gap states Number of channels depends on how many b3s change sign: Up to 4 possible! From zero-field cooling or in thermal equilibrium Look for effects of increased conduction: tunneling, surface conductance, surface reflectance Domain Walls for TCI: 1. 2. 3.

Use surface states of un-magnetized as basis Project into this space, with h ( x) a domain wall configuration Diagonalize projected Hamiltonian a good quantum number 4.aPeriodic b.c. surface Dirac conenumber two DWs per unit cell, per surface good quantum Example: (111) : - (111) 4 b3s change direction Expect 4 x 4 in-gap states Does the presence of in-gap states change the long-range interaction? No.

3-d Coulomb-like interaction among spin gradients. Numerical evidence from graphene: DWs in staggered magnetization L in gap in gap 0 DW DW x in band Expect logarithmic interaction!

in band 1. above and : Low T Ising order DWs: finite energy per unit length 2. Thermal disordering transition in Ising class ~ , above : Low T Ising order Stiffness consistent with

a long-range Ising model surface Dirac cone Thermal disordering transition with mean-field exponents [M.F. Paulos et al., Nuc. Phys. B (2016)] For ~, there are six degenerate directions for ferromagnetic order. Is this a six-state clock model, or two coupled three-state models? Check the domain walls energies! S = 0.01t E = E(L2 L3)- E(L2 L1) always negative S = 0.1t six-state model

3. below , ~: Low T broken 6-fold symmetry Stiffness consistent with a long-range Ising model 4. surface Dirac cone Thermal disordering transition with mean-field exponents? band : Low T broken 6-fold symmetry DWs: finite energy

per unit length Thermal disordering transition in Kosterlitz-Thouless class [Jose et al., 197)7)] Many thanks to Sahinur Reja - Indiana Luis Brey CSIC, Madrid Funding: Shixiong Zhang - Indiana Summary 1. Topological crystalline insulator offers hosts gapless surface states protected

by mirror symmetry. 2. Model of surface states can be formulated, allowing access to explicit forms of surface wavefunctions. 3. Magnetic dopants may order to break the symmetry, lowering total electronic energy. Specific low energy orderings depend on chemical potential and specific surface: 2-fold and 6-fold orderings likely to occur on (111) surface. 4. Long-wavelength stiffnesses sensitive to chemical potential: impacts energetics of domain walls and yields many types of thermal disordering transitions, all in a single system! Sahinur Reja, HAF, Luis Brey, and Shixiong Zhang, PRB 96, 201111 (R) (2017)) Thank you!!