Topological Superconductivity

Annica Black-Schaffer annica.black-schaffer@physics.uu.se

Materials Theory

6th MaNEP winter school, Saas Fee, January 2015

Contents •  Introduction to superconductivity

–  BCS theory –  BdG formulation

•  Unconventional superconductivity

•  Topological matter

•  Topological superconductivity –  p+ip and d+id superconductors –  “Spinless” superconductors – Majorana fermions

Topological Matter

What is it and what is so special about it?

Condensed Matter Physics

Materials

~ 1023 electrons

Relevant degrees of freedom •  Low-energy electrons •  Collective phenomena

Low-Energy Electrons

Metals Insulators Low-energy electrons

(conducting) No low-energy electrons

(insulating)

Semiconductors Regulate current

Collective Phenomena

Ordered states (static) Electrons organize such that a spontaneous order appears

Symmetry breaking

Crystal structure broken translation symmetry

Magnet broken spin symmetry

Superconductor broken gauge symmetry (phase)

Classification •  All forms of matter can be classified according to the

symmetry they break (translation, spin, gauge, time-reversal, …)

•  Except topological matter Quantum Hall effect: 1980, topological insulators: 2005

–  Ordered but no symmetry breaking –  Topology of the wave function

Ribbon Möbius ribbon

Topological Classes

Topologically speaking: coffee cup = donut ≠ bun 1 hole 1 hole 0 holes

Band Theory

How do we know if we have a metal, insulator, or something else, topologically non-trivial?

Assume (almost) free electrons

Assume periodicity (k = crystal momentum)

…

…

…

…

… … k

E

…

+ + + … +

Each electron state gives one band

Band Theory

Metal

Metal: no energy gap

Insulator

Insulator: finite energy gap

Topological Band Theory Anything else than metals and insulators?

+ spin-orbit coupling

! Band inversion

Normal band structure Topological (inverted) band structure

Without spin-orbit coupling With spin-orbit coupling

HgTe & CdTe Semiconductors

Normal Inverted

s-type

p-type (J=3/2)

Bernevig et al., Science 314, 1757 (2006)

Topological Insulators

Topological insulator inside

Vacuum = trivial insulator

Metal on the boundary

2D Topological Insulators

2D

HgTe/CdTe quantum wells

High resistance without edge states

Topological insulator with two edge states

(G = e2/h per state)

König et al., Science 318, 766 (2007)

3D Topological Insulators

3D Edge state ! Dirac cone

Bi1-xSbx Bi2Se3 Bi2Te3 (thermoelectric materials used in wine coolers)

….

ARPES

Hsieh et al., Nature 460, 1101 (2009)

Topological Matter Topological states of matter have

•  Protected boundary states –  At any boundary to a state with other topological order

(vacuum, normal metal, s-wave SC = trivial topological order)

•  Bulk topological invariant –  Number that classifies the topology of the state

(does not change with small perturbations) •  TIs: band inversion (or not)

Bulk-boundary correspondence

# of boundary states = change in topological invariant at boundary

Quantum Hall Effect Invariant

Free electrons in a magnetic field:

Bloch wave function in band m!

Berry connection

Berry flux

Chern number (topological invariant) n = 1 ! 1 chiral edge state

von Klitzing et al., PRL 45, 494 (1980), Thouless et al., PRL 49, 405 (1982)

Topological Insulator Invariant •  Topological insulators

–  Time-reversal invariant (no magnetic field) !

–  Spin-orbit coupling (if Sz conserved) !

Band degeneracy at Γa,b = 0, π/a (Kramer’s degeneracy)

Normal insulator, ν = 0 Topological insulator, ν = 1 Kane and Hasan, RMP 82, 3045 (2010)

Topological Superconductors

Same band theory used for insulators can be used for superconductors

Topological insulators Topological superconductors

Same low-energy excitation spectrum

Condensate of Cooper pairs

+

Topological Classification For non-interacting (single-particle) insulators and superconductors

Schnyder et al., PRB 78, 195125 (2008)

time-reversal

particle-hole

sublattice (chiral) Topological invariants

BdG System Classification Topological classes for BdG systems

Schnyder et al., PRB 78, 195125 (2008), [1]: Tewari et al., PRL 109, 150408 (2012)

In addition: Spinless p+ip-wave in 1D belongs to BDI (not D) because effective TRS [1]

Topological Matter

What is it and what is so special about it?

Classified by topological order = topological invariant

Boundary states crossing the bulk energy gap

Topological Superconductors

Spin-singlet d+id-wave 2D superconductors

Spin-triplet p+ip-wave 2D superconductors

General Occurrence d+id (spin-singlet d(x2-y2)+id(xy)-wave) and p+ip (spin-triplet, mz = 0 p(x)+ip(y)-wave)

occur naturally for 2D irreps

p(x)+ip(y)

d(x2-y2)+id(xy)

Hexagonal lattice Tetragonal lattice

Topological Superconductors

Spin-singlet d+id-wave 2D superconductors

Spin-triplet p+ip-wave 2D superconductors

d-wave SC from Strong Repulsion (Strong) Coulomb repulsion with antiferromagnetic correlations (e.g. Hubbard model):

!  Spin-singlet pairing

!  Double electron site-occupation unfavorable ! no s-wave pairing

!  Spin-singlet d-wave pairing (best state = least number of nodes)

2D hexagonal lattices ! Spin-singlet d(x2-y2)+id(xy) pairing

Graphene Honeycomb lattice:

1st BZ:

sp2 hybridization ⇒ 1pz orbital per C atom left

Tight-binding with nearest neighbor hopping only: (Wallace, 1946)

K

K’

Γ%

NATURE PHYSICS DOI: 10.1038/NPHYS2208 ARTICLES

Ener

gy

DO

S

K’

K’

K

kx

∆3

∆3

∆2

∆2 ∆1

∆1

M3

M3

M2 M1

K

e2

e1

e3

Λ

∆e2i ∆ei

∆e4i ∆e5i

∆∆e3i

¬W W Energy

Van Hove singularityn = 3/8, 5/8

a b cθ

θθ

θ

θ

Saddle point

Dirac point

M1 M2

Figure 1 | Chiral superconductivity arises when graphene is doped to the Van Hove singularity at the saddle point (M points of the Brillouin zone).a, d+ id pairing exhibiting phase winding around the hexagonal Fermi surface, which breaks TRS and parity (⌃ = 2⇡/3). b, Conduction band for monolayergraphene1. At 5/8 filling of the ⇡ band, the Fermi surface is hexagonal, and the DOS is logarithmically divergent (c) at three inequivalent saddle points ofthe dispersion Mi (i= 1,2,3). Their locations are given by ±ei, where 2ei is a reciprocal lattice vector. The singular DOS strongly enhances the effect ofinteractions, driving the system into a chiral superconducting state (a). As the Fermi surface is nested, superconductivity competes with density-waveinstabilities, and a full renormalization group treatment is required to establish the dominance of superconductivity. A hexagonal Fermi surface and logdivergent density of states also arise at 3/8 filling, giving rise to analagous physics.

Competing ordersIn systemswith near-nested Fermi surface, superconductivity has tocompete with charge-density-wave (CDW) and spin-density-wave(SDW) instabilities34. At the first glance, it may seem that a systemwith repulsive interactions should develop a density-wave orderrather than become a superconductor. However, to analyse thisproperly, one needs to know the susceptibilities to the variousorders at a relatively small energy, E0, at which the order actuallydevelops. The couplings at E0 generally differ from their bare valuesbecause of renormalizations by fermions with energies between E0andW . At weak coupling, these renormalizations are well capturedby the renormalization group technique.

Interacting fermionswith a nested Fermi surface and logarithmi-cally divergent DOS have previously been studied on the square lat-tice using renormalization group methods29–31,34, where SDW fluc-tuations were argued to stimulate superconductivity. The analysisalso revealed near degeneracy between superconductivity and SDWorders. The competition between these orders is decided by a subtleinterplay between deviations from perfect nesting, which favoursuperconductivity, and subleading terms in the renormalizationgroup flow, which favour SDW. In contrast, the renormalizationgroup procedure on the honeycomb lattice unambiguously selectssuperconductivity at leading order, allowing us to safely neglect sub-leading terms. The difference arises because the honeycomb latticecontains three saddle points, whereas the square lattice has only two,and the extra saddle point tips the balance seen on the square latticebetween magnetism and superconductivity decisively in favour ofsuperconductivity. A similar tipping of a balance between supercon-ductivity and SDWin favour of superconductivity has been found inrenormalization group studies of Fe-pnictide superconductors35,36.

In previous works on graphene at the M point, various instabil-ities were analysed using the random-phase approximation (RPA)

and mean-field theory. Ref. 4 considered the instability to d-wavesuperconductivity, ref. 5 considered a charge ‘Pomeranchuk’ in-stability to a metallic phase breaking lattice rotation symmetry, andrefs 6–8 considered a SDW instability to an insulating phase.Withinthe framework of mean-field theory, used in the above works, allof these phases are legitimate potential instabilities of the system.However, clearly graphene at theM point cannot be simultaneouslysuperconducting, metallic and insulating. The renormalizationgroup analysis treats all competing orders on an equal footing,and predicts that the dominant weak coupling instability is tosuperconductivity, for any choice of repulsive interactions, even forperfect nesting. Further, the Ginzburg–Landau theory constructednear the renormalization group fixed point favours the d+id state.

The modelWe follow the procedure developed for the square lattice34 andconstruct a patch renormalization group that considers onlyfermions near three saddle points, which dominate the DOS. Thereare four distinct interactions in the low-energy theory, involvingtwo-particle scattering between different patches, as shown in Fig. 2.

The system is described by the low-energy theory

L =3⇤

�=1

†�(�� �⇧k +µ) �� 1

2g4

†�

†� � �

�⇤

� ⇤=⇥

12�g1

†�

†⇥ � ⇥ +g2

†�

†⇥ ⇥ �+g3

†�

†� ⇥ ⇥

⇥(1)

where summation is over patch labels�,⇥=M1,M2,M3.A spin sumis implicit in the above expression, with the spin structure for eachof the four terms being ⇤ ,⌅,⌅,⇤ , where ⇤ and ⌅ label the spin upand down states. Here ⇧k is the tight binding dispersion, expanded

NATURE PHYSICS | VOL 8 | FEBRUARY 2012 | www.nature.com/naturephysics 159

Logarithmically diverging DOS

M

Massless Dirac Fermions (c → vF ~ c/300)

Doped Graphene •  Strong on-site repulsion at moderate doping [1]

•  Weak repulsion at van Hove singularity –  Perturbative RG at low-E [2]

–  Functional RG [3]

+

+

- - +

+ -

- +i

d(x2-y2) +i d(xy) NATURE PHYSICS DOI: 10.1038/NPHYS2208 ARTICLES

En

ergy

DO

S

K’

K’

K

kx

∆3

∆3

∆2

∆2 ∆1

∆1

M3

M3

M2 M1

K

e2

e1

e3

Λ

∆e2i ∆ei

∆e4i ∆e5i

∆∆e3i

¬W W Energy

Van Hove singularityn = 3/8, 5/8

a b cθ

θθ

θ

θ

Saddle point

Dirac point

M1 M2

Figure 1 | Chiral superconductivity arises when graphene is doped to the Van Hove singularity at the saddle point (M points of the Brillouin zone).a, d+ id pairing exhibiting phase winding around the hexagonal Fermi surface, which breaks TRS and parity (⌃ = 2⇡/3). b, Conduction band for monolayergraphene1. At 5/8 filling of the ⇡ band, the Fermi surface is hexagonal, and the DOS is logarithmically divergent (c) at three inequivalent saddle points ofthe dispersion Mi (i= 1,2,3). Their locations are given by ±ei, where 2ei is a reciprocal lattice vector. The singular DOS strongly enhances the effect ofinteractions, driving the system into a chiral superconducting state (a). As the Fermi surface is nested, superconductivity competes with density-waveinstabilities, and a full renormalization group treatment is required to establish the dominance of superconductivity. A hexagonal Fermi surface and logdivergent density of states also arise at 3/8 filling, giving rise to analagous physics.

Competing ordersIn systemswith near-nested Fermi surface, superconductivity has tocompete with charge-density-wave (CDW) and spin-density-wave(SDW) instabilities34. At the first glance, it may seem that a systemwith repulsive interactions should develop a density-wave orderrather than become a superconductor. However, to analyse thisproperly, one needs to know the susceptibilities to the variousorders at a relatively small energy, E0, at which the order actuallydevelops. The couplings at E0 generally differ from their bare valuesbecause of renormalizations by fermions with energies between E0andW . At weak coupling, these renormalizations are well capturedby the renormalization group technique.

Interacting fermionswith a nested Fermi surface and logarithmi-cally divergent DOS have previously been studied on the square lat-tice using renormalization group methods29–31,34, where SDW fluc-tuations were argued to stimulate superconductivity. The analysisalso revealed near degeneracy between superconductivity and SDWorders. The competition between these orders is decided by a subtleinterplay between deviations from perfect nesting, which favoursuperconductivity, and subleading terms in the renormalizationgroup flow, which favour SDW. In contrast, the renormalizationgroup procedure on the honeycomb lattice unambiguously selectssuperconductivity at leading order, allowing us to safely neglect sub-leading terms. The difference arises because the honeycomb latticecontains three saddle points, whereas the square lattice has only two,and the extra saddle point tips the balance seen on the square latticebetween magnetism and superconductivity decisively in favour ofsuperconductivity. A similar tipping of a balance between supercon-ductivity and SDWin favour of superconductivity has been found inrenormalization group studies of Fe-pnictide superconductors35,36.

In previous works on graphene at the M point, various instabil-ities were analysed using the random-phase approximation (RPA)

and mean-field theory. Ref. 4 considered the instability to d-wavesuperconductivity, ref. 5 considered a charge ‘Pomeranchuk’ in-stability to a metallic phase breaking lattice rotation symmetry, andrefs 6–8 considered a SDW instability to an insulating phase.Withinthe framework of mean-field theory, used in the above works, allof these phases are legitimate potential instabilities of the system.However, clearly graphene at theM point cannot be simultaneouslysuperconducting, metallic and insulating. The renormalizationgroup analysis treats all competing orders on an equal footing,and predicts that the dominant weak coupling instability is tosuperconductivity, for any choice of repulsive interactions, even forperfect nesting. Further, the Ginzburg–Landau theory constructednear the renormalization group fixed point favours the d+id state.

The modelWe follow the procedure developed for the square lattice34 andconstruct a patch renormalization group that considers onlyfermions near three saddle points, which dominate the DOS. Thereare four distinct interactions in the low-energy theory, involvingtwo-particle scattering between different patches, as shown in Fig. 2.

The system is described by the low-energy theory

L =3⇤

�=1

†�(�� �⇧k +µ) �� 1

2g4

†�

†� � �

�⇤

� ⇤=⇥

12�g1

†�

†⇥ � ⇥ +g2

†�

†⇥ ⇥ �+g3

†�

†� ⇥ ⇥

⇥(1)

where summation is over patch labels�,⇥=M1,M2,M3.A spin sumis implicit in the above expression, with the spin structure for eachof the four terms being ⇤ ,⌅,⌅,⇤ , where ⇤ and ⌅ label the spin upand down states. Here ⇧k is the tight binding dispersion, expanded

NATURE PHYSICS | VOL 8 | FEBRUARY 2012 | www.nature.com/naturephysics 159

2

��������

���������

��� ��

��������

���� ���� �������

���

���

��

�

���������������

���� ���� �������

���

���

��

�

���������������

���� ����

FIG. 1. (Color online). Schematic phase diagram displayingthe critical instability scale �c � Tc as a function of dop-ing. At the van Hove singularity (VHS, light shaded (orange)area), d + id competes with spin density wave (SDW) (leftflow picture: dominant d + id instability for U0 = 10eV andthe band structure in [5]). Away from the VHS (dark shaded(blue) area), �c drops and whether the d+id or f -wave SC in-stability is preferred depends on the long-rangedness of inter-action (right flow picture: U1/U0 = 0.45 and U2/U0 = 0.15).

investigate in detail how di�erent band structure param-eters a�ect the phase diagram. We find that rather smallvariations of the longer range hopping parameters such asnext nearest (t2) and next-next-nearest (t3) hopping canshift the position of perfect Fermi surface nesting againstthe VHS [Fig. 2], which significantly influences the com-petition between magnetism and SC. Moreover, in par-ticular away from the exact VHS, the reduced screeningof the Coulomb interaction does not justify the assump-tion of a local Hubbard model description. For this case,we find that only a small fraction of longer-ranged Hub-bard interaction [21] can significantly change the phasediagram, as CDW fluctuations become more competitiveto SDW fluctuations, and a triplet SC phase can appear.In particular, we study how the Cooper pairing in thedi�erent SC phases responds to di�erently long-rangedHubbard interactions. Our results suggest that in ex-periment, modifications of the band structure as well aschanging the dielectric environment of the graphene sam-ple would enable the realization of di�erent many-bodystates and possible phase transitions between them.

Model. We consider the ⇥ band structure of grapheneapproximated by a tight binding model including up to3rd nearest neighbors on the hexagonal lattice:

H0 =⇤t1

⇥

⇤i,j⌅

⇥

�

c†i,�cj,� + t2⇥

⇤⇤i,j⌅⌅

⇥

�

c†i,�cj,�

+t3⇥

⇤⇤⇤i,j⌅⌅⌅

⇥

�

c†i,�cj,� + h.c.⌅� µn, (1)

where n =�

i,� ni,� =�

i,� c†i,�ci,�, and c†i,� denotes the

electron annihilation operator of spin ⇤ =⇥, ⇤ at site i.

FIG. 2. (Color online). (a) Band structure of graphene oncefor t1 = 2.8eV (red) and t1 = 2.8, t2 = 0.7, t3 = 0.02eV(black). (b) Brillouin zone displaying the Fermi surface nearthe van Hove point (dashed blue level in (a), 96 patches usedin the FRG and the nesting vector, and the partial nestingvectors. (c) Density of states for both band structures in (a).The inset show the position shift of Fermi surface nesting(dashed vertical lines) versus the VHS peak.

The resulting band structure is a two band model dueto two atoms per unit cell [Fig. 2]. There are certainuncertainties about the most appropriate tight binding fitfor graphene, in particular as it concerns the longer rangehybridization integrals [1, 22]. For dominant t1, the bandstructure features a van Hove singularity (VHS) at x =3/8, 5/8. Constraining ourselves to the electron-dopedcase, the x = 5/8 electron-like Fermi surface is shownin Fig. 2b. As depicted, this is the regime of largelyenhanced density of states which we investigate in thefollowing. For t2 = t3 = 0 [red curve in Fig. 2], the VHScoincides with the partial nesting of di�erent sections ofthe Fermi surface for Q = (0, 2⇥/

⌅3), (⇥,⇥/

⌅3), and

(⇥,�⇥/⌅3). For a realistic band structure estimate with

finite t2 and t3 [5] [black curve in Fig. 2], this gives arelevant shift of the perfect nesting position versus theVHS as well as density of states at the VHS, and a�ectsthe many-body phase found there.We assume Coulomb interactions represented by a long

range Hubbard Hamiltonian [21]

Hint = U0

⇥

i

ni,�ni,⇥ +1

2U1

⇥

⇤i,j⌅,�,�0

ni,�nj,�0

+1

2U2

⇥

⇤⇤i,j⌅⌅,�,�0

ni,�nj,�0 , (2)

where U0...2 parametrizes the Coulomb repulsion scalefrom onsite to the second nearest neighbor interaction.It depends on the density of states how strongly theCoulomb interaction is screened. At the VHS, we as-sume perfect screening and consider U0 only, while awayfrom the VHS, we investigate the phenomenology of tak-

[1]: ABS and Doniach, PRB 75, 134512 (2007), [2]: Nandkishore et al., Nature Physics 8, 158 (2012), [3]: Kiesel et al., PRB 86, 020507 (2012)

Other d+id Superconductors? •  SrPtAs

•  NaxCoO2 " yH2O

•  β-MNCl

•  κ-(BEDT-TTF)2X

•  (111) bilayer SrIrO3

•  In3Cu2VO9

See e.g. recent review: ABS and Honerkamp JPCM 26, 423201 (2014)

Bulk and Edge Properties •  Fully gapped in the bulk

•  Two chiral (co-propagating) edge states per edge

+

+

- - +

+ -

- +i

!

�!"#

!

!"#

π%�π% k!

left edge states right edge states

ABS, PRL 109, 197001 (2012)

Topological Invariant d+id-wave SC breaks TRS ! Chern number invariant ~

Skyrmion number

Counts the unit sphere area spanned by m as k covers the full BZ Non-zero N if fully covered

N

Bottom of band: m ~ -z Top of band: m ~ z !  Non-zero N only if Δ has finite winding along lines of constant ε%

d+id-wave winds twice around Γ ! |N| = 2

ABS and Honerkamp JPCM 26, 423201 (2014)

Topological Superconductors

Spin-singlet d+id-wave 2D superconductors

Spin-triplet p+ip-wave 2D superconductors

Topological Invariant p(x)+ip(y)-wave spin-triplet (d = (0,0,kx+iky)) very similar to the spin-singlet d+id-wave state

•  Fully gapped in the bulk

•  Break TRS ! finite Chern number/Skyrmion winding

++

-- +i

p(x) +i p(y)

p+ip-wave winds once around Γ ! |N| = 1

! One chiral edge state per edge

Sr2RuO4

Sr2RuO4 similar lattice to cuprates (tetragonal symmetry)

Three quasi-2D Fermi surfaces

(Quasi)-2D tetragonal SC with Tc ~ 1.5 K

Sr2RuO4 Spin-triplet SC?

•  Several related SrRuO compounds are ferromagnetic ! spin-triplet SC (from ferromagnetic fluctuations)?

Experimental results: •  Tc decreases with non-magnetic impurities

•  NMR Knight shift

•  Spin-polarized neutron scattering

•  SQUID

•  muon spin relaxation

•  Kerr effect

All consistent with spin-triplet p+ip-wave state

non-s-wave

spin-triplet pairing

odd-parity pairing

broken time-reversal symmetry

Topological Superconductors

Spin-singlet d+id-wave 2D superconductors

Spin-triplet p+ip-wave 2D superconductors

Fully gapped in the bulk Finite Chern numbers, set by phase winding of Δ Chiral edge states crossing the bulk gap, # set by N Breaks TRS, preserves at least Sz symmetry