1. TOPOLOGICAL QUANTUM COMPUTING/MEMORY
Qubit basis |  , |  
| =  |   +  |  
To preserve, need (for “resting” qubit)
in |  , |   basis
on the other hand, to perform (single-qubit) operations, need to impose nontrivial
we must be able to do something Nature can’t.
(ex: trapped ions: we have laser, Nature doesn’t!)
Topological protection:
would like to find d–(>1) dimensional Hilbert space within which (in absence of intervention)
size of system
microscopic length
How to find degeneracy?
Suppose  two operators

2. EXAMPLE OF TOPOLOGICALLY PROTECTED STATE: FQH SYSTEM ON TORUS (Wen and Niu, PR B 41, 9377 (1990))
Reminders regarding QHE:
2D system of electrons, B  plane
Area per flux quantum = (h/eB)  df.
“Filling fraction”  no. of electrons/flux quantum  
“FQH” when  = p/q incommensurate integers
Argument for degeneracy: (does not need knowledge of w.f.)
can define operators of “magnetic translations”
 “magnetic length”
( translations of all electrons through a(b)  appropriate phase factors). In general
In particular, if we choose no. of flux quanta
then commute with b.c.’s (?) and moreover
But the o. of m. of a and b is M·(M /L)osc «M , and  0 for L. Hence to a very good approximation,
must  more than 1 GS (actually q).
Corrections to (*): suppose typical range of (e.g.) external potential V(r) is o, then since |>’s oscillate on scale osc,

3. *plus gap for anyon creation
TOPOLOGICAL PROTECTION AND ANYONS
Anyons (df): exist only in 2D
(bosons:  = 1, fermions:  = ½)
abelian if
nonabelian if 1 2 3
Nonabelian statistics* is a sufficient condition for topological protection: (a) state containing n anyons, n  3:
[not necessary, cf. FQHE on torus]
space must be more than 1D.
(b) groundstate: GS GS
create anyons
annihilate anyons
annihilation process inverse of creation 
*plus gap for anyon creation
GS also degenerate.
Nonabelian statistics may (depending on type) be adequate for (partially or wholly) topologically protected quantum computation

4. p-WAVE FERMI SUPERFLUIDS (in 2D)
Generically, particle-conserving wave function of a Fermi superfluid (Cooper-paired system) is of form
e.g. in BCS superconductor
Consider the case of pairing in a spin triplet, p-wave state (e.g. 3He-A). If we neglect coherence between  and  spins, can write
Concentrate on and redef. N 2N.
suppress spin index
What is ck? KE measured from µ
Standard choice:
real factor
“p+ip”
How does ck behave for k0? For p-wave symmetry, |k| must  k, so
Thus the (2D) Fournier transform of ck is 
and the MBWF has the form
uninteresting factors

5. z-component of ang. momentum
Conclusion: apart from the “single-particle” factor MR ansatz for  = 5/2 QHE is identical to the “standard” real-space MBWF of a (p + ip) 2D Fermi superfluid. Note one feature of the latter: if
then so
z-component of ang. momentum
possesses ang. momentum –N/2, no matter how weak the pairing!
Now: where are the nonabelian anyons in the p + ip Fermi superfluid?
Read and Green (Phys. Rev. B 61, 10217(2000)):
nonabelian anyons are zero-energy fermions bound to cores of vortices.
Consider for the moment a single-component 2D Fermi superfluid, with p + ip pairing. Just like a BCS (s-wave) superconductor, it can sustain vortices: near a vortex the pair wf, or equivalently the gap (R), is given by
COM of Cooper pairs
Since | (R)|2  0 for R  0, and (crudely) bound states can exist in core. In the s-wave case their energy is ~ |o|2 F,   0, so no zero-energy bound states. What about the case of (p + ip) pairing? If we approximate
 mode with u(r) = v*(r), E = 0

6. zero-energy modes are their own antiparticles (“Majorana modes”)
Now, recall that in general
zero-energy modes are their own antiparticles (“Majorana modes”) :
This is true only for spinless particle/pairing of || spins (for pairing of anti || spins, particle and hole distinguished by spin).
Consider two vortices i, j with attached Majorana modes with creation ops.
What happens if two vortices are interchanged?*
Claim: when phase of C. pairs changes by 2, phase of Majorana mode changes by  (true for assumed form of u, v for single vortex). So
more generally, if  many vortices + w df as exchanging i, i + 1, then for |i–j|>1
for |i–j|=1,
braid group!  =
* Ivanov, PRL 86, 268 (2001)

7. How to implement all this?
(a) superfluid 3He-A: to a first approximation,
so prima facie suitable.
Ordinary vortices ( (r) ~  (r) ~ z) well known to occur. Will they do?
Literature mostly postulates half-quantum vortex ( (r) ~z,  (r) = const., i.e. vortex in  spins, none in )
HQV’s should be stable in 3He-A under appropriate conditions (e.g. annular geom., rotation at  ~ c/2, c  /2mR2) sought but not found:
? ?
Additionally, would need a thin slab (how thin?) for it to count as “2D”.
How would we manipulate vortices/quasiparticles (neutral) in 3He-A?
What about charged case (p + ip superconductor)?
Ideally, would like 2D superconductor with pairing in (p + ip) state. Does such exist?

8. STRONTIUM RUTHENATE (Sr2RuO4)*
Strongly layered structure, anal. cuprates  hopefully sufficiently “2D.” Superconducting with Tc ~ 1.5 K, good type-II props. ( “ordinary” vortices certainly exist).
$64 K question: is pairing spin triplet (p + ip)? Much evidence* both for spin triplet and for odd parity (“p not s”).
Evidence for broken T-reversal symmetry: optical rotation (Xia et al. (Stanford), 2006) Josephson noise (Kidwingira et al. (UIUC), 2006)
“topology” of orbital pair w.f. probably (px + ipy).
Can we generate HQV’s in Sr2RuO4?
Problem: in neutral system, both ordinary and HQ vortices have 1/r flow at . HQV’s not specially disadvantaged. In charged system (metallic superconductor), ordinary vortices have flow completely screened out for r  L by Meissner effect. For HQV’s, this is not true:
London penetration depth L L
So HQV’s intrinsically disadvantaged in Sr2RuO4.
*Mackenzie and Maeno, Rev. Mod. Phys. 75, 688 (2003)

9. Problems:
Is Sr2RuO4 really a (p + ip) superconductor? If so, is single-particle bulk energy gap nonzero everywhere on F.S.? Even if so, does large counterflow energy of HQV mean it is never stable?
Non-observation of HQV’s in 3He-A: Consider thin annulus rotating at ang. velocity , and df At exactly, the nonrotating state and the ordinary “vortex” (p-state) with both spins rotating are degenerate.
But a simple variational argument shows that barring pathology, there exists a nonzero range of  close to where the HQV is more stable than either! In a simply connected flat-disk geometry, argument is not rigorous but still plausible.  R
HQV
Yamashita et al. (2008) do experiment in flat-disk geometry, find NO EVIDENCE for HQV!
Possible explanations:
HQV is never stable (Kawakami et al., preprint, Oct 08)
HQV did occur, but NMR detection technique insensitive to it.
HQV is thermodynamically stable, but inaccessible in experiment.
Nature does not like HQV’s. :

10. * D. A. Ivanov, PRL 86, 268 (2001)
‡ Stone & Chung, Phys. Rev. B 73, (2006)

12. HQV1
HQV2
Bog. qp.

13. ⇒ ⇒ An intuitive way of generating MF’s in the KQW:
Kitaev quantum wire ⇒ ⇒

14. Comments on M.F.’s (within standard mean-field approach) (cont.)
(2) The experimental situation
Sr2RuO4: so far, evidence for HQV’s, none for MF’s.
3He-B: circumstantial evidence from ultrasound attenuation
Alternative proposed setup (very schematic)
s-wave
supr. S N S
MF1
MF2
induced supr.
zero-bias anomaly
Detection: ZBA in I-V characteristics
(Mourik et al., 2012, and several subsequent experiments)
dependence on magnetic field, s-wave gap, temperature... roughly right
“What else could it be?”
Answer: quite a few things!

15. Second possibility: Josephson circuit involving induced (p-wave-like) supy.
Theoretical prediction: “4-periodicity” in current-phase relation.
Problem: parasitic one-particle effects can mimic.
One possible smoking gun: teleportation! L e
MF1
MF2
Fermi velocity
Problem: theorists can’t agree on whether teleportation is for real!

16. Majorana fermions: beyond the mean-field approach
Problem: The whole apparatus of mean-field theory rests fundamentally on the notion of SBU(1)S  spontaneously broken U(1) gauge symmetry: *
But in real life condensed-matter physics,
SB U(1)S IS A MYTH!!
This doesn’t matter for the even-parity GS, because of “Anderson trick”:
But for odd-parity states equation ( * ) is fatal! Examples:
(1) Galilean invariance
(2) NMR of surface MF in 3He-B

17. We must replace ( * ) by
creates extra Cooper pairs
Need completely new approach!

18. Main theses of this talk:
For 50 years, almost all theoretical work on inhomogenous Fermi superfluids, including work on “topological superconductors” has been based on the solution of the BdG (mean-field) Hamiltonian, which in turn is justified by appeal to the idea of “spontaneously broken U(1) gauge symmetry” (SBGS)
There is no physical justification for the idea of SBGS, and hence none for the use of the BdG Hamiltonian (at least without considerable caution).
Moreover, simple examples show that it can lead to physically incorrect conclusions.
This is because in the cases of interest the response of the Cooper pairs cannot be ignored.
This consideration drastically affects arguments about the effects of braiding of Majorana fermions, etc. (see poster): it may also affect arguments about their undetectability by local probes.
If so, this could be a disaster for the whole program of using topological superconductors such as Sr2RuO4 for topological quantum computing. (Or, it might help...)
(much of this (1) – (4)) could have been said in 1958! Was it?)

19. L
s-wave energy gap
C. pair radius
*Y. R. Lin and AJL, JETP 119, 1034 (2014)

21. Ans. no. (1): Particle no. is not conserved (e.g. leads) 
Fine for even-parity states, but what about odd-parity ones? Problem is relation of odd-parity states to even-parity ones.

22. B
Brf wL w→

23. Thus the added momentum is
Failure of BdG/MF: a futher trivial (but apparently not well-known) example:
total momentum
standard textbook result
Thus the added momentum is

24. However, by Galilean invariance, for any given condensate number N,
However, by Galilean invariance, for any given condensate number N,
and this result is independent of N (so invoking “spontaneous breaking of U(1) symmetry” in GS doesn’t help!)
So:
Galilean invarance
creates extra Cooper pair (with COM velocity 0)
in accordance with GI argument