Presentation on theme: "Fractionalization in condensed matter systems (pre-Majorana days)"— Presentation transcript:
1 Fractionalization in condensed matter systems (pre-Majorana days) K. SenguptaDepartment of Theoretical Physics, IACS, Kolkata
2 Outline Fermion number fractionalization in 1+1 D field theory Finite sized system with fixed particle numberEdge states in p-wave superconductors: an exact solutionExperimental signature: tunneling conductanceFractional Josephson effect: Theory and experimentConclusion
4 Fermion number fractionalization in field theory R.Jackiw and C.Rebbi,Phys. Rev. D 13, 3398 (1976).Rajaraman, cond-mat/1+1 D coupled field theoryof fermions and bosonsThe bosonic sector in theabsence of the fermionsVacuum sector solution:Soliton sector solution:The soliton sector solution or the kink can not spontaneously decay intothe vacuum sector; kinks and anti-kinks can however annihilate each other
5 Fate of Fermions: Vacuum sector The Lagrangian in the vacuum sectorThis leads to the standardDirac equations in 1+1 DUsual construction of the FermionOperator in terms of bk and dkA consequence of this constriction is that the number(charge) operator always has integer eigenvaluesQ must be an integer.Note that all factors of ½ cancel due tothe existence of paired energy modes
6 Fate of Fermions: Soliton sector The Dirac equation now becomesApart from the standard paired positiveand negative energy solutions withenergies Ek and –Ek, there is an unpairedlocalized zero energy mode which is itsown charge conjugateThe Fermionic operatornow becomesThere are two degenerate groundstates related to the existence ofthe zero energy stateThese degenerate ground states are distinguished by their charge quantum number
7 Number fractionalization Number operator now has fractional eigenvalues due to the presence of the bound statesTwo degenerate ground stateshave different eigenvalues ofnumber operators.First example of Fermion numberfractionalization arising fromdegeneracy.What happens in a real finite solid state sample with N electrons?
8 Finite size version of the J-R solution Imagine that the 1+1D field theoryis put in a finite size 2L with theperiodic boundary conditionIt turns out that there are now two zero energy states at x=0 and LLocalized at the originLocalized at one of the edgesThe Fermion field now becomes
9 There are now four degenerate ground states which correspond to zero or unit filling of a or c quasiparticlesThere is no fractionalization of the total number: the theory is therefore compatiblewith integer number of electronsThe effect of fractionalization can still be seen by local probes which will pick upsignatures from one of the two states at zero energy.Key concept in understanding fractionalization in condensed matter systems
11 Pair Potential Mean-field potential due to pairing of electrons Center of MasscoordinateDirection of spin(for triplet pairing)Relative coordinateVariation around theFermi surfaceGlobal phasefactorDirection of spin(for triplets only)
12 The edge problemConsider a semi-infinite sample occupying x>0having an impenetrable edge at x=0Upon reflection from such an edge, theBdG quasiparticles gets reflected from LTo R on the Fermi surfaceThe right and the left moving quasiparticlessee opposite sign of the pair-potentialThe BdG wavefunction is superposition of the left and the right moving quasiparticlesThe boundary condition for the impenetrable edge
13 Triplet Superonductivity in TMTSF The pair potential for tripletsuperconductivity is given byThe BdG equation for the quasiparticles is given byExperimental inputs suggests that d is a real vector pointing along a; we chooseour spin quantization along d leading to opposite spin-pairing.These are described by a2 component matrixequationis determined by the self-consistencycondition in terms un and vn
14 Exact solution1. Extend the wavefunction from positive semispace to the full space using the mappingx > 0x < 02. This leads to a single BdGequation defined for all x3. The boundary condition of the edge problem translates to continuity of u and v at x = 04. For p-wave, D(x) changes sign at the origin and the problem is exactly mapped onto the1D CDW problem solved by SSH and Brazovski (JETP 1980)5. This allows us to writedown the exact self-consistentsolution for the edge problem
15 Properties and spin response of the edge states The edge states carry zero net chargeThe edge states with momenta kyand –ky are identicalThey have half the number of modes andthus have fractional eigenvaluesThere is one Fermion state foreach (ky,-ky) pair per spinIn the presence of a Zeeman field, one generatea magnetic field of mB/2 per chain end. This isformally equivalent to having Sz=h/4 for these states.These states would be Majorana Fermions in 1D and for spinless (or spin-polarized)Fermions with equal-spin pairing ( Kitaev proposal)
16 Experiments: How to look for edge states eVN-I-N interfaceNormal metal (N)Normal metal (N)Measurement oftunneling conductanceInsulator (I)eVN-I-S interfaceNormal metal (N)Superconductor (S)Insulator (I)
17 NISStrongly suppressed if theinsulating layer provides alarge potential barrier: socalled tunneling limitBasic mechanism of currentflow in a N-I-S junctionAndreev reflection2e charge transferIn the tunneling limit, the tunneling conductancecarries information about the density ofquasiparticle states in a superconductor.Edges with no Midgap StatesEdges with Midgap States
18 Typical tunneling conductance Curves G(E=eV)G(E=eV)Edge withmidgap states.Edge withoutmidgap states.
19 Experiments for cuprates and TMTSF Covington et.al. 1997, Krupkeand Deuscher 1999 ………Naughton et al., unpublishedmevData from Cucolo et al, 2000.Tunneling in a-b plane in YBCOUnpublished data from Naughton et.al
21 Josephson Effect S1 S2 The ground state wavefunctions have different phases for S1 and S2Thus one might expect a currentbetween them: DC Josephson EffectExperiments: Josephson junctions [Likharev, RMP 1979]S1S2S1BS2NS-N-S junctions or weak linksS-B-S or tunnel junctions
22 Josephson effect in conventional tunnel junctions BS2Formation of localized subgapAndreev bound states at thebarrier with energy dispersionwhich depends on the phasedifference of the superconductors.The primary contributionto Josephson current comesfrom these bound states.Kulik-Omelyanchuk limit:Ambegaokar-Baratoff limit:Both Ic and IcRN monotonically decrease as we go from KO to AB limit.
23 Andreev bound states in Josephson junctions Consider two p-wave superconductorsSeparated by a barrier modeled by alocal potential of strength U0 forming aJosephson tunnel unctionLBRb= R,L and s denotes spinThe superconductors acquire a phasedifference f across the junctionSolve the BdG equation across the junction with the boundarycondition and find the subgap localized Andreev bound states
24 Solution for the Andreev states On each side try a solution which is asuperposition of right and left movingquasiparticles (index a denotes + or –for right or left movers) with momentaclose to kFSubstitute expressions for v and uin the boundary condition and demandnon-zero solutions for Ab and BbLeads to 4p periodic JosephsonCurrent for p-p junctionsFractional AC Josephson effect
25 Tunneling Hamiltonian approach Consider two uncoupled 1D superconductors(corresponds to D=0) with two midgap statesfor each transverse momentaThus the projection of the electron operatoron the midgap state is given byNow consider turning on a tunnelingHamiltonian between the left and theright superconductorA little bit of algebra yields theEffective tunneling HamiltonianFor the subgap statesThe tunneling matrix elements vanish at f=p where the states cross
26 Recent experiments on doubling of Shapiro steps Recent experiments in 1DSemconductor wires with proximityinduced superconductivityDoubling of first Shapiro step fromhn/2e to hn/e for B > 2 T.Rokhinson et alNat. Phys (2012)