Topological spin Hall states, charged skyrmions, and superconductivity in two dimensions
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Topological spin Hall states, charged skyrmions, and superconductivity in twodimensions
Tarun Grover and T. Senthil
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (Dated: May 27, 2018)We study the properties of two dimensional topological spin hall insulators which arise throughspontaneous breakdown of spin symmetry in systems that are spin rotation invariant. Such aphase breaks spin rotation but not time reversal symmetry and has a vector order parameter.Skyrmion configurations in this vector order parameter are shown to have electric charge that istwice the electron charge. When the spin Hall order is destroyed by condensation of skyrmionssuperconductivity results. This may happen either through doping or at fixed filling by tuninginteractions to close the skyrmion gap. In the latter case the superconductor- spin Hall insulatorquantum phase transition can be second order even though the two phases break distinct symmetries.
PACS numbers: 75.10.Jm, 71.27.+a, 75.30.Kz
Several recent theoretical and experimental pa-pers have discussed a phenomenon known as the quan-tum spin Hall effect in two dimensional insulators. Insuch an insulator an applied electric field leads to a quan-tized spin current in the transverse direction. Clearlythe effect cannot occur if SU (2) spin rotation symmetryis preserved in the low energy theory of the insulator.Initial discussions of the spin Hall effect focused on situ-ations where the microscopic Hamiltonian has spin-orbitinteractions which are such that only the S z componentof the spin is conserved. Alternately one can contem-plate phases of matter where full SU (2) spin rotationsymmetry is present at the microscopic level but is spon-taneously broken in the ground state in a manner thatenables a spin Hall effect. Such phases were discussedrecently in Ref. 5.In this paper we study various aspects of the sec-ond kind of spin Hall insulator where the spin symme-try is spontaneously broken. The broken spin symmetryis characterized by a vector order parameter which wewill denote ~N . Consequently in two space dimensionsskyrmion topological defects are allowed. We show thatthe quantized spin Hall effect leads (with some restric-tions discussed below) to a quantized electric charge 2 e on the skyrmion. Here e is the electron charge. Thishas some remarkable consequences. First so long asthe skyrmion energy is much smaller than the energy toexcite individual pairs of electrons, the skyrmion num-ber will be conserved due to charge conservation. Thusspace-time hedgehog configurations of ~N are forbidden .Destruction of the spin Hall order by condensation ofskyrmions then very simply leads to a gapped s -wave su-perconductor. This may be done either by doping intothe spin Hall insulator or by closing the skyrmion gap atfixed density by tuning interactions. In the latter casethe resulting superconductor - spin Hall insulator phasetransition can be second order despite their rather differ-ent broken symmetries. This provides a new example ofa Landau-forbidden deconfined quantum critical point .These results are best illustrated in the context of mod-els of of spinful electrons c r σ hopping on a honeycomb lattice at half-filling: H = X < rr ′ > − t ( c † r c r ′ + h.c. ) + H int , (1)where r , r ′ are nearest neighbor sites. H int contains var-ious short-ranged interactions that preserve spin rota-tion and time reversal symmetries. Specific interactionsthat could stabilize the spin Hall insulating phase arediscussed in Ref. 5. In the absence of H int , the band-structure consists of two distinct Fermi points which canbe chosen as K = ( ± π/ √ ,
0) where the lattice con-stant is chosen as unity. Expanding the microscopic elec-tron annihilation operator c r about these nodes in termsof continuum field operators c iασ the low energy DiracHamiltonian is obtained as H = − t Z d k c † ( − τ x µ z k x + τ y k y ) c (2)Here i = L/R is a sublattice index, α = + / − the nodeindex, and σ the spin. The corresponding Pauli matricesare denoted τ , µ and σ respectively. Further the trans-formation c + = τ y ψ + enables writing the the Hamilto-nian in a manifestly symmetric form H = − t Z d k ψ † ( k x τ x + k y τ y ) ψ (3)The corresponding action (in real space) is S = Z d xψ ( − iγ µ ∂ µ ) ψ (4)Here µ = τ, x, y with τ being imaginary time, ψ = − iψ † τ z and γ , γ x , γ y = τ z , τ y , − τ x . We have rescaledspace-time to set the Dirac velocity to 1.Consider a phase where the interactions H int leads toa non-zero expectation value for the operator ψ σ ψ , sayalong the ˆ z direction in the spin-space: < ψ σ ψ > ≡ ~N = N z ˆ z (5)Within a mean field description this leads to a mass forthe low energy Dirac fermions which is opposite for ↑ and ↓ spin: S MF = Z d xψ ( − iγ µ ∂ µ + imσ z ) ψ (6)where the mass m = λN z with λ determined by the in-teractions that lead to this order. At the mean field levelthis is identical to the model discussed in Ref. 1 for thequantum spin hall effect. It breaks spin rotation but nottime reversal symmetry. In a sample with a boundary, anelectric potential difference V applied between the rightand left edge leads to a pair of propagating edge stateswith opposite spin . Each spin species carries a non-vanishing charge current of magnitude e h V transverse tothe electric field in opposite directions. This correspondsto a spin-hall conductivity σ sxy = e h (cid:16) ¯ h/ e (cid:17) = e/ π .It is useful to characterize the spin hall effect in termsof the response to two different external gauge fields A c and A s which couple to spin and charge currents respec-tively. Consider the mean field action in the presence ofthese gauge fields S MF [ A c , A s ] = Z d x ψ ( γ µ ( − i∂ µ + A cµ + σ z A sµ )) ψ + imψσ z ψ (7)in the units e = ¯ h = 1. When the fermions are integratedout, a non-vanishing transverse spin-hall conductivity ofmagnitude 1 / π implies that the low energy effective ac-tion for the gauge fields is given by S eff = i π Z d x ǫ µνλ A cµ ∂ ν A sλ (8)Consider now the effect of fluctuations. These may beusefully discussed by considering the Dirac action in thepresence of a fluctuating unit vector field ˆ N describingthe orientation of the spin Hall order parameter: S = Z d x ψ ( − iγ µ ∂ µ + im~σ. ˆ N ) ψ (9)We may now integrate out the fermions. In the limit oflarge mass m the result is a non-linear sigma model: S = Z d x g (cid:16) ∂ µ ˆ N (cid:17) + .... (10)with ‘stiffness’ g ∼ | m | . The ellipses represent higher or-der terms in the 1 /m expansion. Clearly in the orderedphase there will be two gapless linear dispersing Gold-stone modes associated with the broken spin symmetry.The other class of excitations associated with the vec-tor order parameter ˆ N are skyrmion configurations. Thequantum numbers of a skyrmion whose size is muchbigger than the length scale 1 /m can be convenientlydiscussed within an adiabatic approach where such askyrmion is slowly built up from the ground state. We now show that such a ‘fat’ skyrmion has electric charge2. To see this most simply, lets consider the Dirac actionEqn. 9 in the presence of a static background configura-tion of the ˆ N field that corresponds to a single skyrmion.An example is the configurationˆ N ( r, θ ) = ( sin ( α ( r )) cos ( θ ) , sin ( α ( r )) sin ( θ ) , cos ( α ( r )))(11)with the boundary conditions α ( r = 0) = 0 and α ( r →∞ ) = π . Here ( r, θ ) are polar coordinates for two dimen-sional space. This field configuration corresponds to askyrmion with Pontryagin index one. Rotate the ˆ N vec-tor field to the ˆ z direction by a unitary transformation U at all points in space: U † (cid:16) ~σ. ˆ N (cid:17) U = σ z . Further, onemay define ψ = U ψ ′ so that Eqn. 9 becomes S = Z d x ψ ′ ( γ µ ( − i∂ µ + B µ ) + imσ z ) ψ ′ (12)where B µ = − iU † ∂ µ U . One readily finds that in thefar field limit ~B ( r → ∞ , θ ) → σ z ˆ e θ r where ˆ e θ is the unitvector along θ direction. Therefore a skyrmion with unitPontryagin index induces a spin gauge field ~A s = 2ˆ e θ /r .The total flux of this gauge field is 4 π which remains in-variant under smooth deformations of the ˆ N field (whichdoes not change the skyrmion number).Let us gradually build in a skyrmion configuration bystarting from the ground state. By the above argument,this threads in a spin gauge flux of 4 π for A s . Due to thespin-hall effect of the mean field state described above,this would result in a flow of electric current of magni-tude j c = π ∂A s ∂t in the radial direction. The total chargetransferred from the center of the sample to the boundaryduring the process equals Q c = R C j c rdθ where the con-tour C is a circle located near infinity. This then leads toa charge Q c = 2 e associated with the skyrmion. Thus askyrmion with Pontryagin index one carries an electricalcharge e .This physical argument is confirmed by a more sophis-ticated analysis which carefully integrates out the Diracfermion fields in the presence of an external charge gaugefield. Specifically we consider S [ A c ] = Z d x ψ ( γ µ (cid:0) − i∂ µ + A cµ (cid:1) ψ + imψ~σ. ~N ) ψ (13)Integrating out the fermions and using the large massexpansion yields S = Z d x g ( ∂ µ N ) + 2 iA cµ J Tµ (14)where J Tµ = π ǫ µνλ ~N.∂ ν ~N × ∂ λ ~N is the topological cur-rent whose time component equals the skyrmion densityassociated with the order parameter field . It followsthat the skyrmion carries charge which is given by Q cskyrmion = 2 e Z dxdy π ǫ νλ ~N .∂ ν ~N × ∂ λ ~N (15)Thus one again reaches the conclusion that a skyrmionwith Pontryagin index 1 carries an electric charge two.It is important to emphasize that both the adiabaticargument or the field theoretic derivation above is reallyvalid only for fat skyrmions with size much bigger than1 /m . It is precisely such fat skyrmions that are impor-tant for our considerations below.The electric charge of the skyrmions has profound im-plications for quantum phase transitions out of the spinhall insulator. First if the energy of a fat skyrmion ismuch smaller than the energy of individual pairs of elec-trons, then the skyrmion number is conserved. Thushedgehog configurations of the ˆ N vector in spacetime(which correspond to events that change the Pontrya-gin index) are prohibited. Further in the absence of anytopological terms in the effective action for the ˆ N field,the skyrmions will be bosons . Consider therefore theresult of condensing the skyrmions by tuning the inter-actions in H int while keeping the electron density fixed.Due to the charge on the skyrmions the result will be asuperconductor! This superconductor will have a singleelectron gap, and flux quantization in units of hc e . Indeedit is an ordinary s -wave superconductor.Remarkably the quantum phase transition between thespin hall insulator and the superconductor can be secondorder. This is despite the rather different broken sym-metries in the two phases. This is because the skyrmioncondensation transition can simply be understood as thephase transition of an O (3) vector model in the absenceof hedgehogs. This is conveniently described in the CP formulation of the ˆ N field by writing ˆ N = z † ~σz field with z a two component complex spinon field: S eff = 1 g Z d x | ( − i∂ µ − a µ ) z | (16)Here z = ( z , z ) are coupled to an internal gauge field a µ = − iz † ∂ µ z . The skyrmion number Q is simply givenby Q = 12 π Z d xǫ µν ∂ µ a ν (17)Since the skyrmion number is proportional to the flux ofthe CP gauge field, a µ of Eqn. 16 must be regardedas non-compact, and the phase transition is described bythat in the non-compact CP model ( N CCP ).Exactly the same field theory describes the deconfinedquantum critical point separating Neel and Valence BondSolid (VBS) phases of insulating square lattice quantumantiferromagnets . However there is an important differ-ence: for the Neel-VBS transition quadrupled hedgehogsare actually allowed but have been argued to be irrelevantat the critical fixed point of the N CCP model. They arehowever relevant at the paramagnetic ‘free photon’ fixedpoint. This leads to the appearance of two diverginglength scales at the Neel-VBS transition. In contrast inthe present problem the skyrmion number conservation is an exact symmetry at energy scales below the singleelectron gap which is large and finite in both phases andat the transition. Thus a suitable theory that capturesthe low energy physics of both phases and their transitionshould simply ignore all hedgehogs. Consequently thereis only a single divergent length/energy scale at the de-confined critical point separating the spin hall insulatorfrom the superconductor.The competition between the spin Hall order andthe superconducting order can also be fruitfully de-scribed in a different manner which sheds further lighton the role played by the quantum nature of the topo-logical defects near the phase transition. We com-bine the three component spin Hall order parameter ~N with the two component superconducting order pa-rameter ψ SC into a five component ‘superspin’ vector φ a = ( N x , N y , N z , Reψ SC , Imψ SC ), and impose a unitlength constraint on φ a , i.e P a =1 φ a φ a = 1. What is thestructure of a ‘sigma model’ that describes the dynamicsof φ a ? This question may again be answered by couplingthe underlying Dirac theory to this superspin vector andintegrating out the fermions. Specifically we consider theDirac action S = Z d xψ ( − iγ µ ∂ µ + im~σ. ~N ) ψ + im ( ψ ∗ SC ψ T τ y σ y µ y ψ + c.c )(18)where the last term indicates the coupling to the super-conducting order parameter ψ ∗ SC ∝ (cid:10) ψ T τ y σ y µ y ψ (cid:11) . It ispotent to make a unitary transformation on the fermionsin the ψ basis as e ψ = e iτ x π/ ψ and expand e ψ in its Ma-jorana components e ψ = η + iη . In the η basis thekinetic part of the Dirac action has manifest SO (8) sym-metry under rotations of all 8 components (2 spin, 2 nodeand 2 Majorona indices). The spin-hall order parame-ter is given as (cid:8) η T τ y σ x η, η T τ y σ y ρ y η, η T τ y σ z η (cid:9) while thes-wave SC has the form (cid:8) η T τ y σ y µ y ρ x η, η T τ y σ y µ y ρ z η (cid:9) .Here ρ are Pauli matrices acting on the Majorana in-dex. These five components together transform as a vec-tor under an SO (5) subgroup of SO (8). This algebraicstructure largely determines important features of theeffective sigma model that describes the fluctuations ofthe φ a when the fermions are integrated out. Within thelarge mass expansion the techniques of Ref. 8 lead to an SO (5) action with a Wess-Zumino-Witten(WZW) termat level-1. S = Z d x G ( ∂ i φ a ) + 2 πi Γ h ~φ i (19)The WZW term Γ is defined as follows. The field φ a defines a map from spacetime S to S . Γ is the ratio ofthe volume in S traced out by φ a to the total volumeof S . Specifically let ˆ φ ( x, u ) be any smooth extension ofˆ φ ( x ) such that ˆ φ ( x,
0) = (1 , , , ,
0) and ˆ φ ( x,
1) = ˆ φ ( x ).ThenΓ = ǫ abcde Area( S ) Z d u Z d xφ a ∂ x φ b ∂ y φ c ∂ τ φ d ∂ u φ e (20)The presence of this ‘topological term’ crucially alters thephysics from naive Landau-like descriptions of the com-petition between the two different orders. Parentheticallywe note that the action above must be supplemented withsome anisotropy between the ~N and ψ SC which breaksthe SO (5) symmetry down to SO (3) × U (1). The SO (3)corresponds to spin rotations and the U (1) to charge con-servation symmetry. Exactly the same sigma model withthe WZW term also describes the Neel-VBS transi-tion of insulating quantum magnets in agreement withour previous identification.The physical implications of the WZW term arebrought out by the following instructive calculation.Consider a hc/ e vortex defect in the superconductingstate. In such a configuration the 5-component unit vec-tor will point along the 4 , , , r, θ ) for two dimensional space and τ for time)ˆ φ ( r, θ, τ, u ) = ( sin ( α ( r, u )) ˆ N ( τ, u ) ,cos ( α ( r, u )) cos ( θ ) , cos ( α ( r, u )) sin ( θ ))(21)with ˆ N = 1. Choosing α ( r = 0 , u ) = π/ , α ( r = ∞ , u ) = 0 ∀ u = 0 and α ( r,
0) = π/
2, this configurationindeed describes a static superconducting vortex where ˆ φ points along the ~N direction in the core. We have allowedthe unit vector ˆ N in the core to have time dependence.The integrals defining Γ are readily evaluated for thisconfiguration and lead to the resultΓ = 14 π Z dτ du ˆ N .∂ τ ˆ N × ∂ u ˆ N (22)This is precisely the quantum Berry phase for a spin-1 / close to the transition the super-conducting vortices behave as spinons. Condensing thevortices destroys the superconducting order but at thesame time leads to condensation of the spin Hall order.This is the mechanism for the Landau-forbidden transi-tion. The arguments of Ref. 14 now establish the equiva-lence of the sigma model to the N CCP field theory andthe superconducting vortices are directly identified withthe CP spinons z .Finally we consider the result of doping the spin Hallinsulator focusing again on the situation where the en-ergy of a fat skyrmion is much smaller than the en-ergy gap of individual electrons. Then the doping willbe accommodated by the introduction of a finite den-sity of skyrmions into the system. At very low dop-ing a skyrmion crystal will presumably be stabilized butwith increasing doping a translation invariant skyrmioncondensate is expected. This will be a superconductingphase with no broken spin rotation symmetry.The route to superconductivity discussed in this paperis rather different from two common theoretical mech-anisms - phonon or other boson mediated pairing ofelectrons, or resonating valence bond pairing originat-ing from superexchange in a Mott insulator. Rather itinvolves condensation of charged solitons of a differentbroken symmetry, namely the spin Hall order. Similarphenomena have previously been considered in field the-oretic contexts , and is known as topological supercon-ductivity. Superconductivity arising from the quantumspin Hall state thus provides a nice realization of a certainkind of topological mechanism for superconductivity.This work was supported by NSF Grant DMR-0705255. C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801(2005) C. L. Kane, E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005). B. A. Bernevig and S. C. Zhang, Phys. Rev. Lett. 96,106802 (2006) M. Konig et al, Science 318, 766 (2007) S. Raghu et al, arXiv:0710.0030 Similar ideas are independently being pursued by others inthe context of bilayer quantum Hall systems (G. Murthy,private communication). T. Senthil et al, Science 303, 1490 (2004); T. Senthil, et al,Phys. Rev. B , 144407 (2004) A. G. Abanov, P. B. Wiegmann, Nucl.Phys. B 570, 685(2000) A. G. Abanov, P. B. Wiegmann, JHEP 0110, 030 (2001) Note that a Hopf term will also be generated in the largemass expansion; however its coefficient is θ = πN f where N f is the number of species of fermions and N f = 2 here.It can therefore be dropped for the subsequent discussion.In particular the skyrmion statistics will be bosonic. O. I. Motrunich and A. Vishwanath, Phys. Rev. B 70,075104 (2004) A. Tanaka and X. Hu, Phys. Rev. Lett. 95, 036402 (2005) T. Senthil and M. P. A. Fisher, Phys. Rev. B 74, 064405(2006) M. Levin and T. Senthil, Phys. Rev. B 70, 220403 (2004)15