Charge- 2e Skyrmion condensate in a hidden order state
aa r X i v : . [ c ond - m a t . s t r- e l ] D ec Charge- e Skyrmion condensate in a hidden order state
Chen-Hsuan Hsu and Sudip Chakravarty
Department of Physics and Astronomy,University of California Los AngelesLos Angeles, California 90095-1547 (Dated: August 28, 2018)A higher angular momentum ( ℓ = 2) d -density wave, a mixed triplet and a singlet, interestingly,admits skyrmionic textures. The Skyrmions carry charge 2 e and can condense into a spin-singlet s -wave superconducting state. In addition, a charge current can be induced by a time-dependentinhomogeneous spin texture, leading to quantized charge pumping. The quantum phase transitionbetween this mixed triplet d -density wave and skyrmionic superconducting condensate likely leadsto deconfined quantum critical points. We suggest connections of this exotic state to electronicmaterials that are strongly correlated, such as the heavy fermion URu Si . At the very least, weprovide a concrete example in which topological order and broken symmetry are intertwined, whichcan give rise to non-BCS superconductivity. I. INTRODUCTION
It has become very much in vogue to argue that topo-logical aspects of condensed matter bear no relation tobroken symmetries. In a strict sense this need not be so. One can construct examples where a broken symmetrystate has interesting topological properties and can evenbe protected by the broken symmetry itself. An inter-esting example of a mixed triplet d -density wave and itspossible relevance to one of the many competing phases inthe high temperature cuprate phase digram was recentlydemonstrated, where it was found that the system ex-hibits quantized spin Hall effect even without any explicitspin-orbit coupling. In the present paper we show thatthe system exhibits charge 2 e skyrmions, which can con-dense into a remarkable superconducting state. As weshall discuss, such a mixed triplet d -density wave systemand the resulting superconductivity is potentially rele-vant to the heavy fermion URu Si with hidden order. An early attempt at such a non-BCS mechanism of su-perconductivity was made by Wiegmann, as an exten-sion of Fr¨ohlich mechanism to higher dimension. Morerecently, several interesting papers have led to discussionsof superconductivity in single and bilayer graphenes.Grover and Senthil have provided a mechanism in whichelectrons hopping on a honeycomb lattice can lead toa charge-2 e skyrmionic condensate, possibly relevant tosingle layer graphene. To a certain degree we follow theirformalism; see also the earlier work in Ref. 7 of charge- e skyrmions in a quantum Hall ferromagnet. As to bilayergraphene, a charge-4 e skyrmionic condensate has beensuggested by Lu and Herbut and Moon. In the presentpaper we base our results, instead, on an unusual sponta-neously broken symmetry generated by electron-electroninteraction, not by a given non-interacting band structureof a material, namely the mixed triplet-singlet densitywave state of angular momentum ℓ = 2, and we point outits possible implications to the mysterious hidden orderstate in URu Si , in particular to its superconductivity.It is appropriate to comment on what we mean by “hid-den order”. An order parameter can often be inferred from its macroscopic consequences in terms of certaingeneralized rigidities. Sometimes its direct microscopicsignature is difficult to detect: a direct determination ofsuperconducting order, a broken global U (1) gauge sym-metry, requires subtle Josephson effect, and even antifer-romagnetic order requires microscopic neutron scatteringprobes. Density wave states of higher angular momen-tum, such as the mixed triplet d -density wave, are evenharder to detect. It does not lead to a net charge densitywave or spin density wave to be detected by common s -wave probes. It is further undetectable because it doesnot even break time-reversal invariance. A discussion ofpossible experimental detections of particle-hole conden-sates of higher angular momentum was given in Ref. 10.Thus, it is fair to conclude that the state we considerhere is a good candidate for a hidden order.It is also necessary to remark on the realization ofparticle-hole condensates of higher angular momentum.An effective low energy theory of a strongly correlatedsystem is bound to have a multitude of coupling con-stants, perhaps hierarchically arranged. In such cases, wecan generally expect a phase diagram with a multitudeof broken symmetry states. It is a profound mystery asto why non-trivial examples are so few and far between.A partial reason could be, as stated above, that thesestates are unresponsive to common s -wave probes em-ployed in condensed matter physics and therefore appearto be hidden. The next question is: are these low energy effectiveHamiltonians contrived? If so, it would be of value topursue them. However, simple Hartree-Fock analyseshave shown that they certainly are not: an onsiterepulsion U , a nearest neighbor interaction V , and an ex-change interaction J are sufficient in a single band model.The structure of this paper is as follows: in Sec. II,we construct the low energy effective action of the mixedtriplet and singlet d -density wave system. In Sec. III,we compute the charge and the spin of a skyrmion andverify that the skyrmions in this system are bosons, whichcan lead to a superconducting phase transition. In Sec.IV, we compute the angular momentum of a skyrmion.In Sec. V, we study the charge pumping due to a time-dependent inhomogeneous spin texture that is interestingin its own right. In Sec. VI we discuss mainly the problemof URu Si . In the appendices, the derivation of the non-linear σ -model and the details of computing the Chern-Simons coefficients and charge pumping are provided. II. EFFECTIVE ACTION
In the momentum space the mixed triplet and singlet d -density wave order parameter is ( c and c † are fermonicannihilation and creation operators, respectively, Q =( π, π ), and the lattice constant is set to unity) h c † k + Q,α c k,β i ∝ i ( ~σ · ˆ N ) αβ W k + δ αβ ∆ k , (1)where ˆ N is a unit vector, ~σ are the Pauli matrices actingon spin indices, and the form factors W k ≡ W k x − cos k y ) , (2)∆ k ≡ ∆ sin k x sin k y , (3)correspond to the d x − y and d xy density wave, respec-tively . It is not necessary that d xy and d x − y transi-tions be close to each other, nor are they required to beclose in energy .If we choose the spin quantization axis to be ˆ z , the upspins represent circulating spin currents corresponding tothe order parameter d + id and the down spins to d − id (inan abbreviated notation). So, there are net circulatingspin currents alternating from one plaquette to the nextbut no circulating charge currents. By the choice of thequantization axis we have explicitly broken SU (2), but not U (1), and the coset space of the order parameter S ≡ SU (2) /U (1). Such a state can admit Skyrmionsin two dimensions (ignoring the possibility of hedgehogconfigurations in (2 + 1) dimensions (cf. below).The Hamiltonian is H = X k,α,β ψ † k,α h δ αβ ( τ z ǫ k + τ x ∆ k ) − ( ~σ · ˆ N ) αβ τ y W k i ψ kβ , (4)where the summation is over the reduced Brillouin Zone(RBZ) bounded by k y ± k x = ± π , the spinor is ψ † k,α ≡ ( c † k,α , c † k + Q,α ), and ǫ k ≡ − t (cos k x + cos k y ); additionof longer ranged hopping will not change our conclu-sions . Here τ i ( i = x, y, z ) are Pauli matrices actingon the two-component spinor. It is not necessary butconvenient to construct a low energy effective field the-ory. For this we expand around the points K ≡ ( π , π )and K ≡ ( − π , π ), what would have been the two dis-tinct nodal points in the absence of the d xy term, and K ≡ (0 , π ), what would have been the nodal point inthe absence of the d x − y term. This allows us to de-velop an effective low energy theory by separating thefast modes from the slow modes. After that we make asequence of transformations for simplicity: (1) transformthe Hamiltonian to the real space, which allows us to for-mulate the skyrmion problem; (2) perform a π/ τ y -direction, which allows us to match to thenotation of Ref. 12 for the convenience of the reader; (3)label ψ K i + q,α by ψ iα , since K i is now a redundant no-tation; (4) construct the imaginary time effective action,with the definition ¯ ψ ≡ − iψ † τ z . Finally, after suppress-ing the spin indices, and with the definitions γ ≡ τ z , γ x ≡ τ y , and γ y ≡ − τ x , we obtain the effective action ina more compact notation: S = X j =1 , Z d x ¯ ψ j (cid:20) − iγ ∂ τ − itγ x ( η j ∂ x + ∂ y ) + i W ~σ · ˆ N ) γ y ( − η j ∂ x + ∂ y ) + iη j ∆ (cid:21) ψ j + Z d x ¯ ψ h − iγ ∂ τ − W ( ~σ · ˆ N ) γ y i ψ , (5)where η = 1 and η = − III. THE CHARGE AND SPIN OF ASKYRMION
We will compute the charge of the skyrmions in thesystem by following Grover and Senthil’s adiabatic ar-gument. First, consider the action around K = ( π , π )when the order parameter is uniform (say, ˆ N = ˆ z ). Theresults for K = ( − π , π ) and K = (0 , π ) follow identi- cally. In our previous paper we have shown that in thiscase the non-trivial topology leads to a quantized spinHall conductance in iσd x − y + d xy -density wave state as long as the system is fully gapped. The spin quantumHall effect implies that the external gauge fields A c and A s couple to charge and spin currents, respectively. Inthe presence of these external gauge fields, we add mini-mal coupling in the action by1 i ∂ µ = p µ → p µ + A cµ + σ z A sµ . (6)Then the action is S [ A c , A s ] = Z d x ¯ ψ (cid:20) − iγ ∂ τ + γ ( A cτ + σ z A sτ ) − itγ x ( ∂ x + ∂ y ) + 2 tγ x ( A cx + σ z A sx + A cy + σ z A sy )+ i W σ z γ y ( − ∂ x + ∂ y ) − W σ z γ y ( − A cx − σ z A sx + A cy + σ z A sy ) + i ∆ (cid:21) ψ , (7)where we set e = ~ = 1. The non-vanishing transverse spin conductance implies that the low energy effective actionfor the gauge fields is given by S eff = i π Z d x ǫ µνλ A c µ ∂ ν A s λ , (8)and the charge current is induced by the spin gauge field j cµ = 12 π ǫ µνλ ∂ ν A sλ . (9)Consider now a static configuration of the ˆ N field withunit Pontryagin index in the polar coordinate ( r, θ ):ˆ N ( r, θ ) = [sin α ( r ) cos θ, sin α ( r ) sin θ, cos α ( r )] (10)with the boundary conditions α ( r = 0) = 0 and α ( r →∞ ) = π . Performing a unitary transformation at all points in space such that U † ( ~σ · ˆ N ) U = σ z , and defining ψ = U ψ ′ , and ¯ ψ = ¯ ψ ′ U † , we obtain S = Z d x ¯ ψ ′ (cid:20) − iγ ∂ τ − itγ x ( ∂ x + ∂ y ) + i W σ z γ y ( − ∂ x + ∂ y ) + i ∆ (cid:21) ψ ′ + Z d x ¯ ψ ′ (cid:20) − iγ ( U † ∂ τ U ) − itγ x ( U † ∂ x U + U † ∂ y U ) + i W σ z γ y ( − U † ∂ x U + U † ∂ y U ) (cid:21) ψ ′ (11)To proceed, we write down the explicit form for U ( r, θ ), which is U ( r, θ ) = cos α ( r )2 − sin α ( r )2 e − iθ sin α ( r )2 e iθ cos α ( r )2 ! , (12)In the far field limit, U † ∂ x U = ( − i sin θr ) σ z , and U † ∂ y U = ( i cos θr ) σ z ; substituting into Eq. (11) and introducing f µ = − iU † ∂ µ U , we get S = Z d x ¯ ψ ′ (cid:20) − iγ ∂ τ − itγ x ( ∂ x + ∂ y ) + i W σ z γ y ( − ∂ x + ∂ y ) + i ∆ (cid:21) ψ ′ + Z d x ¯ ψ ′ (cid:20) tγ x ( f x + f y ) + W σ z γ y ( f x − f y ) (cid:21) ψ ′ (13)Equating the above equation and Eq. (7), we obtain inthe far field limit A cx = A cy = 0; A sx = − θr ; A sy = 2 cos θr . In other words, the process of tuning the order parameterfrom σ z to ˆ σ · ˆ N ( r, θ ) is equivalent to adding an externalspin gauge field ~A s = − θr ˆ x + 2 cos θr ˆ y = 2 r ˆ θ. (14) The total flux of this gauge field is clearly 4 π . Supposewe adiabatically construct the Skyrmion configurationˆ N ( r, θ ) from the ground state ˆ z in a very long time period τ p → ∞ . During the process, we effectively thread a spingauge flux of 4 π . The transverse spin Hall conductanceimplies that a radial current j cr will be induced by the 4 π spin gauge flux of ~A s ( t ), which is now time-dependent: ~A s ( t = 0) = 0 and ~A s ( t = τ p ) = ~A s , that is, j cr ( t ) = − π ∂ t A sθ ( t ) . (15)As a result, charge will be transferred from the center tothe boundary, and the total charge transferred is Q c = Z τ p dt Z π rdθj cr ( t ) = − . (16)Therefore, after restoring the unit of charge to e , we ob-tain a Skyrmion with charge 2 e ; its spin is 0.It is important to verify the adiabatic result by a dif-ferent method. This can be done by a computation of theChern number. The charge and spin of the skyrmionsare associated with the coefficients of the Chern-Simonsterms by the following relations: Q skyrmion = C e and S skyrmion = C ~ , where C and C are C = ǫ µνλ π Tr (cid:20)Z d kG ∂G − ∂k µ G ∂G − ∂k ν G ∂G − ∂k λ (cid:21) , (17) C = ǫ µνλ π Tr (cid:20)Z d k ( ~σ · ˆ z ) G ∂G − ∂k µ G ∂G − ∂k ν G ∂G − ∂k λ (cid:21) , (18)where G is the matrix Geen’s function and the trace istaken over the spin index σ and other discrete indices.If the Green’s function matrix is diagonal in the spin in-dex, then the Chern-Simons coefficients for up and downspins can be computed separately. N ( G σ ) = ǫ µνλ π Tr (cid:20)Z d kG σ ∂G − σ ∂k µ G σ ∂G − σ ∂k ν G σ ∂G − σ ∂k λ (cid:21) , (19)and C = N ( G ↑ ) + N ( G ↓ ), C = N ( G ↑ ) − N ( G ↓ ). Fur-thermore, it can be shown (see Appendix B) that for G − σ = iω ˆ I − ˆ τ · ~h σ (20)with ~h σ being the Anderson’s pseudospin vector of theHamiltonian, the Chern-Simons coefficient for spin σ canbe written as N ( G σ ) = − Z d k π ˆ h σ · ∂ ˆ h σ ∂k x × ∂ ˆ h σ ∂k y , (21)where ˆ h σ ≡ ~h σ / | ~h σ | is the unit vector of ~h σ . Here C and C are the total Chern number and the spin Chern num-ber N spin defined in our previous paper, respectively. For iσd x − y + d xy system, we have ~h σ ≡ (∆ k , − σW k , ǫ k ).Explicitly, C = − C = − − −
2; thusthe results are the same as above.Because a Skyrmion in the system carries integerspin, it obeys bosonic statistics and may undergo Bose-Einstein condensate. As a result, the charge-2 e Skyrmioncondensate will lead to a superconducting phase transi-tion. But what about its orbital angular momentum?In the following section, we will prove that it is zero re-sulting in a s -wave singlet state. This is a bit surprisinggiven the original d -wave form factor. IV. THE ANGULAR MOMENTUM OF ASKYRMION
To compute the angular momentum carried by askyrmion in the system, we consider the angular momen-tum density due to the electromagnetic field. For a staticspin texture it is clearly zero, because ~E = 0. For a timedependent texture it is little harder to prove. Consider, N x ( r, θ, t ) = sin α ( r, t ) cos β ( θ, t ) ,N y ( r, θ, t ) = sin α ( r, t ) sin β ( θ, t ) ,N z ( r, t ) = cos α ( r, t ) , where α ( r, t ) and β ( θ, t ) are smooth functions, and α ( r, t )satisfies the boundary conditions α ( r = 0 , t ) = 0 and α ( r → ∞ , t ) = π , for any t , and ∂α ( r,t ) ∂r | r →∞ = ∂α ( r,t ) ∂t | r →∞ = 0 in the far field limit. The unitary matrixis now time dependent. After a little algebra, we obtainthe time-dependent gauge fields in the far field limit tobe A sx ( r, θ, t ) = − θr ∂β ( θ, t ) ∂θ , (22) A sy ( r, θ, t ) = 2 cos θr ∂β ( θ, t ) ∂θ . (23)So, Φ( θ, t ) = A st ( θ, t ) = 2 ∂β ( θ,t ) ∂t and ~A s ( r, θ, t ) = A sx ( r, θ, t )ˆ x + A sy ( r, θ, t )ˆ y = A sθ ( r, θ, t )ˆ θ , where A sθ ( r, θ, t ) = 2 r ∂β ( θ, t ) ∂θ . (24)Therefore, the electric field will have a non-zeroˆ θ − component, ~E = E θ ˆ θ , and the magnetic field will havea non-zero ˆ z − component, ~B = B z ˆ z , where E θ = − r ∂A st ( θ, t ) ∂θ − ∂A sθ ( r, t ) ∂t = − r ∂ β ( θ, t ) ∂θ∂t (25) B z = ∂A sθ ( r, t ) ∂r = − r ∂β ( θ, t ) ∂θ . (26)As a result, the angular momentum density still vanishes, ~L field = 14 πc~r × ( E θ ˆ θ × B z ˆ z ) = 0 . (27)It is possible that superconductivity with non-zero an-gular momentum may be realized when the interactionbetween skyrmions is included, but we do not know howto prove it. It would be interesting to explore what otherkinds of quantum numbers are carried by the topologicaltextures in the model we have studied. V. QUANTIZED CHARGE PUMPING
In Sec.III, we considered a static spin texture and ob-tained charge-2 e skyrmions in the system. If we considera time-dependent spin texture, which has a slow varia-tion in one spatial direction, say, ˆ y , and is uniform in theother, ˆ x , charge will be pumped from one side of the sys-tem to the other along ˆ x . This charge pumping effectcan be understood from the effective gauge action, whichis S eff [ A cµ , A sµ ] = C π Z d x ǫ µνλ A c µ ∂ ν A s λ , (28)where the integral is over the real time, t , instead of theimaginary time, τ . Therefore, the charge current inducedby the spin gauge field will be j cµ = δS eff [ A cµ , A sµ ] δA cµ = C π ǫ µνλ ∂ ν A sλ = C π ǫ µνλ F sνλ , (29)where we define the spin gauge flux F sµν ≡ ∂ µ A sν − ∂ ν A sµ .After some straightforward algebra (see Appendix C), thespin gauge flux can be written in terms of the ˆ N -vector, F sµν = ˆ N · [( ∂ µ ˆ N ) × ( ∂ ν ˆ N )] . (30)As a result, even in absence of the external electromag-netic field, a charge current may be induced by a time-dependent inhomogeneous spin texture because j cµ = C π ǫ µνλ ˆ N · [( ∂ ν ˆ N ) × ( ∂ λ ˆ N )] . (31)To demonstrate the charge response induced by thespin texture, we consider the following configuration withunit Pontryagin index,ˆ N ( y, t ) = [sin θ ( t ) cos φ ( y ) , sin θ ( t ) sin φ ( y ) , cos θ ( t )] , (32)where θ ( t ) and φ ( y ) are smooth functions of t and y ,respectively, with boundary conditions θ ( t = 0) = 0, θ ( t = τ p ) = π , and φ ( y → ±∞ ) = ± π . Therefore, wehave an induced charge current along the ˆ x -direction, j cx = C π ǫ xνλ ˆ N · [( ∂ ν ˆ N ) × ( ∂ λ ˆ N )]= C π ˆ N · [( ∂ y ˆ N ) × ( ∂ t ˆ N )] . (33)Interestingly, we can show that the pumped charge isquantized, Q pumped = Z τ p dt Z ∞−∞ dy j cx = C π Z τ p dt Z ∞−∞ dy ˆ N · [( ∂ y ˆ N ) × ( ∂ t ˆ N )]= C π Z π dθ Z π − π dφ ˆ N · [( ∂ θ ˆ N ) × ( ∂ φ ˆ N )]= C , (34)where we have used that, for the spin texture with unitPontryagin index, Z π dθ Z π − π dφ ˆ N · [( ∂ θ ˆ N ) × ( ∂ φ ˆ N )] = 4 π. (35) After restoring the unit of charge, we have Q pumped = C e . So far we have considered the spin texture withunit Pontryagin index. If the spin texture is generalizedto a general Pontryagin index, N P , then the pumpedcharge will be Q pumped = C N P e .How could we observe this charge pumping experimen-tally? We need to control the direction of the ˆ N -vectorso that it can be the time-dependent inhomogeneous spintexture discussed above. In topological chiral magnets, the ˆ N -vector is the net ferromagnetic moment, whichaligns along the external magnetic field, so one can ap-ply a time-dependent magnetic field ~H ( t ) = H ( t )ˆ x cou-pling to the ˆ N -vector and control the magnitude of ˆ x -component of ˆ N .In the mixed triplet d -density wave, however, the sit-uation is more complicated. In the presence of an ex-ternal magnetic field, there will be a spin flop transitionand the ˆ N -vector will lie in the plane perpendicular tothe external field. In other words, we cannot fully con-trol the direction of ˆ N with a time-dependent magneticfield. Therefore, it would be a challenge to measure thepumped charges in the system.Nevertheless, the charge pumping effect provides, atleast, a different conceptual approach to probe the topo-logical properties of the system in addition to the quan-tized spin Hall conductance. For the quantum spin Halleffect, a spin current is induced by the external electricfield, whereas for the charge pumping effect, a chargecurrent will be induced by the spin texture. It would, ofcourse, be interesting if one can manipulate the ˆ N -vectorexperimentally because the charge current is easier to de-tect than the spin current. VI. DISCUSSION AND APPLICATION TO THEHIDDEN ORDER STATE IN
URu Si There are two points that we have glossed over. Thefirst is rather simple: in the ordered phase at T = 0, thereare also Goldstone modes that can be easily seen by inte-grating out the fermions resulting in a non-linear σ -modelinvolving ˆ N , the form of which is entirely determined bysymmetry. These do not lead to any interesting physics,such as charge-2 e skyrmions that condense into a super-conducting state. At finite temperatures they could leadto a renormalized classical behavior. The second pointis more subtle: we have assumed that the hedgehog con-figurations are absent. This would require, as pointed outby Grover and Senthil , that the energy of the Skyrmion(especially in the limit ∆ →
0) is smaller than indi-vidual pairs of electrons, a question that is likely to bemodel dependent. If this assumption is correct, however,the transition from the the mixed d -density wave stateto the superconducting state will correspond to a decon-fined quantum critical point, which otherwise would havebeen a first order transition, as in Landau theory. We suggest that the superconducting phase driven bythe skyrmion condensate may be realized in the URu Si ,which hosts an exotic hidden order (HO) phase, with bro-ken translatenal symmetry below T HO ≈ . K and asuperconducting phase below T c ≈ . K . Recently, Fu-jimoto proposed a triplet d -density wave with the or-der parameter h c † k, ,α c k + Q , ,β i = ~d ( k ) · ~σ αβ with ~d ( k ) = i (∆ sin ( k x − k y ) √ sin k z , ,
0) to describe this state; here1 and 2 refer to two different bands and Q = (0 , ,
1) isthe nesting vector; even the earlier work in Ref. 19 in-volving circulating spin current is not entirely unrelated.The order parameter considered in Ref. 18 is different buta close cousin of the order parameter considered in ourwork; the circulating staggered spin currents in Ref. 18lie on the diagonal planes instead and the crucial d xy partis missing there. That the currents are in the diagonalplanes instead of being square planar is conceptually notimportant, but is necessary to explain the nematicity ob- served in the experiments. We now discuss the role ofspin-orbit coupling before making our final comments.
Spin-orbit coupling
It will be shown below that the order of magni-tude of the spin-orbit energy E SO ≈ [( ˆ N · ˆ z ) − /W )( W /W ) [1+ O ( W /W ) ], correcting a mistakein Ref. 12. Here Λ is the strength of the spin-orbit cou-pling, given by H SO = X k c † kα ~ Λ( k ) · ~σ αβ c kβ , (36)where ~ Λ( k ) = (Λ / √ x sin k y − ˆ y sin k x ]. In the presenceof spin-orbit coupling, the Hamiltonian is H total = H + H SO = X k Ψ † k ǫ k ∆ k + iN z W k Λ x ( k ) − i Λ y ( k ) iW k ( N x − iN y )∆ k − iN z W k − ǫ k − iW k ( N x − iN y ) − Λ x ( k ) + i Λ y ( k )Λ x ( k ) + i Λ y ( k ) iW k ( N x + iN y ) ǫ k ∆ k − iN z W k − iW k ( N x + iN y ) − Λ x ( k ) − i Λ y ( k ) ∆ k + iN z W k − ǫ k Ψ k , (37)where Ψ † k is the four-component spinor ( c † k, ↑ , c † k + Q, ↑ , c † k, ↓ , c † k + Q, ↓ ). In the absence of spin-orbit coupling, the eigenvaluesare ± E k with E k = p ǫ k + W k + ∆ k . On the other hand when spin-orbit coupling is present, the eigenvalues of theupper and lower bands now become λ up , ± = E k, ± , λ low , ± = − E k, ± , respectively, where E k, ± = r ǫ k + W k + ∆ k + Λ k ± h ( ǫ k + W k )Λ k − W k ( ˆ N · ~ Λ k ) i (38)with Λ k ≡ | ~ Λ k | = Λ x ( k ) + Λ y ( k ). When the d xy component is absent, ∆ k = 0, and the results of Ref. 12 arerecovered. Consider the following two cases separately. ˆ N k ˆ z Since the chemical potential is at the mid-gap, we canfocus on the lower bands. When ˆ N = ˆ z , we have ˆ N · ~ Λ k =0 and λ z low , ± = − q E k + Λ k ± E k Λ k ] = − E k ∓ | ~ Λ k | (39)Assuming that Λ ≪ W , ∆ ≪ W with the electronicbandwidth W = 8 t , the change in the ground state en- ergy will be E SO = X k (cid:2) ( λ z low , + + λ z low , − ) − − E k ) (cid:3) = X k h ( − E k − | ~ Λ k | − E k + | ~ Λ k | ) + 2 E k i = 0 (40) ˆ N ⊥ ˆ z When ˆ N lies in xy -plane, we have ˆ N · ~ Λ k = | ~ Λ k | cos φ k ,where φ k is the angle between ˆ N and ~ Λ k , andcos φ k = ˆ N · ~ Λ k | ~ Λ k | = N x Λ x ( k ) + N y Λ y ( k ) q Λ x ( k ) + Λ y ( k ) . (41)The eigenvalues of the lower bands are now λ xy low , ± = − q E k + Λ k ± E k Λ k − W k Λ k cos φ k ] ≈ − E k ∓ (1 − W k E k ) | ~ Λ k | − W k E k Λ k E k (1 + O ( W k E k )) , (42)where we have used cos φ k ≈ O (1). Notice that the signs of the second order terms for λ xy low , + and λ xy low , − are bothnegative, leading to the net change in the ground state energy, which is opposite to the ˆ N = ˆ z case. Assuming thatΛ ≪ W , ∆ ≪ W , the change in the ground state energy per lattice site will be E SO = X k h ( λ xy low , + + λ xy low , − ) − − E k ) i ≈ − X k Λ k W k E k (cid:20) O ( W k E k ) (cid:21) = − Λ W (cid:18) W W (cid:19) " O (cid:18) W W (cid:19) < , (43)Therefore, ˆ N -vector should lie in the xy -plane in the presence of spin-orbit interaction and the result stated abovefollows.As large as the spin-orbit coupling may be for U atoms, E SO is still a small energy scale. However, if otheranisotropies are absent, the order parameter would be inthe XY -plane, resulting in vortices; exchange anisotropycan also result in an easy-axis anisotropy, in which casespin textures could be Ising domain walls that can trapelectrons. Although skyrmions are finite energy solu-tions, vortices cost infinite energy unless they are boundin pairs. We speculate that charge 2 e -skyrmionic con-densation is a more likely scenario, but the crossover inthe texture is an interesting topic for further research.The following remarks about URu Si are relevant: inboth magnetic field-temperature ( H − T ) and pressure-temperature ( P − T ) phase diagrams, the superconduct-ing phase is enclosed within the HO phase. It impliesthat the superconducting phase is closely related to theHO phase, and is probably induced by it. Throughoutour calculation, ignoring of course skyrmions, we haveassumed that the system is half-filled. The lower band isfilled and the upper band is empty, and the topologicalinvariant is quantized. If this is not the case, then therewill be no quantized spin Hall conductance, but an in-duced superconducting phase from charge 2 e -skyrmioniccondensation; doping will result in conducting mid-gapstates, as in polyacetylene. Of course, such a topolog-ical superconducting phase is very sensitive to disorder.Indeed, this may be supported by the destruction of theHO and SC phases with 4% Rh substitution on the Rusite. To summarize, we can find a rationale for a hiddenorder phase enclosing a superconducting phase at lowertemperatures.
ACKNOWLEDGMENTS
This work is supported by NSF under Grant No.DMR-1004520. We are grateful to E. Abrahams, E.Fradkin, S. Kivelson for useful comments regarding themanuscript. Special thanks are due to S. Raghu for hiscontinued interest in our work and for collaboration atearlier stages. Liang Fu, Tarun Grover, and Igor Her-but have made important suggestions. H. Y. Kee gaveus some confidence in regard to the applicability of ourideas to the hidden order state in URu Si . S. C. alsoacknowledge support from NSF Grant No. PHY-1066293and the hospitality of the Aspen Center for Physics wherethe work germinated. Appendix A: Derivation of the non-linear σ -model To derive the non-linear σ -model, we compute the ef-fective action by integrating out fermions. We start withthe action S = P j =1 S j , where S j ≡ Z d x ¯ ψ j (cid:2) G − j (cid:3) ψ j , (A1)with G − j ≡ G − ,j + Σ j .For j = 1 ,
2, we have G − ,j ≡ − iσ τ z ∂ τ − itσ τ y ( η j ∂ x + ∂ y ) , (A2)Σ j ≡ iη j ∆ σ τ − i W ~σ · ˆ N ) τ x ( − η j ∂ x + ∂ y ) , (A3)and for j = 3, we have G − , ≡ − iσ τ z ∂ τ , (A4)Σ ≡ W ( ~σ · ˆ N ) τ x . (A5)The effective action will be S eff = P j =1 S j,eff with S j,eff = − ln (cid:20)Z D ¯ ψ j Dψ j e − S j (cid:21) = − ln (cid:2) det | G − j | (cid:3) , (A6) where the fermion operators can be integrated out easilysince the Hamiltonian has only bilinear fermion opera-tor terms. Using the mathematical identity ln det | A | =tr ln A with tr being the trace, we have S j,eff = − tr ln G − ,j [1 + G ,j Σ j ]= − tr ln G − ,j − tr [ G ,j Σ j ] + 12 tr [ G ,j Σ j G ,j Σ j ] + · · · , (A7)where we have used ln(1 + x ) = x − x + · · · .The zeroth order term is the effective action for free particles and the first order term vanishes, so our goal isto compute the second order terms: S (2) j,eff ≡
12 tr [ G ,j Σ j G ,j Σ j ]= 12 Z dτ Z dτ ′ Z d x Z d x ′ Tr [ G ,j ( x, τ ; x ′ , τ ′ )Σ j ( x ′ , τ ′ ) G ,j ( x ′ , τ ′ ; x, τ )Σ j ( x, τ )]= 12 X ˜ k, ˜ q Tr h G ,j (˜ k )Σ j (˜ q ) G ,j (˜ k + ˜ q )Σ j ( − ˜ q ) i , (A8)where ˜ k ≡ ( k , k x , k y ), ˜ q ≡ ( q , q x , q y ), and G ,j (˜ k ) canbe obtained by inverting Eq.(A2) and Eq.(A4). Putting all together, taking long wavelength limit (˜ q →
0) and keeping only terms up to the second order deriva-tive, we have, for j = 1 , S (2) j,eff ≈ X ˜ k, ˜ q k + 4 t ( η j k x + k y ) (cid:20) − ∆ + ( W ( − η j q x + q y ) ( ˆ N ˜ q · ˆ N − ˜ q ) (cid:21) , (A9)where terms which are odd in ˜ k and ˜ q are dropped.Using the relation P ˜ q f ˜ q f − ˜ q = R dτ d x | f ( ~x, τ ) | , weobtain S (2)1 ,eff + S (2)2 ,eff ≈ g Z dτ d x (cid:20)(cid:12)(cid:12)(cid:12) ∂ X ˆ N (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ∂ Y ˆ N (cid:12)(cid:12)(cid:12) (cid:21) , (A10)where the constant terms are dropped, ( X, Y ) is the co-ordinate after a π/ g ≡ X ˜ k − W k + 4 t ( k x + k y ) ) (A11)Similarly, for j = 3, we obtain S (2)3 ,eff ≈ − X ˜ k, ˜ q W k ( q k ) ( ˆ N ˜ q · ˆ N − ˜ q )= 1 g Z dτ d x (cid:12)(cid:12)(cid:12) ∂ τ ˆ N (cid:12)(cid:12)(cid:12) , (A12) where 1 g ≡ X ˜ k W k . (A13)Therefore, we obtain the non-linear sigma model, S eff ≈ g Z dτ d x (cid:12)(cid:12)(cid:12) ∂ µ ˆ N (cid:12)(cid:12)(cid:12) , (A14)where the constant terms and higher order terms aredropped, and it is rescaled in order to obtain a famil-iar form. Appendix B: Chern-Simons coefficients
In this appendix we are going to prove that N ( G σ ) = ǫ µνλ π Tr (cid:20)Z d kG σ ∂G − σ ∂k µ G σ ∂G − σ ∂k ν G σ ∂G − σ ∂k λ (cid:21) = − Z d k π ˆ h σ · ∂ ˆ h σ ∂k x × ∂ ˆ h σ ∂k y . (B1) We start by taking ( µ, ν, λ ) to be (0 , x, y ), and obtain G σ ∂G − σ ∂ω = 1( iω ) − | ~h σ | h ( iω ˆ I + ˆ τ · ~h σ ) · ( i ˆ I ) i = 1( iω ) − | ~h σ | ( − ω ˆ I + i ˆ τ · ~h σ ) , (B2)and G σ ∂G − σ ∂k x = 1( iω ) − | ~h σ | h iω ˆ I + ˆ τ · ~h σ i ( − ˆ τ · ∂~h σ ∂k x )= − iω ) − | ~h σ | " ( ~h σ · ∂~h σ ∂k x ) ˆ I + i ˆ τ · ( ω ∂~h σ ∂k x + ~h σ × ∂~h σ ∂k x ) , (B3)where we have used the matrix identity (ˆ τ · ~a )(ˆ τ · ~b ) = ( ~a · ~b ) ˆ I + i ˆ τ · ( ~a × ~b ). Similarly, G σ ∂G − σ ∂k y = − iω ) − | ~h σ | " ( ~h σ · ∂~h σ ∂k y ) ˆ I + i ˆ τ · ( ω ∂~h σ ∂k y + ~h σ × ∂~h σ ∂k y ) . (B4)Therefore, G σ ∂G − σ ∂k x G σ ∂G − σ ∂k y = 1(( iω ) − | ~h σ | ) ( ( ~h σ · ∂~h σ ∂k x )( ~h σ · ∂~h σ ∂k y ) ˆ I + i ˆ τ · " ( ~h σ · ∂~h σ ∂k x )( ω ∂~h σ ∂k y + ~h σ × ∂~h σ ∂k y ) + ( ~h σ · ∂~h σ ∂k y )( ω ∂~h σ ∂k x + ~h σ × ∂~h σ ∂k x ) − " ˆ τ · ( ω ∂~h σ ∂k x + ~h σ × ∂~h σ ∂k x ) ˆ τ · ( ω ∂~h σ ∂k y + ~h σ × ∂~h σ ∂k y ) . (B5)Since we are going to multiply it with the antisym-metric tensor ǫ µνλ , the terms which are symmetric under ( x ↔ y ) will vanish. Therefore, only the last term in thebracket contributes, " ˆ τ · ( ω ∂~h σ ∂k x + ~h σ × ∂~h σ ∂k x ) ˆ τ · ( ω ∂~h σ ∂k y + ~h σ × ∂~h σ ∂k y ) = i ˆ τ · " ω ( ∂~h σ ∂k x × ∂~h σ ∂k y ) + ω~h σ ( ∂~h σ ∂k x · ∂~h σ ∂k y ) − ω ∂~h σ ∂k y ( ~h σ · ∂~h σ ∂k x ) − ω~h σ ( ∂~h σ ∂k x · ∂~h σ ∂k y ) + ω ∂~h σ ∂k x ( ~h σ · ∂~h σ ∂k y ) + ( ~h σ · ∂~h σ ∂k x × ∂~h σ ∂k y ) ~h σ , (B6)where we used the following mathematical identities: ~a × ( ~b × ~c ) = ~b ( ~a · ~c ) − ~c ( ~a · ~b ) , ( ~a × ~b ) × ( ~a × ~c ) = ( ~a · ( ~b × ~c )) ~a. Therefore, after combining with ǫ xy and taking thetrace, we have0 ǫ xy Tr (cid:20) G σ ∂G − σ ∂ω G σ ∂G − σ ∂k x G σ ∂G − σ ∂k y (cid:21) = − iω ) − | ~h σ | ) Tr ( − iω ˆ τ · " ω ( ∂~h σ ∂k x × ∂~h σ ∂k y ) + ( ~h σ · ∂~h σ ∂k x × ∂~h σ ∂k y ) ~h σ − (ˆ τ · ~h σ ) " ˆ τ · [ ω ( ∂~h σ ∂k x × ∂~h σ ∂k y ) + ( ~h σ · ∂~h σ ∂k x × ∂~h σ ∂k y ) ~h σ ] = 2(( iω ) − | ~h σ | ) " ~h σ · [ ω ( ∂~h σ ∂k x × ∂~h σ ∂k y ) + ( ~h σ · ∂~h σ ∂k x × ∂~h σ ∂k y ) ~h σ ] = − iω ) − | ~h σ | ) ( ~h σ · ∂~h σ ∂k x × ∂~h σ ∂k y ) , (B7)where we have used the fact that Pauli matrices are trace-less, so the only contribution will be the term propor-tional to ˆ I .We have six non-zero terms because of the ǫ µνλ tensor,so N ( G σ ) = − · π Z d k iω ) − | ~h σ | ) ( ~h σ · ∂~h σ ∂k x × ∂~h σ ∂k y )= − Z d k π | ~h σ | ( ~h σ · ∂~h σ ∂k x × ∂~h σ ∂k y )= − Z d k π ˆ h σ · ∂ ˆ h σ ∂k x × ∂ ˆ h σ ∂k y , (B8)where the energy integral was done by computing theresidue of the second order pole. Appendix C: Spin gauge flux F sµν in terms of ˆ N In the main text, we obtain the spin gauge field to be f µ = σ z A sµ , (C1) where f µ = − iU † ∂ µ U .Therefore, we can write the spin gauge field in termsof the unitary matrix, A sµ = Tr (cid:20) σ z · σ z A sµ (cid:21) = Tr [ σ z f µ ]= − i Tr (cid:2) σ z U † ∂ µ U (cid:3) , (C2)and we have F sµν = ∂ µ A sν − ∂ ν A sµ = − i Tr (cid:2) σ z ( ∂ µ U † )( ∂ ν U ) − σ z ( ∂ ν U † )( ∂ µ U ) (cid:3) . (C3)Assume that the spin texture has the general form:ˆ N ( ~x, t ) = [sin θ ( ~x, t ) cos φ ( ~x, t ) , sin θ ( ~x, t ) sin φ ( ~x, t ) , cos θ ( ~x, t )] , (C4)where θ ( ~x, t ) and φ ( ~x, t ) can be any function of position and time. Then, we have the unitary matrix U ( ~x, t ) = cos θ ( ~x,t )2 − sin θ ( ~x,t )2 e − iφ ( ~x,t ) sin θ ( ~x,t )2 e iφ ( ~x,t ) cos θ ( ~x,t )2 ! , (C5) ∂ µ U † ( ~x, t ) = (cid:18) − sin θ ∂ µ θ e − iφ ( cos θ ∂ µ θ − i sin θ ∂ µ φ ) e iφ ( − cos θ ∂ µ θ − i sin θ ∂ µ φ ) − sin θ ∂ µ θ (cid:19) , (C6)and ∂ ν U ( ~x, t ) = (cid:18) − sin θ ∂ ν θ e − iφ ( − cos θ ∂ ν θ + i sin θ ∂ ν φ ) e iφ ( cos θ ∂ ν θ + i sin θ ∂ ν φ ) − sin θ ∂ ν θ (cid:19) , (C7)1where we have suppressed the arguments of θ ( ~x, t ) and φ ( ~x, t ).Therefore, we can calculate the product of the last two matrices, and express the spin gauge flux as F sµν = − i (cid:20) i θ ( ∂ µ θ∂ ν φ − ∂ ν θ∂ µ φ ) (cid:21) ×
2= sin θ ( ∂ µ θ∂ ν φ − ∂ ν θ∂ µ φ ) . (C8)In addition, we can also write ˆ N · ( ∂ µ ˆ N × ∂ ν ˆ N ) in terms of θ ( ~x, t ) and φ ( ~x, t ),ˆ N · ( ∂ µ ˆ N × ∂ ν ˆ N )= (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin θ ( ~x, t ) cos φ ( ~x, t ) sin θ ( ~x, t ) sin φ ( ~x, t ) cos θ ( ~x, t )[cos θ ( ~x, t ) cos φ ( ~x, t ) ∂ µ θ ( ~x, t ) [cos θ ( ~x, t ) sin φ ( ~x, t ) ∂ µ θ ( ~x, t ) − sin θ ( ~x, t ) ∂ µ θ ( ~x, t ) − sin θ ( ~x, t ) sin φ ( ~x, t ) ∂ µ φ ( ~x, t )] + sin θ ( ~x, t ) cos φ ( ~x, t ) ∂ µ φ ( ~x, t )][cos θ ( ~x, t ) cos φ ( ~x, t ) ∂ ν θ ( ~x, t ) [cos θ ( ~x, t ) sin φ ( ~x, t ) ∂ ν θ ( ~x, t ) − sin θ ( ~x, t ) ∂ ν θ ( ~x, t ) − sin θ ( ~x, t ) sin φ ( ~x, t ) ∂ ν φ ( ~x, t )] + sin θ ( ~x, t ) cos φ ( ~x, t ) ∂ ν φ ( ~x, t )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , = sin θ ( ∂ µ θ∂ ν φ − ∂ ν θ∂ µ φ ) , (C9)where, again, we suppressed the arguments of θ ( ~x, t ) and φ ( ~x, t ). Finally, we obtain F sµν = ˆ N · ( ∂ µ ˆ N × ∂ ν ˆ N ) . (C10) X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057(2011); M. Z. Hasan and C. L. Kane, ibid . , 3045 (2010). S. Raghu, X.-L. Qi, C. Honerkamp, and S.-C. Zhang, Phys.Rev. Lett. , 156401 (2008); K. Sun, H. Yao, E. Fradkin,and S. A. Kivelson, ibid . , 046811 (2009); B.-J. Yangand H.-Y. Kee, Phys. Rev. B , 195126 (2010). C.-H. Hsu, S. Raghu, and S. Chakravarty, Phys. Rev. B , 155111 (2011). J. A. Mydosh and P. M. Oppeneer, Rev. Mod. Phys. ,1301 (2011). P. B. Wiegmann, Phys. Rev. B , 15705 (1999). T. Grover and T. Senthil, Phys. Rev. Lett. , 156804(2008). S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H.Rezayi, Phys. Rev. B , 16419 (1993). C.-K. Lu and I. F. Herbut, Phys. Rev. Lett. , 266402(2012). E.-G. Moon, Phys. Rev. B , 245123 (2012). C. Nayak, Phys. Rev. B , 4880 (2000). S. Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak, Phys. Rev. B , 094503 (2001). A. A. Nersesyan, G. I. Japaridze, and I. G. Kimeridze, J.Phys.: Condens. Matter , 3353 (1991). V. M. Yakovenko, arXiv:cond-mat/9703195. (1997). P. W. Anderson, Phys. Rev. , 827 (1958). B.-J. Yang and N. Nagaosa, Phys. Rev. B , 245123(2011). S. Chakravarty, B. I. Halperin, and D. R. Nelson, Phys.Rev. B , 2344 (1989). T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, andM. P. A. Fisher, Science , 1490 (2004); T. Senthil,L. Balents, S. Sachdev, A. Vishwanath, and M. P. A.Fisher, Phys. Rev. B , 144407 (2004). S. Fujimoto, Phys. Rev. Lett. , 196407 (2011). H. Ikeda and Y. Ohashi, Phys. Rev. Lett. , 3723 (1998). R. Okazaki, T. Shibauchi, H. J. Shi, Y. Haga, T. D. Mat-suda, E. Yamamoto, Y. Onuki, H. Ikeda, and Y. Matsuda,Science , 439 (2011). A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W. P. Su,Rev. Mod. Phys.60