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Higher-form symmetries and $d-wave$ superconductors from doped Mott insulators

Higher-form symmetries and $d-wave$ superconductors from doped Mott insulators 1 2, 1 Ki-Seok Kim and Yuji Hirono Department of Physics, POSTECH, Pohang, Gyeongbuk 37673, Korea Asia Pacific Center for Theoretical Physics (APCTP), Pohang, Gyeongbuk 37673, Korea (Dated: May 14, 2019) One path to high-temperature cuprate superconductors is doping a Mott insulator. In this paper, we study this system from the view point of higher-form symmetries. On the introduction of slave bosons, the t − J model at a finite hole doping can be written in the form of U(1) gauge theories. After a duality transformation, they can be written as a generalized BF theory, in which the higher- form symmetries are more manifest. We identify the emergent continuous and discrete higher-form symmetries in both s − wave and d − wave superconducting phases, expected to be realized from a doped Mott insulator. The existence of a topological order is tested by examining if there is a spontaneously broken discrete one-form symmetry. We claim that a spontaneous breaking of discrete one-form symmetry may extend to a phase that has massless Dirac fermions in the limit of large number of flavors. We discuss the possibility of a topological phase transition inside the superconducting dome of high T cuprates. I. INTRODUCTION corresponding charged objects are Wilson loops and vor- tex operators. Those two operators obey fractionalized statistics. In the large-loop limit, the expectation values Topological orders provide us with a finer classification of those operators show the perimeter law, which means of gapped quantum phases beyond the Landau-Ginzburg that both one-form symmetries are spontaneously bro- theory based on symmetry breaking patterns [1]. Topo- ken, and the system acquires a topological order. logically ordered states exhibit such properties as frac- In this paper, we study the structure of higher-form tionalized topological excitations, long-range entangle- symmetries of two-dimensional high temperature super- ment, and degeneracy of the ground states that depend conductors. The realization of superconductivity from on the spacetime topology [2, 3]. A conventional s−wave doped Mott insulators is known as a spin-liquid scenario superconductor is an example of topologically ordered for high T cuprates [9]. Resorting to the U(1) slave- phase [4], where the key ingredient is that the Cooper c boson representation of the t − J effective Hamiltonian, pairs have charge 2e. Because of this, if we compactify an effective field theory for d − wave superconductors one spatial direction, we can insert a half magnetic flux is obtained from a spin liquid state, based on the as- in this direction. This state has exactly the same energy sumption of the separation of spin and charge degrees of as the state with no flux, and the ground states become freedom. We take an Abelian duality to see the symme- doubly degenerate. Since the superconductivity does not try structure in a manifest way, and identify the emergent involve symmetry breaking (U(1) gauge symmetry is not higher-form symmetries in the dual effective theories. By a symmetry), the distinction of a superconducting state examining whether those symmetries are spontaneously with a normal state is given by the existence of a topo- broken or not, we clarify the nature of topological order logical order. in such systems. In this analysis, electromagnetic U(1) In certain cases, the appearance of topological order gauge fields are taken to be background potentials. As a can be understood as a consequence of a spontaneous result, ordinary superconductors are identified with su- breaking of higher-form symmetries [5–8], which is a gen- perfluids, that are not topologically ordered. eralization of symmetry. The charged objects of higher- form symmetries are extended objects like line operators When massless Dirac fermions are absent, which corre- or surface operators, and an ordinary symmetry corre- sponds to s−wave superconductors from doped Mott in- sponds to a zero-form symmetry, where the correspond- sulators, we find that the superconducting state has a Z ing charged object is point-like. The concepts such as one-form symmetry. This symmetry turns out to be not Noether theorem and Nambu-Goldstone (NG) theorem spontaneously broken, and there is no topological order are also generalized to higher-form symmetries. For ex- associated with this symmetry. In fact, the Z one-form ample, photons can be understood as NG modes associ- symmetry is shown to be a subgroup of a continuous U(1) ated with the spontaneous breaking of continuous one- one-form symmetry of this state. A generalized version form symmetry. A topological order appears when a of the Coleman-Mermin-Wagner theorem for the contin- discrete higher-form symmetry is spontaneously broken. uous higher-form symmetries prohibits the spontaneous The example of s−wave superconductivity is in this cat- breaking of continuous one-form symmetry in 2 + 1 di- egory. The effective theory can be written as an Abelian mensions, and hence the Z symmetry cannot be broken Higgs model, and after taking an Abelian duality and either. And yet, the Z symmetry is observable in the dropping massive excitations, it can be written as a topo- braiding phase of quasiparticles, as we show in the main logical BF theory. The action has a pair of emergent text. From this observation, we conclude that the topo- Z one-form symmetries (in 2+1 dimensions), and the logical property of s−wave superconducting phase from arXiv:1905.04617v1 [cond-mat.str-el] 12 May 2019 2 U(1) spin liquids is the same as conventional s − wave topological order and superfluidity coexist [12, 13]. The superconductivity from Landau’s Fermi liquids . structure of higher-form symmetries is analyzed, and One subtle issue regarding topological order is that such analysis is useful in classifying quantum phases, there are massless Dirac fermions that statistically in- because a spontaneous breaking of discrete higher-form teract with vortices [14, 15]. In the presence of such symmetries results in a topological order. The formalism coupling to fermions, the discrete one-form symmetry is is applied to the color-flavor locked phase of color super- absent. However, we point out that, if we consider the conductivity of QCD matter. Since this mathematical large−N (number of fermions) limit of d − wave super- structure is also applicable to the present study, let us conductors (this phase has massless Dirac fermions), at briefly review it here in the case of 2 + 1 dimensions. which a conformal fixed point is realized, a Z one-form symmetry is effectively restored and further it is spon- taneously broken. As a result, we claim that d − wave superconductors from doped Mott insulators cannot be smoothly connected to those from Landau’s Fermi liq- uids, because they can be distinguished by a discrete one-form symmetry. This is in contrast to the case of s − wave superconductors. The rest of the paper is organized as follows. In Sec. II, we give a review of a generalized BF theory for the de- scription of superfluids with topological order. In Sec. III, we study the topological properties of d−wave supercon- ductors realized from Landau’s Fermi liquids. In Sec. IV, we study the topological nature of d − wave supercon- ductors from doped Mott insulators. In Sec. V we give concluding remarks. II. GENERALIZED BF THEORY FOR SUPERFLUIDITY WITH TOPOLOGICAL ORDER The starting point is a theory with multiple U(1) sym- metries, in which some parts of them are coupled with A. Effective gauge theory for superfluids with gauge fields through a covariant derivative. At low ener- topological order gies, the only relevant degrees of freedom are the massless NG modes and topological excitations. Such a situation A generalized BF -type theory in 3 + 1 dimensions is can be captured with the following Lagrangian of a gen- studied recently that can describe the situations where eralized Abelian Higgs model, n o ′ ′ ′ 1 1 3 f c f c c c S = d x [H] ′(∂ φ + [K] a )(∂ φ + [K] ′ ′a ) + [G] ′A A , (1) AH ff μ fc μ f c cc μ μ μν μν 2 2 where φ are 2π-periodic scalar fields representing the respectively. phases of Cooper pair fields, a are U(1) gauge fields, To study the existence of topological order, let us take c c c and A ≡ ∂ a − ∂ a are the field strength tensors an Abelian duality transformation so that the higher- μ ν μν ν μ for a . There are multiple scalar and gauge fields, and form symmetries of the system can be seen in a manifest f and c are the indices of the scalar and gauge fields, way. The dual action is written as n o 1 ′ [K] 1 ′ fc 3 −1 f f f c c c S = d x [H ] ′B B + ǫ b ∂ a + [G] ′A A , Dual ff μνλ ν cc μν μν μ λ μν μν 8π 2π 2 (2) A similar situation occurs in the dense QCD matter [10]. There continuity [11]. This observation is based on the fact that those is a scenario that hadronic superfluid phase can be continuously two phases have the same (zero-form) symmetries. The continu- connected to a color superconducting phase, called quark-hadron ity is recently extended to include the topological orders through the analysis of higher-form symmetries [12, 13]. 3 f f f f where b is a one-form gauge field and B = ∂ b −∂ b when integrated over a closed loop C. Under the trans- μ ν μ μν ν μ f f formation (5), the kinetic-energy term for supercurrent is a two-form field strength tensor for b . The field b μ μ b f fluctuations is not affected, since δ[P b ] = 0. The BF is dual to φ , which describe sound modes (supercurrent ff μ term changes as fluctuations). The effective theory is invariant under the gauge transformations, T + 3 c δ S = q [K K] ′ d xǫ λ ∂ a . (8) 1 Dual cc μνλ μ ν c λ c c c f f f 2π a 7→ a + ∂ λ , b 7→ b + ∂ λ , (3) μ μ μ μ μ μ Noting that [K K] ′ is a orthogonal projection matrix cc c f where λ and λ are 2π-periodic parameters. The gauge ⊥ T ⊥ to (kerK) and we have chosen q ∈ (kerK) , we have fields satisfy the standard Dirac quantization condition, T + T q [K K] = q and cc ′ Z Z c f δ S ∈ 2πZ. (9) dS A ∈ 2πZ, dS B ∈ 2πZ, (4) 1 Dual μν μν μν μν Thus, the weight of the path integral is unchanged un- where the integrations are over closed 2-dimensional sub- der this transformation. If this transformation acts on manifolds. The gauge invariance requires that the entries physical operators (vortex operators) nontrivially, this is of [K] are integers. fc a symmetry of the system. The entries of [K ] are cf The final action is obtained by neglecting the kinetic in general fractional numbers, and in that case a discrete terms of the massive sound modes and photons. Since one-form symmetry. The charged objects under this sym- the number of flavor is not necessarily the same as that of metry are vortex operators, color, the matrix K is non-square in general . When the K matrix has a nontrivial cokernel, i.e., dim (cokerK) 6= f V (C) = exp ip dl b , (10) p f μ 0, there are gapless superfluid sound modes. The num- ber of gapless modes is given by dim (cokerK). Those where p is a charge vector and C is a closed one dimen- f b f massless modes can be identified as [b ] = P [b] , 0 ′ μ ff μ sional manifold. This is transformed by the one-form b + ′ ′ where P ≡ δ − [KK ] is the orthogonal projec- ′ ff ff ff symmetry as tion matrix to cokerK, and K is the Moore-Penrose inverse [16] of K. Similarly, when K has a nontriv- V (C) 7→ exp 2πip [K ] V (C). (11) p f cf p ial kernel, i.e., dim (ker K) 6= 0, we have gapless pho- c a c Similarly, one may consider the following transforma- tons, and they are identified as [a ] = P ′[a] , where μ cc μ a + tion, P = δ ′ −[K K] ′ is the projection matrix to kerK. cc cc cc c c + a 7→ a − [K ] p ω , (12) cf f μ μ μ B. Higher-form symmetries ⊥ where p ∈ (cokerK) is an integer vector, and ω is a f μ flat one-form field satisfying the curvature-free condition Now we are ready to discuss higher-form symmetries of ∂ ω − ∂ ω = 0 and is normalized as dl ω = 2πZ. μ ν ν μ μ μ the dual effective theory. Equation (2) is invariant under Here, (cokerK) indicates the orthogonal complement of the following transformation, cokerK. Under this transformation, the action is varied as f f T + b 7→ b + q [K ] λ , (5) cf μ μ μ c + 3 f δ S = − [KK ] ′p d xǫ ∂ b ω . (13) 1 Dual ff f μνλ μ λ where q is an integer vector chosen from the orthogonal 2π complement of kerK, i.e., q ∈ (kerK) , and λ is a c μ Noting that KK is a symmetric matrix and one-form field that satisfies the curvature-free condition, [KK ] ′ p = p ′, we have f f f f ∂ λ − ∂ λ = 0, (6) μ ν ν μ δ S ∈ 2πZ. (14) 1 Dual with the normalization Z The charged objects under this symmetry are the Wilson loops, dl λ ∈ 2πZ, (7) μ μ C Z W (C) = exp iq dl a , (15) q c μ characterized by a charge vector q . 2 c Although a matrix BF theory is used in [12, 13] (which is neces- In addition to discrete one-form symmetries, the effec- sary for 3 + 1 dimensions), in 2 + 1 dimensions, it is also possible to treat all the one-form fields on equal footing and write the tive dual field theory Eq. (2) has the following continuous action in the form of a Chern-Simons theory with matrix coef- U(1) one-form symmetry, ficients. When there is no diagonal term such as a ∧ da, the c c α BF -type description is more economical and we adopt this. a 7→ a + ǫ[C ] λ , (16) c μ μ μ 4 and In this case, the system acquires degeneracy depending on the spacetime topology. If the vortex couples to mass- f f α ¯ b 7→ b + ǫ[D ] λ . (17) f μ less modes, it decays faster than the perimeter law and μ μ the symmetry may not be broken. α α ¯ where ǫ is a continuous parameter, and [C ] and [D ] c f The condition for the existence of topological order can are basis vectors of the kernel and cokernel of the K be summarized as follows: There exists a pair of integer matrix, ⊥ ⊥ vectors (p, q) ∈ (cokerK) × (kerK) such that α α ¯ T + [K] [C ] = 0, [D ] [K] = 0. (18) fc c f fc iq [K ] p cf f e 6= 1. (24) Those vectors are labeled by α and α¯, This condition can be interpreted as that for the presence of mixed ’t Hooft anomalies of one-form symmetries [13]. α = 1, ..., dim (kerK), One way to see this is introducing background two-form α¯ = 1, ..., dim (cokerK). (19) gauge fields for a pair of (discrete) one-form symmetries and considering the partition function. The phase of the partition function may become ambiguous in the pres- C. Statistics of quasi-particles and vortices ence of those background gauge fields, if the factor (24) is different from 1. This indicates the existence of a ’t Hooft In the current setting, not all the Wilson operators or anomaly between those symmetries. In such a case, we vortex operators are topological, because of the presence cannot realize the system in 2 + 1 dimensions in a gauge of massless NG modes. A topological operator is an op- invariant way and we have to couple it to a symmetry- erator invariant under the deformation of the underlying protected topological (SPT) phase in 3 + 1 dimensions. manifold, Then, the ’t Hooft anomaly of the boundary state is can- celled by the anomaly inflow mechanism from the SPT W (C + δC) = W (C), (20) q q bulk phase [17]. Since this anomaly matching structure is invariant under renormalization group, the presence when the deformation does not cross other operators. of the ’t Hooft anomaly precludes a trivial ground state This does not hold if the operator couples to massless for the 2 + 1-dimensional theory. In the present case, modes. Even in that case, it is possible to extract the the anomaly is matched by a topological order in the information about the braiding phase of those operators. infrared. It can be shown that [13] hW (C)V (C )i q p III. TOPOLOGICAL PROPERTY OF d − wave hW (C)ihV (C )i SUPERCONDUCTORS FROM LANDAU’S q p + ′ FERMI LIQUIDS = exp −2πi(q [K ] p ) Lk(C, C ) , (21) c cf f where Lk(C, C ) is the Gauss linking number of the two The dynamics of superfluid fluctuations at low energies world lines C and C . Their mutual statistics is encoded is described by the following effective action T + in q [K ] p . cf f c Z Z cp 2 2 The braiding phase (21) is closely related to the dis- S = dτ d x(∂ φ ) , (25) s−wave μ cp crete one-form symmetries of the system. Suppose a Wil- son loop associated with a charge vector q is topological. where ρ is a phase stiffness parameter proportional to cp Then, the loop C can be contracted to a point, which the Cooper-pair density and φ is a phase field to de- cp means that hW (C)i = 1 (up to the part obeying the scribe sound modes. Here, we consider two space di- perimeter law). When C is singly linked to C , we have mensions in the Euclidean time, where β is the inverse of temperature, set to be infinite, i.e., considering zero ′ −2πi(q [K ] p ) ′ c cf f hW (C)V (C )i = e hV (C )i. (22) q p p temperature. Performing the duality transformation, we obtain an effective dual field theory in terms of vortices cp This is nothing but the definition of the existence of a Φ and superfluid sound modes b , where the partition cp one-form symmetry [7]. Here, W (C) is the generator of function is the symmetry and V (C ) is a charge object. cp s−wave The existence of (discrete) one-form symmetry is not Z = DΦ Db exp − S , (26) s−wave cp Dual enough for a system to be topologically ordered. There has to be a discrete one-form symmetry which is sponta- and the dual effective action is neously broken, to have a topological order. This means Z Z β n s−wave 2 cp 2 2 2 that at large loop C, the corresponding charged object S = dτ d x |(∂ − ib )Φ | + m |Φ | μ cp cp Dual μ cp of the symmetry should behave as u 1 cp 4 cp 2 + |Φ | + (ǫ ∂ b ) . (27) cp μνλ ν hV (C) ≃ exp (−κ perimeter[C]) . (23) p 2 2ρ cp 5 To describe the physics of d−wave superconductors, it for dual photon and statistical photon fields, respectively, is necessary to introduce massless Dirac fermions, which 4π −1 results from the d−wave pairing symmetry. An essential ′ ′ [H ] = δ δ , (32) ff f1 f 1 cp point is that such massless Dirac fermions have statistical interactions with vortices, described by an effective BF for the phase stiffness parameter, term with a statistical angle π, which implies mutual semionic statistics [14, 15]. An effective field theory for −1 [K] = 2δ δ −→ [K ] = , (33) fc f1 c1 11 d − wave superconductors is given by the path integral expression for massless Dirac fermions ψ , Cooper-pair nσ for the K matrix, and cp vortices Φ , superfluid fluctuations b , and emergent cp U(1) statistical gauge fields c , G ′ = δ δ , (34) c1 c 1 cc cp d−wave Z = Dψ DΦ Db Dc exp − S , d−wave nσ cp μ μ Dual for the gauge dynamics. Here, the K matrix is just a number. In this respect it seems that there do not exist (28) any massless modes except for fermion degrees of free- where the effective action is dom. We will discuss this point below more carefully. Z Z It is straightforward to see that this effective dual ac- β n d−wave 2 tion has a discrete one-form symmetry S = dτ d x ψ γ (∂ − ic )ψ nσ μ μ μ nσ Dual f f T −1 b 7→ b + q [K ] λ , 1 i cf μ μ μ c 2 cp cp 2 + (ǫ ∂ c ) + ǫ b ∂ c + |(∂ − ib )Φ | Z μνλ ν λ μνλ ν λ μ cp μ μ 2g π ∂ λ − ∂ λ = 0, dl λ ∈ 2πZ. μ ν ν μ μ μ u 1 cp cp 2 2 4 2 C +m |Φ | + |Φ | + (ǫ ∂ b ) . (29) cp cp μνλ ν cp 2 2ρ cp The curvature-free condition of the one-form gauge field makes the kinetic-energy term be invariant automati- Here, n and σ in ψ represent spin and flavor quantum nσ cally. The BF term is also invariant in the following numbers, respectively, where the flavor number is deter- way mined by the d − wave pairing symmetry, for example, cp n = 2 in a square lattice. The BF term between b and 1 ′ d−wave T −1 3 c δ S = q [K K] ′ d xǫ λ ∂ a c describes the mutual semionic statistics between Φ μ cp 1 cc μνλ μ ν Dual c λ 2π i cp and ψ with θ = π in ǫ b ∂ c . nσ μνλ ν λ ∈ 2πZ. To investigate the topological nature of the supercon- ducting phase, we consider the case when vortices are One can show that the Wilson line of the dual photon gapped. Then, we obtain the following effective field the- field V (C) = exp ip dl b exhibits the perimeter p f μ ory in the d − wave superconducting phase law, implying that this Z global one-form symmetry is Z 2 cp broken for the ground state. The braiding statistics of Z = Dψ Db Dc dSC nσ μ the Wilson loop and the vortex operator is given by Z Z h β n D E ¯ ′ exp − dτ d x ψ γ (∂ − ic )ψ nσ μ μ μ nσ W (C)V (C ) q p D ED E oi 1 i 1 ′ cp 2 cp 2 W (C) V (C ) q p (ǫ ∂ b ) + ǫ b ∂ c + (ǫ ∂ c ) , μνλ ν μνλ ν λ μνλ ν λ λ 2 2ρ π 2g h i cp T −1 ′ = exp − 2πi(q [K ] p ) Lk(C, C ) . cf f (30) T −1 where the phase stiffness parameter ρ acquires some The factor q [K ] p is multiple of 1/2, and the statis- cp cf f renormalization from gapped vortex excitations. Com- tics is semionic. pared with the canonical expression discussed in the pre- However, the above discussion is based on incomplete vious section treatment for massless Dirac fermions, i.e., considering Z Z Z 1 h β n dSC 3 −1 f f ′ ′ ′ ′ cp S = d x [H ] (ǫ ∂ b )(ǫ ∂ b ′) cp 2 2 ff μνλ ν μν λ ν Dual λ λ Z = Db Dc exp − dτ d x (ǫ ∂ b ) 8π μ μνλ ν μ λ 2ρ cp K 1 fc oi f c c c i 1 ′ ′ ′ ′ + ǫ b ∂ a + G (ǫ ∂ a )(ǫ ∂ a ′) μνλ ν cc μνλ ν μν λ ν cp 2 μ λ λ λ + ǫ b ∂ c + (ǫ ∂ c ) , (35) 2π 2 μνλ ν λ μνλ ν λ π 2g +ψ γ (∂ − ia )ψ , nσ μ μ nσ where such Dirac fermions were assumed to be mas- sive through spontaneous “chiral symmetry breaking” we have identification of [18, 19], for example, and thus irrelevant at low ener- f cp c b = b δ , a = c δ , (31) gies. If we consider the limit of large number of flavors, f1 μ c1 μ μ μ 6 we have an effective field theory which describes a confor- given by mal invariant fixed point in three spacetime dimensions x† y† y [20], given by H = ǫ p p + p p . O p i+x/2σ i+x/2σ i+y/2σ i+y/2σ i=(i ,i ) x y 64 1 cp cp √ ′ ′ ′ L = N (ǫ ∂ b ) (ǫ ∂ b ) dSC f μνλ ν μν λ ν ′ (39) λ λ −∂ i 1 1 p (p ) is an electron annihilation operator at cp i+x/2σ i+y/2σ + ǫ b ∂ c + (ǫ ∂ c )√ (ǫ ′ ′∂ ′c ′) . μνλ ν λ μνλ ν λ μν λ ν λ π 16 −∂ the O site i + x/2 (i + y/2), where it acts on the p 2− (p ) orbital for O with its local energy ǫ . O sites are (36) y p in the middle of Cu sites on the square lattice, where The nonlocal expression for the kinetic-energy term of |x| = |y| = 1 with (x, y). Hopping of electrons is realized gauge fields should be understood in the momentum by the hybridization between the Cu d 2 2 orbital and x −y space. Here, the Maxwell dynamics for both kinetic the O p and p orbitals, given by x y energies of gauge fields are irrelevant and neglected in cp this expression. In addition, b has been scaled as H = −t d p + H.c. Cu−O pd iσ i+x/2σ cp N b . We emphasize that this is a self-consistent ef- λ i=(i ,i ) x y fective Lagrangian in the large-flavor limit, which may X † y † +d p + H.c. − t d p + H.c. be regarded to be classical. Although the existence of pd iσ i+y/2σ iσ i−x/2σ i=(i ,i ) massless fermions break the discrete one-form symme- x y try of the theory without fermions, one can see that Z 2 † y +d p + H.c. . (40) iσ i−y/2σ one-form global symmetry emerges in this limit. Based on this critical field theory, one may calculate both Wil- Here, t is strength of the dp hybridization, referred to pd son’s lines and find the perimeter law, which results from as an overlap integral of wave functions. the deconfined nature of both quasiparticles and vortices Based on this effective lattice Hamiltonian, one may involved with their long-range effective interactions. In consider other words, the Z one-form global symmetry is broken spontaneously in the presence of massless Dirac fermions U/t → ∞. (41) pd at least in the large−N limit. Based on this observation, we claim that the system acquires Z topological order 2 This limiting procedure gives rise to the following effec- in this phase. tive Hamiltonian [21], referred to as the t − J model X X H = −t (c˜ c˜ + H.c.) + J S · S − n n . eff jσ i j i j iσ IV. TOPOLOGICAL PROPERTY OF d − wave ij ij SUPERCONDUCTORS FROM DOPED MOTT (42) INSULATORS Here, c˜ = P c P is an electron annihilation operator, iσ G iσ G A. High T cuprates as doped Mott insulators where P = Π (1 − n n ) is the Gutzwiller projection G j i↑ i↓ operator to extract out double occupancy sites. At half 1. Effective UV lattice Hamiltonian filling, the kinetic energy term vanishes identically. In other words, hopping of these electrons can be realized only when inter sites are empty. More precisely, c˜ rep- iσ One way to see high T cuprate superconductors is that resents a doped hole, referred to as a Zhang-Rice singlet they are from doped Mott insulators [21]. To discuss [21]. Since ǫ > ǫ is satisfied for high T cuprates, an p d c this aspect, we introduce an effective lattice Hamiltonian, electron can be extracted out from the O site via hole which consists of Cu and O effective lattice Hamiltonians doping. Such a doped hole turns out to form a singlet pair with the hole of the Cu site for the d configura- H = H + H + H . (37) UV Cu O Cu−O tion. Although the term of the Zhang-Rice singlet ex- Here, the local Hamiltonian for the Cu site is given by plains the UV origin precisely, we just use the term of electrons for convenience. These electrons have superex- X X d d H = ǫ d d + U n n . (38) Cu d iσ change interactions, given by the last Heisenberg term. i↑ i↓ iσ i=(i ,i ) i=(i ,i ) x y x y S = c σ c is a spin operator and n = n + n i αβ iβ i i↑ i↓ iα is a density operator with n = c c . The Gutzwiller iσ iσ iσ d is an electron annihilation operator at the Cu site iσ projection operator guarantees the following constraint 2+ i, where it acts on the d 2 2 orbital for Cu with its x −y local energy ǫ . U is an effective Coulomb interaction n ≤ 1, (43) 2+ d energy for the d 2 2 orbital of Cu . n = d d is an x −y iσ iσ iσ electron density operator of spin σ. i = (i , i ) denotes which implies that the low-energy effective Hamiltonian x y a square lattice. The local Hamiltonian for the O site is is to describe physics of doped Mott insulators. 7 2. U(1) slave-boson representation and effective U(1) χ and χ are inter-site particle-hole composite fields to ij ij lattice gauge theory give effective bands to holons and spinons, respectively. Δ is a link variable to describe d − wave pairing of ij spinon Cooper pairs, which is our central object. ϕ is To solve the double occupancy constraint given by the an effective potential to decompose density-density in- Gutzwiller projection operator, U(1) slave-boson repre- teractions. We refer one way of this construction to Ref. sentation has been introduced [9]. c = b f . (44) iσ iσ i To discuss possible low energy physics of this effective theory, we focus on transverse fluctuations for these “or- Here, the electron annihilation operator is represented by der parameters” given by a composite operator, given by a holon field b to carry b bs ia f fs −ia s iφ ij ij ij χ = χ e , χ = χ e , Δ = Δ e . an electron charge quantum number and a spinon field ij ij ij ij ij ij f to carry an electron spin quantum number. Then, iσ (49) the constraint equation (43) is expressed as Here, a is a phase field for the hopping order parameter, ij † † which plays the role of U(1) gauge fields. These phase b b + f f = 1, (45) i iσ i iσ fluctuations contribute to lowering the ground state en- which allows either one holon or one spinon with spin σ ergy by spontaneous generation of such U(1) internal at the site i. magnetic fluxes in the lattice. This U(1) gauge redun- Now, it is straightforward to construct the path inte- dancy may be regarded to originate from the U(1) slave- gral expression for the partition function boson representation itself, where the electron annihila- Z Z tion operator of Eq. (44) is invariant under U(1) gauge h β n † iθ iθ i i transformation of f → e f and b → e b . φ is a Z = Df Db Dλ exp − dτ f (∂ − µ iσ iσ i i ij iσ i i τ iσ phase field for the d−wave spinon pairing order parame- X X ter, which carries the internal U(1) gauge charge 2e, i.e., −iλ )f + b (∂ − iλ )b + i λ i iσ τ i i i transformed as φ → φ + θ + θ under the U(1) gauge ij ij i j transformation. oi X X † † Considering −t (f b b f + H.c.) + J S · S − n n . i jσ i j i j iσ j s s ij ij iλ = λ + ia , ϕ = ϕ + δϕ , (50) i iτ i i i i (46) where the superscript s means a saddle-point value, and introducing Here, λ is a Lagrange multiplier field to impose the single X X occupancy constraint Eq. (45). µ is an electron chemical J J − ϕ − ia − δϕ → −ia (51) iτ j iτ potential to determine hole concentration given by δ = 2 2 j∈i j∈i hb b i. The density and spin operators are given by into the resulting lattice model, we construct an effective † † lattice gauge theory as follows n = f f , S = f σ f , (47) iσ iσ i αβ iβ iσ iα Z Z = Df Db Da Da Dφ iσ i iτ ij ij respectively, where the constraint equation has been uti- lized. h β n Introducing Eq. (47) into Eq. (46) and performing exp − dτ f ∂ − µ − λ − ia f τ iτ iσ iσ i the Hubbard-Stratonovich decomposition for interactions i into spin singlet channels, we find an effective field theory bs † ia ij −t χ f e f − H.c. jσ ij iσ ij Z = Df Db Dλ Dχ Dχ DΔ Dϕ iσ i i ij i ij ij X s† −iφ ji − Δ e (ǫ f f ) − H.c. αβ iα jβ Z ij h β n X 4 † ij exp − dτ f (∂ − µ − iλ )δ − ϕ f τ i ij j iσ iσ X X † fs† † 0 s iaji ij + b (∂ − λ − ia )b − t χ b e b − H.c. τ iτ i i i i ji j X X i ij † † −t f χ f − H.c. − Δ (ǫ f f ) − H.c. X X X jσ αβ iα jβ iσ ij ij fs s bs +i a + λ + t χ χ + H.c. iτ ij ij i ij ij X X X i i ij † f † + b (∂ − iλ )b − t b χ b − H.c. + i λ τ i i i i  oi i ij j X fs 2 s 2 i ij i + − |χ | + |Δ | . (52) ij ij X X ij b f 2 2 +t χ χ + H.c. + − |χ | + |Δ | ij ij ij ij Here, a is a time-component of the U(1) gauge field. iτ ij ij Based on this U(1) lattice gauge theory, we construct an oi effective field theory below. We refer this procedure to + ϕ ϕ . (48) i j Ref. [9]. ij 8 B. Absence of topological order in s − wave Φ (Φ ) is a vortex field for the spinon Cooper pair sp b sp b superconductors from doped Mott insulators (holon) field and b (b ) is the corresponding superfluid μ μ sound mode. Such spinon pair and holon sound modes are coupled to internal U(1) gauge fields, described by First, we consider the case of s−wave superconductors BF terms with appropriate U(1) charges. from doped Mott insulators, where spinons are gapped to be neglected at low energies. Then, the corresponding effective field theory is given by Z Z β n sp s−wave 2 2 S = dτ d x |∂ φ − 2a | μ sp μ MI ρ 1 2 2 + |∂ φ − a | + (ǫ ∂ a ) . (53) μ b μ μνλ ν λ 2 2g ρ (ρ ) is the phase stiffness parameter for the dynamics sp b of spinon Cooper pairs (holons), given by ρ ∼ |hΔ i| sp sp 2 iφsp iφ (ρ ∼ |hbi| ) with Δ = hΔ ie (b = hbie ). g is b sp sp the unit of the internal U(1) gauge charge. A noticeable point is that the U(1) gauge charge for spinon Cooper pairs is 2g while that of holons is g. Performing the duality transformation, we obtain an effective dual field theory s−wave sp b Z = DΦ DΦ Da Db Db b sp μ μ μ MI Z Z h β n 2 sp 2 exp − dτ d x |(∂ − ib )Φ | μ sp sp 2 2 4 +m |Φ | + |Φ | sp sp sp 1 i 1 sp 2 sp 2 + (ǫ ∂ b ) + ǫ b ∂ a + (ǫ ∂ a ) μνλ ν μνλ ν λ μνλ ν λ 2ρ π 2g sp b 2 2 2 4 +|(∂ − ib )Φ | + m |Φ | + |Φ | μ b b b Here, we focus on the superconducting phase, where μ b oi both spinon pair and holon vortices are gapped. Then, 1 i b 2 b + (ǫ ∂ b ) + ǫ b ∂ a . (54) the effective dual field theory is given by μνλ ν μνλ ν λ λ μ 2ρ 2π Z Z Z h β n 1 1 s−SC sp b 2 sp 2 b 2 Z = Da Db Db exp − dτ d x (ǫ ∂ b ) + (ǫ ∂ b ) μ μνλ ν μνλ ν μ μ λ MI λ 2ρ 2ρ sp b oi i i 1 sp b 2 + ǫ b ∂ a + ǫ b ∂ a + (ǫ ∂ a ) . (55) μνλ ν λ μνλ ν λ μνλ ν λ μ μ π 2π 2g Compared with Eq. (2), we obtain Since the number of superfluid sound modes is larger than that of U(1) gauge fields, it is natural to expect sp the existence of neutral superfluid modes, which decouple 4π H = (56) from U(1) gauge fields. Indeed, it exists, given by 4π f b f for the stiffness parameter, [b ] = P ′[b] , μ ff μ where the projection operator is K = (57) b + ′ ′ P ′ = δ − [KK ] ff ff ff for the K matrix, and with the Moore-Penrose inverse of K ′ ′ G = δ δ (58) cc c1 c 1 + 2 1 K = . (59) 5 5 for the gauge coupling constant. More explicitly, such a neutral superfluid sound mode is 9 given by while that of the holon gauge field is changed as V = exp i dl b b μ f sp b [b ] = (b − 2b )δ . (60) 0 f1 C μ μ μ 1 4 7→ exp × 2πi V = exp − × 2πi V . (67) b b 5 5 Now, let us study the one-form global symmetry of Thus, this symmetry acts as Z phase rotation on those Eq. (55). The dual effective field theory of the s − wave operators. In fact, this phase rotation can also be gener- superconducting phase from a spin liquid state has the ated as a U(1) one-form symmetry (61), following one-form global symmetry sp sp b b b 7→ b + ǫ(−1)ρ , b 7→ b + ǫ(2)ρ (68) μ μ μ μ μ μ f f + c b 7→ b + [K ] λ , by choosing cf μ μ μ c c c ǫ = − . (69) where λ is a flat connection satisfying ∂ λ − ∂ λ = 0 μ ν μ ν μ with normalization dl λ ∈ 2πZ. From the explicit C Hence, the Z one-form symmetry is a subgroup of the form of the Moore-Penrose inverse of K, one can see that U(1) continuous one-form symmetry, if we look at the it is a Z one-form symmetry. transformation property of physical operators. In three This state also has a continuous one-form symmetry, spacetime dimensions, a continuous one-form symme- try cannot be broken, guaranteed by the generalized f f α ¯ b 7→ b + ǫ[D ] ρ , (61) f μ Coleman-Mermin-Wagner theorem for higher-form sym- μ μ metries [7]. Therefore, the Z one-form symmetry cannot where ǫ is a continuous parameter, ρ is a flat one-form be broken either. As a result, the s − wave supercon- α ¯ connection normalized as dl ρ ∈ 2πZ, and D is the ducting phase with one neutral superfluid sound mode, μ μ element of the cokernel of K, resulting from a doped Mott insulating state, does not have a topological order. This implies that conventional α ¯ [D ] = −1 2 . (62) s−wave superconductivity, induced by the condensation of electron Cooper pairs (identified as superfluidity here), can be smoothly connected with the s − wave supercon- Here, the cokernel is one-dimensional. Under the trans- ducting phase with one neutral superfluid sound mode, formation (61), Wilson loop operators are transformed resulting from the spin liquid state. as Even though there is no topological order in this su- α ¯ perconducting state, the mutual statistics between vor- V (C) 7→ exp 2πip ǫ[D ] V (C). (63) p f f p tices and quasiparticles can be exotic. Let us denote W (C) ≡ exp i dl a . The correlation functions of μ μ There is no discrete one-form symmetry to shift the W (C) and V (C), V (C) satisfy sp b gauge field a. Recall that a generic form of discrete one- 4πi ′ ′ − Lk(C,C ) ′ form transformation takes the form hW (C)V (C )i = e hV (C )i, (70) sp sp 2πi c c + ′ − Lk(C,C ) ′ a 7→ a + [K ] p λ , (64) hW (C)V (C )i = e hV (C )i, (71) cf f μ μ μ b b respectively, where we used the fact that Wilson loops where p ∈ (cokerK) . The charge vector p should be f f associated with a are topological, and α ¯ chosen to be orthogonal to D so that the generator is a hW (C)i = 1. (72) topological operator. One choice with minimal integer is The relations (70, 71) indicate that the Wilson loop p = 2 1 . (65) W (C) is the generator of the Z one-form symmetry and q 5 V (C) is the corresponding charged object. Although the Z one-form symmetry is not spontaneously broken, this Although the transformation (64) with this charge vector 5 braiding phase is an observable effect. does not change the action (up to 2πZ), this is not a symmetry of the system, because this acts trivially on the Wilson loop of a. C. d − wave superconductors from doped Mott In the current case, the Z one-form symmetry is a sub- insulators at large−N group of the U(1) continuous one-form symmetry (61). Under the Z one-form symmetry, the Wilson loop oper- Finally, we discuss the topological properties of d − ator for the spinon-pair gauge field is transformed as wave superconductors from doped Mott insulators, based on the perspectives of higher-form symmetries. Introduc- sp V = exp i dl b sp μ ing massless Dirac fermions into the effective dual field theory Eq. (54) for s−wave superconductors from doped 7→ exp × 2πi V (66) Mott insulators, we obtain sp 5 10 Z Z Z h β n d−wave sp b 2 sp Z = Dψ DΦ DΦ Da Db Db Dc exp − dτ d x ψ γ (∂ − ic )ψ + ǫ b ∂ c nσ b sp μ μ nσ μ μ μ nσ μνλ ν λ MI μ μ μ u 1 i 1 sp sp sp 2 2 2 4 2 sp 2 +|(∂ − ib )Φ | + m |Φ | + |Φ | + (ǫ ∂ b ) + ǫ b ∂ a + (ǫ ∂ a ) μ sp sp sp μνλ ν μνλ ν λ μνλ ν λ μ sp μ 2 2ρ π 2g sp oi u 1 i b 2 2 2 4 b 2 b +|(∂ − ib )Φ | + m |Φ | + |Φ | + (ǫ ∂ b ) + ǫ b ∂ a . (73) μ b b b μνλ ν μνλ ν λ μ b λ μ 2 2ρ 2π Here, ψ represents a massless Dirac fermion field re- term with the statistical angle π. In other words, when nσ sulting from the d − wave pairing symmetry of spinon the Dirac fermion turns around the spinon pair vortex, Cooper pairs. Such massless fermions interact with it acquires the Aharonov-Bohm phase of π. This dual spinon pair vortices statistically, described by the BF effective field theory is reduced into Z Z Z h β n dSC sp b 2 Z = Dψ Da Db Db Dc exp − dτ d x ψ γ (∂ − ic )ψ nσ μ μ nσ μ μ μ nσ MI μ μ 1 1 i i i sp 2 b 2 sp b sp + (ǫ ∂ b ) + (ǫ ∂ b ) + ǫ b ∂ a + ǫ b ∂ a + ǫ b ∂ c μνλ ν μνλ ν μνλ ν λ μνλ ν λ μνλ ν λ λ μ μ μ 2ρ 2ρ π 2π π sp b oi 1 1 2 2 + (ǫ ∂ a ) + (ǫ ∂ c ) (74) μνλ ν λ μνλ ν λ 2 2 2g 2g a c in the superconducting phase, where both spinon pair Before we study the topological property of Eq. (74), and holon vortices are gapped. we consider the case that Dirac fermions are gapped, and thus neglected at low energies, described by n o 1 K 1 ′ f f fc dSC−CSB 3 −1 f c c c ′ ′ ′ ′ ′ ′ ′ ′ S = d x [H ] (ǫ ∂ b )(ǫ ∂ b ) + ǫ b ∂ a + G (ǫ ∂ a )(ǫ ∂ a ′) , ff μνλ ν μν λ ν ′ μνλ ν cc μνλ ν μν λ ν μ λ λ λ Dual−MI λ λ 8π 2π 2 where we have correspondences due to “chiral symmetry” breaking [18, 19]. We also note that there is no continuous one-form symmetry in 1 sp 2 b b = b , b = b , μ μ μ μ this case because K is full-rank. As a result, this super- 1 2 conducting state has the Z discrete one-form symmetry. a = a , a = c (75) 2 μ μ μ μ Wilson loops obey the perimeter law and this symmetry for fields, is spontaneously broken. Hence this phase has a Z topo- sp logical order. The mutual statistics between spinon-pair 4π H = (76) b vortices and quasiparticles is semionic, which confirms 4π the Z topological order. for the stiffness matrix, 2 2 0 1 −1 K = → K = (77) 1 0 1/2 −1 for the K matrix, and G = (78) for the gauge coupling-constant matrix. Since we have Now, we introduce the role of massless Dirac fermions dim (ker K) = dim (coker K) = 0, there are no massless into the above discussion. If we consider the limit of degrees of freedom in this “d − wave” superconducting large number of flavors, we have an effective field theory phase, where such massless Dirac fermions are gapped which describes a conformal invariant fixed point in three 11 spacetime dimensions [20], given by that this topological phase transition is a deconfinement- confinement transition involved with the compact U(1) 1 1 dSC gauge field a , where magnetic monopoles as instantons L = N (ǫ ∂ c )√ (ǫ ′ ′∂ ′c ′) f μνλ ν λ μν λ ν λ Dual−MI −∂ in three spacetime dimensions play an essential role. 1 1 Performing the duality transformation for compact √ ′ ′ ′ ′ + (ǫ ∂ a ) (ǫ ∂ a ) μνλ ν λ μν λ ν λ U(1) gauge fields a and taking into account the dilute −∂ 64 1 instanton-gas approximation, we obtain an effective field sp sp + (ǫ ∂ b )√ (ǫ ′ ′∂ ′b ) μνλ ν μν λ ν ′ λ λ 2 theory for instanton-type magnetic monopole excitations −∂ 16 1 b b 1 1 dSC √ ′ ′∂ ′b ) + (ǫ ∂ b ) (ǫ ′ μνλ ν μν λ ν ′ ′ ′ ′ λ λ L = N (ǫ ∂ c )√ (ǫ ∂ c ) 2 f μνλ ν λ μν λ ν λ Dual−MI −∂ 2 −∂ i i i sp b sp 64 1 sp sp + ǫ b ∂ a + ǫ b ∂ a + ǫ b ∂ c . μνλ ν λ μνλ ν λ μνλ ν λ μ μ μ + (ǫ ∂ b )√ (ǫ ′ ′∂ ′b ) μνλ ν μν λ ν λ λ π 2π π 2 −∂ (79) 16 1 i b b sp ′ ′ ′ + (ǫ ∂ b )√ (ǫ ∂ b ′) + ǫ b ∂ c μνλ ν μν λ ν μνλ ν λ λ λ μ sp sp b b 2 π π Here, b → N b and b → N b have been performed. −∂ f f μ μ μ μ Again, we point out that this effective field theory is self- + (∂ ϕ) −∂ (∂ ϕ) − y cosϕ μ μ m consistent and classical in the large number of fermion flavors. All types of superconducting phase fluctuations i 1 sp b + (∂ ϕ) b + b . (80) μ μ and U(1) gauge fields are massless, more precisely, the π 2 dynamics of which is conformal invariant. In spite of Here, ϕ is a scalar magnetic potential field to mediate this quantum criticality, we point out that all the Wilson effective interactions between instanton-type magnetic loop operators show their perimeter-law behaviors, where monopole excitations and y is an effective fugacity given both vortices and quasiparticles are deconfined to have −S inst by y ∼ e with a renormalized instanton action effective Coulomb interactions. For comparison with the S . Doping dependence would be introduced into the inst absence of massless Dirac fermions, we recall the corre- effective instanton action. Although we are not claiming sponding effective field theory that this procedure is rigorously performed in the pres- 1 1 dSC−CSB 2 2 ence of massless Dirac fermions, we can argue that the L = (ǫ ∂ c ) + (ǫ ∂ a ) μνλ ν λ μνλ ν λ Dual−MI 2 2 2g 2g c a condensation transition of magnetic monopoles may be 1 1 a Berezinskii-Kosterlitz-Thouless (BKT) type according sp 2 b 2 + (ǫ ∂ b ) + (ǫ ∂ b ) μνλ ν μνλ ν λ λ to one scenario [22–24]. If the condensation of such mag- 2ρ 2ρ sp b netic monopoles occurs, magnetic scalar potential fluc- i i i sp b sp + ǫ b ∂ a + ǫ b ∂ a + ǫ b ∂ c , tuations are gapped, expected to appear above the op- μνλ ν λ μνλ ν λ μνλ ν λ μ μ μ π 2π π timal doping. Taking into account both magnetic scalar potential fluctuations and “holon” current fluctuations, where all gauge fields are gapped, and thus Z topologi- we find essentially the same effective field theory as Eq. cally ordered. The Maxwell dynamics are replaced with (36), which describes d − wave superconductivity from the conformal invariant kinetic energy due to the pres- Landau’s Fermi liquids. This potential existence of a ence of massless Dirac fermions. The coexistence of the topological phase transition between d− wave supercon- Z topological order and quantum criticality is an essen- ductors in high T cuprates, given by a deconfinement- tial feature for d − wave superconductors in the large confinement transition, is one of the main implications of number of fermion flavors. the current study and it deserves further studies. This analysis suggests that d − wave superconductors from U(1) spin liquids may not be adiabatically con- nected with those from Landau’s Fermi liquids, in con- V. CONCLUSION trast to the case of s − wave superconductors as long as the physics of massless Dirac fermions is governed by the Statistical interactions between vortices and Dirac conformal invariant fixed point. We recall the effective fermions in d − wave superconductors turn out to play field theory of Eq. (36) for d − wave superconductors an important role for the topological order in the per- from Landau’s Fermi liquids spectives of global one-form symmetries when such Dirac 64 1 fermions become gapped due to possible “chiral symme- cp cp ′ ′ ′ L = N (ǫ ∂ b )√ (ǫ ∂ b ) dSC f μνλ ν μν λ ν ′ λ λ 2 try breaking”. On the other hand, the existence of mass- −∂ less Dirac fermions gives rise to the fact that low energy i 1 1 cp + ǫ b ∂ c + (ǫ ∂ c )√ (ǫ ′ ′∂ ′c ′) . μνλ ν λ μνλ ν λ μν λ ν λ fluctuations such as both supercurrent and U(1) gauge- π 16 −∂ field fluctuations become critical in the large number The difference of the K matrix between Eq. (79) of fermion flavors. Interestingly, the discrete one-form fc and Eq. (36) indicates that the topological order of global symmetry still remains in the resulting effective Eq. (79) differs from that of Eq. (36). We speculate field theory of the large−N limit to be spontaneously f 12 broken in the superconducting phase. In other words, a remains as an interesting future project. spontaneous breaking of a discrete one-form symmetry may coexist with quantum criticality in d − wave super- ACKNOWLEDGMENTS conductors. One may criticize that the role of massless Dirac fermions in the notion of topological order for d − wave K. K. was supported by the Ministry of Ed- superconductors is not clarified in a rigorous sense. The ucation, Science, and Technology (No. NRF- coexistence between the spontaneous breakdown of the 2015R1C1A1A01051629 and No. 2011-0030046) of the discrete one-form symmetry and the quantum critical- National Research Foundation of Korea (NRF) and by ity should be verified with a concrete mathematical ma- TJ Park Science Fellowship of the POSCO TJ Park chinery. One possible way of analysis would be the use Foundation. Y. 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A. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Condensed Matter arXiv (Cornell University)

Higher-form symmetries and $d-wave$ superconductors from doped Mott insulators

Kim, Ki-Seok; Hirono, Yuji
Condensed Matter , Volume 2019 (1905) – May 12, 2019
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10.1103/PhysRevB.100.085142
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Abstract

1 2, 1 Ki-Seok Kim and Yuji Hirono Department of Physics, POSTECH, Pohang, Gyeongbuk 37673, Korea Asia Pacific Center for Theoretical Physics (APCTP), Pohang, Gyeongbuk 37673, Korea (Dated: May 14, 2019) One path to high-temperature cuprate superconductors is doping a Mott insulator. In this paper, we study this system from the view point of higher-form symmetries. On the introduction of slave bosons, the t − J model at a finite hole doping can be written in the form of U(1) gauge theories. After a duality transformation, they can be written as a generalized BF theory, in which the higher- form symmetries are more manifest. We identify the emergent continuous and discrete higher-form symmetries in both s − wave and d − wave superconducting phases, expected to be realized from a doped Mott insulator. The existence of a topological order is tested by examining if there is a spontaneously broken discrete one-form symmetry. We claim that a spontaneous breaking of discrete one-form symmetry may extend to a phase that has massless Dirac fermions in the limit of large number of flavors. We discuss the possibility of a topological phase transition inside the superconducting dome of high T cuprates. I. INTRODUCTION corresponding charged objects are Wilson loops and vor- tex operators. Those two operators obey fractionalized statistics. In the large-loop limit, the expectation values Topological orders provide us with a finer classification of those operators show the perimeter law, which means of gapped quantum phases beyond the Landau-Ginzburg that both one-form symmetries are spontaneously bro- theory based on symmetry breaking patterns [1]. Topo- ken, and the system acquires a topological order. logically ordered states exhibit such properties as frac- In this paper, we study the structure of higher-form tionalized topological excitations, long-range entangle- symmetries of two-dimensional high temperature super- ment, and degeneracy of the ground states that depend conductors. The realization of superconductivity from on the spacetime topology [2, 3]. A conventional s−wave doped Mott insulators is known as a spin-liquid scenario superconductor is an example of topologically ordered for high T cuprates [9]. Resorting to the U(1) slave- phase [4], where the key ingredient is that the Cooper c boson representation of the t − J effective Hamiltonian, pairs have charge 2e. Because of this, if we compactify an effective field theory for d − wave superconductors one spatial direction, we can insert a half magnetic flux is obtained from a spin liquid state, based on the as- in this direction. This state has exactly the same energy sumption of the separation of spin and charge degrees of as the state with no flux, and the ground states become freedom. We take an Abelian duality to see the symme- doubly degenerate. Since the superconductivity does not try structure in a manifest way, and identify the emergent involve symmetry breaking (U(1) gauge symmetry is not higher-form symmetries in the dual effective theories. By a symmetry), the distinction of a superconducting state examining whether those symmetries are spontaneously with a normal state is given by the existence of a topo- broken or not, we clarify the nature of topological order logical order. in such systems. In this analysis, electromagnetic U(1) In certain cases, the appearance of topological order gauge fields are taken to be background potentials. As a can be understood as a consequence of a spontaneous result, ordinary superconductors are identified with su- breaking of higher-form symmetries [5–8], which is a gen- perfluids, that are not topologically ordered. eralization of symmetry. The charged objects of higher- form symmetries are extended objects like line operators When massless Dirac fermions are absent, which corre- or surface operators, and an ordinary symmetry corre- sponds to s−wave superconductors from doped Mott in- sponds to a zero-form symmetry, where the correspond- sulators, we find that the superconducting state has a Z ing charged object is point-like. The concepts such as one-form symmetry. This symmetry turns out to be not Noether theorem and Nambu-Goldstone (NG) theorem spontaneously broken, and there is no topological order are also generalized to higher-form symmetries. For ex- associated with this symmetry. In fact, the Z one-form ample, photons can be understood as NG modes associ- symmetry is shown to be a subgroup of a continuous U(1) ated with the spontaneous breaking of continuous one- one-form symmetry of this state. A generalized version form symmetry. A topological order appears when a of the Coleman-Mermin-Wagner theorem for the contin- discrete higher-form symmetry is spontaneously broken. uous higher-form symmetries prohibits the spontaneous The example of s−wave superconductivity is in this cat- breaking of continuous one-form symmetry in 2 + 1 di- egory. The effective theory can be written as an Abelian mensions, and hence the Z symmetry cannot be broken Higgs model, and after taking an Abelian duality and either. And yet, the Z symmetry is observable in the dropping massive excitations, it can be written as a topo- braiding phase of quasiparticles, as we show in the main logical BF theory. The action has a pair of emergent text. From this observation, we conclude that the topo- Z one-form symmetries (in 2+1 dimensions), and the logical property of s−wave superconducting phase from arXiv:1905.04617v1 [cond-mat.str-el] 12 May 2019 2 U(1) spin liquids is the same as conventional s − wave topological order and superfluidity coexist [12, 13]. The superconductivity from Landau’s Fermi liquids . structure of higher-form symmetries is analyzed, and One subtle issue regarding topological order is that such analysis is useful in classifying quantum phases, there are massless Dirac fermions that statistically in- because a spontaneous breaking of discrete higher-form teract with vortices [14, 15]. In the presence of such symmetries results in a topological order. The formalism coupling to fermions, the discrete one-form symmetry is is applied to the color-flavor locked phase of color super- absent. However, we point out that, if we consider the conductivity of QCD matter. Since this mathematical large−N (number of fermions) limit of d − wave super- structure is also applicable to the present study, let us conductors (this phase has massless Dirac fermions), at briefly review it here in the case of 2 + 1 dimensions. which a conformal fixed point is realized, a Z one-form symmetry is effectively restored and further it is spon- taneously broken. As a result, we claim that d − wave superconductors from doped Mott insulators cannot be smoothly connected to those from Landau’s Fermi liq- uids, because they can be distinguished by a discrete one-form symmetry. This is in contrast to the case of s − wave superconductors. The rest of the paper is organized as follows. In Sec. II, we give a review of a generalized BF theory for the de- scription of superfluids with topological order. In Sec. III, we study the topological properties of d−wave supercon- ductors realized from Landau’s Fermi liquids. In Sec. IV, we study the topological nature of d − wave supercon- ductors from doped Mott insulators. In Sec. V we give concluding remarks. II. GENERALIZED BF THEORY FOR SUPERFLUIDITY WITH TOPOLOGICAL ORDER The starting point is a theory with multiple U(1) sym- metries, in which some parts of them are coupled with A. Effective gauge theory for superfluids with gauge fields through a covariant derivative. At low ener- topological order gies, the only relevant degrees of freedom are the massless NG modes and topological excitations. Such a situation A generalized BF -type theory in 3 + 1 dimensions is can be captured with the following Lagrangian of a gen- studied recently that can describe the situations where eralized Abelian Higgs model, n o ′ ′ ′ 1 1 3 f c f c c c S = d x [H] ′(∂ φ + [K] a )(∂ φ + [K] ′ ′a ) + [G] ′A A , (1) AH ff μ fc μ f c cc μ μ μν μν 2 2 where φ are 2π-periodic scalar fields representing the respectively. phases of Cooper pair fields, a are U(1) gauge fields, To study the existence of topological order, let us take c c c and A ≡ ∂ a − ∂ a are the field strength tensors an Abelian duality transformation so that the higher- μ ν μν ν μ for a . There are multiple scalar and gauge fields, and form symmetries of the system can be seen in a manifest f and c are the indices of the scalar and gauge fields, way. The dual action is written as n o 1 ′ [K] 1 ′ fc 3 −1 f f f c c c S = d x [H ] ′B B + ǫ b ∂ a + [G] ′A A , Dual ff μνλ ν cc μν μν μ λ μν μν 8π 2π 2 (2) A similar situation occurs in the dense QCD matter [10]. There continuity [11]. This observation is based on the fact that those is a scenario that hadronic superfluid phase can be continuously two phases have the same (zero-form) symmetries. The continu- connected to a color superconducting phase, called quark-hadron ity is recently extended to include the topological orders through the analysis of higher-form symmetries [12, 13]. 3 f f f f where b is a one-form gauge field and B = ∂ b −∂ b when integrated over a closed loop C. Under the trans- μ ν μ μν ν μ f f formation (5), the kinetic-energy term for supercurrent is a two-form field strength tensor for b . The field b μ μ b f fluctuations is not affected, since δ[P b ] = 0. The BF is dual to φ , which describe sound modes (supercurrent ff μ term changes as fluctuations). The effective theory is invariant under the gauge transformations, T + 3 c δ S = q [K K] ′ d xǫ λ ∂ a . (8) 1 Dual cc μνλ μ ν c λ c c c f f f 2π a 7→ a + ∂ λ , b 7→ b + ∂ λ , (3) μ μ μ μ μ μ Noting that [K K] ′ is a orthogonal projection matrix cc c f where λ and λ are 2π-periodic parameters. The gauge ⊥ T ⊥ to (kerK) and we have chosen q ∈ (kerK) , we have fields satisfy the standard Dirac quantization condition, T + T q [K K] = q and cc ′ Z Z c f δ S ∈ 2πZ. (9) dS A ∈ 2πZ, dS B ∈ 2πZ, (4) 1 Dual μν μν μν μν Thus, the weight of the path integral is unchanged un- where the integrations are over closed 2-dimensional sub- der this transformation. If this transformation acts on manifolds. The gauge invariance requires that the entries physical operators (vortex operators) nontrivially, this is of [K] are integers. fc a symmetry of the system. The entries of [K ] are cf The final action is obtained by neglecting the kinetic in general fractional numbers, and in that case a discrete terms of the massive sound modes and photons. Since one-form symmetry. The charged objects under this sym- the number of flavor is not necessarily the same as that of metry are vortex operators, color, the matrix K is non-square in general . When the K matrix has a nontrivial cokernel, i.e., dim (cokerK) 6= f V (C) = exp ip dl b , (10) p f μ 0, there are gapless superfluid sound modes. The num- ber of gapless modes is given by dim (cokerK). Those where p is a charge vector and C is a closed one dimen- f b f massless modes can be identified as [b ] = P [b] , 0 ′ μ ff μ sional manifold. This is transformed by the one-form b + ′ ′ where P ≡ δ − [KK ] is the orthogonal projec- ′ ff ff ff symmetry as tion matrix to cokerK, and K is the Moore-Penrose inverse [16] of K. Similarly, when K has a nontriv- V (C) 7→ exp 2πip [K ] V (C). (11) p f cf p ial kernel, i.e., dim (ker K) 6= 0, we have gapless pho- c a c Similarly, one may consider the following transforma- tons, and they are identified as [a ] = P ′[a] , where μ cc μ a + tion, P = δ ′ −[K K] ′ is the projection matrix to kerK. cc cc cc c c + a 7→ a − [K ] p ω , (12) cf f μ μ μ B. Higher-form symmetries ⊥ where p ∈ (cokerK) is an integer vector, and ω is a f μ flat one-form field satisfying the curvature-free condition Now we are ready to discuss higher-form symmetries of ∂ ω − ∂ ω = 0 and is normalized as dl ω = 2πZ. μ ν ν μ μ μ the dual effective theory. Equation (2) is invariant under Here, (cokerK) indicates the orthogonal complement of the following transformation, cokerK. Under this transformation, the action is varied as f f T + b 7→ b + q [K ] λ , (5) cf μ μ μ c + 3 f δ S = − [KK ] ′p d xǫ ∂ b ω . (13) 1 Dual ff f μνλ μ λ where q is an integer vector chosen from the orthogonal 2π complement of kerK, i.e., q ∈ (kerK) , and λ is a c μ Noting that KK is a symmetric matrix and one-form field that satisfies the curvature-free condition, [KK ] ′ p = p ′, we have f f f f ∂ λ − ∂ λ = 0, (6) μ ν ν μ δ S ∈ 2πZ. (14) 1 Dual with the normalization Z The charged objects under this symmetry are the Wilson loops, dl λ ∈ 2πZ, (7) μ μ C Z W (C) = exp iq dl a , (15) q c μ characterized by a charge vector q . 2 c Although a matrix BF theory is used in [12, 13] (which is neces- In addition to discrete one-form symmetries, the effec- sary for 3 + 1 dimensions), in 2 + 1 dimensions, it is also possible to treat all the one-form fields on equal footing and write the tive dual field theory Eq. (2) has the following continuous action in the form of a Chern-Simons theory with matrix coef- U(1) one-form symmetry, ficients. When there is no diagonal term such as a ∧ da, the c c α BF -type description is more economical and we adopt this. a 7→ a + ǫ[C ] λ , (16) c μ μ μ 4 and In this case, the system acquires degeneracy depending on the spacetime topology. If the vortex couples to mass- f f α ¯ b 7→ b + ǫ[D ] λ . (17) f μ less modes, it decays faster than the perimeter law and μ μ the symmetry may not be broken. α α ¯ where ǫ is a continuous parameter, and [C ] and [D ] c f The condition for the existence of topological order can are basis vectors of the kernel and cokernel of the K be summarized as follows: There exists a pair of integer matrix, ⊥ ⊥ vectors (p, q) ∈ (cokerK) × (kerK) such that α α ¯ T + [K] [C ] = 0, [D ] [K] = 0. (18) fc c f fc iq [K ] p cf f e 6= 1. (24) Those vectors are labeled by α and α¯, This condition can be interpreted as that for the presence of mixed ’t Hooft anomalies of one-form symmetries [13]. α = 1, ..., dim (kerK), One way to see this is introducing background two-form α¯ = 1, ..., dim (cokerK). (19) gauge fields for a pair of (discrete) one-form symmetries and considering the partition function. The phase of the partition function may become ambiguous in the pres- C. Statistics of quasi-particles and vortices ence of those background gauge fields, if the factor (24) is different from 1. This indicates the existence of a ’t Hooft In the current setting, not all the Wilson operators or anomaly between those symmetries. In such a case, we vortex operators are topological, because of the presence cannot realize the system in 2 + 1 dimensions in a gauge of massless NG modes. A topological operator is an op- invariant way and we have to couple it to a symmetry- erator invariant under the deformation of the underlying protected topological (SPT) phase in 3 + 1 dimensions. manifold, Then, the ’t Hooft anomaly of the boundary state is can- celled by the anomaly inflow mechanism from the SPT W (C + δC) = W (C), (20) q q bulk phase [17]. Since this anomaly matching structure is invariant under renormalization group, the presence when the deformation does not cross other operators. of the ’t Hooft anomaly precludes a trivial ground state This does not hold if the operator couples to massless for the 2 + 1-dimensional theory. In the present case, modes. Even in that case, it is possible to extract the the anomaly is matched by a topological order in the information about the braiding phase of those operators. infrared. It can be shown that [13] hW (C)V (C )i q p III. TOPOLOGICAL PROPERTY OF d − wave hW (C)ihV (C )i SUPERCONDUCTORS FROM LANDAU’S q p + ′ FERMI LIQUIDS = exp −2πi(q [K ] p ) Lk(C, C ) , (21) c cf f where Lk(C, C ) is the Gauss linking number of the two The dynamics of superfluid fluctuations at low energies world lines C and C . Their mutual statistics is encoded is described by the following effective action T + in q [K ] p . cf f c Z Z cp 2 2 The braiding phase (21) is closely related to the dis- S = dτ d x(∂ φ ) , (25) s−wave μ cp crete one-form symmetries of the system. Suppose a Wil- son loop associated with a charge vector q is topological. where ρ is a phase stiffness parameter proportional to cp Then, the loop C can be contracted to a point, which the Cooper-pair density and φ is a phase field to de- cp means that hW (C)i = 1 (up to the part obeying the scribe sound modes. Here, we consider two space di- perimeter law). When C is singly linked to C , we have mensions in the Euclidean time, where β is the inverse of temperature, set to be infinite, i.e., considering zero ′ −2πi(q [K ] p ) ′ c cf f hW (C)V (C )i = e hV (C )i. (22) q p p temperature. Performing the duality transformation, we obtain an effective dual field theory in terms of vortices cp This is nothing but the definition of the existence of a Φ and superfluid sound modes b , where the partition cp one-form symmetry [7]. Here, W (C) is the generator of function is the symmetry and V (C ) is a charge object. cp s−wave The existence of (discrete) one-form symmetry is not Z = DΦ Db exp − S , (26) s−wave cp Dual enough for a system to be topologically ordered. There has to be a discrete one-form symmetry which is sponta- and the dual effective action is neously broken, to have a topological order. This means Z Z β n s−wave 2 cp 2 2 2 that at large loop C, the corresponding charged object S = dτ d x |(∂ − ib )Φ | + m |Φ | μ cp cp Dual μ cp of the symmetry should behave as u 1 cp 4 cp 2 + |Φ | + (ǫ ∂ b ) . (27) cp μνλ ν hV (C) ≃ exp (−κ perimeter[C]) . (23) p 2 2ρ cp 5 To describe the physics of d−wave superconductors, it for dual photon and statistical photon fields, respectively, is necessary to introduce massless Dirac fermions, which 4π −1 results from the d−wave pairing symmetry. An essential ′ ′ [H ] = δ δ , (32) ff f1 f 1 cp point is that such massless Dirac fermions have statistical interactions with vortices, described by an effective BF for the phase stiffness parameter, term with a statistical angle π, which implies mutual semionic statistics [14, 15]. An effective field theory for −1 [K] = 2δ δ −→ [K ] = , (33) fc f1 c1 11 d − wave superconductors is given by the path integral expression for massless Dirac fermions ψ , Cooper-pair nσ for the K matrix, and cp vortices Φ , superfluid fluctuations b , and emergent cp U(1) statistical gauge fields c , G ′ = δ δ , (34) c1 c 1 cc cp d−wave Z = Dψ DΦ Db Dc exp − S , d−wave nσ cp μ μ Dual for the gauge dynamics. Here, the K matrix is just a number. In this respect it seems that there do not exist (28) any massless modes except for fermion degrees of free- where the effective action is dom. We will discuss this point below more carefully. Z Z It is straightforward to see that this effective dual ac- β n d−wave 2 tion has a discrete one-form symmetry S = dτ d x ψ γ (∂ − ic )ψ nσ μ μ μ nσ Dual f f T −1 b 7→ b + q [K ] λ , 1 i cf μ μ μ c 2 cp cp 2 + (ǫ ∂ c ) + ǫ b ∂ c + |(∂ − ib )Φ | Z μνλ ν λ μνλ ν λ μ cp μ μ 2g π ∂ λ − ∂ λ = 0, dl λ ∈ 2πZ. μ ν ν μ μ μ u 1 cp cp 2 2 4 2 C +m |Φ | + |Φ | + (ǫ ∂ b ) . (29) cp cp μνλ ν cp 2 2ρ cp The curvature-free condition of the one-form gauge field makes the kinetic-energy term be invariant automati- Here, n and σ in ψ represent spin and flavor quantum nσ cally. The BF term is also invariant in the following numbers, respectively, where the flavor number is deter- way mined by the d − wave pairing symmetry, for example, cp n = 2 in a square lattice. The BF term between b and 1 ′ d−wave T −1 3 c δ S = q [K K] ′ d xǫ λ ∂ a c describes the mutual semionic statistics between Φ μ cp 1 cc μνλ μ ν Dual c λ 2π i cp and ψ with θ = π in ǫ b ∂ c . nσ μνλ ν λ ∈ 2πZ. To investigate the topological nature of the supercon- ducting phase, we consider the case when vortices are One can show that the Wilson line of the dual photon gapped. Then, we obtain the following effective field the- field V (C) = exp ip dl b exhibits the perimeter p f μ ory in the d − wave superconducting phase law, implying that this Z global one-form symmetry is Z 2 cp broken for the ground state. The braiding statistics of Z = Dψ Db Dc dSC nσ μ the Wilson loop and the vortex operator is given by Z Z h β n D E ¯ ′ exp − dτ d x ψ γ (∂ − ic )ψ nσ μ μ μ nσ W (C)V (C ) q p D ED E oi 1 i 1 ′ cp 2 cp 2 W (C) V (C ) q p (ǫ ∂ b ) + ǫ b ∂ c + (ǫ ∂ c ) , μνλ ν μνλ ν λ μνλ ν λ λ 2 2ρ π 2g h i cp T −1 ′ = exp − 2πi(q [K ] p ) Lk(C, C ) . cf f (30) T −1 where the phase stiffness parameter ρ acquires some The factor q [K ] p is multiple of 1/2, and the statis- cp cf f renormalization from gapped vortex excitations. Com- tics is semionic. pared with the canonical expression discussed in the pre- However, the above discussion is based on incomplete vious section treatment for massless Dirac fermions, i.e., considering Z Z Z 1 h β n dSC 3 −1 f f ′ ′ ′ ′ cp S = d x [H ] (ǫ ∂ b )(ǫ ∂ b ′) cp 2 2 ff μνλ ν μν λ ν Dual λ λ Z = Db Dc exp − dτ d x (ǫ ∂ b ) 8π μ μνλ ν μ λ 2ρ cp K 1 fc oi f c c c i 1 ′ ′ ′ ′ + ǫ b ∂ a + G (ǫ ∂ a )(ǫ ∂ a ′) μνλ ν cc μνλ ν μν λ ν cp 2 μ λ λ λ + ǫ b ∂ c + (ǫ ∂ c ) , (35) 2π 2 μνλ ν λ μνλ ν λ π 2g +ψ γ (∂ − ia )ψ , nσ μ μ nσ where such Dirac fermions were assumed to be mas- sive through spontaneous “chiral symmetry breaking” we have identification of [18, 19], for example, and thus irrelevant at low ener- f cp c b = b δ , a = c δ , (31) gies. If we consider the limit of large number of flavors, f1 μ c1 μ μ μ 6 we have an effective field theory which describes a confor- given by mal invariant fixed point in three spacetime dimensions x† y† y [20], given by H = ǫ p p + p p . O p i+x/2σ i+x/2σ i+y/2σ i+y/2σ i=(i ,i ) x y 64 1 cp cp √ ′ ′ ′ L = N (ǫ ∂ b ) (ǫ ∂ b ) dSC f μνλ ν μν λ ν ′ (39) λ λ −∂ i 1 1 p (p ) is an electron annihilation operator at cp i+x/2σ i+y/2σ + ǫ b ∂ c + (ǫ ∂ c )√ (ǫ ′ ′∂ ′c ′) . μνλ ν λ μνλ ν λ μν λ ν λ π 16 −∂ the O site i + x/2 (i + y/2), where it acts on the p 2− (p ) orbital for O with its local energy ǫ . O sites are (36) y p in the middle of Cu sites on the square lattice, where The nonlocal expression for the kinetic-energy term of |x| = |y| = 1 with (x, y). Hopping of electrons is realized gauge fields should be understood in the momentum by the hybridization between the Cu d 2 2 orbital and x −y space. Here, the Maxwell dynamics for both kinetic the O p and p orbitals, given by x y energies of gauge fields are irrelevant and neglected in cp this expression. In addition, b has been scaled as H = −t d p + H.c. Cu−O pd iσ i+x/2σ cp N b . We emphasize that this is a self-consistent ef- λ i=(i ,i ) x y fective Lagrangian in the large-flavor limit, which may X † y † +d p + H.c. − t d p + H.c. be regarded to be classical. Although the existence of pd iσ i+y/2σ iσ i−x/2σ i=(i ,i ) massless fermions break the discrete one-form symme- x y try of the theory without fermions, one can see that Z 2 † y +d p + H.c. . (40) iσ i−y/2σ one-form global symmetry emerges in this limit. Based on this critical field theory, one may calculate both Wil- Here, t is strength of the dp hybridization, referred to pd son’s lines and find the perimeter law, which results from as an overlap integral of wave functions. the deconfined nature of both quasiparticles and vortices Based on this effective lattice Hamiltonian, one may involved with their long-range effective interactions. In consider other words, the Z one-form global symmetry is broken spontaneously in the presence of massless Dirac fermions U/t → ∞. (41) pd at least in the large−N limit. Based on this observation, we claim that the system acquires Z topological order 2 This limiting procedure gives rise to the following effec- in this phase. tive Hamiltonian [21], referred to as the t − J model X X H = −t (c˜ c˜ + H.c.) + J S · S − n n . eff jσ i j i j iσ IV. TOPOLOGICAL PROPERTY OF d − wave ij ij SUPERCONDUCTORS FROM DOPED MOTT (42) INSULATORS Here, c˜ = P c P is an electron annihilation operator, iσ G iσ G A. High T cuprates as doped Mott insulators where P = Π (1 − n n ) is the Gutzwiller projection G j i↑ i↓ operator to extract out double occupancy sites. At half 1. Effective UV lattice Hamiltonian filling, the kinetic energy term vanishes identically. In other words, hopping of these electrons can be realized only when inter sites are empty. More precisely, c˜ rep- iσ One way to see high T cuprate superconductors is that resents a doped hole, referred to as a Zhang-Rice singlet they are from doped Mott insulators [21]. To discuss [21]. Since ǫ > ǫ is satisfied for high T cuprates, an p d c this aspect, we introduce an effective lattice Hamiltonian, electron can be extracted out from the O site via hole which consists of Cu and O effective lattice Hamiltonians doping. Such a doped hole turns out to form a singlet pair with the hole of the Cu site for the d configura- H = H + H + H . (37) UV Cu O Cu−O tion. Although the term of the Zhang-Rice singlet ex- Here, the local Hamiltonian for the Cu site is given by plains the UV origin precisely, we just use the term of electrons for convenience. These electrons have superex- X X d d H = ǫ d d + U n n . (38) Cu d iσ change interactions, given by the last Heisenberg term. i↑ i↓ iσ i=(i ,i ) i=(i ,i ) x y x y S = c σ c is a spin operator and n = n + n i αβ iβ i i↑ i↓ iα is a density operator with n = c c . The Gutzwiller iσ iσ iσ d is an electron annihilation operator at the Cu site iσ projection operator guarantees the following constraint 2+ i, where it acts on the d 2 2 orbital for Cu with its x −y local energy ǫ . U is an effective Coulomb interaction n ≤ 1, (43) 2+ d energy for the d 2 2 orbital of Cu . n = d d is an x −y iσ iσ iσ electron density operator of spin σ. i = (i , i ) denotes which implies that the low-energy effective Hamiltonian x y a square lattice. The local Hamiltonian for the O site is is to describe physics of doped Mott insulators. 7 2. U(1) slave-boson representation and effective U(1) χ and χ are inter-site particle-hole composite fields to ij ij lattice gauge theory give effective bands to holons and spinons, respectively. Δ is a link variable to describe d − wave pairing of ij spinon Cooper pairs, which is our central object. ϕ is To solve the double occupancy constraint given by the an effective potential to decompose density-density in- Gutzwiller projection operator, U(1) slave-boson repre- teractions. We refer one way of this construction to Ref. sentation has been introduced [9]. c = b f . (44) iσ iσ i To discuss possible low energy physics of this effective theory, we focus on transverse fluctuations for these “or- Here, the electron annihilation operator is represented by der parameters” given by a composite operator, given by a holon field b to carry b bs ia f fs −ia s iφ ij ij ij χ = χ e , χ = χ e , Δ = Δ e . an electron charge quantum number and a spinon field ij ij ij ij ij ij f to carry an electron spin quantum number. Then, iσ (49) the constraint equation (43) is expressed as Here, a is a phase field for the hopping order parameter, ij † † which plays the role of U(1) gauge fields. These phase b b + f f = 1, (45) i iσ i iσ fluctuations contribute to lowering the ground state en- which allows either one holon or one spinon with spin σ ergy by spontaneous generation of such U(1) internal at the site i. magnetic fluxes in the lattice. This U(1) gauge redun- Now, it is straightforward to construct the path inte- dancy may be regarded to originate from the U(1) slave- gral expression for the partition function boson representation itself, where the electron annihila- Z Z tion operator of Eq. (44) is invariant under U(1) gauge h β n † iθ iθ i i transformation of f → e f and b → e b . φ is a Z = Df Db Dλ exp − dτ f (∂ − µ iσ iσ i i ij iσ i i τ iσ phase field for the d−wave spinon pairing order parame- X X ter, which carries the internal U(1) gauge charge 2e, i.e., −iλ )f + b (∂ − iλ )b + i λ i iσ τ i i i transformed as φ → φ + θ + θ under the U(1) gauge ij ij i j transformation. oi X X † † Considering −t (f b b f + H.c.) + J S · S − n n . i jσ i j i j iσ j s s ij ij iλ = λ + ia , ϕ = ϕ + δϕ , (50) i iτ i i i i (46) where the superscript s means a saddle-point value, and introducing Here, λ is a Lagrange multiplier field to impose the single X X occupancy constraint Eq. (45). µ is an electron chemical J J − ϕ − ia − δϕ → −ia (51) iτ j iτ potential to determine hole concentration given by δ = 2 2 j∈i j∈i hb b i. The density and spin operators are given by into the resulting lattice model, we construct an effective † † lattice gauge theory as follows n = f f , S = f σ f , (47) iσ iσ i αβ iβ iσ iα Z Z = Df Db Da Da Dφ iσ i iτ ij ij respectively, where the constraint equation has been uti- lized. h β n Introducing Eq. (47) into Eq. (46) and performing exp − dτ f ∂ − µ − λ − ia f τ iτ iσ iσ i the Hubbard-Stratonovich decomposition for interactions i into spin singlet channels, we find an effective field theory bs † ia ij −t χ f e f − H.c. jσ ij iσ ij Z = Df Db Dλ Dχ Dχ DΔ Dϕ iσ i i ij i ij ij X s† −iφ ji − Δ e (ǫ f f ) − H.c. αβ iα jβ Z ij h β n X 4 † ij exp − dτ f (∂ − µ − iλ )δ − ϕ f τ i ij j iσ iσ X X † fs† † 0 s iaji ij + b (∂ − λ − ia )b − t χ b e b − H.c. τ iτ i i i i ji j X X i ij † † −t f χ f − H.c. − Δ (ǫ f f ) − H.c. X X X jσ αβ iα jβ iσ ij ij fs s bs +i a + λ + t χ χ + H.c. iτ ij ij i ij ij X X X i i ij † f † + b (∂ − iλ )b − t b χ b − H.c. + i λ τ i i i i  oi i ij j X fs 2 s 2 i ij i + − |χ | + |Δ | . (52) ij ij X X ij b f 2 2 +t χ χ + H.c. + − |χ | + |Δ | ij ij ij ij Here, a is a time-component of the U(1) gauge field. iτ ij ij Based on this U(1) lattice gauge theory, we construct an oi effective field theory below. We refer this procedure to + ϕ ϕ . (48) i j Ref. [9]. ij 8 B. Absence of topological order in s − wave Φ (Φ ) is a vortex field for the spinon Cooper pair sp b sp b superconductors from doped Mott insulators (holon) field and b (b ) is the corresponding superfluid μ μ sound mode. Such spinon pair and holon sound modes are coupled to internal U(1) gauge fields, described by First, we consider the case of s−wave superconductors BF terms with appropriate U(1) charges. from doped Mott insulators, where spinons are gapped to be neglected at low energies. Then, the corresponding effective field theory is given by Z Z β n sp s−wave 2 2 S = dτ d x |∂ φ − 2a | μ sp μ MI ρ 1 2 2 + |∂ φ − a | + (ǫ ∂ a ) . (53) μ b μ μνλ ν λ 2 2g ρ (ρ ) is the phase stiffness parameter for the dynamics sp b of spinon Cooper pairs (holons), given by ρ ∼ |hΔ i| sp sp 2 iφsp iφ (ρ ∼ |hbi| ) with Δ = hΔ ie (b = hbie ). g is b sp sp the unit of the internal U(1) gauge charge. A noticeable point is that the U(1) gauge charge for spinon Cooper pairs is 2g while that of holons is g. Performing the duality transformation, we obtain an effective dual field theory s−wave sp b Z = DΦ DΦ Da Db Db b sp μ μ μ MI Z Z h β n 2 sp 2 exp − dτ d x |(∂ − ib )Φ | μ sp sp 2 2 4 +m |Φ | + |Φ | sp sp sp 1 i 1 sp 2 sp 2 + (ǫ ∂ b ) + ǫ b ∂ a + (ǫ ∂ a ) μνλ ν μνλ ν λ μνλ ν λ 2ρ π 2g sp b 2 2 2 4 +|(∂ − ib )Φ | + m |Φ | + |Φ | μ b b b Here, we focus on the superconducting phase, where μ b oi both spinon pair and holon vortices are gapped. Then, 1 i b 2 b + (ǫ ∂ b ) + ǫ b ∂ a . (54) the effective dual field theory is given by μνλ ν μνλ ν λ λ μ 2ρ 2π Z Z Z h β n 1 1 s−SC sp b 2 sp 2 b 2 Z = Da Db Db exp − dτ d x (ǫ ∂ b ) + (ǫ ∂ b ) μ μνλ ν μνλ ν μ μ λ MI λ 2ρ 2ρ sp b oi i i 1 sp b 2 + ǫ b ∂ a + ǫ b ∂ a + (ǫ ∂ a ) . (55) μνλ ν λ μνλ ν λ μνλ ν λ μ μ π 2π 2g Compared with Eq. (2), we obtain Since the number of superfluid sound modes is larger than that of U(1) gauge fields, it is natural to expect sp the existence of neutral superfluid modes, which decouple 4π H = (56) from U(1) gauge fields. Indeed, it exists, given by 4π f b f for the stiffness parameter, [b ] = P ′[b] , μ ff μ where the projection operator is K = (57) b + ′ ′ P ′ = δ − [KK ] ff ff ff for the K matrix, and with the Moore-Penrose inverse of K ′ ′ G = δ δ (58) cc c1 c 1 + 2 1 K = . (59) 5 5 for the gauge coupling constant. More explicitly, such a neutral superfluid sound mode is 9 given by while that of the holon gauge field is changed as V = exp i dl b b μ f sp b [b ] = (b − 2b )δ . (60) 0 f1 C μ μ μ 1 4 7→ exp × 2πi V = exp − × 2πi V . (67) b b 5 5 Now, let us study the one-form global symmetry of Thus, this symmetry acts as Z phase rotation on those Eq. (55). The dual effective field theory of the s − wave operators. In fact, this phase rotation can also be gener- superconducting phase from a spin liquid state has the ated as a U(1) one-form symmetry (61), following one-form global symmetry sp sp b b b 7→ b + ǫ(−1)ρ , b 7→ b + ǫ(2)ρ (68) μ μ μ μ μ μ f f + c b 7→ b + [K ] λ , by choosing cf μ μ μ c c c ǫ = − . (69) where λ is a flat connection satisfying ∂ λ − ∂ λ = 0 μ ν μ ν μ with normalization dl λ ∈ 2πZ. From the explicit C Hence, the Z one-form symmetry is a subgroup of the form of the Moore-Penrose inverse of K, one can see that U(1) continuous one-form symmetry, if we look at the it is a Z one-form symmetry. transformation property of physical operators. In three This state also has a continuous one-form symmetry, spacetime dimensions, a continuous one-form symme- try cannot be broken, guaranteed by the generalized f f α ¯ b 7→ b + ǫ[D ] ρ , (61) f μ Coleman-Mermin-Wagner theorem for higher-form sym- μ μ metries [7]. Therefore, the Z one-form symmetry cannot where ǫ is a continuous parameter, ρ is a flat one-form be broken either. As a result, the s − wave supercon- α ¯ connection normalized as dl ρ ∈ 2πZ, and D is the ducting phase with one neutral superfluid sound mode, μ μ element of the cokernel of K, resulting from a doped Mott insulating state, does not have a topological order. This implies that conventional α ¯ [D ] = −1 2 . (62) s−wave superconductivity, induced by the condensation of electron Cooper pairs (identified as superfluidity here), can be smoothly connected with the s − wave supercon- Here, the cokernel is one-dimensional. Under the trans- ducting phase with one neutral superfluid sound mode, formation (61), Wilson loop operators are transformed resulting from the spin liquid state. as Even though there is no topological order in this su- α ¯ perconducting state, the mutual statistics between vor- V (C) 7→ exp 2πip ǫ[D ] V (C). (63) p f f p tices and quasiparticles can be exotic. Let us denote W (C) ≡ exp i dl a . The correlation functions of μ μ There is no discrete one-form symmetry to shift the W (C) and V (C), V (C) satisfy sp b gauge field a. Recall that a generic form of discrete one- 4πi ′ ′ − Lk(C,C ) ′ form transformation takes the form hW (C)V (C )i = e hV (C )i, (70) sp sp 2πi c c + ′ − Lk(C,C ) ′ a 7→ a + [K ] p λ , (64) hW (C)V (C )i = e hV (C )i, (71) cf f μ μ μ b b respectively, where we used the fact that Wilson loops where p ∈ (cokerK) . The charge vector p should be f f associated with a are topological, and α ¯ chosen to be orthogonal to D so that the generator is a hW (C)i = 1. (72) topological operator. One choice with minimal integer is The relations (70, 71) indicate that the Wilson loop p = 2 1 . (65) W (C) is the generator of the Z one-form symmetry and q 5 V (C) is the corresponding charged object. Although the Z one-form symmetry is not spontaneously broken, this Although the transformation (64) with this charge vector 5 braiding phase is an observable effect. does not change the action (up to 2πZ), this is not a symmetry of the system, because this acts trivially on the Wilson loop of a. C. d − wave superconductors from doped Mott In the current case, the Z one-form symmetry is a sub- insulators at large−N group of the U(1) continuous one-form symmetry (61). Under the Z one-form symmetry, the Wilson loop oper- Finally, we discuss the topological properties of d − ator for the spinon-pair gauge field is transformed as wave superconductors from doped Mott insulators, based on the perspectives of higher-form symmetries. Introduc- sp V = exp i dl b sp μ ing massless Dirac fermions into the effective dual field theory Eq. (54) for s−wave superconductors from doped 7→ exp × 2πi V (66) Mott insulators, we obtain sp 5 10 Z Z Z h β n d−wave sp b 2 sp Z = Dψ DΦ DΦ Da Db Db Dc exp − dτ d x ψ γ (∂ − ic )ψ + ǫ b ∂ c nσ b sp μ μ nσ μ μ μ nσ μνλ ν λ MI μ μ μ u 1 i 1 sp sp sp 2 2 2 4 2 sp 2 +|(∂ − ib )Φ | + m |Φ | + |Φ | + (ǫ ∂ b ) + ǫ b ∂ a + (ǫ ∂ a ) μ sp sp sp μνλ ν μνλ ν λ μνλ ν λ μ sp μ 2 2ρ π 2g sp oi u 1 i b 2 2 2 4 b 2 b +|(∂ − ib )Φ | + m |Φ | + |Φ | + (ǫ ∂ b ) + ǫ b ∂ a . (73) μ b b b μνλ ν μνλ ν λ μ b λ μ 2 2ρ 2π Here, ψ represents a massless Dirac fermion field re- term with the statistical angle π. In other words, when nσ sulting from the d − wave pairing symmetry of spinon the Dirac fermion turns around the spinon pair vortex, Cooper pairs. Such massless fermions interact with it acquires the Aharonov-Bohm phase of π. This dual spinon pair vortices statistically, described by the BF effective field theory is reduced into Z Z Z h β n dSC sp b 2 Z = Dψ Da Db Db Dc exp − dτ d x ψ γ (∂ − ic )ψ nσ μ μ nσ μ μ μ nσ MI μ μ 1 1 i i i sp 2 b 2 sp b sp + (ǫ ∂ b ) + (ǫ ∂ b ) + ǫ b ∂ a + ǫ b ∂ a + ǫ b ∂ c μνλ ν μνλ ν μνλ ν λ μνλ ν λ μνλ ν λ λ μ μ μ 2ρ 2ρ π 2π π sp b oi 1 1 2 2 + (ǫ ∂ a ) + (ǫ ∂ c ) (74) μνλ ν λ μνλ ν λ 2 2 2g 2g a c in the superconducting phase, where both spinon pair Before we study the topological property of Eq. (74), and holon vortices are gapped. we consider the case that Dirac fermions are gapped, and thus neglected at low energies, described by n o 1 K 1 ′ f f fc dSC−CSB 3 −1 f c c c ′ ′ ′ ′ ′ ′ ′ ′ S = d x [H ] (ǫ ∂ b )(ǫ ∂ b ) + ǫ b ∂ a + G (ǫ ∂ a )(ǫ ∂ a ′) , ff μνλ ν μν λ ν ′ μνλ ν cc μνλ ν μν λ ν μ λ λ λ Dual−MI λ λ 8π 2π 2 where we have correspondences due to “chiral symmetry” breaking [18, 19]. We also note that there is no continuous one-form symmetry in 1 sp 2 b b = b , b = b , μ μ μ μ this case because K is full-rank. As a result, this super- 1 2 conducting state has the Z discrete one-form symmetry. a = a , a = c (75) 2 μ μ μ μ Wilson loops obey the perimeter law and this symmetry for fields, is spontaneously broken. Hence this phase has a Z topo- sp logical order. The mutual statistics between spinon-pair 4π H = (76) b vortices and quasiparticles is semionic, which confirms 4π the Z topological order. for the stiffness matrix, 2 2 0 1 −1 K = → K = (77) 1 0 1/2 −1 for the K matrix, and G = (78) for the gauge coupling-constant matrix. Since we have Now, we introduce the role of massless Dirac fermions dim (ker K) = dim (coker K) = 0, there are no massless into the above discussion. If we consider the limit of degrees of freedom in this “d − wave” superconducting large number of flavors, we have an effective field theory phase, where such massless Dirac fermions are gapped which describes a conformal invariant fixed point in three 11 spacetime dimensions [20], given by that this topological phase transition is a deconfinement- confinement transition involved with the compact U(1) 1 1 dSC gauge field a , where magnetic monopoles as instantons L = N (ǫ ∂ c )√ (ǫ ′ ′∂ ′c ′) f μνλ ν λ μν λ ν λ Dual−MI −∂ in three spacetime dimensions play an essential role. 1 1 Performing the duality transformation for compact √ ′ ′ ′ ′ + (ǫ ∂ a ) (ǫ ∂ a ) μνλ ν λ μν λ ν λ U(1) gauge fields a and taking into account the dilute −∂ 64 1 instanton-gas approximation, we obtain an effective field sp sp + (ǫ ∂ b )√ (ǫ ′ ′∂ ′b ) μνλ ν μν λ ν ′ λ λ 2 theory for instanton-type magnetic monopole excitations −∂ 16 1 b b 1 1 dSC √ ′ ′∂ ′b ) + (ǫ ∂ b ) (ǫ ′ μνλ ν μν λ ν ′ ′ ′ ′ λ λ L = N (ǫ ∂ c )√ (ǫ ∂ c ) 2 f μνλ ν λ μν λ ν λ Dual−MI −∂ 2 −∂ i i i sp b sp 64 1 sp sp + ǫ b ∂ a + ǫ b ∂ a + ǫ b ∂ c . μνλ ν λ μνλ ν λ μνλ ν λ μ μ μ + (ǫ ∂ b )√ (ǫ ′ ′∂ ′b ) μνλ ν μν λ ν λ λ π 2π π 2 −∂ (79) 16 1 i b b sp ′ ′ ′ + (ǫ ∂ b )√ (ǫ ∂ b ′) + ǫ b ∂ c μνλ ν μν λ ν μνλ ν λ λ λ μ sp sp b b 2 π π Here, b → N b and b → N b have been performed. −∂ f f μ μ μ μ Again, we point out that this effective field theory is self- + (∂ ϕ) −∂ (∂ ϕ) − y cosϕ μ μ m consistent and classical in the large number of fermion flavors. All types of superconducting phase fluctuations i 1 sp b + (∂ ϕ) b + b . (80) μ μ and U(1) gauge fields are massless, more precisely, the π 2 dynamics of which is conformal invariant. In spite of Here, ϕ is a scalar magnetic potential field to mediate this quantum criticality, we point out that all the Wilson effective interactions between instanton-type magnetic loop operators show their perimeter-law behaviors, where monopole excitations and y is an effective fugacity given both vortices and quasiparticles are deconfined to have −S inst by y ∼ e with a renormalized instanton action effective Coulomb interactions. For comparison with the S . Doping dependence would be introduced into the inst absence of massless Dirac fermions, we recall the corre- effective instanton action. Although we are not claiming sponding effective field theory that this procedure is rigorously performed in the pres- 1 1 dSC−CSB 2 2 ence of massless Dirac fermions, we can argue that the L = (ǫ ∂ c ) + (ǫ ∂ a ) μνλ ν λ μνλ ν λ Dual−MI 2 2 2g 2g c a condensation transition of magnetic monopoles may be 1 1 a Berezinskii-Kosterlitz-Thouless (BKT) type according sp 2 b 2 + (ǫ ∂ b ) + (ǫ ∂ b ) μνλ ν μνλ ν λ λ to one scenario [22–24]. If the condensation of such mag- 2ρ 2ρ sp b netic monopoles occurs, magnetic scalar potential fluc- i i i sp b sp + ǫ b ∂ a + ǫ b ∂ a + ǫ b ∂ c , tuations are gapped, expected to appear above the op- μνλ ν λ μνλ ν λ μνλ ν λ μ μ μ π 2π π timal doping. Taking into account both magnetic scalar potential fluctuations and “holon” current fluctuations, where all gauge fields are gapped, and thus Z topologi- we find essentially the same effective field theory as Eq. cally ordered. The Maxwell dynamics are replaced with (36), which describes d − wave superconductivity from the conformal invariant kinetic energy due to the pres- Landau’s Fermi liquids. This potential existence of a ence of massless Dirac fermions. The coexistence of the topological phase transition between d− wave supercon- Z topological order and quantum criticality is an essen- ductors in high T cuprates, given by a deconfinement- tial feature for d − wave superconductors in the large confinement transition, is one of the main implications of number of fermion flavors. the current study and it deserves further studies. This analysis suggests that d − wave superconductors from U(1) spin liquids may not be adiabatically con- nected with those from Landau’s Fermi liquids, in con- V. CONCLUSION trast to the case of s − wave superconductors as long as the physics of massless Dirac fermions is governed by the Statistical interactions between vortices and Dirac conformal invariant fixed point. We recall the effective fermions in d − wave superconductors turn out to play field theory of Eq. (36) for d − wave superconductors an important role for the topological order in the per- from Landau’s Fermi liquids spectives of global one-form symmetries when such Dirac 64 1 fermions become gapped due to possible “chiral symme- cp cp ′ ′ ′ L = N (ǫ ∂ b )√ (ǫ ∂ b ) dSC f μνλ ν μν λ ν ′ λ λ 2 try breaking”. On the other hand, the existence of mass- −∂ less Dirac fermions gives rise to the fact that low energy i 1 1 cp + ǫ b ∂ c + (ǫ ∂ c )√ (ǫ ′ ′∂ ′c ′) . μνλ ν λ μνλ ν λ μν λ ν λ fluctuations such as both supercurrent and U(1) gauge- π 16 −∂ field fluctuations become critical in the large number The difference of the K matrix between Eq. (79) of fermion flavors. Interestingly, the discrete one-form fc and Eq. (36) indicates that the topological order of global symmetry still remains in the resulting effective Eq. (79) differs from that of Eq. (36). We speculate field theory of the large−N limit to be spontaneously f 12 broken in the superconducting phase. In other words, a remains as an interesting future project. spontaneous breaking of a discrete one-form symmetry may coexist with quantum criticality in d − wave super- ACKNOWLEDGMENTS conductors. One may criticize that the role of massless Dirac fermions in the notion of topological order for d − wave K. K. was supported by the Ministry of Ed- superconductors is not clarified in a rigorous sense. The ucation, Science, and Technology (No. NRF- coexistence between the spontaneous breakdown of the 2015R1C1A1A01051629 and No. 2011-0030046) of the discrete one-form symmetry and the quantum critical- National Research Foundation of Korea (NRF) and by ity should be verified with a concrete mathematical ma- TJ Park Science Fellowship of the POSCO TJ Park chinery. One possible way of analysis would be the use Foundation. Y. 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Published: May 12, 2019

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