Maine Christos,Zhu-Xi Luo,Henry Shackleton,Ya-Hui Zhang,Mathias S. Scheurer,Subir Sachdev
A model of d-wave superconductivity, antiferromagnetism, and charge order on the square lattice
spin liquid,superconductivity,charge order,quantum criticality
Significance The cuprate compounds exhibit high-temperature superconductivity at ambient pressure and a complex phase diagram. Early studies proposed a connection between cuprate superconductivity and spin-liquid states characterized by excitations with fractionalized quantum numbers. But no direct evidence for fractionalization has since emerged, and the low-temperature phase diagram is dominated by a competition between superconductivity and charge-ordered states which break translational symmetry. Our theory uncovers features associated with a particular spin liquid presumed to underlie the higher temperature pseudogap metal phase and shows that it has multiple nearly degenerate instabilities to confinement of fractionalized excitations, leading to antiferromagnetism, d-wave superconductivity, and/or charge order. Our theory provides routes to resolving a number of open puzzles on the cuprate phase diagram.
A model of d-wave superconductivity, antiferromagnetism, and charge order on the square lattice

The cuprate compounds exhibit high-temperature superconductivity at ambient pressure and a complex phase diagram. Early studies proposed a connection between cuprate superconductivity and spin-liquid states characterized by excitations with fractionalized quantum numbers. But no direct evidence for fractionalization has since emerged, and the low-temperature phase diagram is dominated by a competition between superconductivity and charge-ordered states which break translational symmetry. Our theory uncovers features associated with a particular spin liquid presumed to underlie the higher temperature pseudogap metal phase and shows that it has multiple nearly degenerate instabilities to confinement of fractionalized excitations, leading to antiferromagnetism, d-wave superconductivity, and/or charge order. Our theory provides routes to resolving a number of open puzzles on the cuprate phase diagram.

The phase diagram of the hole-doped cuprate compounds has been extensively studied in numerous careful experiments in recent decades, and a remarkably rich picture has emerged of the quantum phases of matter around the dome of high-temperature superconductivity (1). We present a theoretical approach to these phases designed to address the following key puzzles: 1. The pseudogap metal (found at intermediate temperatures and low doping) has a suppressed spin spectral weight and a photoemission gap in the antinodal region of the Brillouin zone. There is a puzzling “Fermi-arc” spectrum in the nodal region of the Brillouin zone (2, 3), not interpretable in terms of band theory. 2. The quantum oscillations observed at low temperatures and high magnetic fields in YBa2Cu3O6.5 (4) appear to have an interpretation in terms of electron pockets induced by charge density wave order (5). However, computations of the reconstruction of the Fermi surface of the Fermi liquid state by charge order also predict additional gapless electronic excitations in the antinodal region of the Brillouin zone (6) which have not been observed. 3. The temperature scales of the d-wave superconductivity and the charge density wave orders are very similar to each other (7), suggesting a common origin. In Fermi liquid theory, the instabilities to such orders are determined by different interactions, and there is no particular reason for them to be similar. Our theory begins (Fig. 1) with the assumption that the Fermi-arc spectrum in the pseudogap arises from an underlying pocket Fermi surface of electron-like particles of charge e and spin-1/2 (12–25) but with anisotropic spectral weight. This pocket Fermi surface encloses a volume which does not equal the free electron Luttinger value, and such a violation requires the presence of a spin liquid with fractionalized excitations, a state called the fractionalized Fermi liquid (FL*) in refs. 26 and 27. We will further assume that the spin liquid underlying the pseudogap metal is the popular π-flux state of fermionic spinons (28). As we will discuss below, although this π-flux state is now known to be ultimately unstable at T = 0, there is significant evidence for its stability over intermediate length scales, and so, it can describe fractionalized excitations at the pseudogap temperatures. Wang et al. (29) have argued that this π-flux state is dual to the critical point of another popular spin-liquid state, that is described by the CP1 model of bosonic spinons (30). The bosonic spinon model is a useful starting point toward studying a confining instability to Néel order (8). Here, we shall exploit the fermionic spinon description to study confinement to charge order and superconductivity, as schematically sketched by arrow 𝔸 in Fig. 1. The π-flux spin liquid is a theory of fermionic spinons coupled as fundamentals to an emergent SU(2) gauge field (31, 32) and moving in a background of gauge-invariant π-flux in the ℤ2 center of SU(2). At low energies, the fermionic spectrum reduces to that of Nf = 2 massless Dirac fermions whose quadratic action has an emergent SO(5)f global symmetry (29, 33–37) (here the subscript f is only an identifying label specifying that the SO(5) acts on fermionic spinons). To obtain the superconducting and charge-ordered states, we will condense a Higgs field B a “chargon” (13–15), which is a fundamental of SU(2) and also carries a unit U(1) charge of electromagnetism (this U(1) is treated as effectively global). The boson B is a spin singlet under the SU(2) global spin rotation, while the fermionic spinons carry spin 1/2. Several earlier works have considered the close relationship between the π-flux spin liquid and the d-wave superconductor (13–15, 31, 38–43). However, they assumed that the analog of the boson B carried all of the doping density and condensed in a spatially uniform manner in the d-wave superconductor. In our approach, the doping density is carried entirely by the electron-like hole pockets responsible for the observed Fermi arcs. This is especially clear in the ancilla formulation of the pseudogap metal phase (8–10, 24, 44), which we discuss in SI Appendix, section 1 (but in principle, as our presentation will show, all the results of the present paper can be obtained without the ancilla method). Consequently, B should be treated as a nearly relativistic Higgs boson or a “slave rotor” (45), rather than a nearly free nonrelativistic boson which undergoes Bose–Einstein condensation. Indeed, the ancilla approach involves a change in perspective on the physical interpretation of B: In earlier approaches (15), B was obtained by fractionalizing the electron into a spinon and a chargon B. In our approach, B is regarded as a composite of the spinon and the physical electron, as in Eq. 6. But, at the level of symmetry and emergent gauge fields, there is no fundamental difference between the two approaches. Furthermore, while the earlier works recognized that B carries a fundamental SU(2) gauge charge, this is a property of the gauge structure crucial to our analysis that has not been accentuated earlier. Like the fermionic spinons, the B bosons also move in a background of π-flux in the ℤ2 center of SU(2). This follows immediately from the facts that B is a composite of the physical electron and a spinon, and the electron cannot experience any emergent flux. This π-flux is SU(2) gauge-invariant, and choosing a gauge in which the π-flux spin liquid is transformed (31, 41, 42) into one with d-wave pairing between the spinons does not remove the ℤ2 flux. A key consequence of the ℤ2 flux is that the dispersion of the B must have at least two degenerate minima (46). (The works refs. 13, 14, and 15 employed a distinct “staggered flux” U(1) spin liquid for the pseudogap at nonzero doping, for which this additional degeneracy does not apply—SI Appendix, 5.) For the simplest case with only two minima, the low-energy theory in the vicinities of these minima yields a continuum theory with Nb = 2 flavors of bosons carrying fundamental SU(2) gauge charges. For reasons similar to the fermionic sector, the static action of this low-energy bosonic theory can have an emergent SO(5)b symmetry (where b is an identifying label to distinguish from the distinct SO(5)f symmetry). Degenerate bosonic minima and a SO(5)b symmetry were also important in the recent work of ref. 47. We note that the global spin rotation symmetry SO(3) ⊂ SO(5)f, while the electromagnetic charge symmetry U(1) ⊂ SO(5)b, and these are the only exact continuous global symmetries of the lattice theory. We determine the physical interpretation of the B bilinears forming the gauge-neutral SO(5)b vector and find the following five orders: (i)+(ii): A d-wave superconductor; this complex order has 2 real components. (iii)+(iv): Site-charge density waves (stripes) at wavevectors (π, 0) and (0, π). (v): The “d-density wave” (48), which has a staggered pattern of circulating charge currents and breaks time-reversal symmetry. The choice between these orders is made by additional terms allowed by the lattice symmetries which break the SO(5)b symmetry. With additional dispersion minima, we can obtain charge density waves at other wavevectors, as we discuss in sections 5 and 6. In the combined theory of the fermionic spinons and bosonic chargons, for the case where the chargon dispersion has two minima, we can now sketch the schematic phase diagram shown in Fig. 2. For r > rc, the Higgs boson B is massive and can be ignored at low energies, where the theory reduces to Nf = 2 massless Dirac fermions coupled to a SU(2) gauge field. The numerical evidence (49–53) indicates that this theory is confining and leads to a phase with SO(5)f global symmetry broken by either Néel or valence bond solid (VBS) order. For r < rc, the Higgs boson B condenses: This quenches the SU(2) gauge field completely and breaks the SO(5)b global symmetry, and so, one of the orders listed in the previous paragraph will be present. At half-filling, r = rc is a possible deconfined critical point (54) described by a conformal field theory with global SO(5)f× SO(5)b symmetry. This CFT is an attractive candidate for describing the transition between the Néel state and the d-wave superconductor numerically observed by Assaad et al. (55) in the particle–hole symmetric half-filled Hubbard model with an additional square-hopping interaction term. We conclude this introduction by noting a few of the many earlier works (56) which have considered the interplay of antiferromagnetism, d-wave superconductivity, and charge order in the context of the cuprates, all in a manner distinct from ours; this discussion may be skipped on a first reading. Zhang (57) considered a SO(5) symmetry mixing antiferromagnetism and superconductivity. This is not related to our SO(5)s, as antiferromagnetism is part of SO(5)f, while d-wave superconductivity is part of SO(5)b. Charge order was obtained in an insulator by the condensation of vortices (58, 59) in the d-wave superconductor. A theory for the transition between an easy-plane Néel state (in contrast to the fully SO(3) symmetric Néel order in our case) and a d-wave superconductor without nodal quasiparticles (our d-wave superconductor can have nodal quasiparticles) was obtained (60) in a dual formulation of vortices in both the Néel order and the superconductor. A direct transition between the Néel state and the d-wave superconductors was described by Raghu et al. (61) in a weak-coupling analysis of the Hubbard model. It is possible that this transition and that in the quantum Monte Carlo study of Assaad et al. (55) are both described by the deconfined critical theory introduced in the present paper. The spin density-wave quantum critical point in a two-dimensional metal was argued to have instabilities to d-wave pairing and charge order with nearly the same strength (62, 63), and theories of the fluctuations of the combined orders have been examined (64–69). There is no fractionalization and no emergent gauge field in these approaches. The studies in refs. 70 and 71, closest in spirit to the present study, examined the condensation of chargons from a pseudogap metal described by a ℤ2 spin liquid. A different fractionalized model for the pseudogap metal was used to study (72, 73) the interplay between spin and charge density wave orders.

We begin by recalling the theory for the π-flux spin liquid on the square lattice. Experimental neutron scattering evidence for the relevance of this state to square lattice antiferromagnets was obtained by Dalla Piazza et al. (74) and Headings et al. (75), and numerical evidence by Hering et al. (76). We express the spin operators Si on site i in terms of fermions fiα, where α = ↑, ↓ is spin index, by . For spin liquids with an emergent SU(2) gauge field, it is useful to introduce the spinor ψi so that the SU(2) gauge transformation acts as ψi → Uiψi, where Ui∈ SU(2). We describe the π-flux spin liquid, in the gauge used by ref. 29, by the quadratic fermion Hamiltonian where i,j are nearest neighbors, J is a real coupling constant of order the antiferromagnetic exchange, and eji = −eij is a fixed element of the ℤ2 center of the gauge SU(2) which ensures π flux per plaquette; we choose where i = (x, y), , . The link field Uij = Uji† is the fluctuating SU(2) lattice gauge field, and the mean-field saddle point of the π-flux phase is obtained by setting Uij = 1. We note that the leading i in Eq. 2 is needed to ensure global SU(2) spin-rotation invariance. At the Uij = 1 saddle point, the dispersion of the fermions in Hf has two Dirac nodes at the Fermi level, as shown in Fig. 3. Linearizing the dispersion at the nodes, we obtain a theory of Nf = 2 relativistic, massless Dirac fermions coupled to a SU(2) gauge field. This theory has a global SO(5)f symmetry (29, 33–37, 77), corresponding to rotations among Néel order and VBS order, which together form a composite order with 5 real components. The Néel-VBS transition has been intensively studied by numerical and bootstrap methods, and the emerging consensus (49–53) is that ground state is ultimately a confining state with SO(5)f broken: This consensus accounts for the r > rc portion of the phase diagram in Fig. 2, where the chargons are massive and unimportant at low energies.

We introduce a chargon field (13–15) on each lattice site, where B1i and B2i are complex boson. We view the chargon as a Higgs field which couples the spinons ψi to the physical electrons . For the doped system, the annihilates an electron-like quasiparticle near the Fermi surface of the hole pockets; for the undoped system, the annihilates an electronic quasiparticle just above the charge gap. After introducing the coupling between the chargons, electrons, and spinons can be written as where εαβ is the unit antisymmetric tensor. The first line in Eq. 6 makes the invariance under gauge SU(2), global spin SU(2), and global charge U(1) transparent. In particular, we have Bi → UiBi under gauge SU(2), Bi → eiϕBi under electromagnetic U(1), and Bi → Bi under global spin SU(2). An explicit microscopic derivation of the form of HH can be obtained in the ancilla model (9, 10), as we describe in SI Appendix, 1. Here, we regard HH as the simplest allowed coupling consistent with the gauge and global symmetries. We now obtain the form of the lattice effective action for Bi by requiring invariance under lattice symmetries and time-reversal under the projective transformations of the π-flux phase. The projective transformations of the fermionic spinons fα have been computed earlier (29), those of the electrons must be trivial, and those of the bosonic charges B then follow from the invariance of Eq. 6. The transformations are listed in Table 1. The key property is the relation which ensures π-flux on both spinons and chargons and at least two degenerate minima for the chargons. With the transformations in Table 1 in hand, we write down the most general effective Lagrangian for the Bi, keeping only terms quadratic and quartic in the Bi, and with only on-site or nearest-neighbor couplings. In this manner, we obtain the Lagrangian (terms with time derivatives will be considered in Section 7) where r, w1 are real Landau parameters, and the quartic terms are in 𝒱(B). The hopping terms in Eq. 8 are identical to the hopping terms for the fermionic spinons in Eq. 2. However, there is a “mass” term, r, present for the chargons, which was not allowed for the spinons—we will use r as the tuning parameter across the transition in which the B condense, as in Fig. 2. The quartic interaction terms in 𝒱(B) are more conveniently expressed in terms of quadratic gauge invariant observables. By examining the transformations in Table 1, we can deduce the following correspondences between bilinears of the B with those of the bilinears of the gauge-neutral electrons (SI Appendix, 1 for the difference between the renormalized quasiparticle operator and bare electron cα): We have checked the correspondences in Eq. 9 in a few cases by computing the expectation values of the cα bilinears in the ancilla theory presented in SI Appendix, 1 and comparing them to the values of B bilinears. Now, we can write an expression for 𝒱(B) by keeping all quartic terms which involve nearest-neighbor sites:

The quadratic form of the chargons in Eq. 8 is identical to that for the spinons in Eq. 2, and so, the dispersion of the chargons is also that shown in Fig. 3. In the low-energy theory for the fermionic spinons, we had to focus on the nodal points in the Brillouin zone at the Fermi level. In contrast, for the bosonic chargons, we have to focus on the minima of the same dispersion. These are at distinct points in the Brillouin zone, and this is a factor in the distinct lattice symmetries of the orders described by the chargons. Specifically, the dispersion of chargons is and the band minima are at and . We can expand the B in terms of the eigenmodes at the minima using fields Bas (a = 1, 2 is the gauge SU(2) index, and s = ± refers to the band minima) This expansion leads to the symmetry transformations in Table 2, which follow from the transformations in Table 1. Obtaining the action of Pxy is a little involved, and it is obtained by requiring and also similar relations when x and y have opposite parity. The relation in Eq. 7 continues to hold in Table 2. We now define the following gauge-invariant order parameters The transformations of these expressions in Table 3 identify them as the labeled orders. Note that Tx and Ty commute for these gauge-invariant order parameters, and Eq. 7 does not apply to Table 3. We can now write down the Landau potential in this continuum limit At v1, 2, 3 = 0, this Higgs potential has an enhanced symmetry also present in the terms displayed in Eq. 8: There is a SO(8) symmetry of rotations among the 8 real components of Bas. After including the coupling to the SU(2) gauge field, we must factor out a SO(3) subgroup, which leaves the advertised SO(5)b symmetry for gauge-invariant order parameters. Indeed, we can now verify that the order parameters in Eq. 13 do indeed correspond to a 5-component order parameter which rotates under SO(5)b, after decomposing Δ into two 2 real components. We numerically minimized Eq. 14 for nonzero v1, 2, 3 and found only solutions which are either some linear combination of the two CDWs, a d-density wave, or a d-wave superconductor, with no coexistence between different orders. Simple ansatzes for these solutions are shown in Table 4. From these ansatzes, we can immediately determine the phase diagram of Eq. 14. The Higgs potential is stable provided all |vi|< u, and the lowest energy state is that associated with the smallest of the vi, i.e., for v1 < v2, 3, the lowest energy state is any linear combination of the x-CDW and y-CDW; for v2 < v1, 3, we obtain the d-density wave with broken time-reversal symmetry, and for v3 < v1, 2, we have a d-wave superconductor. The nature of the nodal Bogoliubov excitations of this superconductor will be similar to that studied in ref. 71.

In this section, we present the results of numerically minimizing the lattice potential for the chargons in Eqs. 8 and 10 on an 8×8 real space lattice. As there are a large number of parameters to minimize over, we will show results for regions of parameter space where the most general states can be found by varying only 2 parameters. Fig. 4 shows the agreement between the continuum and lattice phase diagrams when the lattice parameters are specifically chosen to reproduce the continuum free energy parameters in the low-energy limit as described in SI Appendix, 4. We also choose r to lie near the band minima. In this case, in agreement with analytical expectations, we find that lattice parameters corresponding to v2 < v1 and v2 < 0 lead to a d-density wave state. This state breaks time-reversal symmetry and is characterized by a circulating current pattern as shown in Fig. 4. For v1 < v2 and v1 < 0, we find, exactly as in the continuum theory, that any linear combination of CDW order at (0, π) and (π, 0) is favored. For v1 > 0 and v2 > 0, we find a d-wave superconductor appears where the precise phase boundaries are determined by u and g in Eq. 10. We have verified in these cases that the forms for Bi in our solutions obey Eq. 12 to a very good approximation. We next present a more general phase diagram, with parameters in the lattice model, Eqs. 8 and 10, chosen to be far from the limit where the continuum model applies. To capture a large variety of different ground states, we study both g < 0 and g > 0 with phase diagrams shown in Fig. 5 A and B, respectively, where we further choose a negative J1 and positive value of u in Eq. 10 for stability. Besides the d-density wave and the d-wave superconductor already present in Fig. 4, we also find a CDW with ordering wave vector (π, π), which coexists with either a d-density wave or superconductivity; furthermore, the previous degeneracy of any superposition of x-CDW and y-CDW is lifted; depending on V1, we either find a unidirectional 2-site stripe state or a bidirectional CDW, which preserves the four-fold rotational symmetry of the lattice. These additional states are illustrated in Fig. 5C. In future work, it would be interesting to study whether this coexistence of multiple orders survives the inclusion of SU(2) gauge fluctuations.

When longer-range hoppings of chargons are present, the dispersion can in general have multiple minima. The strategy here is similar to that followed in ref. 78 for the confinement transition out of a ℤ2 spin-liquid model of the pseudogap by condensation of vison. SI Appendix, 2 describes the general structure of the hopping terms in our present SU(2) gauge theory which are compatible with Table 1. The shortest-ranged terms are where w1, w2, w3 are real parameters. Choosing a unit cell containing two neighboring sites separated in the x-direction, the free energy density in momentum space is where τi acts in the sublattice space. The example dispersions associated with Eq. 16 are plotted in SI Appendix, Fig. S2 in Appendix 2. When only w1 is present, the minima are at Q±. Adding a finite w2, the dispersion becomes When w2 < 0, the positions of the minima remain at Q±. When w2 > 0, the minimum has a ring degeneracy since the energy depends only on f(k). When , the minima begin to expand from the two points Q± (or f(k)= − 2) to rings around these points. Upon further increasing w2, the rings will grow and touch when w2 = w1/4 = 0.25w1 (or f(k)=0) and merge to become new rings centered around (0, 0) and (0, π). Then, when w2 dominates, the new rings will shrink until they become points. When w3 is further added, the dispersion relation is complicated. Each ring can split into four minima in axial or diagonal directions, corresponding to the cases of w3 > 0 and w3 < 0, respectively. We will focus on the axial splitting case and for concreteness consider an infinitesimal w3 to split the ring in the regime of . The new incommensurate minima are at where the T, B, L, and R stand for Top, Bottom, Left, and Right, respectively, and q is a number depending on r, w1, w2, and w3. We expand the boson fields in terms of eigenmodes at the 8 minima, Here, the summation runs over α ∈ {L, R, T, B}, vα is a complicated real function of q and thus the parameters r, w1, w2, w3. There is only one independent vα: The vα satisfy vL = vR, vT = vB, and (1 + vT)(1 + vB)=2. When q → 0, the expression Eq. 19 reduces to Eq. 12. The symmetry transformations of the low-energy fields in Eq. 19, the expressions for the gauge-invariant order parameters in terms of these fields, and the allowed quartic terms in the chargon free energy are all discussed in SI Appendix, 2. Here, we write down ansatzes for a few interesting states, along the lines of Table 4 for the commensurate case. Only the nonzero values of Basα and order parameters are shown. x-CDW: Ba + R = (b1, 0), Ba + L = (b2, 0), ρ(nπ + 2q, 0) ∝ b2*b1, ρ(nπ − 2q, 0) = b1*b2, ρ(nπ, 0) ∝ |b1|2 + |b2|2. Here n = 0, 1. y-CDW: , , , , ρ(0, nπ + 2q) = b2*b1, ρ(0, nπ − 2q) ∝ b1*b2, ρ(0, nπ) ∝ |b1|2 + |b2|2. x-CDW and dDW: , , , , ρ(nπ + 2q, 0) = |b|2, ρ(nπ − 2q, 0) ∝ |b|2, D ∝ −2|b|2sin(2qx). d-wave superconductor: Ba + R = (b, 0), Ba − L = (0, b), Δ ∝ b2, ρ(0, 0) ∝ 2|b|2. Pair density wave: Ba + R = (b1, 0), Ba − R = (0, b1), Ba + L = (b2, 0), Ba − L = (0, b2), Δ ∝ 2b1b2 + b12e2iqx + b22e−2iqx, ρ ∝ 2|b1|2 + 2|b2|2 + 2b1*b2e2iqx + 2b2*b1e−2iqx. Note that a spatially uniform d-wave superconductor remains a possible solution even when we include only fields at the incommensurate points in Eq. 19. However, we have been unable to find a solution which is a pure incommensurate charge density wave at wavevectors, say, (π ± 2q). In the examples shown above, there is either an additional charge density wave at (π, 0) or a d-density wave. However, a pure commensurate charge density wave does exist, e.g., at (π/2, 0), for then (π, 0) is an allowed harmonic.

In this section, we will describe additional charge-ordered phases which emerge when we include quartic couplings beyond the nearest neighbor in Eq. 10. More specifically, we will add with Va, b = Vb, a = Va, −b = V−a, b to 𝒱(B) but will not include the further-neighbor terms w2, 3 quadratic in Bi which were already studied in the previous section. As we will see below, this is sufficient to stabilize stripe states with 4-site periodicity and, thus, connect our analysis to the period-4 stripe states found in cuprate experiments (79). Exploring all of parameter space of couplings Va, b is not practical, and so, we will restrict ourselves to just a few additional nonzero couplings out to fourth-nearest neighbors. We find that setting all Va, b, including V1, 0 = V0, 1 = V1, to zero except for V2, 2, V2, −2, V1, 1, and V1, −1 stabilizes period-4 stripe states, as summarized in Fig. 6. We find two types of period-4 stripe states; see Fig. 6B; the first (phase F) is centered on the bonds and coexists with current order with strength which modulates with the density. The second is a site-centered period-4 stripe state which may (phase G) coexist with current order that modulates with the density or appears without any additional current order (phase H). We note a small region (not shown in Fig. 6) at the phase boundary between phases H and G where another state appears with an additional 2-site charge modulation along the y (x) direction with much smaller magnitude compared to the primary period-4 modulation along x (y). Due to the smallness of this additional symmetry breaking compared to the part of this state which is identical to G, we do not separately denote this state on our phase diagram. We also find a region of pure d-density wave for small V2, 2 and small V1, 1 and a region of period 2 unidirectional stripe state for small V2, 2 and large V1, 1. We note that these orderings are all obtained within the chargon mean field theory, and it would be interesting to study their fate after including SU(2) gauge fluctuations.

Let us now collect all the terms in our SU(2) gauge theory for the underdoped cuprates: Hf in Eq. 2 describes the fermionic spinons transforming as a fundamental of SU(2) The chargon Higgs sector is described by ℒ(B) in Eq. 8, along with additional longer-range terms discussed in Sections 5 and 6. The chargon Higgs also transforms as a fundamental of SU(2). The hole pockets in the nodal region of the Brillouin zone are described by Hcg in SI Appendix, Eq. S1 in SI Appendix 1. All the above sectors are coupled by Higgs-fermion coupling HH in Eq. 6, or more specifically by Hfg in SI Appendix, Eq. S2 in SI Appendix 1. The remarkable similarity of the above structure to the Weinberg-Salam SU(2)×U(1) gauge theory of weak interactions (80) may already have been noticed by the alert reader. The electromagnetic U(1) is treated as effectively global in our case, the spinons map to the neutrinos, the electrons and chargons map to the electrons and Higgs bosons, and fermions and bosons are all coupled via the simplest gauge-invariant Yukawa coupling as in Eq. 6. In SI Appendix, 3, we integrate out the fermions and obtain an effective action for chargons. In addition to the terms just summarized above, this leads to terms with time derivatives of B. In general, a linear-time derivative term B†∂τB is allowed, and this will spoil explicit relativistic invariance. However, it remains possible that the B†∂τB term is irrelevant at strongly coupled fixed points which describes a quantum phase transition. With particle–hole symmetry in the cα electron band-structure, the B†∂τB term is absent. We also note that in the low-energy limit of section 2, the Bs†∂τBs term has a global symmetry which is smaller than SO(5)b, but |∂τBs|2 does have the full SO(5)b symmetry. At half-filling, when cα spectrum is gapped, this procedure of integrating out the cα is safe. Assuming particle–hole symmetry, and only two minima in the chargon dispersion, we obtain a relativistic theory for Nf = 2 massless Dirac fermions Ψ, and Nb = 2 scalars Bs (s = ±) both coupled minimally to a SU(2) gauge field with Lagrangian where the scalar potential V is specified in Eq. 14, γμ are the Dirac matrices, and Dμ is a covariant derivative. At v1, 2, 3 = 0, this theory is explicitly invariant under an SO(5)f×SO(5)b global symmetry, which leads to our proposal of a conformal field theory at the r = rc critical point in Fig. 2. We propose this CFT as a description of the phase transition between the antiferromagnet and the d-wave superconductor found in the weak-coupling repulsive Hubbard model by Raghu et al. (61), perhaps extended to strong coupling with additional antiferromagnetic exchange interactions. Depending on the fate of the v1, 2, 3 couplings, as well as possible quartic boson-fermions couplings, there could also be fixed points with a smaller global symmetry. We leave a careful examination of such terms to future work. We also note the study of refs. 81–83 which proposed and obtained numerical evidence for a CFT with a SU(2) gauge field and the same fermionic content as Eq. 21, but with Higgs bosons which were adjoints (and not fundamentals) of SU(2). This CFT described a deconfined critical point between an antiferromagnet and an “orthogonal semimetal” with the topological order of a ℤ2 spin liquid.

The π-flux state with fermionic spinons (28) is one of the earliest versions of a resonating valence bond spin liquid on the square lattice. It was realized early on that fluctuations about this mean-field state are described by a SU(2) gauge theory (13, 14, 31, 32). Furthermore, early work also recognized that doping such a spin-liquid state led naturally to a d-wave superconductor (13, 15, 31, 39–43). This connection is supported by recent numerical evidence (84), for d-wave superconductivity in doped antiferromagnets near the Néel-VBS transition, given the relationship between the π-flux spin liquid and the Néel-VBS transition (85). Here, we have investigated the consequences of a basic feature of the π-flux spin liquid to d-wave superconductor transition (this feature does not apply to the “staggered flux” spin liquid used elsewhere (13–15); SI Appendix, 5). This transition is also a confinement transition of the SU(2) gauge field of the spin liquid. By Higgs-confinement continuity (86), the transition can be implemented by the condensation of a Higgs field which transforms as a fundamental of SU(2). For the confining phase to be a superconductor, the Higgs field must also carry an electromagnetic charge, and these requirements lead essentially uniquely to the Higgs field being the bosonic chargon B (13–15). The basic feature is that the chargon B must also experience π-flux—this follows from the fact that the electron, which is a gauge-invariant combination of the spinon and the chargon, cannot experience any flux of the SU(2) gauge field. In the presence of a π-flux, the chargon dispersion is required (46) to have at least a twofold degeneracy in its low-energy spectrum. By exploiting this degeneracy, we have shown that a variety of competing charge-ordered states also appear naturally as the outcomes of the confinement of the π-flux spin liquid. The minimal SU(2) lattice gauge theory of the spinons and chargons is given by Eqs. 2 and 8, supplemented by the time-derivative terms discussed in SI Appendix, 3. Longer-range terms discussed in sections 5 and 6 can also be included. The phase diagrams in sections 4 and 6 were obtained in a mean-field treatment, in which we set the SU(2) gauge field Uij = 1, and treated Bi as spatially varying complex numbers to be optimized. Closer connections to the cuprate phase diagram require a more complete treatment of the SU(2) gauge fluctuations: We hope that such lattice gauge theory simulations will be carried out. The theory of just the chargons and spinons, and only the second-order time-derivative term in B (SI Appendix, 3), has no sign-problem, and determining its phase diagram will shed considerable light on the cuprate phase diagram. The phases in Figs. 5 and 6 have coexisting broken symmetries which are not required by any conventional symmetry principles and are instead a consequence of the use of a mean-field fractionalized order parameter B. It would be interesting to see whether this coexistence is present in a complete theory which includes SU(2) gauge fluctuations. After the onset of SU(2) confinement at low temperatures along arrow 𝔸 in Fig. 1, it is possible that an effective theory involving only the competing superconducting and charge orders (64–69) will become applicable. However, at higher temperatures, there must be a change to the deconfined characteristics of the pseudogap metal, and the theory presented here is designed to address this transformation. Such a theory also points to resolutions of the key puzzles noted in the introduction: The FL* state with an underlying π-flux spin liquid can fit the photoemission data in the pseudogap metal in both the nodal and antinodal regions of the Brillouin zone, as discussed in earlier work (24). The parent pseudogap metal state already has a gap in the antinodal region of the electronic Brillouin zone. So, it is natural that this gap is preserved when the pseudogap metal undergoes a confinement transition to a charge-ordered state, potentially allowing us to understand the fermiology of the quantum oscillations. The charge-ordered and d-wave superconducting confining states are not distinguished by the leading terms in the continuum static effective action for the chargons B. The degeneracy between these states is broken only by terms quartic in B, such as v1, 2, 3 in Eq. 14. This provides a rationale for the near-equality of the energy scales of charge-ordering and d-wave superconductivity (7).

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