A Time Reversal Symmetry

# Destruction of Néel order in the cuprates by electron doping

## Abstract

Motivated by the evidence (1); (2); (3); (4); (5); (6) in PrCeCuO and NdCeCuO of a magnetic quantum critical point at which Néel order is destroyed, we study the evolution with doping of the quantum phases of the electron doped cuprates. At low doping, there is a metallic Néel state with small electron Fermi pockets, and this yields a fully gapped superconductor with co-existing Néel order at low temperatures. We analyze the routes by which the spin-rotation symmetry can be restored in these metallic and superconducting states. In the metal, the loss of Néel order leads to a topologically ordered ‘doublon metal’ across a deconfined critical point with global O(4) symmetry. In the superconductor, in addition to the conventional spin density wave transition, we find a variety of unconventional possibilities, including transitions to a nematic superconductor and to valence bond supersolids. Measurements of the spin correlation length and of the anomalous dimension of the Néel order by neutron scattering or NMR should discriminate these unconventional transitions from spin density wave theory.

1

## I Introduction

Superconductivity in the cuprates emerges on doping an antiferromagnetic insulator with either holes or electrons. The hole-doped cuprates generally have higher superconducting critical temperatures, but at the same time display a host of complicated phenomena, e.g. incommensurate magnetism and charge order, especially in the La series of compounds. The electron-doped cuprates on the other hand, provide an interesting contrast, where the phenomenology appears to be relatively simple. The superconductivity also has -wave pairing (7), but there is no evidence yet for charge order, and the magnetic correlation remain commensurate even after long-range magnetic order is destroyed. The sharp contrast between electron and hole doping must arise from particle-hole asymmetry in Cu-O planes. The electron-hole asymmetry of the Cu-O plane is evidenced most clearly by photo-emission experiments (8); (9); (10); (11) that show a sharp distinction between the Brillouin zone location of the low-energy fermions in the very lightly hole- [] and electron- [] doped cuprates, see Fig. 1.

A further motivation for the study of the electron-doped cuprates is provided by recent quantum oscillation evidence for the presence of electron pockets in the hole-doped cuprates in a strong magnetic field (12). It seems most natural to us that these electron pockets reside near the . So it seems appropriate to study the physics of the electron pockets where they are already present in zero field: in the electron-doped cuprates. Conversely, as we will see in this paper, the hole pockets near the also play a role in the physics of the electron-doped cuprates. Indeed, in both the electron- and hole-doped cuprates, a central problem is understanding how the hole pockets and the electron pockets reconnect to form a large Fermi surface state after the loss of magnetic order.

A recent neutron scattering study of the Néel correlation length (5) in NdCeCuO  provides evidence for a quantum critical point at , after which the Néel correlation length is finite. Remarkably, even at the optimal doping (at which long range Néel order is lost) a large Néel correlation length is measured; additionally, there is no evidence for incommensurate magnetic order over the entire doping range. The relative stability of the commensurate magnetism in the electron doped cuprates should be contrasted to the La series of the hole-doped cuprates. In the latter, long range magnetic order transforms from the Néel vector to incommensurate ordering vectors before being destroyed at dopings typically three times smaller than in the electron-doped cuprates.

These photoemission and neutron scattering measurements suggest the schematic phase diagram shown in Fig. 2, as a function of temperature () and electron doping ().

The focus of this paper is on the nature of the dynamic spin correlations in the electron-doped cuprates as a function of increasing doping. It is useful to frame our discussion by first recalling the predictions of a conventional spin-density-wave (SDW) theory of the evolution of the Fermi surface as function of electron density and the spontaneous Néel moment (13); (14); (15); (16); (17). We sketch the results of a mean-field computation in Fig. 3.

At very low electron doping (), we have the electron Fermi-pocket states shown in the leftmost panels (AFM Metal), with well established Néel order. When this state goes superconducting at low temperature (), the Fermi surface does not intersect the diagonals along which the pairing amplitude vanishes, and so the resulting -wave superconductor (AFM+SC) is fully gapped. At large electron doping, we have the large Fermi surfaces shown in the rightmost panels, with no Néel order. Now the Fermi surfaces do intersect the diagonals at 4 points, and so the -wave superconductor has 4 nodal points. Examining the evolution of the Fermi surfaces between these two limiting cases in Fig. 3, we observe that there is generically an intermediate Fermi surface configuration, with Néel order, in which the Fermi surfaces intersect the diagonals at the 8 points and , for some small non-zero . The appearance of superconductivity at low will then lead to a -wave superconductor with 8 nodal points in the full Brillouin zone of the square lattice. Thus, in both the metallic and superconducting cases, this intermediate state has 8 zero-energy crossings of the fermion dispersion relation along the diagonals of the full square lattice Brillouin zone.

A further motivation for our study is that the 8 diagonal Fermi points of the intermediate state are not clearly seen in photoemission experiments (8); (9); (10); (11). Fermi surface crossings are seen on only a single point adjacent to the 4 points. We therefore explore here unconventional routes by which the Néel order at low doping, in the AFM metal and the fully gapped -wave superconductor, can be destroyed by increasing hole concentration.

Important aspects of our results on the metallic and superconducting quantum phases and phase transitions are summarized in Figs 4 and 5.

The right panel of Fig. 4 indicates results on a “toy” - model of bosons which we will describe in Section IV. We will see that there is a close analogy between our results for the electronic - model and the toy boson model, with the latter model having the advantage that duality computations of the crossover into confinement can be carried out in explicit detail.

For the metallic case, we find that the quantum transition out of the Néel state with Fermi-pockets (the AFM Metal) is into an exotic ‘doublon metal’ state without magnetic order (see Fig. 4); the nomenclature refers to the sites with double occupancy when the Mott insulator is doped with electrons. The ‘doublon metal’ is the particle-hole conjugate of the ‘holon metal’ state described in recent work (18); (19), and both are examples of ‘algebraic charge liquids’. These states have topological order and no sharp electron-like quasiparticles. However, they are separated from conventional Fermi liquid states by sharp transitions only at ; at there are only crossovers into the Fermi liquid-like regime. As superconductivity always appears as (see Fig. 2), it is these crossovers of the metallic regime which are needed for experimental comparisons. We shall show that the spin excitations near the transition into the doublon metal are described by a quantum field theory with global O(4) symmetry, as indicated in Fig. 4. Further, as we discuss below in Section I.1, spin fluctuations of this O(4) theory have clear experimental signatures. Section IV will show that these metallic phases of the electronic - model also have strikingly similar analogs in the - model of bosons, along with a magnetic ordering transition in the O(4) class.

For the superconducting case, we find a number of distinct possibilities, which are illustrated in Fig. 5. From the AFM superconductor we find 3 distinct classes of transitions:
(a) A transition to a -wave superconductor with full square lattice symmetry, which is in the O(3) universality class. This transition is in the same universality class as conventional SDW theory. It is quite remarkable, and novel, that this SDW transition reappears in our formalism based on fractionalized degrees of freedom.
(b) A transition to a -wave superconductor with co-existing valence bond solid (VBS) order, i.e. to a supersolid. Such a valence bond supersolid was initially discussed in Refs. (20); (21). The pattern of the VBS order is columnar or plaquette (see Figs. 5 and 7), the same as that in the insulator (22); (23) (for rational with odd, other patterns of order are possible, as discussed in Section IV.2.3). This transition is expected to be of the ‘deconfined’ variety, in the CP universality class, similar to the transition in insulating antiferromagnets (24); (22).
(c) The third transition is described by the CP theory, but with an additional ‘doubled monopole’ perturbation allowed, which will be explained in more detail in Sections IV.2. The non-magnetic superconductor does break the square lattice space group symmetry, and the two allowed patterns of symmetry breaking are in Fig. 8. Notice that one them only breaks the rotation symmetry of the square lattice to a rectangular symmetry, leading to a nematic superconductor shown in Fig. 5.

It is interesting to note that the above possibilities match the patterns of transitions found (23) for insulating antiferromagnets as function of the spin . In particular, case (a) occurs for even integer , case (b) for half-odd-integer , and case (c) for odd integer . Here we are considering a antiferromagnet, but with a background of a compressible superconductor. As we shall see, the background density fluctuations in the superconductor are able to modify the spin Berry phases so that the transitions match those for different in insulators.

In addition to cases (a), (b), (c), we note briefly that it is also possible that the AFM+SC state already has density modulations. Then, the transition involving loss of AFM order will lead to modifications in the ordering pattern, as will be discussed in some detail in Section IV.2. An important point is that, in all these cases, the set of allowed periods for the density modulations in the supersolid without AFM order are the same as those characteristic (25) of paired supersolids of density .

An interesting issue, which we shall largely leave open in the present paper, is the nature of the spectrum of the fermionic Bogoliubov quasiparticle excitations in the supersolid or the nematic superconductor. One natural possibility (26); (27); (28) is that these excitations initially remain fully gapped, as in the AFM+SC state. On the other hand, knowing that the supersolid or nematic superconductor has effective density , the structure of the Fermi surfaces in Fig. 3 suggest that such a -wave superconductor should have 4 gapless Dirac points. In the deconfined quantum critical theory, the electron spectral function is fully gapped along the diagonals of the Brillouin zone. If the gapless nodal points do appear in the non-magnetic phase, they would create “Fermi liquid coherence peaks” at the nodal points, with the weight of the coherence peak vanishing as we approach the quantum critical point. This phenomenon would then resemble that in dynamical mean field theory (29); (30), where the Fermi liquid coherence peaks of the metal vanish at the metal-insulator transition, revealing a fully gapped single-particle spectrum at the critical point. This issue will be discussed further in Section III.2.

### i.1 Experimental tests

We note here that neutron scattering or NMR measurements of the spin excitation spectrum can serve as useful experimental probes of whether the Néel order is lost as in a conventional SDW framework, or in a more exotic deconfined critical point . In particular, the temperature dependence of various components of the dynamic structure factor in the quantum critical region can measure two crucial exponents characterizing the transition, the dynamic critical exponent, , and the anomalous dimensions of the Néel order parameter, . In terms of these exponents, we have (31) for , the zero frequency dynamic structure factor at the Néel ordering wavevector (proportional to the elastic neutron scattering cross-section at ):

 SeN∼T(−2+ηN)/z; (1)

for, , the equal-time structure factor at the Néel ordering wavevector (proportional to the energy-integrated neutron scattering cross-section at ):

 SN∼T(−2+z+ηN)/z; (2)

for , the Néel correlation length:

 ξ∼T−1/z; (3)

The present neutron scattering experiment (5) only reports the quantum critical behavior of the spin correlation length, which is consistent with . Although data on exists, a scaling analysis to extract the exponent Eq. (2) has not been carried out. An important test of quantum critical scaling would be to check that the exponent that arises from this analysis should agree with an extraction of the same index by an analysis of the Cu NMR relaxation rate,

 1T1∼TηN/z. (4)

The values of the exponents in the conventional SDW theory depend upon whether the quantum critical region is controlled by a metallic or a superconducting fixed point. For the metallic fixed point, we have the Hertz-Millis-Moriya theory (32) and , while for the superconducting case we have the usual 3D O(3) transition, and .

Our main new experimentally relevant results in this paper are the values of these exponents for the ‘deconfined’ transition at which Néel order is lost. The exponents depend upon whether we are using a superconducting or metallic fixed point, and our results are summarized in Table 1. There are no existing numerical results for the ‘doubled monopole’ transition, and so these are not shown: it may well be that this case has a first-order transition. Note the large values of for the deconfined cases, making them clearly distinguishable from the SDW cases. In particular, with for the metallic case, the equal-time structure factor, has a singular contribution which decreases with decreasing .

We also note that for the superconducting case, the properties of the CP field theory are not fully settled in the literature (36); (37); (38) with a debate on whether the quantum transition is second- or first-order. Nevertheless, there is significant evidence (39); (22) of a crossover into a regime which is described by the CP field theory. Furthermore, even if the transition is first-order, it appears to be only very weakly so, and the simulations of Ref. (36) show a substantial critical scaling regime.

Because the electron-doped cuprates are always superconducting in the proximity of the quantum critical point at low , the superconducting critical theory described above is the correct description at very low- scales. The normal state theory does however apply at temperature scales above the superconducting temperature and hence could be the relevant one for experiments over a large temperature scale. An interesting prediction that arises from this crossover is that the equal-time structure factor, , could have a non-monotonic dependence. It should first decrease with cooling (when the system is controlled by the metallic fixed point with ), and then crossover to increasing with further cooling, when the system is controlled by the superconducting fixed point with .

The outline of the remainder of this paper is as follows: In Sec. II we derive an effective field theory for the electron-doped cuprates in a language well suited to discuss both the magnetic phases and the non-magnetic ones that appear on the destruction of Néel order. In Sec. III we discuss the various possibilities for transitions involving loss of Néel order. The - model of bosons will be introduced in Section IV, along with a complete duality analysis of its phase diagram and its crossover to confining phases. Finally in Sec. V, we conclude with a summary of our results.

## Ii Field Theory at Low Doping

We begin with a symmetry-based derivation of a long wavelength effective action for the electron-doped cuprates. We will use the low energy excitations of the low doping state to build a theory which is valid also at larger doping when spin rotation invariance is restored.

The motion of a small number of charge carriers in a quantum anti-ferromagnet is usually described by the model,

 Ht−J=−∑i,j,αtij(cα†icjα+h.c.)+∑i,jJij→Si⋅→Sj+…, (5)

where destroys an electron with spin on the sites of a square lattice and , with the Pauli matrices. We shall study the case in which the electrons hop on a square lattice. Once extra electrons are doped into the half-filled magnet a constraint must also be included. The constraint,

 ∑αcα†iciα ≥ 1 (6)

is enforced on each site, modeling the large local repulsion between the electrons. It is important to note that our results are more general than a particular - model, and follow almost completely from symmetry considerations. The ellipses in Eq. (5) indicate additional short-range couplings which preserve square lattice symmetry and spin rotation invariance.

Following Ref. (40), but now for the case of electron-doping, we re-write the electron operators in a type lattice model in terms of spinons and ‘doublons’ (for doubly occupied sites). Note that here the site occupation is constrained to be . We use the following representation for the electron operators,

 cα = εαβbβ†g+ (on A) cα = −¯¯b†αg− (on B) (7)

where the constraint is . [We first used on both sub-lattices, then rotated the Schwinger bosons on the B sublattice , like in the Auerbach-Arovas analysis (41)]. Note:

Now we are in a position to write down the transformation of the lattice fields that we have written down under the various square-lattice symmtries and time reversal. We require that the composite fields transform into each other in the usual way under the square lattice symmetries. The implementation of time reversal symmetry is detailed in Appendix A. We thus arrive at the Table 2.

We now proceed to take the continuum limit of the lattice model that we have defined. In order to do so (23), we define fields and and integrate out the massive field. We then arrive at the Lagrangian for the

 Lz=D+μzα∗D−μzα+s|zα|2+u(|zα|2)2+… (8)

where is a spacetime index, , is an emergent U(1) gauge field linked to the local constraint in Eq. (7), and and are couplings which can be tuned to explore the phase diagram. The Néel order parameter is simply .

We also need to take the continuum limit for the charge carrying fermions of this model. As discussed in detail in Ref. (40), fermions that live on opposite sub-lattices carry opposite charges under the gauge field, , and hence must be represented by two distinct continuum fields (both fields are centered at the lattice momentum, ). The lowest derivative term consistent with the symmetry of the is,

 Lg=∑q=±g†q(D¯qτ−μ−D¯q2j2m)gq. (9)

where is the curvature of the fermion bands and . Finally, by requiring consistency with the lattice transformation properties of the continuum fields, presented in Table 3, the lowest allowed derivative term that couples the opposite fermions can be deduced,

 Lz−g = λ εαβ[g†+(D+xg−)zα(D−xzβ)−g†+(D+yg−)zα(D−yzβ)] (10) + εαβ[g†−(D−yg+)zα∗(D+yzβ∗)−g†−(D−xg+)zα∗(D+xzβ∗)]+c.c.

This is the analog for electron-doped cuprates, of the well-known Shraiman-Siggia term (42) in the hole-doped case. Remarkably, this term has two spatial derivatives; there is no term allowed with a single spatial derivative (as is found from a similar analysis in the hole-doped case (40); see also (43)). The extra derivative makes the effect of this term weaker. The weakness of this coupling, which arises because of the BZ location of the low energy fermions, (which in turn is ultimately tied to the p-h asymmetry in the Cu-O layers) is the fundamental reason for the robustness of the commensurate Néel correlations in the electron-doped cuprates as compared to the hole-doped case. These correlations extend at least up to optimal doping (5); (6) and possibly beyond giving us confidence in the present approach.

The complete effective action for the electron-doped antiferromagnet is then . The final term, contains the Berry phases of the monopoles, and has the form

 SB=iπ2∑ȷmȷζȷ (11)

for monopoles with integer charges on the sites on the dual lattice; is fixed at on the four dual sublattices (23).

### ii.1 Néel order and superconductivity

We now discuss the phase diagram of the field theory presented in the previous section. Some of the analysis parallels that presented in Refs. (40) and (18) for the hole-doped case.

The phases are most easily characterized by using a representation for the physical electron annihilation operator in terms of the fields we have introduced above. We first express the electron operator in terms of its components at momenta at and ,

 Ψα(→r)=ei→Q1⋅→rΨ1α(→r)+ei→Q2⋅→rΨ2α(→r). (12)

Then, as in Ref. (40), we use the symmetry transformation properties to deduce the unique bilinear combination of the fermion and and CP fields that transform in the way that the physical electrons should,

 Ψ1,2α = εαβzβ∗g+∓zαg−. (13)

The phases is Fig. 2 can now be characterized in terms of the and degrees of freedom:
(i) AFM metal: This is the Higgs phase of the gauge theory, in which there is Higgs condensate of with . As discussed in Ref. (40), the “Meissner” effect associated with this Higgs condensate ties the gauge charge to the spin quantum number. So for Néel order oriented along the axis, the fermions carry spin and reside in Fermi pockets. The resulting phase is then identical to the AFM metal phase obtained in SDW theory, and shown in the left panel of Fig. 3.
(ii) Doublon metal: This is the particle-hole conjugate of the holon metal, and is a non-Fermi liquid ‘algebraic charge liquid’. We have , and and the phase is described by the gapped quanta and the Fermi pockets interacting via exchange of the gauge force. We observe from Eq. (13) that the physical electron involves a convolution of the propagators of the and , and so will not have Fermi liquid form. We note that the holons and the holon-spinon bound states, discussed in previous work (40); (18) on the holon metal, are also legitimate excitations of the doublon metal. Here they are likely to be gapped, but at will contribute photoemission spectral weight (8) near the points in Fig 1.
(iii) SC phases: As discussed in Ref. (18), the nearest-neighbor hopping term, and the gauge forces, will induce a pairing of the fermions. Let us assume a pairing of the form

 ⟨g+1(k)g−1(−k)⟩=Δ(k). (14)

Then the pairing signature of the electrons can be computed from Eq. (13) and (14): the various possibilities are discussed below. If we also have , then we obtain the AFM+SC phase of Fig. 2. This is a stable phase, because the Higgs condensate quenches the gauge fluctuations and also the monopoles; its physical properties are identical to the AFM+SC phase obtained in the SDW theory noted in Fig. 3. A superconducting phase with is the doublon superconductor, and this is not stable: proliferation of monopoles will lead to confinement, as we shall discuss in Section III.2.

The remainder of this subsection will characterize the symmetry properties of the possible superconducting phases. We also allow long-range Néel order by a condensation of the CP fields with : the Néel order is then polarized in the direction with spontaneous moment . Using these averages, Eq. (14), and the expressions for the physical electron operators, Eq. (13), we can calculate the required anomalous averages,

 ⟨Ψ1α(k)Ψ1β(−k)⟩ = −εαβ[mβΔ(k)+mαΔ(−k)] ⟨Ψ1α(k)Ψ2β(−k)⟩ = εαβ[mβΔ(k)−mαΔ(−k)] ⟨Ψ2α(k)Ψ2β(−k)⟩ = εαβ[mβΔ(k)+mαΔ(−k)] (15)

At the critical point from the AFM+SC state to the SC state the Néel order parameter vanishes, i.e., and the correlator should disappear (this follows from the restoration of full translational invariance), indicating that . Since the superconducting instability arises out of a short-range attractive interaction it is most natural to expect -wave pairing. Remarkably, this naturally leads to pairing for the physical electrons [as can be verified from Eq. (II.1) by substituting ]. However since the underlying particles are in an -wave state, the quasi-particles in this superconductor are fully gapped. We propose that this is the quantum state that describes the phase AFM+SC in Fig. 2 and that is observed in the region of co-existence in NdCeCuO  (5). We note that with increasing , the Néel order is suppressed making the gauge field mediated attraction (that causes superconductivity) stronger, which in turn is expected to result in an increase of , consistent with experimental observations. For the sake of completeness, we present the other symmetry allowed options for pairings (see Fig. 6 second row): corresponds to the case, corresponds to the case and corresponds to the case; all these states have nodal excitations.

Finally, the condition that is satisfied by all the order parameters deduced from Eq. II.1 (and that are illustrated in the second row of Fig. 6) follows quite simply from the fact that the phase factor is on the sublattices and that pairing occurs only between electrons on opposite sub-lattices.

## Iii Quantum criticality

We now turn to our main results on the quantum phase transitions involving loss of Néel order as described by the low energy theory introduced in Section II; the results were summarized in Fig. 4 and 5.

### iii.1 Metallic states

First, let us consider the transition without superconductivity, destroying magnetic order in the AFM metal, leading to the doublon metal. This transition is described directly by the field theory in Section II, and is associated with the condensation of the spinons in the presence of the Fermi surfaces.

At , such a transition between metallic states could be induced by destroying superconductivity by an applied magnetic field. Moreover, even at zero magnetic field, the quantum critical region at temperatures above the superconducting could be controlled by the crossovers of an underlying AFM metal/doublon metal quantum critical point. Monopoles can be ignored in the following analysis because they are suppressed by the gapless excitations at the Fermi surfaces (45) (see Appendix B). The resulting state without antiferromagnetism therefore carries gapless gauge excitations, and as we noted earlier, realizes an algebraic charge liquid which we call a doublon metal.

The theory for this transition follows the analysis of a formally similar transition of bosons and fermions coupled to a U(1) gauge field in Ref. (46). In this previous case, the bosons were spinless and fermions carried spin, whereas here the fermions are spinless while the bosons carry spin. However, for the quantum criticality, the more significant difference is that the quadratic action for the bosons has a relativistic structure, unlike the dispersion in Ref. (46).

The renormalized gauge field propagator is a key ingredient in our analysis. This depends upon the polarizabilities of the fermions and the bosons at the quantum critical point. We evaluate these from the bare actions and , and will confirm later that the same results hold in the fully renormalized critical theory. As usual, the fermion polarizability screens the longitudinal fluctuations, and the only potential singularity arises from the transverse propagator, . In the Coulomb gauge, this has the low momentum and imaginary frequency form (47)

 Dij(k,iω)∼(δij−kikjk2)1k+χ|ω|/k (16)

Here the term in the denominator is the contribution of the fermions, and is present for , where is the Fermi velocity. The term emerges from the critical correlator (it coefficient is proportional to the critical conductivity of the ’s).

Let us now compute the consequence of the overdamped gauge fluctuations in Eq. (16) on the spectral function. At leading order the self energy at criticality is

 Σz(p,iϵ)∼∫dω∫d2k(p2−(p⋅k)2/k2)k+χ|ω|/k1(ω+ϵ)2+(k+p)2. (17)

It is now easy to confirm that this expression for is non-singular at low and , and does not modify the leading behavior of the propagator. In particular, the on-shell self energy has the imaginary part

 ImΣz(p,ϵ=p)∼p3, (18)

which is clearly unimportant to the critical theory. Thus the overdamping of the gauge fluctuations by the fermions strongly suppresses their influence on the excitations. Indeed, in the scaling, the term in the denominator of Eq. (16) can be neglected, and the renormalized action for the transverse component of the gauge field is ; this scales as an anisotropic “mass” term for the gauge boson. Thus we can view this feature as a fermionic version of the Higgs mechanism, in which the low energy excitations of a Fermi surface quench the gauge field fluctuations. We will comment further on this analogy with the Higgs mechanism in Section IV.

In a recent work (47) in a different context, Senthil has computed the consequences of the singular interactions associated with Eqs. (16) and (17) on the spectral function of the fermions, and the associated formation of critical Fermi surfaces (48). All those results apply here too to our theory of the transition from the AF metal to the doublon metal.

At this point, we are now prepared to integrate out the gauge boson and the fermions, and obtain an effective theory for the spinons. Keeping only the terms potentially relevant near the critical point, the resulting effective action has the structure

 Seffz=∫d2rdτ[|∂μzα|2+s|zα|2+u(|zα|2)2]+λ∫d2kdω[|zα|2](−k,−ω)|ω|k[|zβ|2](k,ω) (19)

The last term is a consequence of the compressible fluctuations of the Fermi surfaces, which couple to via a contact term (40). At it is now evident that describes a transition for the loss of Néel order by a conformal field theory in the O(4) universality class. We can therefore ask for the scaling dimension of at this conformal critical point. This follows from a simple scaling argument (49):

 dim[λ]=−3+2ν (20)

The O(4) model has (50) and so is an irrelevant perturbation. Further, when we account for the long-range Coulomb interactions between the fermions, there is an additional factor of in the term, and is then more strongly irrelevant.

We have now established that the transition from the Néel-ordered Fermi-pocket metal to the doublon metal is in the O(4) universality class. The Néel order parameter itself is a quadratic composite of the . It transforms under the symmetric, traceless, second-rank tensor representation of O(4), and the scaling dimension of this composite operator has been computed earlier (34); (35). From the field-theoretic analysis of Calabrese et al. (34) we find , while the Monte Carlo simulations of Isakov et al. (35) we obtain .

### iii.2 Superconducting states

Now we discuss the transition out of the superconducting AFM+SC state with increasing doping. Because the fermions are fully gapped in the superconductor, they can initially be ignored in the analysis of the critical theory. The remaining excitations are described by the CP model. So a natural initial guess is that the critical theory for the loss of Néel order is the same as that in the insulator (24). The presence of superconductivity here does induce additional gapless density fluctuations, but these are irrelevant (44) as long as for neutral systems, and generically unimportant with long-range Coulomb interactions. Further, the paramagnetic state so obtained is not a BCS superconductor, because the fermionic Bogoliubov quasiparticles carry no spin. Rather, as discussed in Ref. (18), it is a “doublon superconductor”. However, once we have moved away from the critical point, there are no gapless excitations which can serve to suppress monopoles in the U(1) gauge field. We expect that the condensation of the monopoles at a large secondary length scale will induce confinement, leading to a generic instability of the doublon superconductor.

We are interested in the nature of the confined state. For the corresponding transition in the insulator (24), the confining state was the valence bond solid (VBS) which was induced by the Berry phases on the monopoles. However, here there is the possibility that the density fluctuations of the superconductor can modify the influence of the Berry phases. Because the fermions are paired in the AFM+SC state and at the critical point, it is plausible that a - model in which the are bosons (and the remain bosons) should have essentially the same properties in its charge and density correlations in their respective paired states: we are merely replacing the internal constituents of the Cooper pairs, but this should not modify the nature of the phase and vortex fluctuations of the superfluid. We will examine such a - model of bosons in Section IV: we are able to carry out an analysis of the influence of monopole condensation in some detail, and find 3 distinct possibilities which were listed earlier in Section I and Fig. 5:
(a) A conventional O(3) transition, as in SDW, theory, to a -wave superconductor with full square lattice symmetry. The monopole Berry phases are precisely cancelled by density fluctuations in the superfluid, and so the monopoles confine the spinons into the vector SDW order parameter.
(b) A ‘deconfined’ CP transition to a valence bond supersolid (20); (21), where the pattern of the VBS order is the same as that in the insulator (22); (23) (see Fig. 7). Here the monopole Berry phases remain as in Eq. (11). For rational with odd, other patterns of order are possible, as discussed in Section IV.2.3.
(c) A direct transition to a -wave superconductor with square lattice symmetry broken as in the states in Fig. 8, one of which is a nematic superconductor. In this case, the monopole Berry phases are only partially compensated by the superfluid modes, so that monopoles with even magnetic charge are allowed at the transition. Little is known about the critical properties of such a ‘doubled monopole’ theory, and it is possible the transition is first order.

The above list exhausts the possible transition out of the AFM+SC state, for the case in which the AFM+SC state does not have density modulations of its own. In Section IV, we consider the further possibility that the AFM+SC state already carries density modulations (so that it is also a supersolid). We classify transition out of such states: monopole condensation modifies the nature of the density modulations in the non-magnetic supersolid, and these will be described in Section IV.2. An interesting feature of the resulting supersolids is that the set of allowed wavevectors for density modulations are the same as those of a model of paired particles (25); (51) of density . Thus, once Néel order is lost, the primary role of the monopoles is to account for the ‘background’ density of one particle per site in the Mott insulator, and to combine this density with the doped particles to yield states which are sensitive to the total density.

This sensitivity to the total density in the supersolid bears some similarity to the constraints placed by Luttinger’s theorem on the volume enclosed by the Fermi surface in Fermi liquid states (46). Let us explain the connection more explicitly. We can view the monopole Berry phases as arising from a filled band of anti-holons in the insulator, and these are extracted into the confining electronic states (19). To see this, let us recall the origin of the monopole Berry phase in Eq. (11). This can be traced to the constraint applied on every site of sublattice A (there are parallel considerations on sublattice B, which we will not write down explicitly), and implemented by a Lagrange multiplier in the effective action with the term

 i∫dτλ(b†αbα+g†+g+−1). (21)

The fluctuations of are on sublattice A ( on sublattice B), and the in the brackets above evaluates (52) to Eq. (11) for a monopole configuration of . Let us now also allow for the gapped holon states(40); (19) by the holon operators , in which case the term in Eq. (21) generalizes to

 i∫dτλ(b†αbα+g†+g++f†+f+−1). (22)

Finally, we perform a particle-hole transformation to anti-holons (which are distinct from the doublons) to obtain

 i∫dτλ(b†αbα+g†+g+−h†+h+). (23)

In this form, there is no in the bracket, but we have a filled band of anti-holons—thus we have an alternative book-keeping in which there is no monopole Berry phase, but we do have to account for the unit density of anti-holons in the Mott insulator. After confinement with spinons, it is this density which contributes to the expansion to the large Fermi surface in the Fermi liquid, and the sensitivity of the supersolid to density .

Next, we consider the structure of the physical electron spectral function in the supersolid. We focus on momenta along the diagonals of the square lattice Brillouin zone. Right at the critical point, the and fermionic excitations are fully gapped, and so the electron spectral function (which is a convolution of these fermion Green’s functions with those of the ) is also fully gapped. Moving on the confining side of the critical point, a natural possibility is that electron spectrum remains fully gapped (26); (27); (28). However, given the fact that the confining supersolid consists of a density (as in a ‘large’ Fermi surface), from Fig. 3 it would seem natural that this state has 4 gapless nodal quasiparticles; if so, the total spectral weight in these low energy fermions would vanish as we approach the critical point, in a manner we expect is related to the scaling dimension of the monopole operators. It is interesting to note that this vanishing of low energy fermionic spectral weight resembles the phenomenon of spectral weight transfer in dynamical mean field theory (30); (29). We also note that the emergence of gapless composite fermions from gapful constituents (here the ‘composite’ electrons consist of gapped spinons and holons) has counterparts in a number of particle theory models (53); (54).

It is useful to discuss this theory in the context of recent ideas by Senthil (48) on ‘critical Fermi surfaces’. In the latter framework for a transition to a -wave superconductor with 4 nodal points, the nodal fermions would be part of the critical theory, and then the deconfined critical theory would not be the CP model. Such a scenario would be realized here if the holon Fermi surfaces formed in the AFM state (this is compatible with current photoemission experiments (8)), and the magnetic disordering transition led to a holon superconductor with gapless Dirac excitations (18). A confinement transition on the holon superconductor would then realize this scenario, but only at ’critical Fermi points’ and not on a ’critical Fermi surface’.

## Iv t-J model of bosons

The Higgs-like suppression of the fluctuations in Section III.1 suggested to us that we examine a toy model of bosons obeying the same - model described here. In other words, we will consider the same theory presented in Section II but now the and the fields are all bosons. We can make quite reliable statements about the phases of this model, including the role of monopoles and Berry phases.

A further motivation to examining this model was noted in Section III.2: this is an efficient way to analyze the paired superconducting states, where we expect pairs of bosons or fermions to have similar properties.

The analogy between the phases of the electronic model and the toy boson model were summarized in Fig. 4 and 5. The parallel of the Higgs-like effects in the metallic phases of Section III.1 appears when we replace the Fermi pockets by Bose condensates of the — the corresponding transitions in the boson model are then in the same universality class as in the metallic electronic model. As indicated in Fig. 5, the boson model also has parallels to the transitions of the superconducting sector of the electronic model which were discussed in Sections III.2. This will be described in more detail below.

First, let us list the phases of the boson - model of interest to us:
(i) AFM boson superfluid: Here both the and the condense with

 ⟨zα⟩≠0  ,  ⟨g±⟩≠0. (24)

The presence of these condensates implies that both and monopole fluctuations are suppressed, as in the AFM metal. Also, by Eq. (13), the physical boson operator also has a non-zero condensate. So this state has AFM order and a flux quantum of .
(ii) Paired boson superfluid: Now spin rotation invariance is restored, with

 ⟨zα⟩=0  ,  ⟨g±⟩≠0. (25)

However, the condensate is sufficient to continue to suppress both the and the monopole fluctuations, making this state the analog of the doublon metal. Two other characteristics of this state reinforce the analogy with the doublon metal: (i) the quanta represent stable, neutral, , gapped excitations, which are also found in the doublon metal, and (ii) the action of an isolated monopole diverges linearly with system size because of the Higgs condensate, and a similar linear divergence appears (55) in an RPA-like estimation (56) of the monopole action in the doublon metal (see Appendix B). With the condensates as in Eq. (25), as discussed in Ref. (51), the only gauge-invariant condensate carries charge , and so the flux quantum is . A comparison of Eqs. (24) and (25) shows that the transition between the AFM boson superfluid and the paired boson superfluid involves criticality of alone. The mode can be ignored and so it is evident that the critical theory is the O(4) model in Eq. (19), but with the last density fluctuation term replaced by the analogous term for a superfluid (44). The latter term is also irrelevant, by an argument similar to that made for the electronic case.
(iii) AFM paired boson superfluid: Now we condense the , but only allow for a paired condensate of the bosons with

 ⟨zα⟩≠0  ,  ⟨g±⟩=0  ,  ⟨g+g−⟩≠0. (26)

There is antiferromagnetic order, and the flux quantum is . The condensate is sufficient to suppress both the and the monopole fluctuations, making this state the analog of the AFM superconductor in the electronic model. In some cases (to be discussed below in Section IV.2) this state will also break translational symmetry, i.e. it will become a supersolid.
(iv) Paired boson supersolid: The only condensate is that associated with the paired bosons:

 ⟨zα⟩=0  ,  ⟨g±⟩=0  ,  ⟨g+g−⟩≠0. (27)

This is the most interesting state here: the and monopole fluctuations are not suppressed, and we expect a crossover to a confining state. The same phenomenon also appeared in the electronic case with the doublon superconductor, which we argued was unstable to confinement to a -wave superconductor. The key advantage of the toy boson model is that we can describe the crossover to confinement in some detail, as will be presented in the following subsections. Our main result will be that there are generally periodic bond/density modulations in this phase, i.e. it is a supersolid; we include here the case of the nematic superconductor, in which only the rotational symmetry of the square lattice is broken. Finally, we will demonstrate that these modulations are characteristic (25); (51) of the total density of bosons, .

### iv.1 Duality and symmetry analysis

We will apply the analog of the duality methods presented in Refs. (24); (25); (51) to this model. These dualities are only operative for abelian symmetry, and so we shall replace the SU(2) spin symmetry by a U(1) symmetry of spin rotations about the axis.

We write the spinons, , and represent them by two angular degrees of freedom , . Similarly we take the (which are now bosons) and write them as . These fields are coupled to a compact U(1) gauge field , with the same charges as in the body of the paper. Finally, the monopoles in are endowed with the Haldane Berry phases (52); (24) in Eq. (11), to properly include the physics of the insulating antiferromagnet.

The simplest model consistent with such a framework is written below. Here we have discretized spacetime onto the sites of direct cubic lattice with sites and is a discrete lattice derivative.

 Z =∏j∫dθ↑jdθ↓jdϕ+jdϕ−jdAjμexp( (28) 1K∑jμcos(Δμθ↑j−Ajμ)+1K∑jμcos(Δμθ↓j−Ajμ) +1L∑jμcos(Δμϕ+j−Ajμ−Bjμ)+1L∑jμcos(Δμϕ−j+Ajμ−Bjμ) +1e2∑□cos(ϵμνλΔνAjλ)−SB).

Apart from the coupling constants, , , , the action contains two fixed external fields. The uniform static external electromagnetic field , where is the chemical potential; the value of is adjusted so that density of each boson species is . The last term accounts for the Berry phases linked to the monopoles in by Eq. (11).

To be complete, we should also add to Eq. (28) a staggered chemical potential which preferentially locates the on opposite sublattices, as has been done in previous work (57); (51). However, this term is not essential for our conclusions here, and so we omit it in the interests of simplicity.

The duality analysis of Eq. (28) is most transparent when the action is written in a Villain (periodic Gaussian) form. We do this by introducing the integer-valued fields, , , , which reside on the links of the direct lattice, and the integer-valued which resides on the links of the dual lattice. The dual lattice sites are labeled by .

 Z =∑{p↑jμ}∑{p↓jμ}∑{n+jμ}∑{n−jμ}∑{qȷμ}∏j∫dθ↑jdθ↓jdϕ+jdϕ−jdAjμexp( (29) −12K∑jμ(Δμθ↑j−Ajμ−2πp↑jμ)2−12K∑jμ(Δμθ↓j−Ajμ−2πp↓jμ)2 −12L∑jμ(Δμϕ+j−Ajμ−Bjμ−2πn+jμ)2−12L∑jμ(Δμϕ−j+Ajμ−Bjμ−2πn−jμ)2 −12e2∑□(ϵμνλΔνAjλ−2πqȷμ)2−iπ2∑ȷζȷΔμqȷμ).

An advantage of this periodic Gaussian form is that we are able to write an explicit expression for the monopole Berry phase (52); the fixed field is the same as that appearing in Eq. (11).

Now we proceed with a standard duality transformation of this action. Initially, this maps the theory onto the integer valued spin currents and , the integer valued charge currents and , and the integer valued fluxes with the partition function

 Zd =∑{J↑jμ}∑{J↓jμ}∑{H−jμ}∑{H−jμ}∑{Qȷμ}δconstraintsexp(−K2∑jμ(J2↑jμ+J2↓jμ) (30) −L2∑jμ(H2+jμ+H2−jμ)−e22∑ȷμ(Qȷμ−14Δμζȷμ)2−i∑jμBjμ(H+jμ+H−jμ)).

The summations in are restricted to integer-valued fields which obey the local constraints

 ΔμJ↑jμ=0  ,  ΔμJ↓jμ=0  ,  ΔμH+μ=0  ,  ΔμH−jμ=0  , ϵμνλΔνQȷλ=J↑jμ+J↓jμ+H+jμ−H−jμ. (31)

We solve these constraints by introducing the dual gauge fields and whose fluxes are the spin currents, the dual gauge fields and whose fluxes are the charge currents, and a height field whose gradients are the fluxes. Finally, we promote these dual discrete fields to continuous fields by introducing the dual matter fields and which annihilate vortices in the spinons, the dual matter fields and which annihilate vortices in the charged bosons, and the corresponding vortex and monopole fugacities. This leads to the dual theory in its unconstrained form

 Zd2 =∏j∫da↑ȷμda↓ȷμdb+ȷμdb−ȷμdhȷdα↑ȷdα↓ȷdβ+ȷdβ−ȷexp( (32) −K2∑□((ϵμνλΔνa↑ȷλ)2+(ϵμνλΔνa↓ȷλ)2) −L2∑□⎛⎝(ϵμνλΔνb+ȷλ−¯¯¯μLδμτ)2+(ϵμνλΔνb−ȷλ−¯¯¯μLδμτ)2⎞⎠ −e22∑ȷμ(Δμhȷ+a↑ȷμ+a↓ȷμ+b+ȷμ−b−ȷμ)2 Missing or unrecognized delimiter for \right +yvc∑ȷμ(cos(Δμβ+ȷ−2πb+ȷμ)+cos(Δμβ−ȷ−2πb−ȷμ)) +ym∑ȷcos(2πhȷ+α↑ȷ+α↓ȷ+β+ȷ−β−ȷ+π2ζȷ)).

The average flux of is and this should equal half the electron density, .

The action in Eq. (32) appears to be of daunting complexity, but its physical interpretation is transparently related to the direct theory. There are 4 vortex matter fields, , , , . These annihilate vortices in , , and respectively. These 4 matter fields carry unit charges under 4 U(1) gauge fields, , , , and respectively. Of these 4 gauge fields, one combination is always Higgsed out by the scalar field (by ‘Higgsed’ we mean that the gauge boson acquires a mass via the Higgs mechanism). The latter is related to the monopole annihilation operator , and the monopoles carry Berry phases .

Let us now make a further simplification of the dual action in Eq. (32). As the gauge field combination is always Higgsed by the field it is convenient to integrate these two fields out, obtaining the dual action

 Sd = K∑□(ϵμνλΔνaλ)2+L∑□(ϵμνλΔνbλ−