Main

Soon after the discovery of high-temperature superconductivity, Anderson6 presented influential ideas on its connection to a novel type of insulator, in which the electron falls apart into emergent fractional particles that separately carry its spin and charge. These ideas have been extensively developed7, and can explain the nodal zero-energy electron states in the superconductor. However, it is now known that the actual cuprate insulators are not of this type, and instead have conventional antiferromagnetic order, with the electron spins aligned in a chequerboard pattern on the square lattice. A separate set of ideas8 take the presence of antiferromagnetic order in the insulator seriously, but require a specific effort to induce zero-energy nodal electrons in the superconductor.

Our theory of algebraic charge liquids (ACLs) uses the recently developed theoretical framework of ‘deconfined quantum criticality’9 to describe the quantum fluctuations of the electrons. We show how this framework naturally combines the virtues of the earlier approaches: we begin with the antiferromagnetic insulator, but obtain electron fractionalization on changing the electron density. A number of experimental observations in the so-called ‘underdoped’ region between the antiferromagnet and the superconductor also fall neatly into place.

A key characteristic of an ACL is the presence of an emergent fractional particle that carries charge e, no spin, and has Fermi statistics. We shall refer to this fermion as a ‘holon’. The holon comes in two species, carrying charges ±1 in its interaction with an emergent gauge field aμ, where μ is a space-time index; this is a U(1) gauge field, similar to ordinary electromagnetism. However, the analogue of the electromagnetic fine structure constant is of order unity for aμ, and so its quantum fluctuations have much stronger effects. We also introduce operators f±† that create holons with charges ±1. From the f± and aμ, we can construct a variety of observables whose correlations decay with a power law as a function of distance or time in an ACL. These include valence-bond-solid and charge-density wave orders similar to those observed in recent scanning-tunnelling microscopy experiments10.

Whereas the f± carry the charge of the electron in the ACL, the spin of the electron resides on another fractional particle, the ‘spinon’, with field operator zα where α=↑,↓ is the spin index. The spinon is electrically neutral, and also carries aμ gauge charge. In an ACL, the spinon is simply related to the antiferromagnetic order parameter; if is the unit vector specifying the local orientation of the chequerboard spin ordering, then , where σ are the Pauli matrices.

The nomenclature ‘ACL’ signals a formal connection to the previously studied insulating ‘algebraic spin liquids’9,11,12,13,14, which have power-law spin correlations, and of which the deconfined critical point is a particular example. However, the observable properties of the ACLs are completely different, with the ‘algebraic’ (or, equivalently ‘critical’) correlations residing in the charge sector.

We begin our more detailed presentation of the ACLs by first considering the simpler state in which long-range antiferromagnetic order is preserved, but a density x of electrons (per Cu atom) have been removed from the insulator. This is the antiferromagnetic metal of Fig. 1. This state has been extensively discussed and is well understood in the literature, and its properties have recently been summarized in ref. 15 using the same language that we use here. Each missing electron creates a charge-e, spin S=1/2 fermionic ‘hole’ (to be distinguished from the spinless ‘holon’) in the antiferromagnetic state. It is known that the momenta of the holes reside in elliptical Fermi pockets centred at the points Ki=(±π/2a,±π/2a) in the square-lattice Brillouin zone (of lattice spacing a)—see Fig. 2. This is a conventional metallic state, not an ACL, with hole-like zero-energy excitations (or equivalently, the hole ‘Fermi surface’) along the dashed lines in Fig. 2. In such a state, both the oscillations of the resistance as a function of applied magnetic field (SdH for Shubnikov–de Haas) and emission of electrons by light (ARPES for angle-resolved photoemission) would indicate zero-energy electron states at the same momenta: along the dashed lines in Fig. 2.

Figure 1: Schematic phase diagram at small x.
figure 1

Here T is the temperature and x is the density of electrons (per Cu atom) that have been removed from the insulator. The phases labelled ‘AF’ have long-range antiferromagnetic order. The AF+dSC state also has d-wave superconductivity and was described in refs 18, 19. For the cuprates, we propose that the ‘normal’ ACL phase above is a holon–hole metal, whereas the superconducting ACL is the holon–hole superconductor. The conventional Fermi-liquid metal and BCS superconductor appear at larger x, and are not shown.

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Figure 2: Square-lattice Brillouin zone containing the ‘diamond’ Brillouin zone (dashed line).
figure 2

In the antiferromagnetic metal, only the black ellipses are present, and they represent Fermi surfaces of S=1/2, charge-e holes that are visible in both ARPES and SdH. In the holon metal, these black ellipses become spinless charge-e holon Fermi surfaces, and are visible only in SdH. The green ‘bananas’ are the hole Fermi surfaces present only in the holon–hole metal, and detectable in both SdH and ARPES.

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Now let us destroy the antiferromagnetic order by considering the vicinity of the deconfined critical point in the insulator9. Arguments were presented in ref. 15 that we obtain a stable metallic state in which the electron fractionalizes into particles with precisely the quantum numbers of the f± and the zα described above. This is the holon metal phase: a similar phase was discussed in early work by Lee16, but its full structure was clarified recently15. Below we will discuss the properties of the holon metal, and of a number of other ACLs that descend from it.

(1) Holon metal. There is full spin-rotation symmetry, and a positive energy (the spinon energy gap) is required to create a zα spinon. The zero-energy holons inherit the zero-energy hole states of the antiferromagnetic metal, and so they reside along the black elliptical pockets in Fig. 2, and these will yield SdH oscillations characteristic of these pockets17. However, the view from ARPES experiments is very different—the distinction between SdH and ARPES views is a characteristic property of all ACLs. The physical electron is a composite of zα and f±, and so the ARPES spectrum will have no zero-energy states, and only a broad absorption above the spinon energy gap. The frequency F of the SdH oscillations is given by the Onsager–Lifshitz relation

where Φ0=h c/(2e) is the flux quantum, and is the area in momentum space enclosed by the fermionic zero-energy charge carriers. For the holons, this area is specified by the Luttinger relation: there are four independent holon pockets, and the area of each pocket is

(2) Holon–hole metal. This is our candidate state for the normal state of the cuprates at low hole density (see Fig. 1). It is obtained from the holon metal when some of the holons and spinons bind to form a charge-e, S=1/2 particle, which is, of course, the conventional hole, neutral under the aμ charge. This binding is caused by the nearest-neighbour electron hopping, and the computation of the dispersion of the bound state is described in the Methods section. Now the metal has both holons and hole charge carriers, and both have independent zero-energy states, that is, Fermi surfaces. These Fermi surfaces are shown in Fig. 2, and both enclosed areas will contribute an SdH frequency via equation (1). The values of the areas depend on specific parameter values, but the Luttinger relation does yield the single constraint

the factor of 2 prefactor of is due to S=1/2 spin of the holes. This relation can offer an explanation of the recent experiment2 that observed SdH oscillations at a frequency of 530T. We propose that these oscillations are due to the holon states. The holon density may then be inferred to be 0.076 per Cu site while the total doping is x=0.1. The missing density resides in the hole pockets, which by equation (3) will also exhibit oscillations at the frequency associated with the area [(2π)2/(4a2)]×0.012. The presence of SdH oscillations at this lower frequency is a key prediction of our theory that we hope will be tested experimentally. ARPES experiments detect only the hole Fermi surface, that is, the ‘bananas’ in Fig. 2. With a finite momentum width due to impurity scattering, and the very small area of each banana, the holon–hole metal also accounts for the arc-like regions observed in current ARPES experiments1,4,5. Furthermore, because the nearest-neighbour electron hopping is suppressed by local antiferromagnetic correlations, we expect the binding to increase with temperature, and this possibly accounts for the observed temperature dependence of the arcs5.

(3) Holon superconductor. On lowering the temperature in the holon metal, the holons pair to form a composite boson that is neutral under the aμ charge, and the condensation of this boson leads to the holon superconductor. Just as we determined the holon dispersion in the holon metal phase by referring to previous work in the proximate antiferromagnetic metal, we can determine the nature of the pair wavefunction by extrapolating from the state with coexisting antiferromagnetism and superconductivity in Fig. 1. The latter state was studied by Sushkov and collaborators18,19, and they found that holes paired with d-wave symmetry. Hence, the AF+dSC state in Fig. 1. We assume that the same pairing amplitude extends into the ACL obtained by restoring spin-rotation symmetry and inducing a spinon energy gap. The resulting holon superconductor is not smoothly connected to the conventional Bardeen, Cooper and Schrieffer (BCS) d-wave superconductor because of the spin gap. The theory describing the low-energy excitations of the holons in the holon superconductor is developed in the Methods section: it is found to be a mathematical structure known as a conformal field theory (CFT). The present CFT has the U(1) gauge field aμ coupled minimally to N=4 species of Dirac fermions, ψi (i=1,…,N), which are descended from the f± holons after pairing. Such CFTs have been well studied in other contexts11,12,13,20. For us, the important utility of the CFT is that it allows us to compute the temperature (T) and x dependence of the density of the superfluid electron, ρs (measured in units of energy through its relation to the London penetration depth, λL, by ρs=ℏ2c2d/(16πe2λL2), where d is the spacing between the layers in the cuprates). For this, we need the coupling of the CFT to the vector potential A of the electromagnetic field: this is studied in the Methods section and has the form j · A, where j is a conserved ‘flavour’ current of the Dirac fermions ψi. The T dependence of ρs is then related to the T dependence of the susceptibility associated with j: such susceptibilities were computed in ref. 21. In a similar manner, we found that as T→0 at small x,

where c1 is a non-universal constant and is a universal number characteristic of CFT. Remarkably, such a ρs is seen in experiments22,23,24 on the cuprates over a range of T and x. The phenomenological importance of such a ρs(x,T) was pointed out by Lee and Wen25, although equation (4) has not been obtained in any earlier theory26. The holon superconductor provides an elegant route to this behaviour, moreover with universal. We analysed the CFT in a 1/N expansion and obtained

We note that in the cuprates, there is a superconductor–insulator transition at a non-zero x=xc, and in its immediate vicinity, distinct quantum critical behaviour of ρs is expected, as has been observed recently27. However, in characterizing different theories of the underdoped regime, it is useful to consider the behaviour as x→0 assuming the superconductivity survives until x=0. In this limit, our present theory is characterized by dρs/dT∼constant, whereas theories of refs 26, 28 have dρs/dT∼x2.

(4) Holon–hole superconductor. This is our candidate state of the superconductor at low hole density (see Fig. 1). It is obtained from a pairing instability of the holon–hole metal, just as the holon superconductor was obtained from the holon metal. It is a modified version of the holon superconductor, which has in addition low-energy excitations from the paired holes. The latter will yield the observed V-shaped spectrum in tunnelling measurements. The holon–hole metal will also have a residual metallic thermal conductivity at low T in agreement with experiment. For ρs(x,T), we will obtain in addition to the terms in equation (4), a contribution from the nodal holes to dρs/dT. This contribution can be computed using the considerations presented in refs 25, 26: it has only a weak x dependence coming from the ratio of the velocities (which can in principle be extracted from ARPES) in the two spatial directions. In particular, the holes carry a current ∼1 and not ∼x; the latter is the case in other theories26,28 of electron fractionalization that consequently have dρs/dT∼x2.

Perhaps the most dramatic implication of these ideas is the possibility that the superconducting ground state of the underdoped cuprates is not smoothly connected to the conventional superconductors described by BCS theory. With the reasonable extra assumption that such a BCS ground state is realized in the overdoped cuprates, it follows that our proposal requires at least one quantum phase transition inside the superconducting dome.

Methods

We represent the electron operator15 on square-lattice site r and spin α=↑,↓ as

where A and B are the two sublattices, and ɛα β is the unit antisymmetric tensor. The fr are spinless charge-e fermions carrying opposite aμ gauge charge ±1 on the two sublattices. They will be denoted f± respectively. The effective action of the doped antiferromagnet on the square lattice has the structure

where K is a momentum extending over the diamond Brillouin zone in Fig. 2. is the lattice action for spinons. describes holons hopping on the same sublattice, preserving the sublattice index s=±1; the holon dispersion εK has minima at the Ki, and μh is the holon chemical potential. The first term in describes an onsite electron chemical potential; the second term describes opposite sublattice electron hopping between nearest neighbours; the coupling κ is expected to be significantly renormalized down by the local antiferromagnetic order from the bare electron hopping element t. We have not included the aμ gauge field in the above action, which is easily inserted by the requirements of gauge invariance and minimal coupling.

For small g, g′, the zα condense, and we obtain the familiar antiferromagnetic metal state with hole pockets (see Fig. 1). For larger g, g′, we reach the holon metal phase15, in which the z are gapped. The term leads to two distinct (but not exclusive) instabilities of the holon metal phase: towards pairing of the f± holons and the formation of bound states between the holons and spinons. These lead, respectively, to the holon superconductor and the holon–hole metal (and to the holon–hole superconductor when both are present). The term will also significantly modify the spin-correlation spectrum, probably inducing incommensurate spin correlations29, but we will not address this here.

Consider, first, the instability due to pairing between opposite gauge charges, so that the order parameter carries physical charge 2e, spin 0 and gauge charge 0. The aμ gauge symmetry remains unbroken. The low-energy theory of the holon superconductor has gapless nodal Dirac holons ψi coupled to the gauge field aμ with the action

Here, μ,ν,λ,… are space-time indices (τ,X,Y) and Dμ=∂μ−i aμ. This describes massless 2+1 dimensional quantum electrodynamics theory with N=4 species of two-component Dirac fermions that flows (within a 1/N expansion) at low energies to a stable fixed point11,12 describing a CFT.

As the superconductivity arises through pairing of holons from a Fermi surface of area ∝x, the superfluid density ρs(x,0)∝x. At T>0, we need to include thermal excitation of unpaired holons, which are coupled to the vector potential A of the physical electromagnetism by

In terms of the Dirac fermions, these are readily seen to correspond to conserved ‘charges’ of equation (5). The susceptibility associated with these charges is ∝T (ref. 21), and so we obtain equation (4) in the main text.

Next, we consider the κ-induced holon–spinon binding in the holon metal that leads to the appearance of the holon–hole metal at low temperatures. is a momentum-dependent attractive contact interaction proportional to κ0−κ γK, with γK=(cos(Kx)+cos(Ky)) between a holon and spinon with centre-of-mass momentum K. We discuss the energy of a bound holon–spinon composite using a non-relativistic Schrödinger equation assuming initially that only a single holon is injected into the paramagnet and ignoring gauge interactions. Consider a single holon valley (say valley 1). Taking a parabolic holon dispersion near K1

with k=K−K1, and a spinon dispersion

centred at (0,0), the energy of the holon–spinon composite will be

where M=mh+Δs. Define

where φ±(r) are the wavefunctions of a composite of a ±-holon and a spinon separated by r and with centre-of-mass momentum k. This satisfies the Schrödinger equation

where ρ=mhΔs/(mh+Δs) and . (σx is a Pauli matrix acting on φ.) For the momenta of interest, this gives a k-dependent hole binding energy

with V0,V1>0. Thus, the dispersion for h1 becomes

The minimum of Eh1 is at a non-zero energy. Furthermore, as a function of kX=(kx+ky)/2, it is shifted along the positive kX direction by an amount 2M V1cos(kY) with kY=(−kx+ky)/2.

Considering both holon and hole bands, it is clear that with increasing doping, both bands will be occupied to get a holon–hole metal. In the full Brillouin zone, h1 is at momentum −K1 so that the hole Fermi surface lies entirely inside the diamond region and has the rough banana shape shown in Fig. 2. The gauge interaction can now be included and leads to the properties discussed in the main text.