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# Quantum dimer model for the pseudogap metal

Contributed by Subir Sachdev, June 23, 2015 (sent for review May 14, 2015; reviewed by Antoine Georges and Masaki Oshikawa)

## Significance

The most interesting states of the copper oxide compounds are not the superconductors with high critical temperatures. Instead, the novelty lies primarily in the higher temperature metallic “normal” states from which the superconductors descend. Here, we develop a simple, intuitive model for the physics of the metal at low carrier density, in the “pseudogap” regime. This model describes an exotic metal that is similar in many respects to simple metals like silver. However, the simple metallic character coexists with “topological order” and long-range quantum entanglement previously observed only in exotic insulators or fractional quantum Hall states in very high magnetic fields. Our model is compatible with many recent observations, and we discuss more definitive experimental tests.

## Abstract

We propose a quantum dimer model for the metallic state of the hole-doped cuprates at low hole density, *p*. The Hilbert space is spanned by spinless, neutral, bosonic dimers and spin *p*. Exact diagonalization, on lattices of sizes up to

The recent experimental progress in determining the phase diagram of the hole-doped Cu-based high-temperature superconductors has highlighted the unusual and remarkable properties of the pseudogap (PG) metal (Fig. 1). A characterizing feature of this phase is a depletion of the electronic density of states at the Fermi energy (1, 2), anisotropically distributed in momentum space, that persists up to room temperature.

Attempts have been made to explain the pseudogap metal using thermally fluctuating order parameters; we argue below that such approaches are difficult to reconcile with recent transport experiments. Instead, we introduce a new microscopic model that realizes an alternative perspective (3), in which the pseudogap metal is a finite temperature (*T*) realization of an interesting quantum state: the fractionalized Fermi liquid (FL*). We show that our model is consistent with key features of the pseudogap metal observed by both transport and spectroscopic probes.

The crucial observation that motivates our work is the tension between photoemission and transport experiments. In the cuprates, the hole density *p* is conventionally measured relative to that of the insulating antiferromagnet (AF), which has one electron per site in the Cu *d* band. Therefore, the hole density relative to a filled Cu band, with two electrons per site, is actually *p*, rather than *p* was indicated directly in Hall measurements (8), whereas other early experiments indicated suppression of the Drude weight (9⇓–11). Although the latter could be compatible with a carrier density of *p* are especially notable: (*i*) the quasiparticle lifetime *c* an order unity constant; and (*ii*) the in-plane magnetoresistance of the pseudogap (13) is proportional to *H*, where *b* is a *T*-independent constant; this is Kohler’s rule for a Fermi liquid.

It is difficult to account for the nearly perfect Fermi liquid-like *T* dependence in transport properties of the pseudogap in a theory in which a large Fermi surface of size *T* dependence of the correlation length of the order.

Moreover, a reasonable candidate for the fluctuating order has not yet been identified. The density wave (DW) order present at lower temperature in the pseudogap regime has been identified to have a *d*-form factor (15⇓⇓–18), and its temperature dependence (19⇓⇓⇓⇓⇓–25) indicates that it is unlikely to be the origin of the pseudogap present at higher temperature. Similar considerations apply to other fluctuating order models (26) based on AF or *d*-wave superconductor.

We are therefore led to an alternative perspective (3), in which the pseudogap metal represents a new quantum state that could be stable down to very low *T*, at least for model Hamiltonians not too different from realistic cuprate models. The observed low-*T* DW order is then presumed to be an instability of the pseudogap metal (27⇓⇓⇓–31). An early discussion (32) of the pseudogap metal proposed a state that was a doped spin liquid with “spinon” and “holon” excitations fractionalizing the spin and charge of an electron: the spinon carries spin

Instead, we need a quantum state that has long-lived electron-like quasiparticles around a Fermi surface of size *p*, even though such a Fermi surface would violate the Luttinger relation of a Fermi liquid. The fractionalized Fermi liquid (FL*) (33) fulfills these requirements.

## Fractionalized Fermi Liquids

The key to understanding the FL* state is the topological nature of the Luttinger relation for the area enclosed by the Fermi surface. For the case of a conventional FL state, Oshikawa (34) provided a nonperturbative proof of the Luttinger relation by placing the system on a torus, and computing the response to a single flux quantum threaded through one of the holes of the torus. His primary assumption about the many-body state was that its only low-energy excitations were fermionic quasiparticles around a Fermi surface. This assumption then points to a route to obtaining a Fermi surface of a different size (35): we need a metal that, in addition to the quasiparticle excitations around the Fermi surface, has global topological excitations nearly degenerate with the ground state, similar to those found in insulating spin liquids (36, 37). In the context of the doped spin liquids noted earlier, we obtain a FL* state when the holon and spinon bind to form a fermionic state with spin *p*, and not

Earlier studies have examined a number of phenomenological and path integral models of FL* theories of the pseudogap (39, 40, 42⇓–44) [and in an ansatz for the pseudogap (47)]. These models contain emergent gauge field excitations, which are needed to provide the global topological states required to violate the Luttinger relation of the FL state. However, they also include spurious auxiliary particle states that are only approximately projected out. The gauge field can undergo a crossover to confinement, but the present models do not keep close track of lattice-scale Berry phases that control the appearance of density wave order in the confining state (48). Here, we propose to overcome these difficulties by a new quantum dimer model that can realize a metallic state that is a FL*. This should open up studies of the photoemission spectrum, density wave instabilities, and crossovers to confinement at low *T* in the pseudogap metal.

## Quantum Dimer Models

Quantum dimer models (49⇓–51) have been powerful tools in uncovering the physics of spin liquid phases, and of their instabilities to conventional confining phases (52⇓–54). Dimer models of doped spin liquids have also been studied (49, 55, 56), but all of these involve doping the insulating models by monomers that carry charge

The Hilbert space of our dimer model is spanned by the close-packing coverings of the square lattice with two species of dimers (Fig. 2), with an additional twofold spin degeneracy of the second species. It can be mapped by an appropriate similarity transform (49) to a truncation of the Hilbert space of the *t*-*J* model.

The first species of dimers are bosons, *t*-*J* model, each boson maps to a pair of electrons in a spin-singlet state:*i* with spin

The second species of dimers are “fermions,” *p*. Each fermionic dimer maps to a bound state of a holon and a spinon, which we take to reside on a bonding orbital between nearest-neighbor sites:

In a three-band model (57, 58), the state

Let us stress our assumption that spinon and holon bind not because of confinement but because of a short-range attraction. Therefore, the bound state (2) can break up at an energy cost of order the antiferromagnetic exchange, and the holon and spinon appear as gapped, free excitations that would contribute two-particle continuum spectra to photoemission or neutron scattering spectra. These fractionalized states can be included in our dimer model by expanding the Hilbert space to include monomers, but we will not do so here because we focus on the lowest energy sector. As a consequence, there is no monomer Fermi surface (42) in the present model of the pseudogap metal.

The states 1 and 2 are precisely those that dominate in the two-site dynamical mean field theory (DMFT) analysis of the Hubbard model by Ferrero et al. (61): they correspond to the *S* and *p* (see also ref. 62). The DMFT analysis captures important aspects of pseudogap physics, but with a coarse momentum resolution of the Brillouin zone. In DMFT, the states on the two-site cluster interact with a self-consistent environment in a mean-field way: the equations have so far only been solved at moderate temperatures and the nature of the ultimate ground state at low doping remains unclear. Our dimer model is a route to going beyond DMFT, and to include the nontrivial entanglement between these states on different pairs of sites in a non-mean field manner. The local constraints between different pairs of dimers are accounted for, allowing for the emergence of gauge degrees of freedom.

The original RK model can be mapped to a compact U(1) lattice gauge theory (50, 52, 53). In the doped dimer models studied earlier, the monomers then carry U(1) gauge charges of

We can now describe our realization of the pseudogap metal. We envisage a state where the confinement length scale of the compact U(1) gauge field is large, and specifically, larger than the spacing between the *p*, thus realizing a FL* state. The confinement scale becomes large near the solvable RK point in the RK model (63, 64), near a Higgs transition to a

We present results below for the following Hamiltonian, illustrated in Fig. 3, acting on the dimer Hilbert space described above:*J*. A perturbative estimate of the dimer hopping amplitudes *SI Appendix*. Note that all such terms must preserve the dimer close-packing constraint on every site, and we have chosen three terms with short-range hopping; longer-range hopping terms for the fermonic dimers are also possible, but expected to decay with distance, and are omitted for simplicity. Finally,

## Results

We now present results for the dispersion and quasiparticle residue of a single fermion described by *p*, the interactions between the fermionic dimers can be treated by a dilute gas expansion in *p*, whereas the dominant contributions to the quasiparticle dispersion and residue arise from the interaction between a single fermion and the close-packed sea of bosonic dimers. We computed the latter effects by exactly diagonalizing the singe fermion Hamiltonian on lattice sizes up to *SI Appendix*.

Our numerical study explored the dispersion of a single fermion over a range of values of the hopping parameters. We show in Fig. 4 the dispersion *t*-*J* model appropriate for the cuprates at the RK point *SI Appendix* has similar results for additional parameter values.

The minima of the fermion dispersion were found at different points in the Brillouin zone, but there was a wide regime with minima near momenta

Our numerical results also yield interesting information on the quasiparticle residue of the electron operator. This is nontrivial even for the case of a single fermionic dimer, because, unlike a free electron, a fermionic dimer can only move by “resonating” with the background of bosonic dimers, as is clear from Fig. 3. In the presence of a finite density of fermionic dimers, there will be an additional renormalization from the interaction between the fermions that we will not compute here. We do not expect this to have a significant **1** and **2**, the electron annihilation operator on site *i* has the same matrix elements as the following:

It is also possible to study the system in perturbation theory in *SI Appendix*. The fermion hoppings at nonzero *SI Appendix*. This perturbative dispersion is found to be in good agreement with our exact diagonalization results only for

## Discussion

In this article, we develop a new class of doped dimer models featuring coherent electronic quasiparticle excitations on top of a spin-liquid ground state. The scenario considered here is based on the assumption that spinons and holons form bound states on nearest-neighbor sites. These fermionic bound states with spin *p* and are observable as electronic quasiparticles in experiments. Such a Fermi sea realizes a topological quantum state called the “fractionalized Fermi liquid” (33), whose Fermi surfaces encloses an area distinct from the Luttinger value in a conventional Fermi liquid.

The undoped RK model on the square lattice features a deconfined spin liquid ground state only at the special RK point **3**) features a FL* phase in an extended parameter range. However, similar to the RK model, we also expect a wide parameter regime where our model has a ground state with broken lattice symmetries. We leave the computation of the phase diagram of our model for future study.

The main implication of our model of the pseudogap metal (in zero applied magnetic field and at moderate *T* below _{2} layers requires to break either fermonic or bosonic dimers in our model, which naturally accounts for the observed gap in *c*-axis optical conductivity.

Experiments that involve removing one electron from the system (such as photoemission) have difficulty observing the back sides of the pockets because of the small (but nonzero) quasiparticle residue

Our theory can be loosely summarized by “the electron becomes a dimer in the pseudogap metal,” as in Eq. **5**: with a spin-liquid background present, there can be no single-site state representing an electron, and a dimer is the simplest possibility.

The main advantage of our quantum dimer model over previous treatments (39, 40, 42⇓–44) of fractionalized Fermi liquids (FL*) is that it properly captures lattice-scale dispersions, quasiparticle residues, and Berry phases: all of these are expected to play crucial roles in the crossovers to confinement and associated symmetry breaking at low *T* (48, 52, 53). Given the elongated dimer and dipolar nature of the electron, Ising-nematic order (71) is a likely possibility; the *d*-form factor density wave (15, 16) is then a plausible instability of such a nematic metal. The interplay between the monopole-induced crossovers to confinement (52, 53) and the density wave instabilities of the hole pockets (30, 31) can also be examined in such dimer models. The onset of superconductivity will likely require additional states, such as a spinless, charge

## Acknowledgments

We thank J. Budich, D. Chowdhury, J. C. Davis, D. Drew, E. Fradkin, A. Georges, S. A. Kivelson, A. Läuchli, A. Millis, and E. Sorensen for valuable discussions. K. Fujita and J. C. Davis provided the phase diagram, which was adapted to produce Fig. 1. We thank R. Melko and D. Hawthorn for central processing unit time at the University of Waterloo. This research was supported by the National Science Foundation under Grant DMR-1360789, the Templeton Foundation, and Multidisciplinary University Research Initiative Grant W911NF-14-1-0003 from Army Research Office. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. M.P. is supported by the European Research Council Synergy Grant Ultracold Quantum Matter and Sonderforschungsbereich Foundations and Application of Quantum Science of the Austrian Science Fund, as well as the Nano Initiative Munich.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: sachdev{at}g.harvard.edu.

Author contributions: M.P., A.A., and S.S. performed research; and M.P., A.A., and S.S. wrote the paper.

Reviewers: A.G., College de France; and M.O., Institute for Solid State Physics, University of Tokyo.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1512206112/-/DCSupplemental.

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