Topological order in the pseudogap metal Mathias S. Scheurera,1, Shubhayu Chatterjeea, Wei Wub,c, Michel Ferrerob,c, Antoine Georgesb,c,d,e, and Subir Sachdeva,f,g,1 aDepartment of Physics, Harvard University, Cambridge MA 02138; bCentre de Physique Theorique,´ Ecole´ Polytechnique, CNRS, Universite´ Paris-Saclay, 91128 Palaiseau, France; cInstitut de Physique, College` de France, 75005 Paris, France; dCenter for Computational Quantum Physics, Flatiron Institute, New York, NY 10010; eDepartment of Quantum Matter Physics, Universite´ de Geneve,` CH-1211 Geneva, Switzerland; fPerimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5; and gDepartment of Physics, Stanford University, Stanford, CA 94305 Contributed by Subir Sachdev, March 9, 2018 (sent for review November 27, 2017; reviewed by Gabriel Kotliar and Andre-Marie Tremblay) We compute the electronic Green’s function of the topologically 15) with the suppression of “hedgehog” defects in the spacetime ordered Higgs phase of a SU(2) gauge theory of fluctuating anti- configuration of the fluctuating AF order. ferromagnetism on the square lattice. The results are compared We will present a mean-field computation of the electronic with cluster extensions of dynamical mean field theory, and quan- Green’s function across the entire Brillouin zone in the U(1) tum Monte Carlo calculations, on the pseudogap phase of the Higgs phase of the SU(2) gauge theory. Such results allow for strongly interacting hole-doped Hubbard model. Good agree- a direct comparison with numerical computations on the Hub- ment is found in the momentum, frequency, hopping, and doping bard model. One of our main results will be that for a reasonable dependencies of the spectral function and electronic self-energy. range of parameters in the SU(2) gauge theory, both the real and We show that lines of (approximate) zeros of the zero-frequency imaginary parts of the electron Green’s function of the gauge electronic Green’s function are signs of the underlying topological theory with topological order closely resemble those obtained order of the gauge theory and describe how these lines of zeros from the dynamical cluster approximation (DCA), a cluster appear in our theory of the Hubbard model. We also derive a mod- extension of dynamical mean field theory (DMFT) (16–19). ified, nonperturbative version of the Luttinger theorem that holds While DCA allows us to study the regime of strong correla- in the Higgs phase. tions down to low temperature, it has limited momentum-space resolution. For this reason, we have also performed determi- pseudogap j topological order j Hubbard model j nant quantum Monte Carlo (DQMC) calculations and find high-temperature superconductors self-energies that, in the numerically accessible temperature range, agree well with the gauge theory computations. Addi- he pseudogap metal is a novel state of electronic matter tional results on the comparison between the SU(2) gauge Tfound in the hole-doped, cuprate high-temperature super- theory and the DCA and DQMC computations, as a function conductors (1). It exhibits clear evidence of electrical transport of doping and second-neighbor hopping, appear in a companion with the temperature and frequency dependence of a conven- paper (20). tional metal obeying Fermi liquid theory (2, 3). However, a In several discussions in the literature (21–29), violations of long-standing mystery in the study of the cuprates is that pho- the Luttinger theorem have been linked to the presence of lines toemission experiments do not show the “large” Fermi surface of zeros (in two spatial dimensions) in the electron Green’s func- that is expected from the Luttinger theorem of Fermi liquid the- tion on a “Luttinger surface.” The conventional perturbative ory (4). [Broken square lattice translational symmetry can allow proof of the Luttinger theorem yields an additional contribution “small” Fermi surfaces, but there is no sign of it over a wide range to the volume enclosed by the Fermi surface when the electron of temperature and doping over which the pseudogap state is Green’s function has lines of zeros: It was therefore argued that present (1, 5), and we will not discuss states with broken sym- metallic states with small Fermi surfaces are permitted, even in metry here.] There are nonperturbative arguments (6–10) that the context of this perturbative proof of the Luttinger theorem. deviations from the Luttinger volume are only possible in quan- tum states with topological order. But independent evidence for Significance the presence of topological order in the pseudogap has so far been lacking. The copper oxide-based high-temperature superconductors In this paper, we use a SU(2) gauge theory of fluctuating display a mysterious “pseudogap” metal phase at tempera- antiferromagnetism (AF) in metals (11, 12) to describe the pseu- tures just above the critical temperature in a regime of low dogap metal. Such a gauge theory describes fluctuations in the hole density. Extensive experimental and numerical studies orientation of the AF order, while preserving a local, nonzero have yielded much information on the nature of the electron magnitude. The emergent gauge fields of the theory indicate the corrections, but a fundamental theoretical understanding has long-range quantum entanglement of the topologically ordered been lacking. We show that a theory of a metal with topolog- phase. An alternative, semiclassical treatment of fluctuations of ical order and emergent gauge fields can model much of the the AF order parameter has been used to describe the electron- numerical data. Our study opens up a route to a deeper under- doped cuprates (13), but this remains valid at low temperatures standing of the long-range quantum entanglement in these (T ) only if the AF correlation length ξAF diverges as T ! 0. superconductors and to the direct detection of the topological We are interested in the case where ξAF remains finite at T = 0, characteristics of the many-body quantum state. and then a gauge theory formulation is required to keep proper PHYSICS track of the fermionic degrees of freedom in the background of Author contributions: M.S.S., S.C., W.W., M.F., A.G., and S.S. performed research, and the fluctuating AF order. Such a gauge theory can formally be M.S.S. and S.S. wrote the paper. derived from a lattice Hubbard model, as we will outline in the Reviewers: G.K., Rutgers University; and A.-M.T., University of Sherbrooke. next section. The SU(2) gauge theory yields a pseudogap metal The authors declare no conflict of interest. with only small Fermi surfaces when the gauge group is “Hig- Published under the PNAS license. gsed” down to a smaller group. We will describe examples of 1 To whom correspondence may be addressed. Email: [email protected] or Higgsing down to U(1) and Z2, and these will yield metallic states [email protected]. with U(1) and Z2 topological order. See SI Appendix, A for a def- This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. inition of topological order in gapless systems; for the U(1) case 1073/pnas.1720580115/-/DCSupplemental. we primarily consider, the topological order is associated (14, Published online April 2, 2018. www.pnas.org/cgi/doi/10.1073/pnas.1720580115 PNAS j vol. 115 j no. 16 j E3665–E3672 Downloaded by guest on September 27, 2021 But this argument appears problematic because the real part of (30–32). To this end, we decompose the electronic fields in terms the Green’s function (and hence the positions of the zeros) can of the spin and charge degrees of freedom: be changed by modifying the spectral density at high frequen- y y y cies, and so it would appear that high-energy excitations have ci (τ) = Ri (τ) i (τ), ci (τ) = i (τ)Ri (τ): [5] an undue influence on the low-energy theory. Indeed, specific computations of higher order corrections (21) do find that lines Here, the unitary 2 × 2 matrices, Ri (τ), are the bosonic spinon of zeros in metallic Green’s functions do not contribute to the and i (τ), the two-component fermionic chargon operators. volume enclosed by the Fermi surface. This parameterization introduces an additional redundancy We maintain that violations of the Luttinger theorem in met- leading to an emergent local SU(2) gauge invariance (11): als cannot appear in states that are perturbatively accessible from y the free electron state but are only possible in nonperturbative Ri (τ) ! Ri (τ)Vi (τ), i (τ) ! Vi (τ) i (τ): [6] metallic states with topological order (6–9). While the SU(2) 6 gauge theory is shown to violate the conventional Luttinger the- By design, the transformation in Eq. leaves the electron field c (τ) orem, we derive a modified sum rule on the Fermi surfaces of the operators i invariant and hence is distinct from spin rotation. metallic states with topological order. Note that the chargon field, , does not carry spin. In contrast, R We also find that the nonzero-temperature SU(2) gauge the- carries spin 1/2, and global spin rotations act via left multipli- R 6 ory has lines of approximate zeros of the electron Green’s cation on (in contrast to the right multiplication in Eq. ). Inserting the transformation (Eq. 5) into the electron–boson function in a suitable regime. These lines are remnants of lines of x y z coupling Sint and introducing the Higgs field H = (H , H , H ) zeros in the mean-field Green’s functions of fractionalized parti- y cles (“chargons”) in the theory. So we claim that the approximate via σ · Hi (τ) = Ri (τ)σRi (τ) · Φi (τ), we obtain (11) zeros can be interpreted as heralds of the underlying topological Z β order.