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arXiv:1402.4807 · Physical Review B

Density wave instabilities in a correlated two-dimensional metal

Andrea Allais, Johannes Bauer, Subir Sachdev A

Published Oct 10, 2014• 86 citations
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Abstract

Motivated by recent experimental evidence of charge order in the pseudogap phase of cuprates, we perform a variational analysis of spin-singlet density wave ordering in metals with antiferromagnetic interactions on the square lattice, using a wave function with double occupancy projected out. We examine ordering with and without time-reversal symmetry, with an arbitrary wave vector and a tunable form factor. Depending on parameters, we find $d$-form factor density wave ordering, with a wave vector either parallel to the lattice generators or diagonally oriented, or a ground state which carries a time-reversal breaking pattern of spontaneous currents.

DOI
10.1103/physrevb.90.155114

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Motivated by recent experimental evidence of charge order in the pseudogap phase of cuprates, we perform a variational analysis of spin-singlet density wave ordering in metals with antiferromagnetic interactions on the square lattice, using a wave function with double occupancy projected out. We examine ordering with and without time-reversal symmetry, with an arbitrary wave vector and a tunable form factor. Depending on parameters, we find ddd-form factor density wave ordering, with a wave vector either parallel to the lattice generators or diagonally oriented, or a ground state which carries a time-reversal breaking pattern of spontaneous currents.
Recent experiments in the underdoped regime of the hole-doped cuprates have found evidence for an incommensurate charge density-wave state. We present an analysis of the charge ordering instabilities in a metal with antiferromagnetic correlations, where the electronic excitations are coupled to the fractionalized excitations of a quantum fluctuating antiferromagnet on the square lattice. The resulting charge density-wave state emerging out of such a fractionalized Fermi liquid (FL*) has wave vectors of the form $(\ifmmode\pm\else\textpm\fi{}{Q}_{0},0),(0,\ifmmode\pm\else\textpm\fi{}{Q}_{0})$, with a predominantly ddd-form factor, in agreement with experiments on a number of different families of the cuprates. In contrast, as previously shown, the charge density-wave instability of a nearly antiferromagnetic metal with a large Fermi surface, interacting via short-range interactions, has wave vectors of the type $(\ifmmode\pm\else\textpm\fi{}{Q}_{0},\ifmmode\pm\else\textpm\fi{}{Q}_{0})$. Our results show that the observed charge density-wave appears as a low-energy instability of a fractionalized metallic state linked to the proximity to an antiferromagnetic insulator, and the pseudogap regime can be described by such a metal at least over intermediate length and energy scales.
We present an unrestricted Hartree-Fock computation of charge-ordering instabilities of two-dimensional metals with antiferromagnetic exchange interactions, allowing for arbitrary ordering wave vectors and internal wave functions of the particle-hole pair condensate. We find that the ordering has a dominant d symmetry of rotations about lattice points for a range of ordering wave vectors, including those observed in recent experiments at low temperatures on YBa2Cu3O(y). This d symmetry implies the charge ordering is primarily on the bonds of the Cu lattice, and we propose incommensurate bond order parameters for the underdoped cuprates. The field theory for the onset of Néel order in a metal has an emergent pseudospin symmetry which "rotates" d-wave Cooper pairs to particle-hole pairs [M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075128 (2010)]; our results show that this symmetry has consequences even when the spin correlations are short ranged and incommensurate.
We study a one-dimensional (1D) electron liquid embedded in a 2D antiferromagnetic insulator, and coupled to it via a weak antiferromagnetic spin-exchange interaction. We argue that this model may qualitatively capture the physics of a single charge stripe in the cuprates on length and time scales shorter than those set by its fluctuation dynamics. Using a local mean-field approach we identify the low-energy effective theory that describes the electronic-spin sector of the stripe as that of a sine-Gordon model. We determine its phases via a perturbative renormalization-group analysis. For realistic values of the model parameters we obtain a phase characterized by enhanced spin density and composite charge-density-wave correlations, coexisting with subleading triplet and composite singlet-pairing correlations. This result is shown to be independent of the spatial orientation of the stripe on the square lattice. We argue that slow transverse fluctuations of the stripes tend to suppress the density correlations, thus promoting the pairing instabilities. The largest amplitudes for the composite instabilities appear when the stripe forms an antiphase domain wall in the antiferromagnet. For twisted spin alignments the amplitudes decrease and leave room for a new type of composite pairing correlation, breaking parity but preserving time-reversal symmetry.
Pair-density waves (PDWs) are superconducting states that spontaneously break translation symmetry in systems with time-reversal symmetry (TRS). Evidence for PDWs has been seen in several recent experiments, as well as in the pseudogap regime in cuprates. Theoretical understanding of PDWs has been largely restricted to phenomenological and numerical studies, while microscopic theories typically require strong coupling or fine tuning. In this work, we provide a symmetry-based mechanism under which PDWs emerge as a weak-coupling instability of a two-dimensional TRS metal. Combining mean-field and renormalization-group analyses, we identify a weak-coupling instability towards a triplet PDW realized in the $\ensuremath{\pi}$-flux square lattice model with on-site repulsion and moderate nearest-neighbor attraction when the Fermi level crosses Van Hove singularities at 1/4 and 3/4 fillings. This PDW is protected by the magnetic translation symmetries characteristic of Hofstadter systems, of which the $\ensuremath{\pi}$-flux lattice is a special time-reversal-symmetric case.
We provide, for the first time, in a doped strongly correlated system (two-leg ladder), a controlled theoretical demonstration of the existence of a state in which long-range ordered orbital currents are arranged in a staggered pattern, coexisting with a charge density wave. The method used is the highly accurate density-matrix renormalization group technique. This brings us closer to recent proposals that this order is realized in the enigmatic pseudogap phase of the cuprate high temperature superconductors.
The pseudogap metal phase of the hole-doped cuprate superconductors has two seemingly unrelated characteristics: a gap in the electronic spectrum in the "antinodal" region of the square lattice Brillouin zone and discrete broken symmetries. We present a SU(2) gauge theory of quantum fluctuations of magnetically ordered states which appear in a classical theory of square lattice antiferromagnets, in a spin-density wave mean field theory of the square lattice Hubbard model, and in a CP^{1} theory of spinons. This theory leads to metals with an antinodal gap and topological order which intertwines with the observed broken symmetries.