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arXiv:1303.2114 · Physical Review Letters

Bond Order in Two-Dimensional Metals with Antiferromagnetic Exchange Interactions

Subir Sachdev A, Rolando L. La Placa

Published Jul 9, 2013• 197 citations
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Abstract

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.

DOI
10.1103/physrevlett.111.027202

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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.
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.
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.
We investigate the interplay between charge order and superconductivity near an antiferromagnetic quantum critical point using sign-problem-free Quantum Monte Carlo simulations. We establish that, when the electronic dispersion is particle-hole symmetric, the system has an emergent SU(2) symmetry that implies a degeneracy between d-wave superconductivity and charge order with d-wave form factor. Deviations from particle-hole symmetry, however, rapidly lift this degeneracy, despite the fact that the SU(2) symmetry is preserved at low energies. As a result, we find a strong suppression of charge order caused by the competing, leading superconducting instability. Across the antiferromagnetic phase transition, we also observe a shift in the charge order wave vector from diagonal to axial. We discuss the implications of our results to the universal phase diagram of antiferromagnetic quantum-critical metals and to the elucidation of the charge order experimentally observed in the cuprates.
Recent scanning tunneling microscopy (STM) experiments on underdoped cuprates have displayed modulations in the local electronic density of states, which are centered on a Cu-O-Cu bond [Kohsaka et al., Science 315, 1380 (2007)]. As a paradigm of the pinning of such bond-centered ordering in strongly correlated systems, we present the theory of valence bond solid (VBS) correlations near a single impurity in a square lattice antiferromagnet. The antiferromagnet is assumed to be in the vicinity of a quantum transition from a magnetically ordered N\'eel state to a spin-gap state with long-range VBS order. We identify two distinct classes of impurities: (i) local modulation in the exchange constants and (ii) a missing or additional spin, for which the impurity perturbation is represented by an uncompensated Berry phase. The ``boundary'' critical theory for these classes is developed. In the second class, we find a VBS pinwheel around the impurity, accompanied by a suppression in the VBS susceptibility. Implications for numerical studies of quantum antiferromagnets and for STM experiments on the cuprates are noted.
We show that a two-dimensional (2D) metal in a superconducting (SC) state with wave-vector-dependent gap behaves as a quantum-spin liquid characterized by a collective mode of resonance type. Formally the mode exists whatever is a form of Fermi surface (FS). However, when the latter is spherical the effect is rather formal: any factor leading to an even weak electron scattering, like finite T, impurity, etc., easily kills the mode. On the contrary, for the type of FS observed in the high-Tc{T}_{c}Tc​ cuprates (characterized by a proximity to the saddle-point wave vectors) the resonance mode exists in a true sense being well separated from the electron-hole continuum and well defined. The effect is due to the effective 1D dimension in a proximity of electronic topological transition quantum critical point. The physical nature of the collective mode varies progressively between two possibilities. In the case of a weak interaction, the resonance mode is a typical effect of a two-particle bound state that appears slightly below the bottom of the two-particle continuum. The effect exists even in the case of infinitesimally weak interaction (of relevant sign). In the case of a strong or intermediate interaction, it is a critical mode existing as a precursor of spin-density wave (SDW) instability. The gap of the collective mode is not related directly to the value of the SC gap (being, however, limited by the latter from above) but rather to a proximity to SDW instability. We predict also a second collective mode lying inside the two-particle continuum. The existence of these two modes and the particular form of their dispersion in the case corresponding to the underdoped cuprates ${(d}_{{x}^{2}\ensuremath{\sim}{y}^{2}}$ symmetry of SC gap, exchange interaction of antiferromagnetic sign, etc.) allows one to understand the main features observed experimentally in the underdoped cuprates, namely the so-called ``41 meV resonance peak'' and ``24 meV incommensurability'' and their two-dimensional wave vector dependences, the doping evolution of the spin dynamics and many other features. The theory also predicts new features to be tested experimentally.
Interaction-induced charge orders with electronic origin occur as states of spontaneously broken symmetry in several materials platforms. An electronic mechanism for charge order requires an attractive component in the effective charge vertex. We put forward such a mechanism for the formation of unconventional charge density waves in a metal. These states result from the condensation of particle-hole pairs with finite wave vector and nonzero angular momentum and correspond to bond or loop current order on a lattice. The mechanism we describe can be viewed as Kohn Luttinger analysis in the particle-hole channel with finite transferred momentum. It incorporates one-loop spin and pairing corrections, which are then used as an input for a summation in the charge channel triggering an instability. We extend our analysis to a spin-fluctuation approach, where the effective charge interaction is dressed by the particle-hole ladder with exchanged momentum. We argue that this mechanism works for weakly interacting metals with a nested Fermi surface and a large number of fermion flavors. We apply the Kohn-Luttinger-like approach to square–PLXINSERT-, and triangular-lattice Hubbard models with SU ( N f ) flavour symmetry and show that it leads to different types of p -wave charge density waves. We also study effects beyond weak coupling at and away from Van Hove filling in terms of a phenomenological model with additional exchange interaction. In the vicinity of Van Hove filling, we obtain d -wave charge density waves with wave vectors determined by nesting as leading instabilities. In addition, we find another charge density wave with wave vector K / 4 on the triangular lattice on both sides of the Van Hove filling. We demonstrate that this K / 4 instability can win the competition against pairing for N f = 4 via an unbiased functional renormalization group calculation.