Orbital order driven quantum criticality

Charge, spin, and orbital degrees of freedom underlie the physics of transition metal compounds. Much work has revealed quantum critical points associated with spin and charge degrees of freedom in many of these systems. Here we illustrate that the simplest models that embody the orbital degrees of freedom the twoand three-dimensional quantum orbital compass models exhibit an exact quantum critical behavior on diluted square and cubic lattices (with doping δ = 1/4 and δ = 1/2 respectively). This raises the possibility of quantum critical points triggered by the degradation of orbital order upon doping (or applying pressure to) such transition metal systems. We prove the existence of an orbital spin glass in several related systems in which the orbital couplings are made non-uniform. Moreover, a new orbital Larmor precession (i.e., a periodic change in the orbital state) is predicted when uniaxial pressure is applied. Introduction. The interplay between charge, spin, and orbital degrees of freedom is a key ingredient underlying the physics of transition metal compounds. The electronic orbital degrees of freedom often allow for cooperative effects leading to well defined spatial orderings. Generally, crystal field effects split the five n = 3 dwave orbital wavefunctions into a triplet (t2g orbitals of the xy ≡ (Y2,−2 − Y2,2)/ √ 2, yz ≡ (Y2,−1 + Y2,1)/ √ 2 or xz ≡ (Y2,−1 − Y2,1)/ √ 2 types) and an eg doublet (spanned by orbitals whose angular dependence is of the 3z − r ≡ Y2,0 and x − y ≡ (Y2,−2 + Y2,2)/ √ 2 forms). Colossal magnetoresistance (CMR) – the sharp decrease of resistivity with magnetic field – occurs in some of the compounds that display orbital orders, e.g. the manganite LaMnO3 [1] and its derivatives. In LaMnO3, the orbital order is of a simple character: the state of the single outermost electron in one ion may be that of, say, the n = 3 d-wave state of the 3z − r type while it is 3x − r on a neighboring ion and so on in a staggered fashion within each plane. Other prominent systems that exhibit various types of orbital order are found among the vanadates (e.g., V2O3 [2], LiVO2 [3] and LaVO3) [4] and cuprates (e.g. KCuF3) [5]. Orbital ordering can be observed via orbitalrelated magnetism and lattice distortions or by resonant X-ray scattering techniques [6]. Although there are numerous works on quantum criticality in systems with various electronic phases [7, 8] there is little prior work [9–12] on systems in which the orbital ordering temperature veered to zero. We wish to motivate the following question: If quantum critical points are associated with the degradation of magnetic/charge/superconducting orders in numerous systems why can these not appear in orbitally ordered systems? Such quantum critical points are associated with the degradation of orbital order. This path has not been followed despite the success in finding quantum critical points and novel low-T transitions in spin and other electronic systems. We demonstrate the existence of exact quantum criticality driven by orbital order in the simplest of all orbital only models: the D = 2 and 3 dimensional quantum Orbital Compass Models (OCMs). These systems rigorously exhibit order in their classical limit [13–15]. This order is expected to be fortified by quantum effects. Indeed, numerically orbital order was detected in the undiluted D = 2 OCM [11]. We will show that, at a prescribed doping (dilution) of magnitude δ = 1/4, this system displays quantum critical correlations. This quantum critical phase may be driven away by applying pressure/strain (i.e. varying the size of the exchange constants corresponding to different spatial directions). Similarly, we show that a particular set of anisotropic couplings of the D = 3 OCM leads to quantum criticality on a lattice of doping δ = 1/2. In the particular case that the orbital exchange couplings become random (i.e., non-uniform), we