Session W21: Density Waves in Superconductors and Other Systems

Charge ordering in three-band models of the cuprates

We examine trends in the wavevectors and form-factors of charge density wave instabilities of three-band models of the underdoped cuprates. For instabilities from a high temperature state with a large Fermi surface, we extend a study by Bulut et al. (Phys. Rev. B 88, 155132 (2013)) to include a direct antiferromagnetic exchange coupling between the Cu sites. As in previous work, we invariably find that the primary instability has a diagonal wavevector $(\pm Q_0, \pm Q_0)$ and a $d$-form factor. The experimentally observed wavevectors along the principal axes $(\pm Q_0,0)$, $(0, \pm Q_0)$ have higher energy, and their form factor is found to be predominantly $d$. Next, we gap out the Fermi surface in the anti-nodal regions of the Brillouin zone by including static, long-range antiferromagnetic order at the wavevector $(\pi, \pi)$: this is a simple model of the pseudogap in which we assume the antiferromagnetic order averages to zero by 'renormalized classical' thermal fluctuations in its orientation, valid when the antiferromagnetic correlation length is large. The charge density wave instabilities of this pseudogap state are found to have the optimal wavevector $(\pm Q_0,0)$, $(0, \pm Q_0)$, with the magnitude of the $d$-form factor decreasing with increasing magnetic order.   

`Hot-spot' Charge-Density Wave Nesting in Cuprates

Recent experiments have found evidence for at least short-range charge-density wave (CDW) order in several families of cuprates. Theories of this order fall into two categories: intermediate coupling theories where the transition involves Fermi surface nesting and coupling to a lattice distortion[1], and strong-coupling theories of purely electronic CDWs where the ordering vector is a form of `hot-spot' nesting associated with underlying antiferromagnetic fluctuations[2]. Surprisingly, both models find the same nesting vectors[3]. We will discuss ways of distinguishing the models, and of the relationship of this CDW order with the pseudogap phenomena in underdoped cuprates. 1. R.S. Markiewicz, J. Lorenzana, G. Seibold, and A. Bansil, arXiv:1207.5715. 2. K.B. Efetov, H. Meier, and C. P\'epin, Nature Phys. 9, 442 (2013); R. La Placa and S. Sachdev, Phys. Rev. Lett. 111, 027202 (2013); Y. Wang and A.V. Chubukov, Phys. Rev. B 90, 035149 (2014). 3. R.S. Markiewicz, J. Lorenzana, G. Seibold, and A. Bansil, Phys. Rev. B81, 014509 (2010).   

Charge-order in the underdoped cuprates: a window into the normal state

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 [1]. The resulting charge density wave state emerging out of such a fractionalized Fermi-liquid (FL*) is remarkably similar to the one observed in experiments on a number of different families of the cuprates [2]. 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 also describe the transition from a Fermi-liquid with a large Fermi-surface to a FL* around optimal doping via a Higgs-transition of a SU(2) gauge-theory. The implications of such Higgs criticality in two-dimensional metals on the physics of strange metal will be discussed.\\[4pt] [1] D. Chowdhury and S. Sachdev, arXiv:1409.5430. \\[0pt] [2] K. Fujita et al., PNAS 111, E3026 (2014)   

dc Resistivity at the Onset of Spin Density Wave Order in Two-dimensional Metals

The theory for the onset of spin density wave order in a metal in two dimensions flows to strong coupling, with strong interactions not only at the ``hot spots,'' but on the entire Fermi surface. We advocate the computation of dc transport in a regime where there is rapid relaxation to local equilibrium around the Fermi surface by processes which conserve total momentum. The dc resistivity is then controlled by weaker perturbations which do not conserve momentum. We consider variations in the local position of the quantum-critical point, induced by long-wavelength disorder, and find a contribution to the resistivity which is linear in temperature (up to logarithmic corrections) at low temperature. Scattering of fermions between hot spots, by short-wavelength disorder, leads to a residual resistivity and a correction which is linear in temperature.   

Are there quantum oscillations in an incommensurate charge density wave?

Because a material with an incommensurate charge density wave (ICDW) is only quasi-periodic, Bloch's theorem does not apply and there is no sharply defined Fermi surface. We will show that, as a consequence, there are no quantum oscillations which are truly periodic functions of 1/B (where B is the magnitude of an applied magnetic field). For a weak ICDW, there exist broad ranges of 1/B in which approximately periodic variations occur, but with frequencies that vary inexorably in an unending cascade with increasing 1/B. For a strong ICDW, e.g. in a quasi-crystal, no quantum oscillations survive at all. Rational and irrational numbers really are different. The duality between quasi-periodic systems in different dimensions can be straightforwardly generalized beyond the quantum oscillations and provides an accurate and efficient perspective.   

Short-range incommensurate $d$-wave charge order from a two-loop renormalization group calculation of the fermionic hot spot model

The two-loop renormalization group (RG) calculation is considerably extended here for a two-dimensional (2D) fermionic effective field theory model, which includes only the so-called ``hot spots'' that are connected by the spin-density-wave (SDW) ordering wavevector on a Fermi surface generated by the 2D $t-t'$ Hubbard model at low hole doping. We compute the Callan-Symanzik RG equation up to two loops describing the flow of the single-particle Green's function, the corresponding spectral function, the Fermi velocity, and some of the most important order-parameter susceptibilities in the model at lower energies. As a result, we establish that -- in addition to clearly dominant SDW correlations -- an approximate (pseudospin) symmetry relating a short-range \emph{incommensurate} $d$-wave charge order to the $d$-wave superconducting order indeed emerges at lower energy scales, which is in agreement with recent works available in the literature addressing the 2D spin-fermion model. We derive implications of this possible electronic phase in the ongoing attempt to describe the phenomenology of the pseudogap regime in underdoped cuprates.\\[4pt] Reference: V. S. de Carvalho and H. Freire, Annals of Physics 348, 32 (2014).   

Chiral Charge Density Wave and Superconductivity in Cu$_x$TiSe$_2$ Single Crystals

We investigate atomic scale scanning tunneling microscopy and spectroscopy in Cu$_x$TiSe$_2$ single crystals at low temperatures. We map the CDW and superconducting phase diagram as a function of copper doping. STM measurements reveal coexistence of chiral charge density wave and superconductivity. In case of optimally doped and overdoped cases we find that the amplitude of charge density wave modulation is strongly suppressed with respect to strongly underdoped case ($x<0.06$) with the chiral domain size remaining approximately the same. Superconductivity exhibits BCS character at variety of dopings with $2\Delta/kT_c\sim 3.6\div3.7$ indicating an intermediate coupling strength. Application of the external magnetic field introduces the Abrikosov vortex lattice that is weakly pinned. The size of the vortex core extracted from vortex images corresponds to the one extracted from the magnetization measurements. Our results suggest that, if charge density wave quantum critical point exist, it should be well above the optimal copper concentration of x=0.08.   

Chiral Symmetry Breaking and Mott Physics from Gauge/Gravity Duality

We use holographic techniques to address the origin of the Mott gap and Fermi arcs in the cuprates. We first show that dynamically generated gaps of the Mott kind arise in holographic settings from a bulk coupling that breaks chiral symmetry. We then explore a bulk coupling which breaks rotational symmetry but preserves chiral symmetry and show that Fermi arcs arise in the dual system at the boundary. We draw further lessons for the cuprates through the unambiguous interpretation of chiral symmetry as a combination of particle-hole and time reversal symmetry on the lattice, and suggest that the interplay of these symmetries may be the key to understanding the transition between the Mott insulating phase and the pseudogap.   

Quantum Critical Transitions in Spin and Charge Ordered Systems

This talk will focus on search and discovery of novel forms of quantum order in ferroelectric and multiferroic systems. Materials tuned to the neighbourhood of a zero temperature phase transition often show the emergence of novel quantum phenomena. Much of the effort to study these new emergent effects, like the breakdown of the conventional Fermi-liquid theory in metals has been focused in narrow band electronic systems. But Spin or Charge ordered phases in insulating systems can also be tuned to absolute zero. Close to such a zero temperature phase transition, physical quantities like susceptibility change into unconventional forms due to the fluctuations experienced in this region giving rise to new kinds ordered states.   

Fermiology of the undoped cuprate superconductor Pr$_{2}$CuO$_{4}$

Recent advances in molecular beam epitaxy growth and preparation of cuprate thin films indicate that annealing can be employed to minimize apical oxygen defects. For Pr$_{2}$CuO$_{4}$ the resulting square planar coordinated structure exhibits a 25~K superconducting transition in the absence of doping. This calls into question whether a Mott insulating groundstate is the relevant description of the square-planar parent phase of the electron-doped cuprate superconductors. We present high field (\textgreater 90~T) measurements of magnetic quantum oscillations --~the first observation of it's kind for a cuprate thin film. The oscillation frequency and effective mass are consistent with the reconstructed Fermi surface of the electron-doped cuprate Nd$_{2-x}$Ce$_{x}$CuO$_{4}$. The combination of a reconstructed bandstructure and the occurrence of metalicity at zero doping is consistent with a Slater picture of band magnetism, indicating that the ``doped Mott insulator'' paradigm may not apply in this system.   

Influence of Ti doping on the incommensurate charge density wave in 1T-TaS$_2$

Using x-ray scattering and transport, we studied the temperature dependence of the transition between incommensurate (IC) and nearly commensurate (NC) phases of the charge density wave (CDW) in Ti doped 1$T$-TaS$_2$. Our results showed a first order phase transition from IC to NC-CDW phase in all doping levels with decreased transition temperature, as Ti doping was increased. During this transition, the angle of the CDW in the basal plane rotates from 11.9$^o$ at 0$\%$ Ti doping to 16.4$^o$ at 12$\%$ Ti doping, while the in-plane component of the CDW wave vector does not change significantly. In addition, we observed that at 8$\%$ Ti doping, the CDW diffraction peak position and resistivity resemble that of pure TaS$_2$ in its commensurate CDW state. With our data, we revisit the resistive anomaly originally observed by DiSalvo [F. J. DiSalvo et al., Phys. Rev. B 12, 2220 (1975)] at 8$\%$ doping. DeSalvo explained this anomaly as arising from the pinning of the CDW on the crystal lattice. Our study shows that the commensuration effects in the NC phase is the cause of this anomaly.   

Scanning Tunneling Spectroscopy study of the Charge Density Wave driven Mott Insulator 1$T$-TaS$_{2}$

Exotic ground states can be generated by competition or interplay between various interactions, such as electron-phonon ($e-ph$), spin-orbit ($s-o$) and electron-electron ($e-e$) coupling. 1$T$-TaS$_{2}$ is a prime example to study such interplay since a Mott gap coexists with charge density waves (CDW) at low temperatures. Our scanning tunneling spectroscopy measurements with high spatial and energy resolution determine the CDW and the Mott gap as 0.20 -- 0.24 eV and 0.32 eV, respectively, by analyzing the phase difference between the real space electron densities across multiple energy gaps. In addition, we observe a peculiar reduction of the Mott gap in the vicinity of defect sites. The effect of competition between $e-ph$ and $e-e$ coupling on the Mott gap size will be discussed within the Hubbard-Holstein picture.   

Wavevector dependent electron$-$phonon coupling drives the CDW formation in TbTe$_3$

The charge density wave (CDW) transition in the rare-earth tritellurides RTe$_{3}$ (R$=$rare earth) is commonly assumed to originate from a textbook Fermi surface nesting instability. Contrary to this weak coupling scenario, our investigation of the soft phonon mode in TbTe$_{3}$ provides direct evidence that the periodicity of the CDW superstructure in this canonical compound is defined by a strong momentum dependence of the electron-phonon coupling. Our high-resolution inelastic x-ray measurements reveal a renormalization of the soft-phonon energy and a strong broadening of the soft-phonon linewidth over a large part of reciprocal space adjacent to the CDW ordering vector. Our detailed theoretical calculations reproduce these observations very well and show that the position in reciprocal space of the phonon renormalization, and with it the CDW order wavevector, is not related to Fermi surface nesting. Our results demonstrate the importance of strongly momentum dependent electron-phonon coupling in defining the CDW order, which could also be relevant to many other systems.   

Anisotropic symmetry breaking in two-dimensional charge density waves of ErTe$_{3}$ investigated by femtosecond electron crystallography

Electron-phonon interactions can give rise to various charge-ordered states, especially at low dimensions, where Fermi surface is more prone to form nesting. Rare earth tritellurides compound ErTe$_{3}$ develops charge density waves (CDW) along two perpendicular directions at different temperatures. By directly probing the order parameters of the two CDWs using femtosecond electron crystallography under different temperatures and driving photonic energy, we investigated the emergences of competing CDW orders in a dynamical phase diagram. The anisotropic symmetry breaking and the role of electron-phonon coupling, and photo-doping effect are discussed in reference to other CDW systems.   

Electronic band structure of Charge Density Wave PdxHoTe3

The origin of superconductivity and interplay between superconductivity and different ground states remains challenging. The Pd-intercalated HoT$_3$, suppresses the charge density wave (CDW) order and leads to the superconductivity. Here we report the detailed Angle-resolved photoemission spectroscopy (ARPES) study of the electronic structure on Pd$_x$HoT$_3$. In the CDW parent phase (HoT$_3$), we found out the Fermi surface topology, CDW gap symmetry have 2 fold symmetry, with one CDW vector. With further Pd-intercalations, the system evolves from 2-fold symmetry to 4-fold symmetry with two CDW vectors, and eventually into superconducting state. The evolution of the CDW gap symmetry, gap size and CDW caused shadow bands are discussed at different phases.