Stripe Order Research Articles

Quantum critical point for stripe order: An organizing principle of cuprate superconductivity

A spin density-wave quantum critical point (QCP) is the central organizing principle of organic, iron-pnictide, heavy-fermion and electron-doped cuprate superconductors. It accounts for the superconducting Tc dome, the non-Fermi-liquid resistivity, and the Fermi-surface reconstruction. Outside the magnetically ordered phase above the QCP, scattering and pairing decrease in parallel as the system moves away from the QCP. Here we argue that a similar scenario, based on a stripe-order QCP, is a central organizing principle of hole-doped cuprate superconductors. Key properties of La1.8−xEu0.2SrxCuO4, La1.6−xNd0.4SrxCuO4 and YBa2Cu3Oy are naturally unified, including stripe order itself, its QCP, Fermi-surface reconstruction, the linear-T resistivity, and the nematic character of the pseudogap phase.

Theory of a Continuous Stripe Melting Transition in a Two-Dimensional Metal: A Possible Application to Cuprate Superconductors

We construct a theory of continuous stripe melting quantum phase transitions in two-dimensional metals and the associated Fermi surface reconstruction. Such phase transitions are strongly coupled but yet theoretically tractable in situations where the stripe ordering is destroyed by proliferating doubled dislocations of the charge stripe order. The resulting non-Landau quantum critical point has strong stripe fluctuations which we show decouple dynamically from the Fermi surface even though static stripe ordering reconstructs the Fermi surface. We discuss connections to various stripe phenomena in the cuprates. We point out several puzzling aspects of old experimental results [G. Aeppli et al., Science 278, 1432 (1997)] on singular stripe fluctuations in the cuprates, and provide a possible explanation within our theory. These results may thus have been the first observation of non-Landau quantum criticality in an experiment.

Coherentc-axis transport in the underdoped cuprate superconductor YBa2Cu3Oy

The electrical resistivity rho_c of the underdoped cuprate superconductor YBCO was measured perpendicular to the CuO_2 planes on ultra-high quality single crystals in magnetic fields large enough to suppress superconductivity. The incoherent insulating-like behavior of rho_c at high temperature, characteristic of all underdoped cuprates, is found to cross over to a coherent regime of metallic behavior at low temperature. This crossover coincides with the emergence of the small electron pocket detected in the Fermi surface of YBCO via quantum oscillations, the Hall and Seebeck coefficients and with the detection of a unidirectional modulation of the charge density as seen by high-field NMR measurements. The low coherence temperature is quantitatively consistent with the small hopping integral t_perp inferred from the splitting of the quantum oscillation frequencies. We conclude that the Fermi-surface reconstruction in YBCO at dopings from p = 0.08 to at least p = 0.15, attributed to stripe order, produces a metallic state with 3D coherence deep in the underdoped regime.

Structural aspects of materials with static stripe order

A growing number of cuprate superconductors show evidence of an intrinsic spatial electronic inhomogeneity on a nanometer length scale. Corresponding observations were made with a variety of probes sensitive to the sample’s surface or bulk as well as to its average or local structure. Charge and spin stripe correlations constitute one class of inhomogeneity initially derived from neutron scattering data on La-based cuprates. By now magnetic excitation spectra with very similar dispersions were observed in several systems, suggesting that incommensurate spin fluctuations are a universal property of the cuprates. In contrast, it proves much harder to obtain a unified picture for the charge stripe correlations as observed with scattering and tunneling techniques. This article reviews the evidence of charge stripe correlations and their interactions with the crystal lattice in the La-based compounds, which are presently the only cuprates with confirmed static spin and charge stripe order. Particular emphasis is put on the relationship between stripe order and average and local structure properties, isotope effects, phonon anomalies, and doping dependencies. A second important topic concerns new insights obtained in studies of stripes under extreme conditions such as high magnetic fields and high pressure. Furthermore, the tilting of stripes in orthorhombic lattices will be discussed, as well as the possibility of diagonal charge stripes below the metal insulator transition, and alternative models for the stripe phase.

Neutron scattering studies of stripe correlations in curate oxide superconductors

In the last quarter of a century, neutron-scattering measurements have provided important clues for understanding the novel magnetism of carrier doped cuprate oxides. Discovery of evidence for charge-density-wave (CDW) and spin-density-wave (SDW) orders in La1.6−xNd0.4SrxCuO4 with x∼1/8 has attracted much attention on the universality of inhomogeneous carrier distribution in CuO2 planes, namely stripe correlations, and its relevance to the superconductivity. Here we review the nature of charge and spin stripe orders in a prototypical superconducting La-214 system, which has been investigated mainly by neutron scattering measurements. After introducing the hole concentration range, where the each order can be observed, characteristic features of low-energy spin fluctuations in the stripe ordered and disordered La2−xBaxCuO4 with x∼1/8 will be discussed in terms of fluctuating stripes. The impact of magnetic field and impurity substitution on the orders is also presented, followed by a comparative study on electron-doped cuprates. Finally, we will give a brief summary of experimental trends.

Stripe, checkerboard, and liquid-crystal ordering from anisotropicp-orbital Fermi surfaces in optical lattices

We study instabilities of single-species fermionic atoms in the p-orbital bands in two-dimensional optical lattices at noninteger filling against interactions. Charge density wave and orbital density wave orders with stripe or checkerboard patterns are found for attractive and repulsive interactions, respectively. The superfluid phase, usually expected of attractively interacting fermions, is strongly suppressed. We also use field theory to analyze the possible phase-transitions from orbital stripe order to liquid-crystal phases and obtain the phase diagram. The condition of nearly-perfect Fermisurface nesting, which is key to the above results, is shown robustly independent of fermion fillings in such p-orbital systems, and the $(2k_F,\pm2k_F)$ momentum of density wave oscillation is highly tunable. Such remarkable features show the promise of making those exotic orbital phases, which are of broad interest in condensed-matter physics, experimentally realizable with optical lattice gases.

Stripes and electronic quasiparticles in the pseudogap state of cuprate superconductors

This article is devoted to a discussion of stripe and electron-nematic order and their connection to electronic properties in the pseudogap regime of copper-oxide superconductors. We review basic properties of these symmetry-breaking ordering phenomena as well as proposals which connect them to quantum-oscillation measurements. Experimental data indicate that these orders are unlikely to be the cause of the pseudogap phenomenon, implying that they occur on top of the pseudogap state which itself is of different origin. Specifically, we discuss the idea that the non-superconducting pseudogap ground state hosts electron-like quasiparticles which coexist with a spin liquid, realizing a variant of a fractionalized Fermi liquid. We speculate on how stripe order in such a pseudogap state might offer a consistent description of ARPES, NMR, quantum-oscillation, and transport data.

Detecting incipient stripe order in the high-temperature superconductor Bi2Sr2CaCu2O8+x

We review recent spectroscopic mapping measurements on Bi2Sr2CaCu2O8+x that establish experimental signatures of incipient stripe order. Model calculations show how to disentangle the effects of band structure and incipient order and allow us to model details of the energy-dependence of the observed modulations associated with incipient stripe order. Comprehensive doping and temperature studies reveal the correlation between the onset of the pseudogap and incipient stripe order in Bi2Sr2CaCu2O8+x, yet they also show that stripes are most likely the consequence of the pseudogap rather than its cause.

Formula omitted]SR studies of the interplay of magnetic spin stripe order with superconductivity in transition metal oxides

In this work we review muon spin relaxation experiments on the layered La2−xSrxNiO4 nickelate as well as La2−xBaxCuO4 and La2−xSrxCuO4 cuprate systems to examine spin stripe order. In particular, the interplay of stripe order with superconductivity in Nd and Eu doped La2−xSrxCuO4 cuprates is discussed. Detailed studies of the electronic phase diagrams as well as the magnetic and superconducting order parameters for different rare-earth and Sr doping levels in La2−x−yREySrxCuO4 revealed the strong correlation of static spin stripe order with the structural distortion in the low temperature tetragonal (LTT) phase and the competition with the superconducting ground state. High magnetic field studies demonstrate the nearly degenerate ground state energy of the different electronic phases. Slow transverse fluctuations of the charge stripes are found in nickelates and cuprates at low temperatures.

Resonant soft X-ray scattering, stripe order, and the electron spectral function in cuprates

We review the current state of efforts to use resonant soft X-ray scattering (RSXS), which is an elastic, momentum-resolved, valence band probe of strongly correlated electron systems, to study stripe-like phenomena in copper-oxide superconductors and related materials. We review the historical progress including RSXS studies of Wigner crystallization in spin ladder materials, stripe order in 214-phase nickelates, 214-phase cuprates, and other systems.One of the major outstanding issues in RSXS concerns its relationship to more established valence band probes, namely angle-resolved photoemission (ARPES) and scanning tunneling microscopy (STM). These techniques are widely understood as measuring a one-electron spectral function, yet a relationship between RSXS and a spectral function has so far been unclear. Using physical arguments that apply at the oxygen K edge, we show that RSXS measures the square modulus of an advanced version of the Green’s function measured with STM. This indicates that, despite being a momentum space probe, RSXS is more closely related to STM than to ARPES techniques.Finally, we close with some discussion of the most promising future directions for RSXS. We will argue that the most promising area lies in high magnetic field studies, particularly of edge states in strongly correlated heterostructures, and the vortex state in superconducting cuprates, where RSXS may clarify the anomalous periodicities observed in recent quantum oscillation experiments.

Nuclear magnetic and quadrupole resonance studies of the stripes materials

Nuclear Magnetic and Quadrupole Resonance (NMR/NQR) is a powerful tool to probe electronic inhomogeneities in correlated electron systems. Its local character allows for probing different environments due to spin density modulations or inhomogeneous doping distributions emerging from the correlations in these systems. In fact, NMR/NQR is not only sensitive to magnetic properties through interaction of the nuclear spin, but also allows to probe the symmetry of the charge distribution and its homogeneity, as well as structural modulations, through sensitivity to the electric field gradient (EFG). We review the results of NMR and NQR in the cuprates from intrinsic spatial variations of the hole concentration in the normal state to stripe order at low temperatures, thereby keeping in mind the influence of doping induced disorder and inhomogeneities. Finally, we briefly discuss NQR evidence for local electronic inhomogeneities in the recently discovered iron pnictides, suggesting that electronic inhomogeneities are a common feature of correlated electron systems.

Transport properties of stripe-ordered high Tc cuprates

Transport measurements provide important characterizations of the nature of stripe order in the cuprates. Initial studies of systems such as La1.6−xNd0.4SrxCuO4 demonstrated the strong anisotropy between in-plane and c-axis resistivities, but also suggested that stripe order results in a tendency towards insulating behavior within the planes at low temperature. More recent work on La2−xBaxCuO4 with x=1/8 has revealed the occurrence of quasi-two-dimensional superconductivity that onsets with spin-stripe order. The suppression of three-dimensional superconductivity indicates a frustration of the interlayer Josephson coupling, motivating a proposal that superconductivity and stripe order are intertwined in a pair-density-wave state. Complementary characterizations of the low-energy states near the Fermi level are provided by measurements of the Hall and Nernst effects, each revealing intriguing signatures of stripe correlations and ordering. We review and discuss this work.

Probing the connections between superconductivity, stripe order, and structure in La1.905Ba0.095Cu1−yZnyO4

The superconducting system La${}_{2\ensuremath{-}x}$Ba${}_{x}$CuO${}_{4}$ is known to show a minimum in the transition temperature ${T}_{c}$ at $x=\frac{1}{8}$ where maximal stripe order is pinned by the anisotropy within the CuO${}_{2}$ planes that occurs in the low-temperature-tetragonal (LTT) crystal structure. For $x=0.095$, where ${T}_{c}$ reaches its maximum value of 32 K, there is a roughly coincident structural transition to a phase that is very close to LTT. Here, we present a neutron scattering study of the structural transition, and demonstrate how features of it correlate with anomalies in the magnetic susceptibility, electrical resistivity, thermal conductivity, and thermoelectric power. We also present measurements on a crystal with 1$%$ Zn substituted for Cu, which reduces ${T}_{c}$ to 17 K, enhances the spin stripe order, but has much less effect on the structural transition. We make the case that the structural transition correlates with a reduction of the Josephson coupling between the CuO${}_{2}$ layers, which interrupts the growth of the superconducting order. We also discuss evidence for two-dimensional superconducting fluctuations in the normal state, analyze the effective magnetic moment per Zn impurity, and consider the significance of the anomalous thermopower often reported in the stripe-ordered phase.

Uniaxial linear resistivity of superconducting La1.905Ba0.095CuO4induced by an external magnetic field

We present an experimental study of the anisotropic resistivity of superconducting La(2-x)Ba(x)CuO(4) with x=0.095 and transition temperature Tc=32 K. In a magnetic field perpendicular to the CuO(2) layers, H(perp), we observe that the resistivity perpendicular to the layers, \rho(perp), becomes finite at a temperature consistent with previous studies on very similar materials; however, the onset of finite parallel resistivity, \rho(par), occurs at a much higher temperature. This behavior contradicts conventional theory, which predicts that \rho(perp) and \rho(par) should become finite at the same temperature. Voltage vs. current measurements near the threshold of voltage detectability indicate linear behavior perpendicular to the layers, becoming nonlinear at higher currents, while the behavior is nonlinear from the onset parallel to the layers. These results, in the presence of moderate H(perp), appear consistent with superconducting order parallel to the layers with voltage fluctuations between the layers due to thermal noise. In search of uncommon effects that might help to explain this behavior, we have performed diffraction measurements that provide evidence for H(perp)-induced charge and spin stripe order. The field-induced decoupling of superconducting layers is similar to the decoupled phase observed previously in La(2-x)Ba(x)CuO(4) with x=1/8 in zero field.

Scattering from incipient stripe order in the high-temperature superconductor Bi2Sr2CaCu2O8+δ

Recently we have used spectroscopic mapping with the scanning tunneling microscope to probe modulations of the electronic density of states in single crystals of the high temperature superconductor Bi2Sr2CaCu2O8+d (Bi-2212) as a function of temperature [C. V. Parker et al., Nature (London) 468, 677 (2010)]. These measurements showed Cu-O bond-oriented modulations that form below the pseudogap temperature with a temperature-dependent energy dispersion displaying different behaviors in the superconducting and pseudogap states. Here we demonstrate that quasiparticle scattering off impurities does not capture the experimentally observed energy- and temperature-dependence of these modulations. Instead, a model of scattering of quasiparticles from short-range stripe order, with periodicity near four lattice constants (4a), reproduces the experimentally observed energy dispersion of the bond-oriented modulations and its temperature dependence across the superconducting critical temperature, Tc. The present study confirms the existence of short-range stripe order in Bi-2212.

Two-orbital quantum spin model of magnetism in the iron pnictides

We study a two-orbital spin model to describe (pi,0) stripe antiferromagnetism in the iron pnictides. The "double-spin" model has an on-site Hunds's coupling and inter-site interactions extending to second neighbors (inter- and intra-orbital) on the square lattice. Using a variational method based on a cluster decomposition, we optimize wave functions with up to 8 cluster sites (up to 2^16 variational parameters). We focus on the anomalously small ordered moments in the stripe state of the pnictides. To account for it, and large variations among different compounds, we show that the second-neighbor cross-orbital exchange constant should be ferromagnetic, which leads to "partially hidden" stripe order, with a moment that can be varied over a large range by small changes in the coupling constants. In a different parameter region, we confirm the existence of a canted state previously found in spin-wave theory. We also identify several other phases of the model.

On the coexistence of different types of long-range order in the strongly frustrated two-dimensional Heisenberg model

In the spherically symmetric self-consistent approach to spin–spin Green’s functions, we consider a two-dimensional strongly frustrated J1−J2−J3 quantum S=1/2 antiferromagnet. In the classical limit S≫1, the phase diagram of the model demonstrates two triple points. We show that, in the quantum limit S=1/2 for J3<0, a quadruple point of the quantum phase transition appears. At this point, the four coexisting phases are a disordered spin-liquid phase and three phases with different types of long-range order. Two of them are well-known “checkerboard” Néel order and stripe order; the last one corresponds to a non-trivial state with two coexisting mutually penetrating long-range orders.

Absence of static stripes in the two-dimensionalt−Jmodel determined using an accurate and systematic quantum Monte Carlo approach

We examine the two-dimensional $t\ensuremath{-}J$ model by using variational approach combined with well established quantum Monte Carlo techniques [S. Sorella et al., Phys. Rev. Lett. 88, 117002 (2002).] that are used to improve systematically the accuracy of the variational ansatz. Contrary to recent density-matrix renormalization group and projected entangled-pair state calculations [P. Corboz et al., Phys. Rev. B 84, 041108(R) (2011).], a uniform phase is found for $J/t=0.4$, even when the calculation is biased with an ansatz that explicitly contains stripe order. Moreover, in the small hole doping regime, that is, $\ensuremath{\delta}\ensuremath{\lesssim}0.1$, our results support the coexistence of antiferromagnetism and superconductivity.

Effects of Zn on superconductivity, stripe order, and pseudogap correlations in YBa2(Cu1−yZny)3O7−δ

The effect of Zn substitution in the CuO2 plane on the superconducting transition temperature, Tc, was studied for the YBa2(Cu1−yZny)3O7−δ compounds over a wide range of hole concentration, p, and Zn content (y). Zn induced rate of suppression of Tc, dTc(p)/dy, was found to be strongly p-dependent and showed a monotonic variation with p, except in the region of the 60K plateau where the charge/spin stripe correlations are believed to be at their strongest in Y123. The magnitude of dTc(p)/dy decreased significantly around this composition, implying that Zn suddenly became less effective in degrading Tc. This is somewhat counterintuitive since static stripe ordering itself degrades superconducting order. We have discussed the possible scenarios that can give rise to such an anomalous dTc(p)/dy at the Tc(δ) plateau. We have also looked at the p-dependent characteristic pseudogap energy scale, εg(p), which shows a quasi-linear decrease with increasing p without any noticeable feature in the plateau region. Moreover, there is no appreciable effect of Zn content on the value of εg(p). All these observations are indicative of a complex and possibly competing interplay among the superconducting, pseudogap, and stripe correlations in YBa2(Cu1−yZny)3O7−δ compounds.

Ashkin-Teller Criticality and Pseudo-First-Order Behavior in a Frustrated Ising Model on the Square Lattice

We study the challenging thermal phase transition to stripe order in the frustrated square-lattice Ising model with couplings J(1) < 0 (nearest-neighbor, ferromagnetic) and J(2) > 0 (second-neighbor, antiferromagnetic) for g = J(2)/|J(1| > 1/2. Using Monte Carlo simulations and known analytical results, we demonstrate Ashkin-Teller criticality for g ≥ g*; i.e., the critical exponents vary continuously between those of the 4-state Potts model at g = g* and the Ising model for g → ∞. Thus, stripe transitions offer a route to realizing a related class of conformal field theories with conformal charge c = 1 and varying exponents. The transition is first order for g < g* = 0.67 ± 0.01, much lower than previously believed, and exhibits pseudo-first-order behavior for |g* ≤ g </~1.