The geometric order of stripes and Luttinger liquids

Z_{2} Parton Phases in the Mixed-Dimensional t-J_{z} Model

We study the interplay of spin and charge degrees of freedom in a doped Ising antiferromagnet, where the motion of charges is restricted to one dimension. The phase diagram of this mixed-dimensional t-J_{z} model can be understood in terms of spinless chargons coupled to a Z_{2} lattice gauge field. The antiferromagnetic couplings give rise to interactions between Z_{2} electric field lines which, in turn, lead to a robust stripe phase at low temperatures. At higher temperatures, a confined meson-gas phase is found for low doping whereas at higher doping values, a robust deconfined chargon-gas phase is seen, which features hidden antiferromagnetic order. We confirm these phases in quantum Monte Carlo simulations. Our model can be implemented and its phases detected with existing technology in ultracold atom experiments. The critical temperature for stripe formation with a sufficiently high hole concentration is around the spin-exchange energy J_{z}, i.e., well within reach of current experiments.

Giant electron-phonon coupling of the breathing plane oxygen phonons in the dynamic stripe phase of [Formula: see text]

Doped antiferromagnets host a vast array of physical properties and learning how to control them is one of the biggest challenges of condensed matter physics. [Formula: see text] (LSNO) is a classic example of such a material. At low temperatures holes introduced via substitution of La by Sr segregate into lines to form boundaries between magnetically ordered domains in the form of stripes. The stripes become dynamic at high temperatures, but LSNO remains insulating presumably because an interplay between magnetic correlations and electron-phonon coupling localizes charge carriers. Magnetic degrees of freedom have been extensively investigated in this system, but phonons are almost completely unexplored. We searched for electron-phonon anomalies in LSNO by inelastic neutron scattering. Giant renormalization of plane Ni-O bond-stretching modes that modulate the volume around Ni appears on entering the dynamic charge stripe phase. Other phonons are a lot less sensitive to stripe melting. Dramatic overdamping of the breathing modes indicates that dynamic stripe phase may host small polarons. We argue that this feature sets electron-phonon coupling in nickelates apart from that in cuprates where breathing phonons are not overdamped and point out remarkable similarities with the colossal magnetoresistance manganites.

String patterns in the doped Hubbard model

Understanding strongly correlated quantum many-body states is one of the most difficult challenges in modern physics. For example, there remain fundamental open questions on the phase diagram of the Hubbard model, which describes strongly correlated electrons in solids. In this work, we realize the Hubbard Hamiltonian and search for specific patterns within the individual images of many realizations of strongly correlated ultracold fermions in an optical lattice. Upon doping a cold-atom antiferromagnet, we find consistency with geometric strings, entities that may explain the relationship between hole motion and spin order, in both pattern-based and conventional observables. Our results demonstrate the potential for pattern recognition to provide key insights into cold-atom quantum many-body systems.

Time-resolved observation of spin-charge deconfinement in fermionic Hubbard chains

Elementary particles carry several quantum numbers, such as charge and spin. However, in an ensemble of strongly interacting particles, the emerging degrees of freedom can fundamentally differ from those of the individual constituents. For example, one-dimensional systems are described by independent quasiparticles carrying either spin (spinon) or charge (holon). Here, we report on the dynamical deconfinement of spin and charge excitations in real space after the removal of a particle in Fermi-Hubbard chains of ultracold atoms. Using space- and time-resolved quantum gas microscopy, we tracked the evolution of the excitations through their signatures in spin and charge correlations. By evaluating multipoint correlators, we quantified the spatial separation of the excitations in the context of fractionalization into single spinons and holons at finite temperatures.

Tuning from failed superconductor to failed insulator with magnetic field

Do charge modulations compete with electron pairing in high-temperature copper oxide superconductors? We investigated this question by suppressing superconductivity in a stripe-ordered cuprate compound at low temperature with high magnetic fields. With increasing field, loss of three-dimensional superconducting order is followed by reentrant two-dimensional superconductivity and then an ultraquantum metal phase. Circumstantial evidence suggests that the latter state is bosonic and associated with the charge stripes. These results provide experimental support to the theoretical perspective that local segregation of doped holes and antiferromagnetic spin correlations underlies the electron-pairing mechanism in cuprates.

Stripe order in the underdoped region of the two-dimensional Hubbard model

Competing inhomogeneous orders are a central feature of correlated electron materials, including the high-temperature superconductors. The two-dimensional Hubbard model serves as the canonical microscopic physical model for such systems. Multiple orders have been proposed in the underdoped part of the phase diagram, which corresponds to a regime of maximum numerical difficulty. By combining the latest numerical methods in exhaustive simulations, we uncover the ordering in the underdoped ground state. We find a stripe order that has a highly compressible wavelength on an energy scale of a few kelvin, with wavelength fluctuations coupled to pairing order. The favored filled stripe order is different from that seen in real materials. Our results demonstrate the power of modern numerical methods to solve microscopic models, even in challenging settings.

Remarkable Stability of Charge Density Wave Order in La_{1.875}Ba_{0.125}CuO_{4}

The occurrence of charge-density-wave (CDW) order in underdoped cuprates is now well established, although the precise nature of the CDW and its relationship with superconductivity is not. Theoretical proposals include contrasting ideas such as that pairing may be driven by CDW fluctuations or that static CDWs may intertwine with a spatially modulated superconducting wave function. We test the dynamics of CDW order in La_{1.825}Ba_{0.125}CuO_{4} by using x-ray photon correlation spectroscopy at the CDW wave vector, detected resonantly at the Cu L_{3} edge. We find that the CDW domains are strikingly static, with no evidence of significant fluctuations up to 2 ¾  h. We discuss the implications of these results for some of the competing theories.

Direct observation of dynamic charge stripes in La2-xSrxNiO4

The insulator-to-metal transition continues to be a challenging subject, especially when electronic correlations are strong. In layered compounds, such as La2-xSrxNiO4 and La2-xBaxCuO4, the doped charge carriers can segregate into periodically spaced charge stripes separating narrow domains of antiferromagnetic order. Although there have been theoretical proposals of dynamically fluctuating stripes, direct spectroscopic evidence of charge-stripe fluctuations has been lacking. Here we report the detection of critical lattice fluctuations, driven by charge-stripe correlations, in La2-xSrxNiO4 using inelastic neutron scattering. This scattering is detected at large momentum transfers where the magnetic form factor suppresses the spin fluctuation signal. The lattice fluctuations associated with the dynamic charge stripes are narrow in q and broad in energy. They are strongest near the charge-stripe melting temperature. Our results open the way towards the quantitative theory of dynamic stripes and for directly detecting dynamical charge stripes in other strongly correlated systems, including high-temperature superconductors such as La2-xSrxCuO4.

Superconducting pairing and the pseudogap in the nematic dynamical stripe phase of La2-xSrxCuO4

Fully absorption coefficient corrected Raman spectra were obtained in La2-xSrxCuO4. The B1g spectra have a Fleury-Loudon type two-magnon peak (resonant term) whose energy decreases from 3180 cm(-1) (394 meV) to 440 cm(-1) (55 meV) on increasing the carrier density from x = 0 to 0.25, while the B2g spectra have a 1000-3500 cm(-1) (124-434 meV) hump (hill) whose lower-edge energy increases from x = 0 to 0.115 and then stays constant to x = 0.25. The B2g hump is assigned to the electronic scattering (non-resonant term) of the spectral function with magnetic self-energy. The completely different carrier density dependence arises from anisotropic magnetic excitations of spin-charge stripes. The B1g spectra were assigned to the sum of k ∥  and k⊥ stripe excitations and the B2g spectra to k⊥ stripe excitations according to the calculation by Seibold and Lorenzana (2006 Phys. Rev. B 73 144515). The k ∥  and k⊥ stripe excitations in fluctuating spin-charge stripes were separately detected for the first time. The appearance of only k⊥ stripe excitations in the electronic scattering arises from the charge hopping perpendicular to the stripe. This is the same direction as the Burgers vector of the edge dislocation in metal. The successive charge hopping in the Burgers vector direction across the charge stripes may cause Cooper pairs as predicted by Zaanen et al (2004 Ann. Phys. 310 181). Indeed, this is supported by the experimental fact that the superconducting coherent length coincides with the inter-charge stripe distance in the wide carrier density range. The one-directional charge hopping perpendicular to the stripe causes the flat Fermi surface and the pseudogap near (π,0) and (0,π), but the states around (π/2,π/2) cannot be produced. The low-energy Raman scattering disclosed that the electronic states at the Fermi arc around (π/2,π/2) are coupled to the A1g soft phonon of the tetragonal-orthorhombic phase transition. This suggests that the Fermi arc is produced by the electron-phonon interaction. All the present Raman data suggest that Cooper pairs are formed at moving edge dislocations of dynamical charge stripes.

Evidence for short-range-ordered charge stripes far above the charge-ordering transition in La1.67Sr0.33NiO4

The temperature evolution of structural effects associated with charge order (CO) and spin order in La1.67Sr0.33NiO4 has been investigated using neutron powder diffraction. We report an anomalous shrinking of the c/a lattice parameter ratio that correlates with T(CO). The sign of this change can be explained by the change in interlayer Coulomb energy between the static-stripe-ordered state and the fluctuating-stripe-ordered state or the charge-disordered state. In addition, we identify a contribution to the mean-square displacements of Ni and in-plane O atoms whose width correlates quite well with the size of the pseudogap extracted from the reported optical conductivity, with a non-Debye-like component that persists below and well above T(CO). We infer that dynamic charge-stripe correlations survive to T∼2T(CO).

Magneto-optical measurements of a cascade of transitions in superconducting La1.875Ba0.125CuO4 single crystals

Recent experiments on the original cuprate high-temperature superconductor, La(2-x)Ba(x)CuO4, revealed a remarkable sequence of phase transitions. Here we investigate such crystals with the polar Kerr effect, which is sensitive to time-reversal-symmetry breaking. Concurrent birefringence measurements accurately locate the structural phase transitions from high-temperature tetragonal to low-temperature orthorhombic, and then to lower-temperature tetragonal, at which temperature strong Kerr signal onsets. Hysteretic behavior of the Kerr signal suggests that time-reversal symmetry is already broken well above room temperature, an effect that was previously observed in high quality YBa2Cu3O(6+x) crystals.

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.

Quantum order-by-disorder near criticality and the secret of partial order in MnSi

The vicinity of quantum phase transitions has proven fertile ground in the search for new quantum phases. We propose a physically motivated and unifying description of phase reconstruction near metallic quantum-critical points using the idea of quantum order by disorder. Certain deformations of the Fermi surface associated with the onset of competing order enhance the phase space available for low-energy, particle-hole fluctuations and self-consistently lower the free energy. Applying the notion of quantum order by disorder to the itinerant helimagnet MnSi, we show that, near the quantum critical point, fluctuations lead to an increase of the spiral ordering wave vector and a reorientation away from the lattice-favored directions. The magnetic ordering pattern in this fluctuation-driven phase is found to be in excellent agreement with the neutron-scattering data in the partially ordered phase of MnSi.

Spontaneous symmetry breaking by charge stripes in the high pressure phase of superconducting La1.875Ba0.125CuO4

In those cases where charge-stripe order has been observed in cuprates, the crystal structure is such that the average rotational symmetry of the CuO2 planes is reduced from fourfold to twofold. As a result, one could argue that the reduced lattice symmetry is essential to the existence of stripe order. We use pressure to restore the average fourfold symmetry in a single crystal of La1.875Ba0.125CuO4, and show by x-ray diffraction that charge-stripe order still occurs. Thus, electronically driven stripe order can spontaneously break the lattice symmetry.

Charge excitations in the stripe-ordered La5/3Sr1/3NiO4 and La2-x(Ba,Sr)xCuO4 superconducting compounds

Charge excitations in stripe-ordered 214 compounds La_{5/3}Sr_{1/3}NiO_{4} and 1/8-doped La2-x(Ba or Sr)xCuO4 are studied using resonant inelastic x-ray scattering in the hard x-ray regime. We observe = or approximately 1 eV excitation with a momentum transfer corresponding to the charge stripe spatial period both for the diagonal (nickelate) and parallel (cuprates) stripes. They are interpreted as collective stripe excitations or anomalous softening of the charge excitonic modes of the in-gap states.

Dispersive excitations in the high-temperature superconductor La2-xSrxCuO4

High-resolution neutron scattering experiments on optimally doped La2-xSrxCuO4 (x=0.16) reveal that the magnetic excitations are dispersive. The dispersion is the same as in YBa2Cu3O6.85, and is quantitatively related to that observed with charge sensitive probes. The associated velocity in La2-xSrxCuO4 is only weakly dependent on doping with a value close to the spin-wave velocity of the insulating (x=0) parent compound. In contrast with the insulator, the excitations broaden rapidly with increasing energy, forming a continuum at higher energy and bear a remarkable resemblance to multiparticle excitations observed in 1D S=1/2 antiferromagnets. The magnetic correlations are 2D, and so rule out the simplest scenarios where the copper oxide planes are subdivided into weakly interacting 1D magnets.

Stripes, topological order, and deconfinement in a planar t-Jz model

We determine the quantum phase diagram of a two-dimensional bosonic t-Jz model as a function of the lattice anisotropy gamma, using a quantum Monte Carlo loop algorithm. We show analytically that the low-energy sectors of the bosonic and the fermionic t-Jz models become equivalent in the limit of small gamma. In this limit, the ground state represents a static stripe phase characterized by a nonzero value of a topological order parameter. This phase remains up to intermediate values of gamma, where there is a quantum phase transition to a phase-segregated state or a homogeneous superfluid with dynamic stripe fluctuations depending on the ratio Jz/t.

NMR evidence for a two-step phase separation in Nd1.85Ce0.15CuO4-delta

By Cu NMR we studied the spin and charge structure in Nd(2-x)Ce(x)CuO(4-delta). For x=0.15, starting from a superconducting sample, the low temperature magnetic order in the sample reoxygenated under 1 bar oxygen at 900 degrees C reveals a peculiar modulation of the internal field, indicative of a phase characterized by large charge droplets ("blob" phase). By prolonged reoxygenation at 4 bars the blobs break up and the spin structure changes to that of an ordered antiferromagnet. We conclude that the superconductivity in the n-type systems competes with a genuine type I Mott-insulating state.

Quantum magnetic excitations from stripes in copper oxide superconductors

In the copper oxide parent compounds of the high-transition-temperature superconductors the valence electrons are localized--one per copper site--by strong intra-atomic Coulomb repulsion. A symptom of this localization is antiferromagnetism, where the spins of localized electrons alternate between up and down. Superconductivity appears when mobile 'holes' are doped into this insulating state, and it coexists with antiferromagnetic fluctuations. In one approach to describing the coexistence, the holes are believed to self-organize into 'stripes' that alternate with antiferromagnetic (insulating) regions within copper oxide planes, which would necessitate an unconventional mechanism of superconductivity. There is an apparent problem with this picture, however: measurements of magnetic excitations in superconducting YBa2Cu3O6+x near optimum doping are incompatible with the naive expectations for a material with stripes. Here we report neutron scattering measurements on stripe-ordered La1.875Ba0.125CuO4. We show that the measured excitations are, surprisingly, quite similar to those in YBa2Cu3O6+x (refs 9, 10) (that is, the predicted spectrum of magnetic excitations is wrong). We find instead that the observed spectrum can be understood within a stripe model by taking account of quantum excitations. Our results support the concept that stripe correlations are essential to high-transition-temperature superconductivity.

Partial order in the non-Fermi-liquid phase of MnSi

Only a few metallic phases have been identified in pure crystalline materials. These include normal, ferromagnetic and antiferromagnetic metals, systems with spin and charge density wave order, and superconductors. Fermi-liquid theory provides a basis for the description of all of these phases. It has been suggested that non-Fermi-liquid phases of metals may exist in some heavy-fermion compounds and oxide materials, but the discovery of a characteristic microscopic signature of such phases presents a major challenge. The transition-metal compound MnSi above a certain pressure (p(c) = 14.6 kbar) provides what may be the cleanest example of an extended non-Fermi-liquid phase in a three-dimensional metal. The bulk properties of MnSi suggest that long-range magnetic order is suppressed at p(c) (refs 7-12). Here we report neutron diffraction measurements of MnSi, revealing that sizeable quasi-static magnetic moments survive far into the non-Fermi-liquid phase. These moments are organized in an unusual pattern with partial long-range order. Our observation supports the existence of novel metallic phases with partial ordering of the conduction electrons (reminiscent of liquid crystals), as proposed for the high-temperature superconductors and heavy-fermion compounds.

Nonuniversal ordering of spin and charge in stripe phases

We study the interplay of topological excitations in stripe phases: charge dislocations, charge loops, and spin vortices. In two dimensions these defects interact logarithmically on large distances. Using a renormalization-group analysis in the Coulomb-gas representation of these defects, we calculate the phase diagram and the critical properties of the transitions. Depending on the interaction parameters, spin and charge order can disappear at a single transition or in a sequence of two transitions (spin-charge separation). These transitions are nonuniversal with continuously varying critical exponents. We also determine the nature of the points where three phases coexist.