Pseudogap Phase Research Articles

Signatures of pairing and spin-orbit coupling in correlation functions of Fermi gases

We derive expressions for spin and density correlation functions in the (greatly enhanced) pseudogap phase of spin-orbit coupled Fermi superfluids. Density-density correlation functions are found to be relatively insensitive to the presence of these Rashba effects. To arrive at spin-spin correlation functions we derive new $f$-sum rules, valid even in the absence of a spin conservation law. Our spin-spin correlation functions are shown to be fully consistent with these $f$-sum rules. Importantly, they provide a clear signature of the Rashba band-structure and separately help to establish the presence of a pseudogap.

The origins of macroscopic quantum coherence in high temperature superconductivity

A new, theoretical approach to macroscopic quantum coherence and superconductivity in the p-type (hole doped) cuprates is proposed. The theory includes mechanisms to account for e-pair coupling in the superconducting and pseudogap phases and their inter relations observed in these materials.Electron pair coupling in the superconducting phase is facilitated by local quantum potentials created by static dopants in a mechanism which explains experimentally observed optimal doping levels and the associated peak in critical temperature. By contrast, evidence suggests that electrons contributing to the pseudogap are predominantly coupled by fractal spin waves (fractons) induced by the fractal arrangement of dopants.On another level, the theory offers new insights into the emergence of a macroscopic quantum potential generated by a fractal distribution of dopants. This, in turn, leads to the emergence of coherent, macroscopic spin waves and a second associated macroscopic quantum potential, possibly supported by charge order. These quantum potentials play two key roles. The first involves the transition of an expected diffusive process (normally associated with Anderson localization) in fractal networks, into e-pair coherence. The second involves the facilitation of tunnelling between localized e-pairs. These combined effects lead to the merger of the super conducting and pseudo gap phases into a single coherent condensate at optimal doping. The underlying theory relating to the diffusion to quantum transition is supported by Coherent Random Lasing, which can be explained using an analogous approach. As a final step, an experimental program is outlined to validate the theory and suggests a new approach to increase the stability of electron pair condensates at higher temperatures.

Checkerboard order in vortex cores from pair-density-wave superconductivity

We consider competing pair-density-wave (PDW) and $d$-wave superconducting states in a magnetic field. We show that PDW order appears in the cores of $d$-wave vortices, driving checkerboard charge-density-wave (CDW) order in the vortex cores, which is consistent with experimental observations. Furthermore, we find an additional CDW order that appears on a ring outside the vortex cores. This CDW order varies with a period that is twice that of the checkerboard CDW and it only appears where both PDW and $d$-wave order co-exist. The observation of this additional CDW order would provide strong evidence for PDW order in the pseudogap phase of the cuprates. We further argue that the CDW seen by nuclear magnetic resonance at high fields is due to a PDW state that emerges when a magnetic field is applied.

Polar Kerr effect from a time-reversal symmetry breaking unidirectional charge density wave

We analyze the Hall conductivity ${\ensuremath{\sigma}}_{xy}(\ensuremath{\omega})$ of a charge ordered state with momentum $\mathbf{Q}=(0,2Q)$ and calculate the intrinsic contribution to the Kerr angle ${\ensuremath{\Theta}}_{K}$ using the fully reconstructed tight-binding band structure for layered cuprates beyond the low energy hot spot model and particle-hole symmetry. We show that such a unidirectional charge density wave (CDW), which breaks time-reversal symmetry, as recently put forward by Wang and Chubukov [Phys. Rev. B 90, 035149 (2014)], leads to a nonzero polar Kerr effect, as observed experimentally. In addition, we model a fluctuating CDW via a large quasiparticle damping of the order of the CDW gap and discuss possible implications for the pseudogap phase. We can qualitatively reproduce previous measurements of underdoped cuprates, but making quantitative connections to experiments is hampered by the sensitivity of the polar Kerr effect with respect to the complex refractive index $n(\ensuremath{\omega})$.

Chiral Mott insulators, Meissner effect, and Laughlin states in quantum ladders

We introduce generic bosonic models exemplifying that chiral Meissner currents can persist in insulating phases of matter. We first consider interacting bosons on a two-leg ladder. The total density sector can be gapped in a bosonic Mott insulator at odd-integer filling, while the relative density sector remains superfluid due to interchain hopping. Coupling the relative density to gauge fields yields a pseudospin Meissner effect. We show that the same phase arises if the bosons are replaced by spinful fermions confined in Cooper pairs, and find a dual fermionic Mott insulator with spinon currents. We prove that by tuning the mean density the Mott insulator with Meissner currents turns into a low-dimensional bosonic $\nu = \frac{1}{2}$ Laughlin state for strong enough repulsive interactions across the ladder rungs. We finally discuss extensions to multileg ladders and bilayers in which spinon superfluids with Meissner currents become possible. We propose two experimental realizations, one with ultracold atoms in the setup of Atala et al., Nat. Phys. \textbf{8}, 588 (2014) and another with Josephson junction arrays. We also address a Bose-Fermi mixture subject to a magnetic field in connection with the pseudo-gap phase of high-Tc cuprates.

A model of magneto-electric multipoles

A long-known Hamiltonian of electrons with entangled spin and orbital degrees of freedom is re-examined as a model of magneto-electric multipoles (MEs). In the model, a magnetic charge and simple quantum rotator are tightly locked in action, some might say they are enslaved entities. It is shown that MEs almost perfectly accord with those inferred from an analysis of magnetic neutron diffraction data on a ceramic superconductor (YBCO) in the pseudo-gap phase. Nigh on perfection between Stone's model and inferred MEs is achieved by addition to the original model of a crystal-field potential appropriate for the magnetic space group used in the published data analysis. An impression of thermal properties of multipoles is sought from a molecular-field model.

Two-component energy spectrum of cuprates in the pseudogap phase and its evolution with temperature and at charge ordering.

In the search for mechanisms of high-temperature superconductivity it is critical to know the electronic spectrum in the pseudogap phase from which superconductivity evolves. The lack of angle-resolved photoemission data for every cuprate family precludes an agreement as to its structure, doping and temperature dependence and the role of charge ordering. Here we show that, in the entire Fermi-liquid-like regime that is ubiquitous in underdoped cuprates, the spectrum consists of holes on the Fermi arcs and an electronic pocket. We argue that experiments on the Hall coefficient identify the latter as a permanent feature at doped hole concentration x > 0.08–0.10, in contrast to the idea of the Fermi surface reconstruction via charge ordering. The longstanding issue of the origin of the negative Hall coefficient in YBCO and Hg1201 at low temperature is resolved: the electronic contribution prevails as mobility of the latter (evaluated by the Dingle temperature) becomes temperature independent, while the mobility of holes scattered by the short-wavelength charge density waves decreases.

Emergent loop current order from pair density wave superconductivity

There is evidence that the pseudogap phase in the cuprates breaks time-reversal symmetry. Here we show that pair density wave (PDW) states give rise to a translational invariant nonsuperconducting order parameter that breaks time-reversal and parity symmetries, but preserves their product. This secondary order parameter has a different origin, but shares the same symmetry properties as a magnetoelectric loop current order that has been proposed earlier in the context of the cuprates to explain the appearance of intracell magnetic order. We further show that, due to fluctuations, this secondary loop current order, which breaks only discrete symmetries, can preempt PDW order, which breaks both continuous and discrete symmetries. In such a phase, the emergent loop current order coexists with spatial short-range superconducting order and possibly short-range charge density wave (CDW) order. Finally, we propose a PDW phase that accounts for intracell magnetic order and the Kerr effect, has CDW order consistent with x-ray scattering and nuclear magnetic resonance observations, and quasi-particle properties consistent with angle-resolved photoemission scattering.

Exploring quasiparticles in high-Tc cuprates through photoemission, tunneling, and x-ray scattering experiments

One of the key challenges in the field of high-temperature superconductivity is understanding the nature of fermionic quasiparticles. Experiments consistently demonstrate the existence of a second energy scale, distinct from the d-wave superconducting gap, that persists above the transition temperature into the ‘pseudogap’ phase. One common class of models relates this energy scale to the quasiparticle gap due to a competing order, such as the incommensurate ‘checkerboard’ order observed in scanning tunneling microscopy (STM) and resonant elastic x-ray scattering (REXS). We develop a minimal phenomenological model that allows us to quantitatively describe STM and REXS experiments and discuss their relation with photoemission spectroscopy. Experimental signatures of the incommensurate order are explained in terms of scattering of short-lived quasiparticles from local impurities. We identify the unknown second energy scale with the inverse lifetime of the quasiparticles, refocusing questions about the nature of the pseudogap phase to the study of the origin of inelastic scattering.

Stability of Staggered Flux State in d-p Model Studied Using Variational Monte Carlo Method

The stability of a staggered flux (SF) or d-density wave state is studied in a d-p model using a variational Monte Carlo (VMC) method. This state possesses a local circular current and possibly causes the pseudogap phase in high-Tc cuprate superconductors. We introduce into the trial function a configuration dependent phase factor, which was recently shown to be indispensable to stabilize current-carrying states. We pay attention to the energy gain as a function of adjacent O-O hopping ▪ and Cu-site coulomb repulsion U. It is found the SF state becomes stable for hole doping rates of ▪ and ▪, owing to the reduction in kinetic energy, but seems fragile because the energy gain rapidly decays on doping holes for any ▪. The loop current tends to increase as ▪ increases.

Charge order in the pseudogap phase of cuprate superconductors

Charge ordering instabilities are studied in a multiorbital model of cuprate superconductors. A known, key feature of this model is that the large local Coulomb interaction in the Cu orbitals generates local moments with short range antiferromagnetic (AF) correlations. The strong simplifying ansatz that these moments are static and ordered allows us to explore a regime not generally accessible to weak-coupling approaches. The AF correlations lead to a pseudogap-like reconstruction of the Fermi surface. We find that the leading charge instability within this pseudogap-like state is to a phase with a spatially modulated transfer of charge between neighbouring oxygen px and py orbitals accompanied by weak modulations of the charge density on the Cu orbitals. As a prime result of the AF Fermi-surface reconstruction, the wavevectors of the charge modulations are oriented along the crystalline axes with a periodicity that agrees quantitatively with experiments. This suggests a resolution to a discrepancy between experiments, which find axial order, and previous theoretical calculations, which find modulation wavevectors along the Brillouin zone diagonal. The axial order is stabilized by hopping processes via the Cu4s orbital, which is commonly not included in model analyses of cuprate superconductors. The main implication of our results is that charge order emerges from the pseudogap state, and is not the primary source of the pseudogap.

Exact correlation functions in the cuprate pseudogap phase: Combined effects of charge order and pairing

There is a multiplicity of charge-ordered, pairing-based, or pair density wave theories of the cuprate pseudogap, albeit arising from different microscopic mechanisms. For mean-field schemes (of which there are many) we demonstrate here that they have precise implications for two-body physics in the same way that they are able to address the one-body physics of photoemission spectroscopy. This follows because the full vertex can be obtained exactly from the Ward-Takahashi identity. As an illustration, we present the spin response functions, finding that a recently proposed pair density wave (Amperean pairing) scheme is readily distinguishable from other related scenarios.

Duality between zeroes and poles in holographic systems with massless fermions and a dipole coupling

We discuss the zeroes and poles of the determinant of the retarded Green function ($\det G_R$) at zero frequency in a holographic system of charged massless fermions interacting via a dipole coupling. For large negative values of the dipole coupling constant $p$, $\det G_R$ possesses only poles pointing to a Fermi liquid phase. We show that a duality exists relating systems of opposite $p$. This maps poles of $\det G_R$ at large negative $p$ to zeroes of $\det G_R$ at large positive $p$, indicating that the latter corresponds to a Mott insulator phase. This duality suggests that the properties of a Mott insulator can be studied by mapping the system to a Fermi liquid. Finally, for small values of $p$, $\det G_R$ contains both poles and zeroes (pseudo-gap phase).

Thermodynamic properties of Rashba spin-orbit-coupled Fermi gas

We investigate the thermodynamic properties of a superfluid Fermi gas subject to Rashba spin-orbit coupling and effective Zeeman field. We adopt a $T$-matrix scheme that takes beyond-mean-field effects, which are important for strongly interacting systems, into account. We focus on the calculation of two important quantities: the superfluid transition temperature and the isothermal compressibility. Our calculation shows very distinct influences of the out-of-plane and the in-plane Zeeman fields on the Fermi gas. We also confirm that the in-plane Zeeman field induces a Fulde-Ferrell superfluid below the critical temperature and an exotic finite-momentum pseudogap phase above the critical temperature.

Pseudogap, charge order, and pairing density wave at the hot spots in cuprate superconductors

We address the timely issue of the presence of charge ordering at the hot-spots in the pseudo-gap phase of cuprate superconductors in the context of an emergent SU(2)-symmetry which relates the charge and pairing sectors. Performing the Hubbard-Stratonovich decoupling such that the free energy stays always real and physically meaningful we exhibit three solutions of the spin-fermion model at the hot spots. A careful examination of their stability and free energy shows that, at low temperature, the system tends towards a co-existence of charge density wave (CDW) and the composite order parameter made of diagonal quadrupolar density wave and pairing fluctuations of Ref. [Nat. Phys. $\bf{9}$, 1745 (2013)].The CDW is sensitive to the shape of the Fermi surface in contrast to the diagonal quadrupolar order, which is immune to it. SU(2) symmetry within the pseudo-gap phase also applies to the CDW state, which therefore admits a pairing density pave counterpart breaking time reversal symmetry.

Spin and charge order in Hg1201

We describe the Hg1201 cuprate in the pseudogap (PG) phase from a 3-band Fermi liquid point of view. The PG is modeled by a k-dependent gap parameter Δ(k), conforming with experiment. Susceptibilities are evaluated in the low-temperature metallic state without further renormalisation by interactions. The effect of strong correlations in the 2D metal is studied by coherent kinematic factors, originating from setting the Hubbard Ud→∞[1].The magnetic susceptibility shows a Y-shaped transverse spin response much like the recent experimental results [2] in the PG phase of Hg1201. The response is suppressed below the commensurate peak at ω=2Δ0. The strongly anisotropic gap Δ(k) extends the non-dispersive vertical part of the Y-shape, as seen in experiment. (Using an isotropic gap yields a very short vertical part, making the response more V-shaped.) At higher frequencies it displays isotropic upward branches from interband excitations.The SDW gap Δ(k), assumed to open at T⁎≈300K, reconstructs the Fermi surface, leaving only a pocket around the nodal points. The bands at TCDW≪T⁎nest to give a qCDW of around 0.28 r.l.u., as observed [3]. Such a gapped Fermi liquid scenario suffices to understand the charge and spin responses of underdoped Hg1201 in the experimentally observed hierarchy. The kinematic correlations caused by a large Ud change the Y-response towards the “hourglass” X-shape, thus replacing the peak at ω=2Δ0 and q=QAF with incommensurate downward branches. This result indicates that effects of strong-coupling correlations at the Fermi surface are more significant in the lanthanum than in the mercury cuprates.

Tilted loop currents in cuprate superconductors

The paper briefly surveys theoretical models for the polar Kerr effect (PKE) and time-reversal symmetry breaking in the pseudogap phase of cuprate superconductors. By elimination, the most promising candidate is the tilted loop-current model, obtained from the Simon–Varma model by tilting one triangular loop up and another one down toward the apical oxygens. The model is consistent with the PKE, spin-polarized neutron scattering, and optical anisotropy measurements. Spontaneous currents in this model flow between the in-plane and apical oxygens in such a manner that each oxygen belongs to one current loop. This loop-current pattern is similar to the spin order in the magnetoelectric antiferromagnet Cr2O3, where the PKE is observed experimentally. By analogy, it should be possible to train the PKE sign in the cuprates magnetoelectrically. Several experiments are proposed to confirm the loop-current order: the magnetic-field-induced polarity, the nonlinear anomalous Hall effect, and the second-harmonic generation.

On the Boson–Fermion resonant model on a lattice

We review briefly the properties of a mixture of mutually interacting bosons (bound electron pairs) and itinerant fermions on a lattice (the boson–fermion model). The calculations of the superconducting phase transition temperature () and the phase diagram are the main concern. The self-consistent -matrix method is applied to determine the superconducting critical temperature from a pseudogap phase. The method takes into account the pairing fluctuations effects. The -matrix results for are given for a three-dimensional (3-D) cubic lattice with tight-binding dispersion of electrons and standard bosons, and they are also compared with those of the BCS- mean-field approximation. Our results describe the BCS-Bose-Einstein condensation crossover in the boson–fermion mixture with resonant interaction. The energy scales involved in the pseudogap formation are also analysed.

Evidence for a common physical origin of the Landau and BEC theories of superfluidity.

There are two renowned theories of superfluidity in liquid (4)He, quite different and each with specific domains of application. In the first, the Landau theory, superflow follows from the existence of a well-defined collective mode supported by dense liquid (4)He, the phonon-roton mode. In the second, superflow is a manifestation of Bose-Einstein condensation (BEC) and phase coherence in the liquid. We present combined measurements of superfluidity, BEC and phonon-roton (P-R) modes in liquid (4)He confined in the porous medium MCM-41. The results integrate the two theories by showing that well-defined P-R modes exist where there is BEC. The two are common properties of a Bose condensed liquid and either can be used as a basis of a theory of superfluidity. In addition, the confinement and disorder suppresses the critical temperature for superfluidity, Tc, below that for BEC creating a localized BEC "phase" consisting of islands of BEC and P-R modes. This phase is much like the pseudogap phase in the cuprate superconductors.

Are preformed Cooper pairs the cause for the pseudogap in superconductors?

A weak-coupling scenario wherein bosonic preformed electron pairs emerge upon cooling from two-electron correlations can explain the pseudogap phase consisting of segments where a Bogoliubov-like energy-momentum relation gapped spectrum alternates with a normal ungapped one. Bose–Einstein condensation (BEC) of preformed pairs interacting with the background fermions leads to either - or -wave-like superconducting gaps, the result being sensitive to the magnitude of the total number density of pairs at which BEC occurs and becomes possible already for a moderately anisotropic -wave pairing of fermions repelling each other via isotropic coulombic forces. The present model is compatible with the coexistence of pseudogap and of superconductivity phenomena.