Local particle-hole pair excitations by SU(2) symmetry fluctuations

Understanding the pseudo-gap phase which opens in the under-doped regime of cuprate superconductors is one of the most enduring challenges of the physics of these compounds. A depletion in the electronic density of states is observed, which is gapping out part of the Fermi surface, leading to the formation of mysterious lines of massless excitations- the Fermi arcs. Here we give a new theoretical account of the physics of the pseudo-gap phase in terms of the emergence of local patches of particle-hole pairs generated by SU(2) symmetry fluctuations. The proliferation of these local patches accounts naturally for the robustness of the pseudo-gap phase to disturbances like disorder or magnetic field and is shown to gap out part of the Fermi surface, leading to the formation of the Fermi arcs. Most noticeably, we show that these patches induce a modulated charge distribution on the Oxygen atoms, in remarkable agreement with recent X-ray and STM observations.

The concept of symmetries governing the behaviour of physical states is maybe the most robust in theoretical physics. From the formation of nuclei to the Higgs-Boson it has been instrumental in the determination of every emerging state in high energy physics. It would be quite remarkable if a phenomenon as complex as high temperature superconductivity would be governed as well by an overall emergent symmetry. Suggestions about the existence of a pseudo-spin symmetry in the background of the physics of cuprates have been introduced since the early days of these compounds 1 and has been revived over the years in different contexts. In all cases, the main simple idea is that one can rotate the d-wave superconducting (SC) state towards another state of matter quasi-degenerate in energy, like anti-ferromagnetism (AF) 2, 3 , a nematic state 4 , or else alternating loop-currents of d-density wave 5 or π-flux phases 6,7 . The physics is then controlled solely by the powerful constraint of the emergent symmetry which produces a vast region of the phase diagram where the fluctuations between those two states are dominant. Here, we argue that an emerging SU(2) symmetry connecting the d-wave SC state with a d-wave charge order (CO) is the main ingredient of the physics of the under-doped (UD) region, and that the very specific fluctuations associated with this symmetry are responsible for the opening of a gap in the anti-nodal (AN) zone -i.e. (0, π) region-of the first Brillouin zone (BZ) leading to the formation of Fermi arcs in the nodal (N) zone i.e. the (π, π) region initially observed by a depletion in the electronic density of states 8 . A few theoretical proposals for the pseudogap (PG), have led to interesting investigations where the axial CDW is associated with axial pair density wave (PDW) pairing order, a finite momentum superconductivity [9][10][11] , or stabilized by short-range AF fluctuations 12,13 . A state where charge and spin degrees of freedom are separated (fractionalized Fermi liquid) has also been proposed to explain the PG phase 14 .
The SU(2) symmetry is realized explicitly in two microscopic models: the t-J model at half filling 1 (for AF wave vector) and also the spin-fermion "hotspot" model 15,16 with a linearized electron dispersion. In both models, AF interactions are at the origin of the emergent SU(2) symmetry-which is likely to be the generic case for UD cuprates. For the purpose of this study, the strong-coupling character of the first model, or the vicinity of an AF quantum critical point in the second one are not crucial contrary to the additional symmetry induced in both. Generically, the degeneracy, or quasi-degeneracy in energy of two states can be either accidental or controlled by a symmetry here a pseudo-spin SU(2) symmetry which rotates from a d-wave SC state to a d-wave CO state (The definition of the SU(2) symmetry is detailed in the SI). The SU(2) fluctuations, typically captured within the O(4) non-linear σ-model 3,[16][17][18] , involve phase fluctuations within each state but also between the two pseudo-spin states. The presence of the SU (2) Here, ψ represents spin σ fermions with bare propagator , where ξ k represents the dispersion with subtracted chemical potential and S 1 [ψ] accounts for the electron-SC interaction mediated through bosonic field . For simplicity we focus on the pairing part of the SU(2) propagator and its effect onto the formation of patches of charge modulations. The reciprocal effect of the charge sector will create a PDW, and will be treated in a more detailed version of this paper.
This bosonic field corresponds to SU(2) pairing fluctuations of a small momentum q, with the form typical of the O(4) non linear σ-model describing the thermal fluctuations between d-wave SC and CO states [15][16][17][18] : where π 0 , J 1 , a k 0, are non-universal parameters and v k the Fermi velocity. The influence of the curvature to the mass is modeled by a contribution to a 0 (Eqn. (3)) of the form ~ξ ξ + , where Q 0 is the diagonal wave vector relating two-hot spots on opposite Fermi surface accross the zone edge 16,19 . The SU(2) symmetry is realized , which favours the AN zone. Integrating out the bosonic field Δ k,q yields a new effective two body electron-electron interaction of the form:

Linear response
The form of the effective interaction Eqn. (4) leads to an attractive coupling between the electrons. Its effect on the linear response of the charge susceptibility is visualized in Fig. 1. We observe a drastic increase of the response at zero momentum, accompanied by a continuous series of maxima in a locus of wave vectors following the Fermi surface at 2k F in the anti-nodal region of the Brillouin Zone (BZ). The zero momentum peak is a signature that the system is on the brink of a nematic transition, whereas the continuum of peaks at momentum 2k F signals a new kind of instability of the particle-hole response.

Excitonic pairing mechanism
To get a simple insight, let consider the electron-hole wavefunction χ r,r′ : i i r r P k P r r k r r k P , , , , P the modulation wave vector at P = {2k F } which connects two opposite sides of the Fermi surface and χ  k P , assumed as a constant. Inserting the electron-hole wave function χ ′ r r , (equation (5)) into the two particles Schrödinger equation, we find r r r r r r r r elsewhere. Eqn. (6) can then easily be solved, leading to the bonding energy The formation of particle-hole pairs at multiple 2k F wave vectors is a logarithmic instability of the Fermi liquid in the presence of an attractive potential. In the standard BCS theory, the coupling between density and phase fluctuations is weak. In some specific cases, however, like the attractive Hubbard model, density and phase couple strongly and our model likewise predicts the emergence of s-wave particle-hole pair patches (see the SI for more details). Within the SU(2) scenario for Cuprates, the typical scale associated the pairing fluctuations is strong, of order of the formation of the SU(2) dome, and can naturally be associated with the PG scale T * .

Solution of the gap equation
Eqn. (4) can now be decoupled in the charge channel. Upon introducing the collective field , , n representing a generalized charge order at wave vectors k and ′ k , the self-consistent mean-field equation yields Eq. (9) is solved numerically to find the stability region for the particle-hole pair at wave vectors k and k′, and depicted in Fig. 2.
The (k, k′)-points for which χ k,k′ is non zero have the peculiarity that the outgoing wave vector k′ is related to the ongoing wave vector k through the relation k′ − k = 2p F , which leads to the formation of a local particle-hole pair mode-called Resonant Excitonic State (RES), analogous to a soliton. The fact that the preferred wave vectors 2p F are peaked on the Fermi surface comes from the denominator of Eqn. (9), which is minimal for those wave vectors. Note the non-linearity in the field χ in Eqn. (9), which is a crucial ingredient for stabilizing the local solution. The situation is similar to electrons interacting with a wide range of bosonic modes leading to the formation of local polarons 20  The momentum dependence of the order parameter χ , is fully determined by the equation (9). The form factor F k measures the phase space allocated to each P modulation, with ≤  F k k , as shown in Fig. 2b). The natural local structure of the particle-hole field χ ′ k k , is due to the summation on the multiple P-vectors, which forces the center of mass of the particle-hole pair to rest at x = 0 and makes it similar to a soliton solution of the nonlinear Eq. (9). This local object, or patch-, then has the tendency to proliferate, and while interacting with its surroundings will induce phase separation. From another viewpoint, the non-linear sigma-model, which describes non-linear interactions between the charge and superconducting sectors, produces such local modes in the form of topological defects. The relationship between the particle-hole patches and topological defects will be studied in a further work. Translation symmetry is broken, but in contrast to the standard charge ordering, it is broken locally, which does not imply global periodicity.  (2) pairing fluctuations. The charge susceptibility is maximal at p = 0 which is the signature of a nematic instability and is non zero along the 2p F vectors. Note that it exhibits local maxima along p x and p y axes that correspond to the 2p F vectors of the zone edge. Inset: we draw the Feynam diagram of the charge susceptibility and the mass a 0 in the positive quarter of the first BZ (the dashed line represents the Fermi surface). The real part of the charge susceptibility is determined after integration over internal Matsubara frequencies at zero temperature and the summation over internal momentum has been performed numerically in the first Brillouin zone with meshes of 200 × 200 at zero frequency. Regarding the bosonic propagator written in equation (3), the main contribution to charge susceptibility is raised for q ≈ 0.
Remarkably, each patch resembles very much a superconductor, but made of electron-hole pairs instead of Cooper pairs (electron-electron pairs). The typical size of the pair -also called "coherence length" in the SC analogy, is typically small with ξ πχ ∼ v /( )  (with χ AN the typical amplitude of the AN gap), which leads to the preferential formation of pair on nearest neighbor Cu-bonds. The second typical length corresponds to the size of the patches, which we call L p . It is controlled by the spread S p of the P wave vectors as shown in Fig. 2b). The typical patch size of the RES  π ∼ L S 2 / P P is of order of a few lattice sites. with d ~ 0.3 have been observed in this doping range through various experimental probes [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36] (with a 0 the elementary cell paramter of the square lattice). A priori there is no obvious relation between the incipient charge order and the PG phase of the Cuprates. However, a recent set of data obtained using the Scanning Tunnelling Microscope (STM) argue otherwise 37 . A careful study of the form factor associated to the charge signal has allowed to distinguish the s-wave part from the d-wave part of the charge modulation located on the Oxygen atoms, both from STM, NMR and X-ray experiments 32, 36-38 . The signal, shows a very striking behaviour as a function of the applied voltage. Its s-wave part (10% of the signal at T * ) shows a typical energy scale of the order of the superconducting gap. Its d-wave part (90%), however, is strongly peaked at the PG energy scale. This result links in a very unexpected way the physics of the PG with the modulations observed in the charge sector, as if they were coherent only below a certain energy scale, but their pure d-wave part persists up to the PG scale in a very incoherent manner, detectable only by local probes like STM.

Implications for an STM experiment
We present here a simple explanation for this remarkable observation. We evaluate the charge density on the oxygen O(2p) orbitals of the the CuO 2 lattice induced by the RES, which writes  , in the AN zone (π, 0), as a function of the ordering wave vector q c (k′ = k − q c ) (normalized to the maximum value). The amplitude of the electron-hole pair is maximum in the AN region selecting a set of degenerate solutions at 2p F wave vectors. It is suppressed in the nodal region (π, π). We represent the quasidegenerate 2p F wave vectors of the electron-hole pairs on opposed Fermi surfaces in the AN region, that give rise to the RES state. The vector Q 0 is the diagonal ordering vector (red vector). The momentum spread is S p . (b) Density plot of the particle-hole pair amplitude χ | | ′ k k , for k′ = k + 2p F (k) (normalized to the maximum value). The size of the gapped zone depends on the magnitude of the interaction π 0 as demonstrated in the Figure S2. Here, the figure are presented for π = 1 0 , the other parameters are exposed in the SI. (c) Feynman diagram representing the χ ′ k k , mean field equation (9). sity on O orbitals is presented in Fig. 3(a,b) where one can notice the checkerboard structure with preponderant axial wave vectors Q x and Q y . The signal is found to be 98% d-wave, in good agreement with the experimental data. The d-wave form factor originates the diagonal part of the charge order modulation while the s′-wave form factor accounts for non-diagonal modulations. Note that a pure diagonal charge order would produce a pure d-wave form factor 16 . We then Fourier transform the d-wave component and report the result in Fig. 4. Rather large spots centered on = | | = .
π Q Q 0 237 a x y 2 0 are found, which agree in a remarkable way with the observation of ref. 37 that the pure d-wave incoherent part of the axial charge modulation extends up to the PG energy. This internal modulation of each patch comes from the summation over the ordering vectors P in equation (10). As shown in Fig. 2(a), the modulation spreading is on the y-direction for the (0, π)region and respectively in the  Fig. 3(a,b) have been determined by evaluating the formula (10). The summation over the momentum k has been performed in the FBZ while the summation over P vectors has been done over two-hundred ordering vectors P. We choose a distribution of vectors P around the Fermi surface. The extension in momentum of these distribution is the one presented in Fig. 2(a). The calculation has been done considering the electronic dispersion for Bi2212 compounds used in refs 33, 37 at hole doping p = 0.08. The coherence length of the particle hole pair is of order of the length between two Copper site ξ ≈ a 0 while the size of the patch is larger, L P ≈ 10a 0 .  Fig. 3(a) in the first Brillouin Zone. The charge order is organized preferentially along Q x and Q y that are parallel to k x and k y respectively.
x-direction in the (π, 0)-region. This leads to the typical checkerboard form, with two average modulation wave vectors around Q x = ±2π/a 0 (d, 0) and Q y = ±2π/a 0 (0, d) with d ~ 0.3 (see Fig. 2(b), which are expected to be visible through local probes like STM. The key feature in our model is that diagonal wave vectors are much disfavoured compared to parallel ones, because they have very small phase space in the extension zone F k where the RES order parameter χ RES is non zero (see Fig. 2). Hence very remarkably, our proposal for the particle-hole pair state has a preferential structure in k-space dominated by incoherent wave vectors centered on Q x and Q y rather than on the diagonal. We are aware that the STM experiments show patches with strong x/y anisotropy accompanied with a C4 symmetry breaking. We have not focused here on this aspect of the experiment, preferring to explain how to use the underlying SU(2) symmetry to account for the emergence of charge modulations at the PG energy scale. The theory is easily generalized to the uniaxial case, with a set of SU(2) operators (defined in the SI) for each wave vector Q x or Q y .

The global phase diagram of high-Tc superconductors
We propose a real space picture which describes how the proliferation of patches is responsible for the opening of the PG (see Fig. 5). First, the proliferation of local objects like particle-hole pair patches leads to a state very robust to perturbations by strong disorder 8 or magnetic field 30,36,39,40 , which puts strong restrictions on theoretical proposals. Then, our theory belongs to the class of "one-gap" scenarios, for which the coherent SC state is destroyed at T c by fluctuations [41][42][43] , but the class of fluctuations at play here have SU(2)-character, as opposed to only the U(1)-type preformed pair scenario. Proliferation of particle-hole pair patches destabilizes the AN part of the Fermi surface, leading to the formation of Fermi arcs whose lengths is increasing with T up to T * , as reported in Angle Resolved Photo Emission (ARPES) (see e. g. refs [43][44][45]. The temperature at which the patches start to proliferate is proportional to the difference between the binding energies of particle-hole pairs and Cooper pairs T prolif ~ Max[E p − E C , 0] (see Fig. 5). SU(2) pairing fluctuations vanish around optimal doping, leading to a maximal T prolif ~ T c , in contrast to heavily underdoped compounds (x < 0.12) where, in the vicinity of the Mott localization, particle-hole pairing beats Cooper pairing (E P − E C < 0), and the proliferation of patches occurs at almost zero temperature. In this regime the whole anti-nodal region of the BZ is gapped out by the particle-hole pair patches, leading to a fractionalization of the Fermi surface into two gaps.
A typical evolution with temperature is as follows. Below T c (0 < T < T c ) as depicted in Fig. 5(a), phase fluctuations destroy the coherent SC state. These fluctuations manifest themselves as local defects of the SC density, that are similar to vortex-antivortex pairs in classical superconductors, but with the difference that in our case, the normal phase is made of local particle-hole pair patches which can be seen inside the vortices. An important point is the one of global phase coherence. Below T c the global phase of modulations inside each particle-hole pair patch can get locked with the SC one, so that the intrinsic checkerboard modulation of the patches acquires a global phase coherence. This phenomenon enhances the coherent charge density wave (CDW)-like signal experimentally observed at zero magnetic field, and has been recently reported in another STM study 46 . (c) For T 0 < T < T * , the patches modulations are incoherent and electron scattering through the "particle-hole pair soup" leads to dissipation which accounts for the increase of the Fermi arcs with T. The temperature of proliferation of the patches T prolif is presented in yellow.
Scientific RepoRts | 7: 3477 | DOI:10.1038/s41598-017-01538-1 At T = T c , the gap around the node disappears, causing the loss of SC coherence, and at an intermediate temperature T 0 , such that T c < T 0 < T * the standard SC fluctuations are lost, as observed for example, by Josephson effect 47 , Nernst effect 48 or by study of the resistivity 49 . The transition towards the SC state can thus be described by a phase correlation length λ  c , which diverges at T c and becomes very small at T * . λ −  c 1 can be understood as the typical scale of all the phase fluctuations in the system (in the present SU(2) scenario there are three types of phase fluctuations: SC, CO, and the angle between these two). For T c < T < T 0 (Fig. 5(b), the global phase coherence of the Cooper pairs and modulated particle-hole pair patches is lost. The SC superfluid density n s vanishes at T 0 . In the regime, T 0 < T < T * shown in Fig. 5(c), the RES manifests itself as an incoherent set of particle-hole pair patches. Electronic scattering through the RES induces a finite lifetime, which accounts for the size of the Fermi arcs increasing with T 43 . Above T * , the RES disappears.

Concluding remarks
We propose a new mechanism for gapping out the AN region of the Fermi surface, leading to the formation of Fermi arcs below T * . In our model, an SU(2) symmetry governs the physics of the UD region of the phase diagram, and SU(2) pairing fluctuations allow for the emergence of new local excitations in the form of particle-hole pair patches which possess intrinsic checkerboard modulations. The proliferation of these patches opens a gap in the anti-nodal region of the Brillouin zone, leading to the formation of Fermi arcs. Very importantly, it is accompanied (see Fig. 1) with the formation of a q = 0 order in the nematic sector, intensely discussed at the moment 50 . The polarizable structure of the patches can also lead to the formation of intra unit cell orders 51,52 . One striking feature of our theory, is that it reconciles the one-gap vs. two gap scenarios: fluctuations destroy the SC phase at T c , creating Fermi arcs, as in the one-gap scenario 7,43 , but on the other hand the AN and nodal regions behave very differently upon raising the temperature (with particle-hole pair patches forming in the AN region), which is reminiscent of two-gaps. Note that typical two gaps scenarios produce Fermi pockets instead of Fermi arcs in the nodal region 53,54 . The SU(2) symmetry is identically verified in a very simplified model where the Fermi surface is reduced to eight points or "hot spots" and where the dispersion of the electrons has been linearized 15,16 , and the electrons interact through long range anti-ferromagnetic (AF) modes within the spin-fermion model 55 . The SU(2) paradigm, though, is more robust than AF quantum critical point -or a zero temperature phase transition with long range fluctuations of the AF modes, in the sense that the symmetry can be in the background of the under-doped region even though the cuprates show only short range AF modes. Recent pump-probe experiments exhibiting phase fluctuating SC/CDW composite order, give support to this idea 56 . The importance of anti-ferromagnetism though, should not be underestimated, as it supports the SU(2) pairing fluctuations predominantly in the anti-nodal region of the BZ 57 and originates the d-wave nature of both SC and charge order states. The recent findings that the modulation q-space are intrinsically related to the formation of the PG 37,45 are one of the most unexpected developments in the history of the physics in cuprates. The new particle-hole pair state -RES-proposed here possesses intrinsically this feature, and this can only let us wonder whether such a particle-hole pair state could be the solution to the mystery of the PG. The absence of electron pockets in the AN zone of the first BZ in the overdoped part of the phase diagram could provide a check of our scenario and could be performed by ARPES. Moreover, one possible specific signature of such new state could be the presence of photoluminescence signal.