Kuramoto synchronization of quantum tunneling polarons for describing the dynamic structure in cuprate superconductors

A major open topic in cuprates is the interplay between the lattice and electronic dynamics and the importance of their coupling to the mechanism of high-temperature superconductivity (HTSC). As evidenced by extended x-ray absorption fine structure (EXAFS) experiments, anharmonic structural effects are correlated with the charge dynamics and the transition to a superconducting phase in different HTSC compounds. Here we describe how structural anharmonic effects can be coupled to electronic and lattice dynamics in cuprate systems by performing the exact diagonalization of a prototype anharmonic many-body Hamiltonian on a relevant 6-atom cluster and show that the EXAFS results can be understood as a Kuramoto synchronization transition between coupled internal quantum tunneling of polarons associated with the two-site distribution of the copper–apical oxygen ($\mathrm{Cu}\text{−}{\mathrm{O}}_{\mathrm{ap}}$) pair in the dynamic structure. The transition is driven by the anharmonicity of the lattice vibrations and promotes the pumping of charge, initially stored at the apical oxygen reservoirs, into the copper-oxide plane. Simultaneously, a finite projection of the internal quantum tunneling polaron extends to the copper–planar oxygen ($\mathrm{Cu}\text{−}{\mathrm{O}}_{\mathrm{pl}}$) pair. All these findings allow an interpretation based on an effective quantum-mechanical triple-well potential associated with the oxygen sites of the 6-atom cluster, which accurately represents the phase synchronization of apical oxygens and lattice-assisted charge transfer to the ${\mathrm{CuO}}_2$ plane.

Figure 1

Relevant structures. (a) The ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7/{\mathrm{YSr}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7$ crystal structure shows the features of multilayered cuprates with higher $T_c$'s: two conducting ${\mathrm{CuO}}_2$ planes [blue Cu(2) and magenta ${\mathrm{O}}_{\mathrm{pl}}$ atoms] bridged by the intervening Y (gray), the charge reservoir consisting in this class of materials of Cu-O chains [turquoise Cu(1) and orange O], and the dielectric layer composed of the ${\mathrm{O}}_{\mathrm{ap}}$ (red) and Sr (green). (b) The original three-atom IQTP derived from the two-site $\mathrm{Cu}\left(1\right)\text{−}{\mathrm{O}}_{\mathrm{ap}}$ distribution found in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7$ showing its oscillations between its two configurations denoted by the location of the extra hole ($+\delta$) and expanded Cu-O distance [17]. (c) The atoms circled in black in (a) form the six-atom cluster used here. Its excess charge and displacement tunnel between the two $\mathrm{Cu}\left(2\right)\text{−}{\mathrm{O}}_{\mathrm{ap}}$ pairs through the ${\mathrm{O}}_{\mathrm{pl}}$ charge transfer bridge. (d) For the three-atom cluster the potential energy corresponds to a double-well structure [21]. (e) For the 6-atom cluster, the potential energy corresponds to a triple-well structure.

Figure 2

(a) The ground state energy of the 6-atom cluster Hamiltonian as a function of reduced anharmonicity, $K/K_c$, for the nonadiabatic, intermediate $\lambda$ coupling regime where lateral polarons are already formed: ground state (black), 10th excited state (green), 20th excited state (red), 40th excited state (blue). (b) The electronic occupations for left/right ${\mathrm{O}}_{\mathrm{ap}}$ (black/blue squares/crosses) as well as for the central ${\mathrm{O}}_{\mathrm{pl}}$ (red circles), as a function of reduced anharmonicity, $K/K_c$, of the ground state. (c) The phonon occupations for left/right sites (black/blue square/cross) as well as for the central oxygen site (red circles), as a function of reduced $K/K_c$ anharmonicity, of the ground state. Insets: As we increase anharmonicity, the central electronic (b) and phonon (c) occupations surpass the lateral ones.

Figure 3

(a) The solution to Kuramoto's order parameter Eq. (7) as a function of the reduced coupling in the ground state, showing a first-order synchronization phase transition. (b) Displacement $\langle u\rangle =\sqrt{\hslash /2m\omega }(b+b^{†})$ of the ${\mathrm{O}}_{\mathrm{ap}}$ equilibrium position for both the left and right positions in the cluster. (c) The polaronic tunneling frequency.

Figure 4

(a) For $K<K_c$, the potential (black) depth, the symmetric (blue) and antisymmetric (orange) components of the lateral ground state wave function, and a central excited state (green). (b) The absence of tunneling of the polarons in this scenario is demonstrated by its fixed position over time. (c) Ground state wave function coefficients for the unsynchronized, localized polaron phase. (d) for $K>K_c$, the triple-well potential (black), the symmetric (blue) and antisymmetric (orange) solutions of the wave function, and the excited state (green). (e) Due to the ASAP, the IQTP is now observed to tunnel from left to right through the center as time evolves. (f) Ground state wave function coefficients for the synchronized IQTP phase. The states in (c) and (f) are represented in the electronic occupation representation for the three holes in the five charge transfer sites.