Flux Phase in Bilayer $t-J$ Model: Time-Reversal Symmetry Breaking Surface State without Spontaneous Magnetic Field (1502.02904v3)

Abstract: We study surface states of high-$T_C$ cuprate superconductor YBCO using the bilayer $t-J$ model. Calculations based on the Bogoliubov de Gennes method show that a flux phase that breaks time-reversal symmetry (${\cal T}$) may arise near a (110) surface where the $d_{x^2-y^2}$-wave superconductivity is strongly suppressed. It is found that the flux phase in which spontaneous magnetic fields in two layers have opposite directions may be stabilized in a wide region of doping rate, and split peaks in the local density of states appear. Near the surface, spontaneous magnetic field may not be observed experimentally, because the contributions from two layers essentially cancel out. This may explain the absence of local magnetic filed near the (110) surface of YBCO, for which the sign of ${\cal T}$ violation has been detected.

• The paper demonstrates that a flux phase induced at the (110) surface of YBCO breaks time-reversal symmetry without producing a detectable spontaneous magnetic field.
• It utilizes a bilayer t-J model and the Bogoliubov de Gennes method to analyze the spatial variations of superconducting and bond order parameters across different doping levels.
• The study explains how opposing fluxes in the two layers cancel each other, reconciling experimental observations of missing magnetic fields on the YBCO surface.

Flux Phase in Bilayer $t-J$ Model: Time-Reversal Symmetry Breaking Surface State without Spontaneous Magnetic Field

Introduction

The investigation presented encompasses the surface states of the high-$T_C$ cuprate superconductor YBCO via a bilayer $t-J$ model approach. Central to this paper is the emergence of a flux phase at a (110) surface, which breaks time-reversal symmetry (${ \cal T }$) without generating a spontaneous magnetic field observable at the surface. The Bogoliubov de Gennes method underpins this analysis, demonstrating the conditions under which contrasting spontaneous magnetic fields in two layers offset one another.

Model and Methodology

The model is constructed on a square lattice bilayer system, represented by the effective Hamiltonian $H = H_1 + H_2 + H_\perp$, where $H_i$ describes the intra-layer interactions and $H_\perp$ covers inter-layer coupling. The Hamiltonian incorporates electron operators obeying the no-double-occupancy condition within the slave-boson framework and accounts for transfer integrals and superexchange interactions.

In this context, the Bogoliubov de Gennes equations are deployed for the decoupling of mean-field Hamiltonian components, allowing numerical resolution of the eigenstates and eigenfunctions across doping rates. The precise configuration of OPs—including the SC, interlayer, and bond OPs—is derived iteratively until convergence, thereby elucidating the spatial variation of these parameters near the (110) surface.

Surface State Results

Numerical simulations reveal significant suppression of the $d$-wave SCOP in the vicinity of the (110) YBCO surface, a condition facilitating the local manifestation of a ${ \cal T }$-breaking flux phase. Remarkably, whereas traditional manifestations could predict a spontaneous local magnetic field, this paper posits that the magnetic fields of opposing layers cancel out, resulting in null overall field detection. This reconciles previous experimental conundrums wherein the violation of ${ \cal T }$ symmetry had been detected absent accompanying magnetic fields.

The bilayer model predicts two flux phase types: type A and type B, distinguished by the alignment of flux lines in layered oppositions or correspondences. Numerical outcomes reveal that the type B flux phase predominates at low doping, persisting more robustly against modifications beyond values traditionally sustained in uniform systems, reaching up to $\delta \sim 0.3$.

Local Density of States Analysis

The local density of states (LDOS) elucidates a split-peak structure consonant with experimental measures. This split in zero-energy state peaks is ascribed primarily to the surface flux-phase order rather than bulk SC contributions, affirming the unique role of the flux phase as observed through LDOS behaviors in both surface and bulk scenarios. This further fortifies the theoretical construct that surface properties in cuprates are significantly modulated by unconventional ordering absent in the bulk.

Conclusion

This paper uncovers the nuance and complexity of the bilayer $t-J$ model’s applicability to high-$T_C$ superconductors, particularly in explaining the experimentally observed absence of local magnetic fields at the (110) surfaces of YBCO despite ${ \cal T }$ violation. It proposes a mechanism by which the degeneracy of opposite spontaneous fields conceals their presence, potentially stimulating reevaluation of assumptions in cuprate surface physics. Future explorations may venture into more refined doping level impacts and interlayer coupling variations, offering a broader insight into the interplay of surface and bulk properties in complex oxide superconductors.