Intertwined Orders and Electronic Structure in Superconducting Vortex Halos

We present a comprehensive study of vortex structures in $d$-wave superconductors from large-scale renormalized mean-field theory of the square-lattice $t$-$t'$-$J$ model, which has been shown to provide a quantitative modeling for high-$T_c$ cuprate superconductors. With an efficient implementation of the kernel polynomial method for solving electronic structures, self-consistent calculations involving up to $10^5$ variational parameters are performed to investigate the vortex solutions on lattices of up to $10^4$ sites. By taking into account the strong correlation of the model, our calculations shed new lights on two puzzling results that have emerged from recent scanning tunneling microscopy (STM) experiments. The first concerns the issue of the zero-biased-conductance peak (ZBCP) at the vortex core for a uniform $d$-wave superconducting state. Despite its theoretical prediction, the ZBCP was not observed in most doping range of cuprates except in heavily over-doped samples at low magnetic field. The second issue is the nature of the checkerboard charge density waves (CDWs) with a period of about 8 unit cells in the vortex halo at optimal doping. Although it has been suggested that such bipartite structure arises from low-energy quasiparticle interference, another intriguing scenario posits that the checkerboard CDWs originate from an underlying bidirectional pair-density wave (PDW) ordering with the same period. We present a coherent interpretation of these experimental results based on systematic studies of the doping and magnetic field effects on vortex solutions with and without a checkerboard structure. The mechanism of the emergent intertwined orders within the vortex halo is also discussed.Comment: 19 pages, 7 figure

Generating Function for Tensor Network Diagrammatic Summation

The understanding of complex quantum many-body systems has been vastly boosted by tensor network (TN) methods. Among others, excitation spectrum and long-range interacting systems can be studied using TNs, where one however confronts the intricate summation over an extensive number of tensor diagrams. Here, we introduce a set of generating functions, which encode the diagrammatic summations as leading order series expansion coefficients. Combined with automatic differentiation, the generating function allows us to solve the problem of TN diagrammatic summation. We illustrate this scheme by computing variational excited states and dynamical structure factor of a quantum spin chain, and further investigating entanglement properties of excited states. Extensions to infinite size systems and higher dimension are outlined.Comment: v1: 6 pages, 2 figures. v2: published versio

Field-induced easy-axis softening of quantum ferromagnet with cubic anisotropy

In this work, we study the two-dimensional $S=2$ model that can lead to the well-known cubic single-ion anisotropy along with a ferromagnetic Heisenberg interaction under a perpendicular magnetic field. When the Heisenberg term ($J$) is ignored, this model can be exactly solved with a reshuffled local Hilbert space. A three-fold (two-fold) eigenstate degeneracy is found at $h=0$ ($h=2K$), where $h$ stands for the field strength and $K$ is the strength of cubic anisotropy. Around the second degeneracy point there is a huge gap that separates two low-energy states from the rest three. As we turn on the $J$ term, its leading-order contribution boils down to an effective hard-core bosonic model as $K,h\gg J$. By using the two-dimensional tensor network ansatz we reveal that indeed various ground-state ansatz with different magnetic easy axes possess competing energies. Our finding demonstrates that the easy axes can be "softened" as we turn on the magnetic field for such quantum model with cubic anisotropy, which goes beyond the mean-field analysis. Such easy-axis softening might provide a new measure to control the magnetic orientation for spinful semiconductors.Comment: 10 pages, 5 figures, 1 Tabl

Field-induced Bose-Einstein condensation and supersolid in the two-dimensional Kondo necklace

The application of an external magnetic field of sufficient strength to a spin system composed of a localized singlet can overcome the energy gap and trigger bosonic condensation and so provide an alternative method to realize exotic phases of matter in real materials. Previous research has indicated that a spin Hamiltonian with on-site Kondo coupling may be the effective many-body Hamiltonian for $\text{Ba}_2\text{NiO}_2\text{(AgSe)}_2$ (BNOAS) and here we study such a Hamiltonian using a tensor network ansatz in two dimensions. Our results unveil a phase diagram which indicates the underlying phases of BNOAS. We propose, in response to the possible doping-induced superconductivity of BNOAS, a fermionic model for further investigation. We hope that our discovery can bring up further interest in both theoretical and experimental researches for related nickelate compounds.Comment: 11 pages, 6 figures, 2 table