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Diagrammatic summation is a common bottleneck in modern applications of projected entangled-pair states, especially in computing low-energy excitations of a two-dimensional quantum many-body system. To solve this problem, here we extend the generating-function approach for tensor-network diagrammatic summation, a scheme previously proposed in the context of matrix product states. Taking the form of a one-particle excitation, we show that the excited state can be computed efficiently in the generating-function formalism, which can further be used in evaluating the dynamical structure factor of the system. Our benchmark results for the spin-1/2 transverse-field Ising model and Heisenberg model on the square lattice provide a desirable accuracy, showing good agreement with known results. We then study the spin-1/2J_{1}-J_{2} model on the same lattice and investigate the dynamical properties of the putative gapless spin liquid phase. We conclude with a discussion on generalizations to multiparticle excitations.

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 light on two puzzling results that have emerged from recent scanning tunneling microscopy 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 eight 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. Due to the small size of Cooper pairs, the vortex core has a radius of about three unit cells, which results in a strong spatial dependence on pairing fields. This may be an important mechanism for the formation of PDW states inside the vortex core.

We propose a simple and generic construction of the variational tensor network operators to study the quantum spin systems by the synergy of ideas from the imaginary-time evolution and the variational optimization of trial wave functions. By applying these operators to simple initial states, accurate variational ground-state wave functions with extremely few parameters can be obtained. Furthermore, the framework can be applied to spontaneously study symmetry-breaking, symmetry-protected topological, and intrinsic topologically ordered phases, and we show that symmetries of the local tensors associated with these phases can emerge directly after the optimization without any gauge fixing. This provides a universal way to identify quantum phase transitions without prior knowledge of the system.

The Hofstadter model describes noninteracting fermions on a lattice in the presence of an external magnetic field. Motivated by the plethora of solid-state phases emerging from electron interactions, we consider an interacting version of the Hofstadter model, including a Hubbard repulsion U . We investigate this model in the large- U limit corresponding to a t - J Hamiltonian with an external (orbital) magnetic field. By using renormalized mean-field theory supplemented by exact diagonalization calculations of small clusters, we find evidence for competing symmetry-breaking phases, exhibiting (possibly coexisting) charge, bond, and superconducting orders. Topological properties of the states are also investigated, and some of our results are compared to related experiments involving ultracold atoms loaded on optical lattices in the presence of a synthetic gauge field. [ABSTRACT FROM AUTHOR]