We speed up thermal simulations of quantum many-body systems in both one-(1D) and two-dimensional (2D) models in an exponential way by iteratively projecting the thermal density matrixρ = e −βĤ onto itself. We refer to this scheme of doubling β in each step of the imaginary time evolution as the exponential tensor renormalization group (XTRG). This approach is in stark contrast to conventional Trotter-Suzuki-type methods which evolveρ on a linear quasi-continuous grid in inverse temperature β ≡ 1/T . As an aside, the large steps in XTRG allow one to swiftly jump across finite-temperature phase transitions, i.e., without the need to resolve each singularly expensive phase transition point right away, e.g., when interested in low-energy behavior. A fine temperature resolution can be obtained, nevertheless, by using interleaved temperature grids. In general, XTRG can reach low temperatures exponentially fast, and thus not only saves computational time but also merits better accuracy due to significantly fewer truncation steps. For similar reasons, we also find that the series expansion thermal tensor network (SETTN) approach benefits in both efficiency and precision, from the logarithmic temperature scale setup. We work in an (effective) 1D setting exploiting matrix product operators (MPOs) which allows us to fully and uniquely implement non-Abelian and Abelian symmetries to greatly enhance numerical performance. We use our XTRG machinery to explore the thermal properties of Heisenberg models on 1D chains and 2D square and triangular lattices down to low temperatures approaching ground state properties. The entanglement properties, as well as the renormalization group flow of entanglement spectra in MPOs, are discussed, where logarithmic entropies (approximately ln β) are shown in both spin chains and square lattice models with gapless towers of states. We also reveal that XTRG can be employed to accurately simulate the Heisenberg XXZ model on the square lattice which undergoes a thermal phase transition. We determine its critical temperature based on thermal physical observables, as well as entanglement measures. Overall, we demonstrate that XTRG provides an elegant, versatile, and highly competitive approach to explore thermal properties, including finite temperature thermal phase transitions as well as the different ordering tendencies at various temperature scales for frustrated systems.

The self-assembly of anisotropic gold nanorods (GNRs) into ordered phases remains a challenge. Herein, we demonstrated the fabrication of symmetric circular- or semicircular-like self-assembled superlattices composed of multilayers of standing GNRs by fine-tuning the repulsive interactions among GNRs. The repulsive force is tailored from electrostatic interaction to steric force by replacing the surface coating of cetyltrimethylammonium bromide (CTAB) (ζ potential of 20-50 mV) with an OH-terminated hexa(ethylene glycol) alkanethiol (here termed as EG(6)OH, ζ potential of -10 mV). The assembly mechanism is discussed via theoretical analyses of the major interactions, and an effective balance between the repulsive steric and attractive depletion interactions is the main driving force for the self-assembly. The real-time observations of solution assembly (UV-vis-NIR absorption spectroscopy) supports the mechanism that we suggested. The superlattices obtained here not only enrich the categories of the self-assembled structures but more importantly deepen the insight of the self-assembly process and pave the way for various potential applications.

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Bottom-up self-assembly is able to constitute a variety of structures and has been thought to be a promising way for advanced nanofabrication. Droplet evaporation, as the simplest method, has been used in various self-assemblies. However, the assembled area is not large enough and the order is still not well controlled. Here we show a facile and controllable two-stage droplet evaporation method by adjusting the humidity and temperature of the evaporating droplet. Taking the highly monodispersed gold nanorods (GNRs) as an example, large-area, self-assembly monolayer arrays are reproducibly achieved. To understand the self-assembly mechanism, we adopted simplified models to analyze the interactions between the nanorods. The results show that a metastable state of secondary-energy-minimum exists, especially in the latter stage of the assembly process, leading to the ordered arrays. A large electrostatic barrier between the assembled arrays prevents the formation of the multilayer structures and thereby leads to the preferential monolayers. Moreover, we predict possibilities of different types of assemblies of the nanorods, and a schematic phase diagram is finally given. The results here may offer a way toward high-quality self-assembled nanoparticles superlattices for use in enhanced spectroscopy, sensors, or nanodevices.

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By carefully controlling the reacting conditions, including atmosphere, temperature, and time, we directly acquire the nanowires of γ-Fe 2 O 3 from the nanowires of R-Fe 2 O 3 in a reduced atmosphere. X-ray diffraction, X-ray photoelectron spectroscopy, Raman spectrum, and transmission electronic microscope analyses demonstrate that the nanowires are single-crystalline γ-Fe 2 O 3 . The nanowires have a diameter ranging from 50 to 90 nm and their typical lengths are in the range of 10∼20 µm. The optical property of the nanowires is observed by photoluminescence spectrum. The magnetic behavior of it is investigated by a magnetic property measurement system. The blocking temperature is found to be about 200 K. In addition, the mechanism of the transformation from R-Fe 2 O 3 nanowires to γ-Fe 2 O 3 nanowires is preliminarily studied by ab initio approach. It is found that the role of H 2 is to change the Fe-O bonds in R-Fe 2 O 3 nanowires, and then R-Fe 2 O 3 is transformed into γ-Fe 2 O 3 .

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