The record superconducting transition temperature (T(c)) for the iron-based high-temperature superconductors (Fe-HTS) has long been 56 K. Recently, in single-layer FeSe films grown on SrTiO3 substrates, indications of a new record of 65 K have been reported. Using in situ photoemission measurements, we substantiate the presence of spin density waves (SDWs) in FeSe films--a key ingredient of Fe-HTS that was missed in FeSe before--and we find that this weakens with increased thickness or reduced strain. We demonstrate that the superconductivity occurs when the electrons transferred from the oxygen-vacant substrate suppress the otherwise pronounced SDWs in single-layer FeSe. Beyond providing a comprehensive understanding of FeSe films and directions to further enhance its T(c), we map out the phase diagram of FeSe as a function of lattice constant, which contains all the essential physics of Fe-HTS. With the simplest structure, cleanest composition and single tuning parameter, monolayer FeSe is an ideal system for testing theories of Fe-HTS.

We propose a novel coarse graining tensor renormalization group method based on the higherorder singular value decomposition. This method provides an accurate but low computational cost technique for studying both classical and quantum lattice models in two-or three-dimensions. We have demonstrated this method using the Ising model on the square and cubic lattices. By keeping up to 16 bond basis states, we obtain by far the most accurate numerical renormalization group results for the 3D Ising model. We have also applied the method to study the ground state as well as finite temperature properties for the two-dimensional quantum transverse Ising model and obtain the results which are consistent with published data.

We have proposed a novel numerical method to calculate accurately the physical quantities of the ground state with the tensor-network wave function in two dimensions. We determine the tensor network wavefunction by a projection approach which applies iteratively the Trotter-Suzuki decomposition of the projection operator and the singular value decomposition of matrix. The norm of the wavefunction and the expectation value of a physical observable are evaluated by a coarse grain renormalization group approach. Our method allows a tensor-network wavefunction with a high bond degree of freedom (such as D = 8) to be handled accurately and efficiently in the thermodynamic limit. For the Heisenberg model on a honeycomb lattice, our results for the ground state energy and the staggered magnetization agree well with those obtained by the quantum Monte Carlo and other approaches.The application of the density matrix renormalization group (DMRG) proposed by White[1] has achieved great success in one dimension in both zero and finite temperatures [2,3,4]. However, in two dimensions, the application of the DMRG in both the real and momentum space [5,6] has been limited only to small lattices. The error resulting from the DMRG truncation increases extremely fast with increasing size of lattice. To resolve this problem, the tensor-network state, which is an extension of the matrix product in one dimension [7], was proposed [8]. In a tensor network state, a local tensor is directly entangled with the other local tensors on all the directions of the lattice. This leads to two problems in the treatment of the tensor network state. First, it is difficult to determine accurately all the elements of local tensors by any variational approach since the total degree of freedom of a local tensor increases exponentially with the dimension of the tensor. Second, it is even more difficult to calculate the expectation value of any physical observable even if we know the expression of the tensor network wave function, since the number of summations over the basis configurations increases exponentially with lattice size.In this paper, we propose a novel method to handle the tensor-network wave function in two dimensions. We will show that the tensor network wavefunction |Ψ of the ground state can be accurately determined by applying an iterative projection approach. This approach is similar to the time-evolving block decimation method that was used to determine the matrix product wave function of the ground state in one dimension[9, 10] Then we will generalize the classic coarse grain renormalization group approach proposed by Levin and Nave[11] to the quantum system, and use it to calculate the norm of the wave function and the expectation value of any physical observable. This provides an accurate and efficient tool to determine the expectation values of physical quantitise from the tensor-network wavefunction of the ground state.Below, we will take the S=1/2 Heisenberg model on a honeycomb latticeas an example to show how the method works...