Derivative Formula and Applications for Degenerate Diffusion Semigroups

By using the Malliavin calculus and solving a control problem, Bismut type derivative formulae are established for a class of degenerate diffusion semigroups with non-linear drifts. As applications, explicit gradient estimates and Harnack inequalities are derived.Comment: 18 page

Asymptotic correlation functions and FFLO signature for the one-dimensional attractive Hubbard model

We study the long-distance asymptotic behavior of various correlation functions for the one-dimensional (1D) attractive Hubbard model in a partially polarized phase through the Bethe ansatz and conformal field theory approaches. We particularly find the oscillating behavior of these correlation functions with spatial power-law decay, of which the pair (spin) correlation function oscillates with a frequency $\Delta k_F$ ($2\Delta k_F$). Here $\Delta k_F=\pi(n_\uparrow-n_\downarrow)$ is the mismatch in the Fermi surfaces of spin-up and spin-down particles. Consequently, the pair correlation function in momentum space has peaks at the mismatch $k=\Delta k_F$, which has been observed in recent numerical work on this model. These singular peaks in momentum space together with the spatial oscillation suggest an analog of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state in the 1D Hubbard model. The parameter $\beta$ representing the lattice effect becomes prominent in critical exponents which determine the power-law decay of all correlation functions. We point out that the backscattering of unpaired fermions and bound pairs within their own Fermi points gives a microscopic origin of the FFLO pairing in 1D.Comment: 26 pages, 4 figures, published version, a series of study on the 1D attractive Hubbard model, few typos were corrected, references were added, also see arXiv:1708.07784 and arXiv:1708.0777

FFLO correlation and free fluids in the one-dimensional attractive Hubbard model

In this Rapid Communication we show that low energy macroscopic properties of the one-dimensional (1D) attractive Hubbard model exhibit two fluids of bound pairs and of unpaired fermions. Using the thermodynamic Bethe ansatz equations of the model, we first determine the low temperature phase diagram and analytically calculate the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing correlation function for the partially-polarized phase. We then show that for such a FFLO-like state in the low density regime the effective chemical potentials of bound pairs and unpaired fermions behave like two free fluids. Consequently, the susceptibility, compressibility and specific heat obey simple additivity rules, indicating the `free' particle nature of interacting fermions on a 1D lattice. In contrast to the continuum Fermi gases, the correlation critical exponents and thermodynamics of the attractive Hubbard model essentially depend on two lattice interacting parameters. Finally, we study scaling functions, the Wilson ratio and susceptibility which provide universal macroscopic properties/dimensionless constants of interacting fermions at low energy.Comment: In this Letter we analytically study FFLO pairing correlation and the universal nature of the FFLO-like state. More detailed studies of this model will be presented in arXiv:1710.08742 and arXiv:1708.0778

Well-Supported vs. Approximate Nash Equilibria: Query Complexity of Large Games

In this paper we present a generic reduction from the problem of finding an epsilon-well-supported Nash equilibrium (WSNE) to that of finding an Theta(epsilon)-approximate Nash equilibrium (ANE), in large games with n players and a bounded number of strategies for each player. Our reduction complements the existing literature on relations between WSNE and ANE, and can be applied to extend hardness results on WSNE to similar results on ANE. This allows one to focus on WSNE first, which is in general easier to analyze and control in hardness constructions. As an application we prove a 2^{Omega(n/log n)} lower bound on the randomized query complexity of finding an epsilon-ANE in binary-action n-player games, for some constant epsilon>0. This answers an open problem posed by Hart and Nisan and Babichenko, and is very close to the trivial upper bound of 2^n. Previously for WSNE, Babichenko showed a 2^{Omega(n)} lower bound on the randomized query complexity of finding an epsilon-WSNE for some constant epsilon>0. Our result follows directly from combining Babichenko\u27s result and our new reduction from WSNE to ANE

Boundary effect and dressed states of a giant atom in a topological waveguide

The interaction between the quantum emitter and topological photonic system makes the photon behave in exotic ways. We here study the properties of a giant atom coupled to two sites of a one-dimensional topological waveguide, which is described by the Su-Schrieffer-Heeger (SSH) chain. We find that the giant atom can act as an effective boundary and induce the chiral zero modes, which are similar to those in the SSH model with open boundary, for the waveguide under the periodical boundary. Except for the boundary effect, we also find that the giant atom can lift energy degeneracy inside the energy bands of the SSH chain and adjust spatial symmetry of the photon distributions for the states of the dressed giant atom and waveguide. That is, the giant atom can be used to change the properties of the topological environment. Our work may stimulate more studies on the interaction between matter and topological environment.Comment: 7 Pages, 4 Figure

Appin Colliery explosion reassessed

The Judicial Inquiry into the Appin Colliery explosion on 23rd July 1979 made a specific finding that the fan starter box was the location of the initial ignition of gas. The subsequent Coronial Inquiry recognized the possibility of the fan starter box being the source but also recognized the possibility that the Deputy’s flame safety lamp could have been the trigger. Neither Inquiry detailed the mistakes that must have been made for either of those options to be the cause. It is contended that the flow of factual information at such Inquiries can be influenced by the high-profile legal representation used by the interested parties. It is important to ask whether an Inquiry before an independent technical expert would give the industry a better explanation of the factors that led to and caused the event

Coexistence of multi-photon processes and longitudinal couplings in superconducting flux qubits

In contrast to natural atoms, the potential energies for superconducting flux qubit (SFQ) circuits can be artificially controlled. When the inversion symmetry of the potential energy is broken, we find that the multi-photon processes can coexist in the multi-level SFQ circuits. Moreover, there are not only transverse but also longitudinal couplings between the external magnetic fields and the SFQs when the inversion symmetry of potential energy is broken. The longitudinal coupling would induce some new phenomena in the SFQs. Here we will show how the longitudinal coupling can result in the coexistence of multi-photon processes in a two-level system formed by a SFQ circuit. We also show that the SFQs can become transparent to the transverse coupling fields when the longitudinal coupling fields satisfy the certain conditions. We further show that the quantum Zeno effect can also be induced by the longitudinal coupling in the SFQs. Finally we clarify why the longitudinal coupling can induce coexistence and disappearance of single- and two-photon processes for a driven SFQ, which is coupled to a single-mode quantized field.Comment: 11 pages, 6 figure