Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGenerality of Liebs Concavity Theorem
285
0
0.0
(
0
)
Authors
De Huang
Ask ChatGPT about the research
No Arabic abstract
We show that Liebs concavity theorem holds more generally for any unitarily invariant matrix function $phi:mathbf{H}^n_+rightarrow mathbb{R}$ that is monotone and concave. Concretely, we prove the joint concavity of the function $(A,B) mapstophibig[(B^frac{qs}{2}K^*A^{ps}KB^frac{qs}{2})^{frac{1}{s}}big] $ on $mathbf{H}_+^mtimesmathbf{H}_+^n$, for any $Kin mathbb{C}^{mtimes n},sin(0,1],p,qin[0,1], p+qleq 1$.
rate research
Read More
We show that Liebs concavity theorem holds more generally for any unitary invariant matrix function $phi:mathbf{H}_+^nrightarrow mathbb{R}_+^n$ that is concave and satisfies Holders inequality. Concretely, we prove the joint concavity of the function $(A,B) mapstophibig[(B^frac{qs}{2}K^*A^{ps}KB^frac{qs}{2})^{frac{1}{s}}big] $ on $mathbf{H}_+^ntimesmathbf{H}_+^m$, for any $Kin mathbb{C}^{ntimes m}$ and any $s,p,qin(0,1], p+qleq 1$. This result improves a recent work by Huang for a more specific class of $phi$.
We introduce the notion of $k$-trace and use interpolation of operators to prove the joint concavity of the function $(A,B)mapstotext{Tr}_kbig[(B^frac{qs}{2}K^*A^{ps}KB^frac{qs}{2})^{frac{1}{s}}big]^frac{1}{k}$, which generalizes Liebs concavity theorem from trace to a class of homogeneous functions $text{Tr}_k[cdot]^frac{1}{k}$. Here $text{Tr}_k[A]$ denotes the $k_{text{th}}$ elementary symmetric polynomial of the eigenvalues of $A$. This result gives an alternative proof for the concavity of $Amapstotext{Tr}_kbig[exp(H+log A)big]^frac{1}{k}$ that was obtained and used in a recent work to derive expectation estimates and tail bounds on partial spectral sums of random matrices.
We outline a simple proof of Hulanickis theorem, that a locally compact group is amenable if and only if the left regular representation weakly contains all unitary representations. This combines some elements of the literature which have not appeared together, before.
The noninteracting electronic structures of tight binding models on bipartite lattices with unequal numbers of sites in the two sublattices have a number of unique features, including the presence of spatially localized eigenstates and flat bands. When a emph{uniform} on-site Hubbard interaction $U$ is turned on, Lieb proved rigorously that at half filling ($rho=1$) the ground state has a non-zero spin. In this paper we consider a `CuO$_2$ lattice (also known as `Lieb lattice, or as a decorated square lattice), in which `$d$-orbitals occupy the vertices of the squares, while `$p$-orbitals lie halfway between two $d$-orbitals. We use exact Determinant Quantum Monte Carlo (DQMC) simulations to quantify the nature of magnetic order through the behavior of correlation functions and sublattice magnetizations in the different orbitals as a function of $U$ and temperature. We study both the homogeneous (H) case, $U_d= U_p$, originally considered by Lieb, and the inhomogeneous (IH) case, $U_d eq U_p$. For the H case at half filling, we found that the global magnetization rises sharply at weak coupling, and then stabilizes towards the strong-coupling (Heisenberg) value, as a result of the interplay between the ferromagnetism of like sites and the antiferromagnetism between unlike sites; we verified that the system is an insulator for all $U$. For the IH system at half filling, we argue that the case $U_p eq U_d$ falls under Liebs theorem, provided they are positive definite, so we used DQMC to probe the cases $U_p=0,U_d=U$ and $U_p=U, U_d=0$. We found that the different environments of $d$ and $p$ sites lead to a ferromagnetic insulator when $U_d=0$; by contrast, $U_p=0$ leads to to a metal without any magnetic ordering. In addition, we have also established that at density $rho=1/3$, strong antiferromagnetic correlations set in, caused by the presence of one fermion on each $d$ site.
Lins theorem states that for all $epsilon > 0$, there is a $delta > 0$ such that for all $n geq 1$ if self-adjoint contractions $A,B in M_n(mathbb{C})$ satisfy $|[A,B]|leq delta$ then there are self-adjoint contractions $A,B in M_n(mathbb{C})$ with $[A,B]=0$ and $|A-A|,|B-B|<epsilon$. We present full details of the approach in arXiv:0808.2474, which seemingly is the closest result to a general constructive proof of Lins theorem. Constructive results for some special cases are presented along with applications to the problem of almost commuting matrices where $B$ is assumed to be normal and also to macroscopic observables.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
