Equilibrium Statistical Mechanics of Fermion Lattice Systems

We study equilibrium statistical mechanics of Fermion lattice systems which require a different treatment compared with spin lattice systems due to the non-commutativity of local algebras for disjoint regions. Our major result is the equivalence of the KMS condition and the variational principle with a minimal assumption for the dynamics and without any explicit assumption on the potential. It holds also for spin lattice systems as well, yielding a vast improvement over known results. All formulations are in terms of a C*-dynamical systems for the Fermion (CAR) algebra with all or a part of the following assumptions: (I) The interaction is even with respect to the Fermion number. (Automatically satisfied when (IV) below is assumed.) (II) All strictly local elements of the algebra have the first time derivative. (III) The time derivatives in (II) determine the dynamics. (IV) The interaction is lattice translation invariant. A major technical tool is the conditional expectation from the total algebra onto the local subalgebra for any finite subset of the lattice, which induces a system of commuting squares. This technique overcomes the lack of tensor product structures for Fermion systems and even simplifies many known arguments for spin lattice systems.Comment: 103 pages, no figure. The Section 13 has become simpler and a problem in 14.1 is settled thanks to a referee. The format has been revised according to the suggestion of this and the other referee

A tight analysis of Kierstead-Trotter algorithm for online unit interval coloring

Kierstead and Trotter (Congressus Numerantium 33, 1981) proved that their algorithm is an optimal online algorithm for the online interval coloring problem. In this paper, for online unit interval coloring, we show that the number of colors used by the Kierstead-Trotter algorithm is at most $3 \omega(G) - 3$, where $\omega(G)$ is the size of the maximum clique in a given graph $G$, and it is the best possible.Comment: 4 page

Maximally entangled fermions

Fermions play an essential role in many areas of quantum physics and it is desirable to understand the nature of entanglement within systems that consists of fermions. Whereas the issue of separability for bipartite fermions has extensively been studied in the present literature, this paper is concerned with maximally entangled fermions. A complete characterization of maximally entangled quasifree (gaussian) fermion states is given in terms of the covariance matrix. This result can be seen as a step towards distillation protocols for maximally entangled fermions.Comment: 13 pages, 1 figure, RevTex, minor errors are corrected, section "Conclusions" is adde

Entropy growth of shift-invariant states on a quantum spin chain

We study the entropy of pure shift-invariant states on a quantum spin chain. Unlike the classical case, the local restrictions to intervals of length $N$ are typically mixed and have therefore a non-zero entropy $S_N$ which is, moreover, monotonically increasing in $N$. We are interested in the asymptotics of the total entropy. We investigate in detail a class of states derived from quasi-free states on a CAR algebra. These are characterised by a measurable subset of the unit interval. As the entropy density is known to vanishes, $S_N$ is sublinear in $N$. For states corresponding to unions of finitely many intervals, $S_N$ is shown to grow slower than $(\log N)^2$. Numerical calculations suggest a $\log N$ behaviour. For the case with infinitely many intervals, we present a class of states for which the entropy $S_N$ increases as $N^\alpha$ where $\alpha$ can take any value in $(0,1)$.Comment: 18 pages, 2 figure

The χ2\chi^2 - divergence and Mixing times of quantum Markov processes

We introduce quantum versions of the $\chi^2$-divergence, provide a detailed analysis of their properties, and apply them in the investigation of mixing times of quantum Markov processes. An approach similar to the one presented in [1-3] for classical Markov chains is taken to bound the trace-distance from the steady state of a quantum processes. A strict spectral bound to the convergence rate can be given for time-discrete as well as for time-continuous quantum Markov processes. Furthermore the contractive behavior of the $\chi^2$-divergence under the action of a completely positive map is investigated and contrasted to the contraction of the trace norm. In this context we analyse different versions of quantum detailed balance and, finally, give a geometric conductance bound to the convergence rate for unital quantum Markov processes

Universal fermionization of bosons on permutative representations of the Cuntz algebra O2{\cal O}_{2}

Bosons and fermions are described by using canonical generators of Cuntz algebras on any permutative representation. We show a fermionization of bosons which universally holds on any permutative representation of the Cuntz algebra ${\cal O}_{2}$. As examples, we show fermionizations on the Fock space and the infinite wedge.Comment: 12 page