March 7, 2014
Location: Achter de Dom 22 (Utrecht), room 102
Probabilistic aspects of quantum spin systems
I will discuss ground states and low-temperature states of so-called gapped quantum spin systems. To guide thoughts, the Hamiltonian of these systems can be (roughly) compared to the generator of a Glauber dynamics of an Ising model above the critical temperature, and the ground state then corresponds to the invariant state of the dynamics.
Questions that interest us are: Do these states satisfy a large deviation principle? Are they Gibbsian in an appropriate sense? In fact, I will show that sometimes the answer to the second question is negative. Time permitting, I will also discuss the occurence of so-called 'Long-Range Localizable Entanglement'. This is a property that is perhaps a more natural analogue of 'non-gibbsianness', and that at some points had been linked to 'topological phase transitions'. In any case, I will build up the talk in such a way that no familiarity with quantum mechanics is required.
Part of the talk is based on joint work with Maes, Netočný, Schütz, reported in arXiv:1312.4782.
Some wonderful conjectures (but very few theorems) at the boundary between analysis, combinatorics and probability
Many problems in combinatorics, statistical mechanics, number theory and analysis give rise to power series (whether formal or convergent) of the form
| ∞ | ||||
| f(x, y) | = | ∑ | an(y) | xn |
| n=0 |
where {an(y)} are formal power series or analytic functions satisfying an(0)≠0 for n=0,1 and an(0)=0 for n≥2. Furthermore, an important role is played in some of these problems by the roots xk(y) of f(x,y) — especially the "leading root" x0(y), i.e. the root that is of order y0 when y→0. Among the interesting series f(x,y) of this type are the "partial theta function" Θ0(x,y) = ∑n≥0 xn yn(n-1)/2 which arises in the theory of q-series, and the "deformed exponential function" F(x,y) = ∑n≥0 xn yn(n-1)/2 / n! which arises in the enumeration of connected graphs. These two functions can also be embedded in natural hypergeometric and q-hypergeometric families.
In this talk I will describe recent (and mostly unpublished) work concerning these problems — work that lies on the boundary between analysis, combinatorics and probability. In addition to explaining my (very few) theorems, I will also describe some amazing conjectures that I have verified numerically to high order but have not yet succeeded in proving. My hope is that one of you will succeed where I have not!
Further information is available at http://www.maths.qmul.ac.uk/~pjc/csgnotes/sokal/
The slides of this talk are available here.