Physics 525, Winter 2025

STATISTICAL MECHANICS 2

Marcel den Nijs Office: PAB B429 Department of Physics University of Washington Seattle, WA 98195 (206)-543-7305 london@uw.edu Tu and Th 9:30 am - 10:50 am We 9:30-10:20 am PAA A114

OUTLINE:

This is the second quarter of the graduate statistical mechanics course sequence.

The focus of this course will be on equilibrium and non-equilibrium critical phenomena, in particular, scale invariance, renormalization theory, topological phenomena with non-local order parameters, transfer matrices, and time permitting quantum phase transitions.

The purpose of this course is to provide graduate students essential background and core materials that build directly on research from the 2nd half of the last century. Current applications and extensions are found in, e.g., two dimensional quantum materials, biophysics, quantum information, and non-equilibrium processes.

We start with an overview of scale invariance as it emerged around 1960 in experimental work on phase transitions. Scale invariance is the consequence of divergent correlation lengths by strongly fluctuating degrees of freedom. Scale invariance was first observed in experimental data and theoretical series expansions and then explained about 1965 by Kadanoff in terms of "block spin invariance", followed in 1971 by the formulation of renormalization theory (RT) by Wilson and Fisher, while merging with similar ideas and phenomena in particle theory.

Scale invariance is linked to fractal type geometric structures, and RT's can be viewed in retrospect as recursive reformulations of partition functions and correlation functions based on the definition of fractal dimensions, such that exact and approximative methods give the values of the scaling dimensions. We will review RT from this geometric perspective.

Molecular field theory methods date back to van der Waals. They fail fundamentally in describing strongly fluctuating collective phenomena and scale invariant systems. These methods are still with us today in, e.g., Landau theory, density functional theory, and effective field theory. We will review the reasons for why they fail for strongly fluctuating phenomena.

Topological phase transitions lack local order-parameters, and are often said to elude the Landau like classification of phase transitions. Kosterlitz-Thouless (KT) type phase transitions in two dimensional equilibrium critical phenomena are the classic example. The transition is driven by vortices excitation unbinding as realized in, e.g., adsorbed He films. Long range-order is represented by a so-called non-local string order parameter associated with the vortex-vortex correlation function. Under so-called duality the KT transition describes crystalline surface roughening, with a genuine local order parameter.

Kramer's transfer matrix links classical equilibrium statistical mechanics in D=d+1 dimensions with quantum phase transitions in the ground state of d dimension quantum systems. The equivalence of the D=2 Ising model and d=1 free fermion electrons is the classic examples. Another example are VBS type phases in quantum spin chains that represent quantum fluids with topological long range order and are equivalent to so-called disordered flat equilibrium phases in D=2 surfaces.

Time permitting we will review transport theory and the Onsager reciprocal relations in the context of electrons in metals and the Boltzmann equation. Driven stochastic processes lead to novel types of stationary states. Scale invariance in the KPZ process is one of the basic examples.

CANVAS

HOMEWORK, TESTS, EXAMS, and CREDIT:

OFFICE HOURS:

ACCOMODATION POLICIES: