exercises and tutorials for
Mathematical Statistical Physics

  spring term 2013 (SoSe 2013)

(This page is not being updated)

wiki page of the course

weekly schedule:







C 111



office hours:

Robert Helling

by appointment office B-339
Wolfram Bauer
2-4 p.m. or by appointment
office B-412
Alessandro Michelangeli Friday
10-11 a.m.
office B-335
Benedikt Rehle
by appointment
Johannes Knebel

by appointment office A-335
Christoph Fischbacher

by appointment

Markus Furtner

by appointment

lecture notes from class:

further material:

homework assignments:

homework + tutorial sessions:
  • JUVE group --> Tuesday. CHELSEA group --> Wednesday
  • Participation is not mandatory.
  • The material to discuss is the same for both sessions.
  • If the participation decreases significantly, one slot will be terminated.
  • A selection of the homework exercises will be discussed in detail in the homework part. (Die Lösungen werden in den Übungen besprochen.)
  • The tutorial part is meant for catching up with missing background material and for more details left open in the lectures or in the homework. It will be in the form of a frontal presentation, or on-the-fly problems, or an interactive discussion.

  • Final exam (Endklausur): Fri 19 July 2013, 10-14, lecture room B-004.

                     Final test                 Solutions                Results

weekly diary of homework / tutorial sessions:

week 11
  • A number of Fock-space computations require exponentiating functions of a and a* which are only essentially self-adjoint, namely self-adjoint upon operator closure (see, e.g., Exercise 41). A quick recap on operator closure can be find here (from FA2 2011/2012).
  • Discussion of the homework assignment 11:
    • E41 - Coherent states in the bosonic Fock space.
    • E42 - Group of Bogoliubov transformations for the ideal Bose gas.
    • E43 - Distributional formalism for the (bosonic) Fock space.
    • E44 - Two-point function for a bosonic Gibbs state.

week 10
  • Further comments on BEC.
  • Discussion of the homework assignment 10:
    • E37 - Perierl's argument -- part II.
    • E38 - Application of Renormalisation Group: a quantum flute – or a bosonic string.
    • E39 - Fermi-Dirac ideal gases: computation of thermodynamic quantities.
    • E40 - Variance of the particle number for a Fermi-Dirac ideal gases.

week 9
  • Discussion on Peierls' argument.
  • Discussion on Schramm-Loewner evolution.
  • Discussion of the homework assignment 08:
    • E29 - Explicit block-matrix computations.
    • E30 - Computation of the free energy density.
    • E31 - Peierls’ argument -- Part I.
  • Discussion of the homework assignment 09:
    • E33 - Asymptotics of the complete elliptic integral of the first kind.
    • E34 - Computation of thermodynamic quantities.
    • E35/E36 - Renormalisation Group analysis of the Sierpinski gasket.

week 8
  • discussion of circulant matrices and plane wave ansatz to obtain their eigenvectors and eigenvalues
  • real-space RG in Ising 1D, exact spin block transformation, decimation procedure and RG map
  • real-space RG in Ising 2D, higher order interaction terms occur during spin block transformation, approximate solution
  • real-space RG in Ising on the Sierpinski gasket, exact spin block transformation because of the fractal geometry; outline of how to carry out computations in exercise 35 and 36

week 7
  • Embedding of Ising model into the class of O(n) vector models and applications of these models to experimental observations.
  • Overview about different methods to analyse the critical behavior of the Ising model in different dimensions (transfer matrix, Peierls' argument, renormalization group, mean field theory).
  • Connection of mean field theory to phenomenological Ginzburg-Landau theory of phase transitions.
  • Discussion of Ising 2D lattice simulations (videos): criticality and scale invariance at the critical point.
  • Discussion of the correlation function. Mean field theory means neglecting the correlations of the fluctuations in the spin variable (in thermodynamic equilibrium) around the mean value (= magnetization/site). At the critical point, this assumptions is wrong: at the critical temperature, fluctuations in the spin variable around the mean are correlated over all length scales (i.e., correlation length diverges).
  • Discussion of the homework assignment 07:
    • E25 - Representations of the Cliff ord relations.
    • E26 - Spin representation of rotations.
    • E27 - Maximum Entropy Principle.
    • E28 - Ising model in mean field approximation.

week 6
  • Phenomenology of phase transitions.
  • Smple liquid and ferromagnet.
  • Phase diagrams, order parameter, discontinuous and continuous phase transitions, criticality.
  • Concept of universality, scaling functions, critical exponents.
  • Recapitulation of KMS
  • Discussion of the homework assignment 06:
    • E21 - A clustering state w.r.t. an asymptotic abelian dynamics exhibits return to equilibrium in its folium.
    • E22 - Construction of KMS states as limit of KMS states (applicable for a KMS state on a quasi-local algebra).
    • E23 - 1D Ising: partition function (via transfer matrix), free energy, magnetisation. Absence of phase transition at T>0.
    • E24 - Tensor product of matrices.

week 5

week 4
  • Generalisation from class: prescribing that the map A-->Tr(qA) realised with a trace class operator q is a state on the C*-algebra of bounded operators on a complex Hilbert space, implies that q is self-adjoint, positive, and with trace equal to one.
  • Revision: group of automorphisms and C*-dynamical systems. In the GNS representation of a C*-algebra with respect to a stationary state, the dynamics is implemented by a strongly continuous unitary group that leaves the vacuum invariant. Equation of motion at the level of the algebra and at the level of the GNS representation.
  • Construction of a stationary state: in a C*-dynamical system a stationary state always exists. The problem of distinguishing a stationary state from a KMS state.
  • Discussion of the homework assignment 04:
    • E13 - Quasi-local CAR algebra.
    • E14 - Equivalent conditions for a representation to be faithful. A simple algebra has always faithful representations.
    • E15 - Proof of irreducibility of the Schroedinger representation of the Weyl algebra.
    • E16 - The GNS representation of C[0,1] on L^2[0,1]

week 3

  • Recap on representations of a C*-algebra. Automatically continuous. Faithful representations. Irreducible => Cyclic => Non-degenerate. The GNS representation relative to a given state is irreducible iff the state is pure. The representation of a simple algebra is always faithful. Construction of the Weyl C*-algebra A_W: von Neumann's theorem states that all the regular and irreducible representation of A_W are unitarily equivalent to the customary Schroedinger representation on R^d. (Homework E10/E11 discusses inequivalent representations instead.)
  • Recap on Quasi-local Algebras. Physical meaning. Quasi-local algebra for an infinite quantum spin system. Note that we defined a quasi-loc algebra A giving the algebra and its family of local sub-algebras, whose union is dense in A. Conversely, one can start from the family of local sub-algebras, recognise that their union has all properties of a C*-algebra except for completeness, then we take the "inductive limit" (i.e., the completion!) to construct A. Completion is an intrinsic, "automatic" construction: see some standard facts on completeness from Problem 15, Problem 19, Exercise 13, Exercise 15, Exercise 16, Exercise 17, Exercise 19(ii), Exercise 21, Exercise 34, and Problem 28 from FA SoSe2012.
  • Discussion of the homework assignment 03:
    • E09 - Boundedness of creation/annihilation operators in the fermionic Fock space. Irreducibility of the CAR algebra.
    • E10 / E11 - Another representation of the anti-commutation relation, inequivalent to the Schroedinger representation. (The groups are not strongly continuous any longer!)
    • E12 - Characterisation of positive elements in a C*-algebra. They are a closed and convex cone.

week 2

  • Recap on bosonic/fermionic Fock space: see this excerpt from Bratteli-Robinson.
  • Further outlook:
    • Heisenberg group and its representations
    • The Weyl algebra
    • in which sense the Schrödinger representation is unique (von Neumann's theorem)
    • the Bargmann-Fock-Segal representation.
    Reference on these topics: this chapter from Folland; see also this informal note for an outlook.
  • Discussion of the homework assignment 02:
    • E05 - A unital, commutative Banach algebra with ||A^2||=||A||^2 for every A is necessarily commutative. Proof through series expansion and holomorphic functions methods.
    • E06 - An example of unital Banach algebra that contains nilpotent elements. The Gelfand transform is in this case non-injective. Classification of its characters. 
    • E07 - Weakly-* closure of the space of characters.
    • E08 - *-isomorphism representing the commutative C*-algebra generated by 1,A,A* onto the continuous functions on the spectrum of A.

week 1
  • Revision of the algebraic setting for the commutative case. The characters (=non-zero multiplicative functionals) of a unital commutative Banach algebra form a weakly-* compact Hausdorff space X. A commutative C*-algebra is isomorphic to C(X) (Gelfand-Naimark) ---> ordinary CLASSICAL MECHANICS description is thus retrieved.
  • Revision of weak-* topology in the dual B* of a Banach space B. It is indeed weaker then the norm topology, sufficiently weaker that the unit ball in the dual B* is weakly-* compact (Banach-Alaoglu), but still reach enough to be a Hausdorff topology.
  • Mention of the scenario in the non-commutative case, non-commutative Gelfand-Naimark theorem ---> ordinary QUANTUM MECHANICS is thus retrieved (needs heavier representation theory, see e.g. what we did last semester in MQM).
  • Additional problems set 01, providing further examples and applications to the notions presented in class.
  • Discussion of the homework assignment 01:
    • E01 - Another example of a Banach-* algebra that fails to be C*: the case of \ell^1(Z).
    • E02 - A unital commutative Banach algebra that is simple (i.e., no proper ideals) is necessarily the complex numbers.
    • E03 - The standard mechanism of adjoining a unit to a non-unital C*-algebra so to get a larger and unital C*-algebra.
    • E04 - The commutative C*-algebra C(X), X compact: the spectrum of f in the algebra is nothing but the range of f.

week 0