Abstract
This script is based on the notes the author prepared to give a set of six lectures at the Les Houches School ``Integrability in Atomic and Condensed Matter Physics'' in the summer of 2018. The responsibility for the selection of the material is partially with the organisers, Jean-Sebastien Caux, Nikolai Kitanine, Andreas Klümper and Robert Konik. The school had its focus on the application of integrability based methods to problems in non-equilibrium statistical mechanics. My lectures were meant to complement this subject with background material on the equilibrium statistical mechanics of quantum spin chains from a vertex model perspective. I was asked to provide a minimal introduction to quantum spin systems including notions like the reduced density matrix and correlation functions of local observables. I was further asked to explain the graphical language of vertex models and to introduce the concepts of the Trotter decomposition and the quantum transfer matrix. This was basically the contents of the first four lectures presented at the school. In the remaining two lectures I started filling these notions with life by deriving an integral representation of the free energy per lattice site for the Heisenberg-Ising chain (alias XXZ model) using techniques based on non-linear integral equations.Up to small corrections the following sections 1-6 display the six lectures almost literally. The only major change is that the example of the XXZ chain has been moved from section 5 to 2. During the school it was not really necessary to introduce the model, since other speakers had explained it before. But for these notes I thought it might be useful to introduce the main example rather early. I also supplemented each lecture with a comment section which contains additional references and material of the type that was discussed informally with the participants.
Highlights
In the following we shall focus on quantum-spin chains with Hilbert space H2L and with local interactions h ∈ End( d )⊗m, where m ∈ {2, . . . , 2L} will be called the range of the interaction
After preparation in an experiment any large quantum-spin system will rather be in a state described by a density matrix ρL ∈ End H2L with properties ρL = ρ+L, ρL ≥ 0, tr ρL = 1
Transients and long relaxation times are possible, for integrable quantum-spin systems, and stationary non-equilibrium ensembles may be realised in driven systems
Summary
Spin systems are the simplest conceivable quantum mechanical systems. In nature the spin occurs in first place as an internal degree of freedom of elementary particles. In certain experiments on such systems, e.g. on Mott insulators at low temperatures or on ultra-cold atomic gases trapped in optical lattices, it is possible to create ‘pure spin excitations’ Such systems are well described, in a certain energy range, by (generalised) Hubbard or Heisenberg models [1], which are in the class of models to be considered in these notes. All many-body quantum physics can be phrased in terms of spin systems of sufficiently general type This should provide enough motivation to thoroughly study their statistical mechanics. As far as the general theory is concerned we shall introduce and explain what we call the ‘quantum transfer matrix approach’ to quantum spin systems In our understanding this approach is a clever and systematic way of attaching an equilibrium statistical operator with any type of local interaction. This construction cannot only be used to calculate the free energy per lattice site of such models, but seems to be optimised as well for the calculation of their correlation functions [6]
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