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Course Description

The main focus of this course will be on describing states of matter where the combination of strong quantum fluctuations and interactions lead to new quantum phases. We will be most interested in `strongly correlated phases', ones that are not simply described in terms of the original degrees of freedom (eg. in terms of electrons). Describing these phases requires conceptually new ideas (eg. topological order) to complement the traditional description of phases in terms of order parameters and fermi liquid theory. It also needs new mathematical techniques, such as dualities, that are non-perturbative in terms of the original variables.

The simplest many body systems that display such interesting physics are quantum spin systems which will be the subject of this course. In particular we will focus on spin chains (1+1D systems), where several theoretical tools can be applied and where strong correlation physics is ubiquitous. They will provide the simplest setting where several conceptually important issues like:

  • quantum number fractionalization
  • `topological' order
  • role of instantons, and Berry phases (theta terms) associated with them.

can be discussed. Also several non-perturbative theoretical and mathematical techniques can be brought to bear on them, and consistency between very different techniques provides a check on some of these results. These include:

  • deriving field theories (long distance descriptions) from microscopic lattice models.
  • exact solutions
  • bosonization
  • dualities

some of these methods and all the conceptual ideas generalize to higher dimensions and more complicated systems.

During the course I will also spend some time describing how quantum spin systems are realized in nature, such as in transition metal oxides, and the origin of various terms in the spin Hamiltonian. The importance of this problem to the understanding the physics of several interesting materials, like the high temperature cuprate superconductors will be emphasized.

This course is intended for students of condensed matter (theory and experiment) who are interested in strongly correlated electronic systems. It would also be of interest more generally to students who are curious to learn of microscopic realizations of several abstract ideas of field theory.



1.   Quantum Spin Systems in Nature, and ‘Nature’ in Quantum Spin Systems

·       Landau’s description of phases and phase transitions

·       Experimental signatures of unconventional phases

·       Spin systems as microscopic models with emergent `sound’ (phonons), `light’ (photons) and `matter’ (fermions).

·       Origin of magnetic moments and magnetic interactions in insulators.

·       Double exchange and super-exchange.

·       Origin of Magnetic anisotropies.


2.     Ordered States

·       Semiclassical theory  (large S) of magnetic ordering

·       Holstein-Primakoff representation of quantum spins.

·       Spin waves in ferromagnets and collinear antiferromagnets.

·       Order and disorder at T=0 and finite temperatures in Ferromagnets and collinear antiferromagnets from spin wave theory.

·       Relations to superfluid and solid phases.

·       Rigorous definition of symmetry breaking. The Anderson tower.

·       Some theorems - Marshall sign.


3.     Ising Spin Chain in a Transverse Field

·       Structure factor and susceptibility

·       Weak and strong coupling expansions

·       Mean field theory.

·       Duality

·       Exact solution via Jordan Wigner transformation


4.     Experimental probes of Magnetism

·       Susceptibility and correlations. Spectral representation.

·       Neutron Scattering

·       ESR, μSR and NMR


5.     D=1 Spin Systems with Continuous Spin Symmetry

·       Mermin-Wagner Theorem and absence of order.

·       Lieb-Shultz-Mattis theorem

·       Exact results: S=1/2, XX spin chain – solution via free fermions. Majumdar-Ghosh Hamiltonian and dimerized states.

·       Rotor models with Berry phases. Duality.

·       The anisotropic S=1/2 (XXZ ) chain. Solution via bosonization.

·       Recovering SU(2) invariance.


6.     Haldane Gap and Haldane Conjecture

·       Path integral representation of spins

·       Nonlinear σ model representation of antiferromagnetic spin chains

·       AKLT point and valence bond solid state. Topological order.

·       Analysis of the Nonlinear σ model with Berry phase (theta) term. Gaplessness of half integer spin chains.


7.     Phase diagrams of the S=1/2 and S=1 chains.

·       Novel phase transitions driven by domain walls with spin.


Time permitting, and depending on class interest, some of the following topics may also be covered.


8.     D=2 Quantum Spin Systems

·       Magnetically ordered states and valence bond crystals.

·       Idea of Resonating Valence Bond states. Implementation via deconfined gauge fields and spinons.

·       Exactly soluble spin model with emergent fermions and deconfined Z2 gauge fields.


9.   Hubbard Model and Correlated Electrons