## Quantum Magnetism Lecture Notes Homework Helper

**Prof. Matthew Fisher **

email: mpaf@kitp.ucsb.edu ;

Office: 1118 Kohn Hall

Lectures: Tuesday, Thursday 12:30-1:45 pm, Phelps Hall 1448

Office Hours: Monday 2:00-3:00 pm, Wednesday 2:00-3:00 pm

**SYLLABUS: **

Physics 217A will be an introduction to field theory methods, especially as used in condensed matter physics.

(I) Basic Field Theory Formalism: Illustrated via Single Particle Quantum Mechanic

A. Quantum Dynamics and Statistical Mechanics

B. Linear Response theory - Interaction and spectral representations, fluctuation-dissipation theorem

C. Imaginary time dynamics, "thermal" Green's function and analytic continuation

D. Path integral - brief review

E. Perturbation theory: Wicks thm, Feynman diagrams, self-energy and irreducible vertex

(II) Quantum Ising model as a Field theory

A. Single spin-dynamics - tunneling in a double-well potential

B. Phi-4 Scalar Field Theory - d+1 space-time dimensions

C. Quantum Phase Transition: Symmetry breaking, fluctuaitons and the renormalization group

(III) Interacting Boson Systems

A. Wave functions, 2nd quantization and field operators

B. Coherent State Path integral

C. Superfluidity; symmetry breaking, Goldstone modes, density-phase uncetainty

D. XY spin-models: path integral, Berry's phase for s=1/2, Phi-4 complex field theory for s=1

(IV) Quantum Magnetism: Heisenberg model

A. Spin-wave theory for ferro-antiferro manget

B. Path integral for a single spin; Berry's phase

C. Non-linear sigma model for spin-s antiferromagnet on hypercubic lattice, in d=1,2,3

(V) Duality for 1d and 2d Quantum Spin models

A. 1d Quantum Ising model; self-dual

B. 1d quantum XY model: Dual to Since-Gordon theory

C. 2d Quantum Ising model dual to Z2 gauge theory

D. 2d Quantum XY model: dual to U(1) Gauge theory

**Textbooks**: No official textbook. Some books you might wish to check out:

- "Quantum Many-particle systems", by Negele and Orland (quite formal, but very precise)
- "Interacting Electrons and Quantum Magnetism", by A. Auerbach (especially good on magnetism)
- "Quantum Phase Transitions", by S. Sachdev (great, but challenging)
- "Quantum Field theory of Many-body systems", by X.G.Wen (Wen's unique perspective - check it out)
- "Condensed Matter Field Theory", by A.Atland and B. Simons (newest, and closest to a textbook)

**Homework**: Roughly 5-7 homeworks throughout the quarter, available on the course web site. Homework solutions also posted online. You are encouraged to work together on the homeworks.

**Exams**: None.

Help/feedback: In addition to my office hours and Ryans' office hours, I would be happy to schedule a mutually convenient time to chat about homework or other topics (best arranged by e-mail). I would greatly appreciate feedback on any aspect of the course, anytime throughout the quarter.

If quantum cosmologists can reconcile quantum mechanics with Einstein's Theory of General Relativity, we may be in for a wild ride, as scientists will finally have a single theory with the potential to describe all aspects of time and space in the universe. In short, we will be on our way to knowing the unknowable. Quite exciting, to say the least. At the heart of quantum theory lies such ideas as the particle nature of waves, the wave-like properties of particles, the quantized energy levels of atoms, and the Heisenberg Uncertainty Principle. It is from these fundamental assumptions about the natural world that we may be able to unlock its secrets through the development of a unified field theory. A good course in quantum physics will take students through the following areas:

- The wave function
- Time-independent Schrodinger equation
- Formalism
- Quantum mechanics in three dimensions
- Identical particles
- Time-independent perturbation theory
- The variational principle
- The WKB approximation
- Time-dependent perturbation theory
- The adiabatic approximation
- Scattering

There are plenty of books on quantum physics to be found on Google and Amazon.com. There is also a nice collection of books and journals devoted to this area which students can find at the Springer website.

**The variational principle**

If a quantum system has a Hamiltonian independent of time and one chooses a normalized “well behaved” function (i.e. a continuous function for which the first and second derivatives exist) then

where is the lowest energy eigenvalue of the Hamiltonian (bound energy).

In other words, if we chose a random test function the bounded energy will be minimum of the expression

**Question 1**

Consider a one-dimensional system of a particle with a potential energy:

Make a graph of V vs *X*. Use the test function to determine the energy of the ground level. Note that there is no variational parameter. Note that is given by the equation (why?):

**Solution**:

If you consider a test function having no variational parameter (that is having no constant coefficients inside) then the eigenvalue of the bound level can be approximated by

Since

We can write:

since we have:

**Question 2**

Consider you have a test function:

where f_{1} and f_{2} are real functions. Find the energy of the bound level in terms of the integrals

For the special case when:

find also the rapport of the coefficients:

**Solution**:

If you have a test function of the type

Then the energy of the bound level is given by (like above):

Which for:

gives the rapport of coefficients:

To have a solution of the above system of two equations one needs to have the characteristic determinant = 0. In other words the condition for existence of E states is:

or

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