Quantum Phase Transitions and the characterization of Gapped Quantum Phases are once again a hot topic. Solving the classification problem of gapped ground states is important to better understand the possibilities and challenges of implementing quantum computation and information processing with topological quantum ground state phases. The critical points separating different gapped ground state phases (quantum phase transitions) have fascinating mathematical properties. A case in point is the conjectured $E_8$ symmetry of the critical quantum Ising chain, experimental evidence for which was recently reported in the literature (Science, 8 January 2010, **327** (no. 5962) pp. 177-180; see also arXiv:1012.5407). In this RFG, participants who are new to the topic will become familiar with the main techniques to study these topics, with an emphasis on those that have been or might be amenable to mathematical analysis. As needed, more advanced participants will organize tutorials on relevant topics of statistical mechanics, functional analysis, representation theory, Feynman-Kac-type formulas, etc.. For context, there will be a reading course on Quantum Information Theory. In Spring 2012 the RFG will concentrate on topics of current research and there will be a special topics course (MAT280) on quantum phase transitions.

Check out the slides of a short talk introducing the the topic of the RFG at the *Davis Math Conference 2011*.

Register for MAT-290-083; CRN: 69645. Sign up for 1 or 2 units unless you plan to take a more active role, including presentations, in which case you should talk to me (Bruno) to make arrangements.

Typically one hour will be devoted to Tutorials and Introductory Topics and the other hour will be a reading course on Quantum Information Theory (likely text: Mark M. Wilde, From Classical to Quantum Shannon Theory).

We meet again on Mondays, 3-5pm in Rm 1147. The first meeting was on Monday, January 9. Mondays this quarter

are impacted by holidays and special events. Therefore, there will be some additional meetings on Wednesdays

or Fridays 3:10-4pm.

Register for MAT-290-083; CRN: 40265. Sign up for 1 or 2 units unless you plan to take a more active role,

including presentations, in which case you should talk to me (Bruno) to make arrangements. Typically one hour will be devoted to the discussion of basic models of phase transitions and the other hour will be a reading course on Quantum Information Theory, led by Anna (text: Mark M. Wilde, From Classical to Quantum Shannon Theory). The QIT reading course

schedule is here: http://www.math.ucdavis.edu/~aver/Quantum%20Information%20Theory.html.

Topics, readings, and schedule for the phase transitions seminar:

- The Ising model: http://www.math.ucdavis.edu/~bxn/m266W10-week3_IsingModel.pdf (Eric Brattain & Amanda Young)

- The XY model: http://www.math.ucdavis.edu/~bxn/lieb1961.pdf(Ricky Kwok)

- The Ising chain in a transverse field: http://www.math.ucdavis.edu/~bxn/pfeuty1970.pdf , http://www.math.ucdavis.edu/~bxn/dziarmaga2005.pdf (Bruno)

- Heisenberg models and graphical representationshttp://www.math.ucdavis.edu/~bxn/aizenman1994.pdf, http://www.math.ucdavis.edu/~bxn/ueltschi2011.pdf (Sven)

- Bose-Einstein condensation (Zagrebnov)

- Localization and the Anderson model (Tim Wertz)

- The Bose-Hubbard model (?)

General background: Notes on Statistical Mechanics

**Schedule:**

Monday, 01/09: 3-4; Intro, 4-5: QIT (MSB 1147)

Wednesday, 01/18: 3-4: Introduction to Statistical Mechanics (Bruno) (MSB 3106)

Friday, 01/20: 3-4: QIT (MSB 3106)

Wednesday, 01/25, The Ising model (Amanda/Eric) 3-4 (MSB 3106) & 4-5 (MSB 3106 or 1147)

Monday, 1/30, 3-5: The XY model (Ricky) (MSB 1147)

Wednesday, 2/1: 3-4: QIT (MSB 3240)

Monday, 2/6: 3-4: Ising chain in transverse field, 4-5: QIT (MSB 1147)

Monday, 2/13: 3-5 QIT (MSB 1147)

Friday, 2/17: 3-4 QIT (MSB 3106)

Monday, 2/27: 3-5 QPT (Sven/David) (MSB 1147)

Monday, 3/5: 3-5 QPT (Sven/David) (MSB 1147)

Wednesday, 3/7: 3-4 QIT (MSB 3106)

Monday, 3/19: 3-5 QIT (last meeting) (MSB 1147)

- Current research and open problems. The seminar will meet on Mondays, 4:10 - 5:00pm, in MSB 1147, CRN: 79774.

4/2 Organizational meeting

4/9 rescheduled

4/16 Tim Wertz (Anderson Localization)

4/23 Anna Vershynina (Bloch Coherent States)

4/30 Anna Vershynina (Bloch Coherent States and the classical limit Notes on Coherent States)

5/7 Anna Vershynina (Bloch Coherent States and the thermodynamic limit)

5/14 Sven Bachmann (PVBS models) Meeting in Rm instead of 1147

5/21 Dirk Deckert (The Bohmian debate)

5/29 Ricky Kwok (An introduction to Bose-Einstein Condensation) Note: meeting on Tuesday 5/29 at 10:30am in MSB 1147

6/4 Michael Krug, Experiments on Bell's Inequalities

6/5-14: Series on Bose-Einstein Condensation by Valentin Zagrebnov. Abstracts of the 5 lectures,
and Lecture Notes (slides).

The schedule of this lecture serties,
organized by Anna Vershynina and Amanda Young, is as follows:
Introductory Seminar: June 6th 1:40-3 pm, First Lecture: June 8th 11-11:50 am, Second Lecture: June 11th 11-11:50 am, Third Lecture: June 12th 11-11:50 am, Fourth Lecture: June 13th 11-11:50 am. All lectures are in MSB 1147.

sponsored by VIGRE)

- MAT280 *Quantum Phase Transitions* (CRN: 79747, Time: Tuesdays and Thursdays 10:30 - 11:50am, Room: MSB 1147).

First class meeting: 4/3. Enrolled students write a paper on the topic of two lectures; a write-up of lecture notes with details filled

in is acceptable. We will have a special lecture series on Bose-Einstein condensation during finals week (See above).

4/3, Introduction, the variational principle

4/5, The quantum Ising model (meeting in 3106)

4/10, 4/12 rescheduled

4/17 Correlation functions of the quantum Ising model (Sven)

4/19 Quantum Spin Systems, general framework

4/24 Lieb-Robinson Bounds

4/26 Existence of infinite system dynamics and applications to ground states

5/1 Frustration-free spin chains

5/3 The AKLT model

5/8, 5/10 rescheduled

5/15 General Properties of Quantum Spin Chains. Overview

5/17,The Exponential Clustering Theorem

5/22, Ground state projections, factorization property etc.

5/24, The commutation property

5/29, 5/31 Spectral gap estimates. rescheduled

6/5 Bose-Einstein condensation I

6/7 Bose-Einstein condensation II

Finals week (6/11-14), Bose-Einstein condensation III & IV

Contact: Bruno Nachtergaele