**Statistical Mechanics **

**Professor: Fernando Brandao**

office: 201 Annenberg

e-mail: fbrandao@caltech.edu

**Teching Assistants: **

Belinda Pang bpang@caltech.edu 357 Cahil

**Office hours (TA):**

8pm Mondays at 319 Cahill (TAPIR interaction room)

**References:**

O. Motrunich, Lecture notes on Statistical Mechanics, attached below

M. Kardar, "Statistical Physics of Fields”.

J. B. Kogut, "An introduction to lattice gauge theory and spin systems" Rev. Mod. Phys. 51, 659 (1979).

Rev. Mod. Phys. 51, 659 (1979)S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar, "Continuous quantum phase transitions",

Rev. Mod. Phys. 69, 315 (1997).N. Goldenfeld, "Lectures on Phase Transitions and the Renormalization Group".

J. Cardy, "Scaling and Renormalization in Statistical Physics".

I. Herbut, "A Modern Approach to Critical Phenomena".

S.-k. Ma, "Modern Theory of Critical Phenomena".

K. Huang, "Statistical Mechanics".

X.-G. Wen, "Quantum field theory of many-body systems."

**Grading:**

100%: homework

Homeworks are due before 5pm Wednesday (**drop of:** Mailbox in 350 Cahill)

- You can work in groups and discuss with other students. But you should write

your own solutions by youeself.

- Each student is entitled to one late extension of up to one week.

- Any further late homework decreases the grade by half each day delayed (total/2 if one day late, total/4 if two days late, …)

**Syllabus:**

Lattice models; phases and phase transitions; quantum-classical mapping. Spin waves. Quasi-long-range order in the 2D XY model. Quantum Thermodynamics. Foundations Statistical Mechanics

**Homework:**

Homework 1 Due: April 18

Homework 2 Due: May 02

Homework 3 Due: May 18

**Lecture Notes: **

Lecture Notes 1:

Lattice models and non-perturbative effects. Overview.

(**suggested reading**: Sections I, II from Kogut’s RMP; Lectures 13, 14 from Kadar Lectures)

Lecture Notes 2:

Quantum-classical mapping (Euclidean path integrals); quantum Ising model and mapping to classical Ising model. Connection with Feynman path integrals in quantum mechanics.

(**suggested reading**: Sections III,IVA from Kogut's RMP (it may be easier to read these "backwards" from Hamiltonians to path integrals). Section II of S. L. Sondhi et al, Rev. Mod. Phys. 69, 315 (1997). has a very clear discussion of the path integral for the quantum XY model)

Lecture Notes 3:

Classical Ising Model: pictures of the phases and series expansions: low-temperature series and high-temperature series. Duality in the 2d classical Ising model.

(**suggested reading**: Kardar Chapter 7 or Lectures 15,16 from Kardar lectures; Sections III, IV from Kogout’s RMP)

Lecture Notes 4: (and Lecture Notes 4b and Lecture Notes 4c for the text we followed in class)

Quantum Ising model: pictures of the phases (ground states and excitations). Self-duality in the 1+1D Ising.

(**suggested reading:** Sections III,IV from Kogut's RMP)

Lecture Notes 5: part a, part b

quantum thermodynamics

(**suggested reading:** https://arxiv.org/abs/1505.07835)

Lecture Notes 6:

foundations stat mech 1

(**suggested reading: **https://arxiv.org/pdf/quant-ph/0511225.pdf**)**

Lecure Notes 7:

quantum thermodynamics

Lecture Notes 8:

foundations stat mech 2

(**suggested reading: **https://arxiv.org/pdf/0812.2385.pdf)

Lecture Notes 9:

Eigenstate Thermalization Hypothesis

(**suggested reading: **https://arxiv.org/abs/1509.06411)

Out-of-time-ordered Correlators

(**suggested reading: **https://arxiv.org/abs/1511.04021, https://arxiv.org/abs/1111.6580)