(Some)%20Concepts%20of%20Condensed%20Matter%20Physics

1
(Some) Concepts of Condensed Matter Physics
Anatoli Polkovnikov, Boston University
AFOSR
2
Condensed matter physics deals with properties of
interacting many-particle systems.
P. Anderson more is different (Science, 1972).
The ability to reduce everything to simple
fundamental laws does not imply the ability to
start from those laws and reconstruct the
universe. The behavior of large and complex
aggregates of elementary particles, it turns out,
is not to be understood in terms of a simple
extrapolation of the properties of a few
particles. Instead, at each level of complexity
entirely new properties appear, and the
understanding of the new behaviors requires
research which I think is as fundamental in its
nature as any other the whole becomes not
only more than the sum of but very different from
the sum of the parts
3
Two examples.
Single neuron relatively easy to characterize.
Is more fundamentally different or just more
complicated?
4
From single particle physics to many particle
physics.
Classical mechanics Need to solve Newtons
equation.
Single particle
Many particles
Instead of one differential equation need to
solve n differential equations, not a big deal!?
With modern computers we can simulate thousands
or even millions of particles.
5
Quantum mechanics Need to solve Schrödinger
equation.
Instead of two numbers need to know the whole
complex field.
6
M
Use specific numbers M200, n100.
n
Fermions
Bosons
A computer built from all known particles in
universe is not capable to exactly simulate even
such a small system.
7
Other issues.
Typical level spacing for our system
Gravitation field of the moon on electron
Typical time scales needed to resolve these
energy levels (e.g. to prepare the system in the
pure state)
Required accuracy of the theory (knowledge of
laws of nature) 10-80-10-50.
Many-body physics is fundamentally different from
single particle physics. It can not be derived
purely from microscopic description.
8
Are there ways out?
1. Statistical physics in equilibrium
Takes care of many irrelevant noise sources.
2. Use collective coordinates phonons, magnons,
quasiparticles, Bogoliubov excitations,
3. Use phenomenological methods Ginzburg-Landau
theory of superconductivity. 4. Renormalization
group approach gradually eliminate unimportant
degrees of freedom and follow the important
ones. 5. Exactly solvable models. Generic
considerations about many-body energy levels do
not work because of many conservation laws. 6.
Numerical (e.g. Monte-Carlo) methods.
Non-equilibrium dynamics ???
9
This course.

1. Non-interacting particles in periodic potential
   crystals and reciprocal lattices, Bloch theorem,
   fermions in solids
2. Broken symmetry and phase transitions in many
   particle systems mean field and variational
   approaches.
3. Failure of mean field theory in low dimensions,
   possible alternatives.

10
Crystals.
Wigner-Seiz cell
Translational symmetries and point group
symmetries define Bravais lattices (5 in 2D, 14
in 3D).
11
Reciprocal Lattice
Choose the basis for G b1, b2 such that
The set of G forms reciprocal lattice.
Wigner-Seitz cell of the reciprocal lattice is
the Brillouin zone.
12
Electrons moving in a crystal constantly scatter
from ions.
Atoms are heavy, electrons are light. To a good
approximation atoms do not move and form a
periodic lattice.
Need to find eigenstates of the single particle
Schrödinger equation
13
Bloch Theorem.
Suppose ?n(r) is a (non-degenerate) solution
14
Then
Therefore
kn- lattice momentum, Bloch momentum,
quasi-momentum, crystal momentum, It is defined
in the first Brillouin zone.
Bloch momentum plays the role of momentum in
crystals.
15
Consider a wave packet.
m is the effective mass.
Semiclassical equations of motion in a smooth
external field
more details in Ashcroft-Mermin book.
16
Motion in a weak periodic potential.
17
Free particle dispersion, conventional vs. Bloch
pictures.
Image from I. Bloch, Nature Physics 1, 23 - 30
(2005)
18
Second order perturbation theory
19
Keep only two terms in the wave function
Need to solve the secular equation.
20
(No Transcript)
21
(No Transcript)
22
Tight binding approximation.
Very strong atomic potential. Different atomic
orbitals barely overlap.
23
(No Transcript)
24
(No Transcript)
25
Example square lattice
26
Evolution of the single-particle spectrum
free particle spectrum
tight-binding regime
27
Many-electron system.
Two fermions can not occupy the same state due to
Pauli principle! We can have two electrons per
each (lattice) momentum state due to spin
degeneracy.
Free electrons, zero temperature
28
(No Transcript)
29
Fermi distribution function.
30
(No Transcript)
31
Electrons in solids.
32
Crystals with filled bands are typically
insulators. Crystals with half-filled bands are
usually metals.
Structure of Brilluoin zones
I. Bloch, Nature Physics 1, 23 - 30 (2005)
33
Fermi surface E(k)Ef
34
Interacting systems. Mean field theory and
broken symmetry. (following notes by S. Girvin,
Yale University)
Start with a classical Ising model.
35
(No Transcript)
36
Spin ordering is a collective effect!
37
Self consistent method.
Many nearest neighbors. (large dimensions).
38
(No Transcript)
39
Which of the solutions is right?

1. By continuity. The answer is obvious at zero
   temperature.
2. Need to compare free energies for both solutions.

40
(No Transcript)
41
(No Transcript)
42
(No Transcript)
43
Short summary from this simple example.

1. In interacting many-particle systems the symmetry
   can be spontaneously broken.
2. This phenomenon can be described within
   variational approach and mean-field approach.
   These descriptions are equivalent.
3. Mean field approach particles are moving in the
   effective field created by other particles, which
   is to be determined self consistently.
4. Variational approach find a simple solvable
   Hamiltonian H0 and minimize

How to find the right mean field approach or
H0? How accurate are these methods?
44
Quantum Mean Field Theory.
45
(No Transcript)
46
Fermi liquid. No broken symmetries.
47
(No Transcript)
48
(No Transcript)
49
(No Transcript)
50
Some possible broken symmetries and phases.
1. Translational symmetry (charge density wave).
51
Broken time reversal symmetry Ferromagnets.
52
Broken gauge symmetry superfluids.
53
(No Transcript)
54
Weakly repulsive Bose gas.
55
Choose trial Hamiltonian.
56
Keep only k0 term.
57
BCS theory of superconductivity.
58
Sketch of the BCS theory.
59
(No Transcript)
60
(No Transcript)
61
(No Transcript)
62
Failures of mean-field theories. Low-dimensional
systems.
Ising model in 1D.
63
(No Transcript)
64
More dimensions or more nearest neighbors?
Periodic boundary conditions z4, like in 2D.
Long range order is destroyed in 1D (quasi 1D)
because of special correlated defects.Mean
field theory does not capture such defects.
65
Systems with continuous symmetries.
Mermin-Wagner theorem.
Consider planar spins in d-dimensions (xy-model)
66
(No Transcript)
67
Fluctuations around mean field solution destroy
long range order in 1D at all temperatures.
68
Correlation functions
69
(No Transcript)
70
(No Transcript)
71
There must be a phase transition between
algebraic and exponential regimes
(Berezinsky)-Kosterlitz-Thouless transition. It
occurs due to vortex unbinding.
No long range order again in thermodynamic limit.
72
Mermin-Wagner theorem (also known as
Mermin-Wagner-Hohenberg theorem or Coleman
theorem) states that continuous symmetries cannot
be spontaneously broken at finite temperature in
one and two dimensional theories.

• The reason is exactly which we observed above
• Broken symmetry implies massless Goldstone
  bosons.
• In one and two dimensions, because of high
  density of low energy states, these bosons always
  destroy long range order.

Other examples crystals, ferromagnets and
anti-ferromagnets, superfluids,
superconductors This theorem assumes there are
no long range interactions.
73
Quantum systems.
74
(No Transcript)
75
How to proceed if the mean field description is
wrong?

1. Effective low energy field theory descriptions
   universality of physics at long distances, low
   energies.
2. Exactly solvable models. (Onsager solution of 2D
   Ising model, Lieb-Liniger solution of 1D
   interacting Bose gas).
3. Renormalization group Coarse-graining the system
   and following evolution of the parameters of the
   model.
4. Numerical methods.

76
Sketch of the solution for the Ising model in 1D.
77
(No Transcript)
78
(No Transcript)
79
Onsager solution for 2D Ising model (1944).
80
Another example of solvable model attractive 1D
Bose gas.
81
This Hamiltonian has a special property. The
scattering of two particles on each other does
not depend on the presence of other particles.
Three particle scattering factorizes into the
product of two particle scatterings (Young Baxter
Conditions).
In general this is not true.
Repulsive case Lieb-Liniger solution. Confirmed
experimentally in cold atoms (Kinoshita et. al.,
2004).
82
Repulsive Bose gas.
Also, B. Paredes1, A. Widera, V. Murg, O. Mandel,
S. Fölling, I. Cirac, G. V. Shlyapnikov, T.W.
Hänsch and I. Bloch, Nature 277 , 429 (2004)
T. Kinoshita, T. Wenger, D. S. Weiss ., Science
305, 1125, 2004
83
Energy vs. interaction strength experiment and
theory.
No adjustable parameters!
Kinoshita et. Al., Science 305, 1125, 2004
84
Local pair correlations.
Kinoshita et. Al., Science 305, 1125, 2004
85
(No Transcript)
86
In the continuum this system is equivalent to an
integrable KdV equation. The solution splits into
non-thermalizing solitons Kruskal and Zabusky
(1965 ).
87
Qauntum Newton Craddle.
T. Kinoshita, T. R. Wenger and D. S. Weiss,
Nature 440, 900 903 (2006)
No thermalization during collisions of two
one-dimensional clouds of interacting
bosons. Fast thermalization if the clouds are
three dimensional.
Quantum analogue of the Fermi-Pasta-Ulam problem.
88
Kondo problem and the renormalization group.
(Poor mans scaling)
Hard to solve the problem exactly (though
possible). Idea try to eliminate electrons with
high kinetic energy since they barely interact
with the spin.
89
(No Transcript)
90
(No Transcript)
91
After integrating out high energies we find
92
Not perturbative!
93
Why cold atoms?

• Highly tunable and controlled Hamiltonians
• experimental quantum simulations of many-body
  systems
• unambiguous tests of theoretical methods.
• Nearly perfect isolation from environment
• Low external decoherence
• Possibility to study quantum many-particle
  dynamics.
• Applications to quantum information.