Some Concepts of Condensed Matter Physics Anatoli Polkovnikov
- Slides: 93
(Some) Concepts of Condensed Matter Physics Anatoli Polkovnikov, Boston University AFOSR
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…”
Two examples. Single neuron – relatively easy to characterize. ~1010 neurons ? ? ? NAND gate Computers are complicated, but we understand them Is “more” fundamentally different or just more complicated?
From single particle physics to many particle physics. Classical mechanics: Need to solve Newton’s 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.
Quantum mechanics: Need to solve Schrödinger equation. Instead of two numbers need to know the whole complex field. Simplify the problem: Let x to be discrete : x={x 1, …, x. M}, M is the size of the Hilbert space. Now we need to solve M (complex) ordinary differential equations. n particles. How many equations do we need to solve? Need to compute the Hilbert space size.
M n Use specific numbers: M=200, n=100. Fermions: Bosons: A computer built from all known particles in universe is not capable to exactly simulate even such a small system.
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 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.
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 ? ? ?
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.
Crystals. Wigner-Seiz cell Unit cell – fills the space when translated by all possible r Translational symmetries and point group symmetries define Bravais lattices (5 in 2 D, 14 in 3 D). Oblique Rectangular Rect. FC Square Hexagonal
Reciprocal Lattice Choose the basis for G: b 1, b 2 such that The set of G forms reciprocal lattice. Wigner-Seitz cell of the reciprocal lattice is the Brillouin zone.
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:
Bloch Theorem. Suppose n(r) is a (non-degenerate) solution: Note
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.
Consider a wave packet. k k’ m* is the effective mass. Semiclassical equations of motion in a smooth external field: more details in Ashcroft-Mermin book.
Motion in a weak periodic potential.
Free particle dispersion, conventional vs. Bloch pictures. Image from I. Bloch, Nature Physics 1, 23 - 30 (2005) Now add a weak periodic potential. First order perturbation theory:
Second order perturbation theory: Perturbation theory diverges near the edges of the Brillouin zone! We have to use the degenerate perturbation theory.
Keep only two terms in the wave function Need to solve the secular equation.
Tight binding approximation. Very strong atomic potential. Different atomic orbitals barely overlap.
Example: square lattice
Evolution of the single-particle spectrum tight-binding regime free particle spectrum
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:
Fermi distribution function.
Electrons in solids.
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)
Fermi surface: E(k)=Ef
Interacting systems. Mean field theory and broken symmetry. (following notes by S. Girvin, Yale University) Start with a classical Ising model.
Spin ordering is a collective effect!
Self consistent method. Many nearest neighbors. (large dimensions).
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. Alternative approach. Variational method.
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 H 0 and minimize How to find the right mean field approach or H 0? How accurate are these methods?
Quantum Mean Field Theory.
Fermi liquid. No broken symmetries.
Self consistent method.
Some possible broken symmetries and phases. 1. Translational symmetry (charge density wave). CDW in Na. Se 2, K. Mc Elroy, U. of Colorado, Boulder
Broken time reversal symmetry: Ferromagnets. Broken time reversal + translational symmetry: Antiferromagnets.
Broken gauge symmetry: superfluids.
Weakly repulsive Bose gas.
Choose trial Hamiltonian.
Keep only k=0 term.
BCS theory of superconductivity.
Sketch of the BCS theory.
Failures of mean-field theories. Low-dimensional systems. Ising model in 1 D.
Two (and more) dimensions
More dimensions or more nearest neighbors? Periodic boundary conditions: z=4, like in 2 D. Long range order is destroyed in 1 D (quasi 1 D) because of special correlated defects. Mean field theory does not capture such defects.
Systems with continuous symmetries. Mermin-Wagner theorem. Consider planar spins in d-dimensions (xy-model)
Fluctuations around mean field solution destroy long range order in 1 D at all temperatures.
Correlation functions:
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. In 3 D and above long range order is not destroyed by small fluctuations. Mean field approach is good.
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 antiferromagnets, superfluids, superconductors… This theorem assumes there are no long range interactions.
Quantum systems.
Ground state No long range order in 1 D at zero temperature due to quantum fluctuations. True for superfluids, antiferromagnets. Not true for ferromagnets.
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 2 D Ising model, Lieb-Liniger solution of 1 D interacting Bose gas). 3. Renormalization group: Coarse-graining the system and following evolution of the parameters of the model. 4. Numerical methods.
Sketch of the solution for the Ising model in 1 D.
Onsager solution for 2 D Ising model (1944). 3 D: NP complete problem. Can be solved only numerically
Another example of solvable model: attractive 1 D Bose gas.
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).
Repulsive Bose gas. T. Kinoshita, T. Wenger, D. S. Weiss. , Science 305, 1125, 2004 Also, B. Paredes 1, A. Widera, V. Murg, O. Mandel, S. Fölling, I. Cirac, G. V. Shlyapnikov, T. W. Hänsch and I. Bloch, Nature 277 , 429 (2004)
Energy vs. interaction strength: experiment and theory. No adjustable parameters! Kinoshita et. Al. , Science 305, 1125, 2004
Local pair correlations. Kinoshita et. Al. , Science 305, 1125, 2004
In the continuum this system is equivalent to an integrable Kd. V equation. The solution splits into non-thermalizing solitons Kruskal and Zabusky (1965 ).
Qauntum Newton Craddle. T. Kinoshita, T. R. Wenger and D. S. Weiss, Nature 440, 900 – 903 (2006) No thermalization during collisions of two onedimensional clouds of interacting bosons. Fast thermalization if the clouds are three dimensional. Quantum analogue of the Fermi-Pasta-Ulam problem.
Kondo problem and the renormalization group. (Poor man’s 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.
After integrating out high energies we find:
Not perturbative! Antiferromagnetic coupling J>0 – relevant perturbation, grows under RG. Ferromagnetic coupling J<0 – irrelevant perturbation, goes to zero under RG.
Why cold atoms? 1. Highly tunable and controlled Hamiltonians: • experimental quantum simulations of manybody systems • unambiguous tests of theoretical methods. 2. Nearly perfect isolation from environment • Low external decoherence • Possibility to study quantum many-particle dynamics. • Applications to quantum information.
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