ntroduction to Condensed matter theory Ehud Altman Weizmann

ntroduction to Condensed matter theory Ehud Altman Weizmann

Essence of (quantum) Condensed Matter physics 1. Take a piece of junk: 2. Cool it down 3. measure something: e. g transport sij (T, w) , k(T ) or scattering intensity S(q, w)

And a miracle occurs Example: the quantum hall effect Hall resistivity quantized to amazing precision: (with n integer) Transition of rxx on going from plateau to step: How to understand such beautiful universal data given the complicated mess that is the sample?

The underlying microscopic theory of everything in condensed matter is known Isn’t condensed matter simply a complicated and messy exercise in quantum mechanics?

Let’s solve the exercise and move on … Simplified Hamiltonian: n=100 particles on M=200 sites What is the dimension of the Hilbert space? Fermions: Bosons: Need to store vectors larger than particle number in the universe ! In generic cases Hmic is fundamentally insoluble!

What is condensed matter ? Rather than solve a horrendous hamiltonian CM aims to uncover the organizing principles and emergent properties of matter at large scales. Because these properties cannot be directly derived from the fundamental forces they are, in a sense, also fundamental. A few Iron atoms are paramagnetic A chunk of iron is a permanent magnet “More is different” P. W. Anderson, Science (1972)

Framework for analyzing emergent phenomena Fundamentally insoluble ! More modest question: How does the system appear to a probe with low resolution ? If we stand close we can see every grain of sand. But if we can see only sand dunes we might be able to explain their shapes using simpler effective dynamics ! Don’t need to know the trajectory of every grain!

Framework for analyzing emergent phenomena Renormalization Effective low energy, long wave-length theory Fermi liquid Broken symmetry Quantum-phases = Stable fixed points: Systems with different microscopic interactions appear the same when probed over sufficiently long length and time scales. Universality ( Quantum phase transitions = Unstable fixed points ) ?

Fundamental principles that can guide us in explaining properties of the phases of matter • Broken symmetry (order) and rigidity • Fermi surface • Topology

Solid order Liquid Solid Crystals are ordered (periodic) – Broken translational symmetry This is how X-rays tell the difference between solid and liquid: Order parameter:

Solids are rigid This is how penguins tell the difference between solid and liquid Perfect transmitter of shear force!

Low energy effective theory Expand in small displacements around a Broken symmetry configuration: Fourier transform to obtain normal modes (independent oscillators)

Phonons are a particular example of Goldstone Bosons A concequence of broken (continuous) translational symmetry q=0 : uniform translation of the solid (Symmetry operation) Eel=0 q→ 0 : close local approximation to a uniform translation Eel ~ q Argument breaks down in case of long range interactions (e. g. coulomb).

Another example of Broken symmetry: Superfluidity of the interacting Bose gas Macroscopic occupation of a single-particle wave-function : Broken U(1) symmetry (subtle, more on this later …) What is the analogue of rigidity? Something that even penguins can feel …

Phase rigidity “Elastic” energy cost: Phase stiffness (rigidity): What is the perfectly transmitted quantity (analogue of the force in a solid)?

Phase rigidity → Macroscopic persistant current Despite the energy cost, current cannot decay. Topologically protected ! Note: single valuedness of y requires integer winding.

Solid Comparison table Superfluid

Quantized vortices ? Irrotational flow ? True except at possible point singularities of j , (topological defects): n=1 n = -1 Rotation is concentrated at points and quantized (integer phase winding n). Compare classical rigid rotation: (Uniform and non-quantized)

Image of real vortices in a rotating Bose gas MIT 2001

Vortices provide a mechanism for current decay A vortex transversing the sample can unwind the twist (or vortexantivortex pair generated in the middle and taken to the edges) What are the analogues of vortices in a rigid solid? Dislocation lines - A solid yields due to motion of dislocation lines - A superfluid yields (dissipates current) when vortices start to flow

More on broken symmetry Noether’s theorem: For every symmetry of the Hamiltonian there is an associated conserved quantity which is the generator of that symmetry. Example – Translation operator Translation generator: Translational invariance Momentum conservation P is not conserved in a solid, where translation symmetry is broken! (Only crystal-momentum which is the generator of the discrete lattice translation group is conserved in a crystal).

Nature of the broken symmetry in a superfluid All terms in H have the same number of b and b+. Total particle number conservation. The same property ensures invariance of H under global U(1) transformations: Hence the conserved number N is the generator of the global U(1) symmetry. Global “phase” operator conjugate to N: Broken of U(1) symmetry Total number not conserved! What does it mean?

Effetive low energy theory Expand to quadratic order in the fluctuations n and j Quantize with the local commutator: Quantum Hydrodynamics Note: We can neglect the last term in Heff compared to the second only if we resolve scales larger than a healing length This is the short distance cutoff of theory.

Collective modes Fourier transform the hydrodynamic theory to obtain decoupled oscillators (phonons): These are the Goldstone modes associated with broken U(1) symmetry Nir Davidson’s group PRL 2002 High momentum cutoff of the low energy theory

Universality of the low energy spectrum Superfluid Helium T ~ 1 °K n ~ 10 -23 cm-3 (Neutron scattering) Henshaw & Woods, 1961 Rubidium condensate T ~ 10 -6 °K n ~ 10 -13 cm-3 (Bragg spectroscopy) Davidson group PRL 1961

Last example of broken symmetry: the antifferromagnet Local order parameter : Broken su(2) symmetry La 2 Cu. O 4 Order parameter dynamics: Linearize to obtain spin-wave spectrum Hayden etal, PRL 91

Electrons in a Crystal Periodic potential: Single electron energy bands: (Bloch bands) Many electrons (neglecting interactions): Fill bands up to chemical potential

Fermi surface Empty particle Full hole • All low energy excitations: particle hole pairs near Fermi surface • For the low energy excitations band structure is important only for determining the shape and topology of the Fermi surface.

For all this we assumed non interacting electrons ! ky Empty Full kx Does the concept of a Fermi surface survive in the presence of interactions between the electrons?

Consider a Fermi gas with one extra particle: k’-q k Is this an exact eigenstate in the presence of interactions? k’ k+q Pauli principle and energy conservation restrict the summation over initial and final states to narrow bands. Fermi gas ~ perturbatively stable w. r. t interaction Fermi liquid theory

Momentum distribution in a Fermi liquid Non interacting: Fermi gas Interacting: Fermi liquid

Fermi Liquid theory of metals Quasi-particle Essentially non interacting fermions at low energy and low temperature. Pottassium

Kammerling Onnes From Nobel lecture (1913): r Superconductivity ! T [K] Fermi liquid theory must be unstable to something!

Superconductors also expell magnetic fields: Meissner effect

Pairing instability Consider interaction in a particular channel: between a time reversed pair of electrons.

Poor man’s RG 2 nd order perturbation theory: Pairing instabillity

Pairing instability For attractive int. there is an energy scale for which the denominator vanishes and the perturbative approach fails This is the binding energy of electron pairs

Electron pairs behave like charged bosons Superfluid of charged bosons is a superconductor! Apply the following gauge transformation (to gauge away the phase): Electromagnetic field (photon) becomes gapped (Higgs mechanism) Meissner effect !

Frontiers in CM physics • Are there quantum phases of spins or bosons that do not involve symmetry breaking? • Are there conducting states of Fermions that are not described by Fermi liquid theory?

“S ta nd de ls” Strongly correlated quantum systems Breakdown of the standard models

Failure of Fermi-Liquid theory Normal state of the cuprates (High Tc superconductors) ARPES spectra Linear in T resistivity T<Tc q. p. peak T>Tc no q. p. peak A long standing puzzle ! Non Fermi liquid behavior in heavy fermion materials, Mn. Si …

Failure of Landau theory Spin-½ AFM on the Kagome lattice: OH)6 Cl 2)Zn. Cu 3 Highly frustrated magnet Helton etal cond-mat/0610539 No magnetic order down to lowest T ! Quantum spin liquid state?