Quantum Field Theory and Condensed Matter Physics: making the vacuum concrete. Fabian Essler (Oxford)

Transcription

1 Quantum Field Theory and Condensed Matter Physics: making the vacuum concrete Fabian Essler (Oxford) Oxford, June 2013

2 Lev Landau This work contains many things which are new and interesting. Unfortunately, everything that is new is not interesting, and everything which is interesting, is not new.

3 The Field is real! hbar>0!

4 The Field is real! hbar>0!

5 Here: Relativistic QFT as an emergent (rather than fundamental) phenomenon.

6 Length/Energy Scales: Organizing Principle of Physics high energies short distances low energies large distances Think of this in terms of probing a physical system by e.g. light: to resolve what happens at length scale a, the wavelength λ must be smaller than a. short distances small λ high energies as E=hc/λ

7 Length Classical Mechanics Quantum Mechanics Standard Model ToE?

8 A Reductionist View of the World Quantum Field Theory Quantum Mechanics Chemistry If we understand QFT/ToE, we understand everything.

9 A Reductionist View of the World Quantum Field Theory Quantum Mechanics Chemistry If we understand QFT/ToE, we understand everything. Depends on what understand means...

10 Following 30 minutes: Quantum Field Theory Quantum Mechanics Quantum Field Theory interesting perspective on QFT (and perhaps the reductionist view)

11 Example: Lattice Vibrations in Crystals Crystal: atoms in a periodic array Interaction between the atoms (e.g. Coulomb) causes oscillations of atoms Crystal mean ( equilibrium ) positions V ( ~ R 1, ~ R 2,..., ~ R N ) ~R (0) 1, ~ R (0) 2,...

12 Lattice Vibrations of a Linear Chain equilibrium positions: R (0) j = ja 0... a 0 a 0 a 0 a 0 oscillations: r j = R j R (0) j... r 1 r 2 r 3 deviations from equilibrium positions

13 Main interactions between nearest neighbours Oscillations around equilibrium positions typically small: r j a 0 V (R 1,R 2,...,R N )=V 0 + apple 2 X (r j r j+1 ) N 1 j=1 V 0 = V (R (0) 1,R(0) 2,...,R(0) N ) The linear terms add up to zero equilibrium positions... small cubic, quartic etc anharmonic terms

14 H = NX l=1 {z} p 2 l 2m + apple 2 Kinetic energy X (r l r l+1 ) N 1 l=1 {z} Potential energy N coupled harmonic oscillators! Quantum mechanically: p l l

15 Solution of the Classical Problem Newton s equations: r 2 = apple(r l+1 r l ) apple(r l r l 1 ) Ansatz: Works if r l (t) =A cos(kla 0!t + ) periodicity in k periodicity of the lattice

16 finite number of atoms finite number of normal modes

17 Quantum Mechanics Each ω(k) gives a simple harmonic oscillator (in an appropriate coordinate) E x quantized energies:

18 Ground State ( Vacuum ): zero-point energy: wave function: (r 1,...,r N ) / exp 1 2 X j,k 1 r j M jk r k A 3.0 Probability distr. of 2 nd atom

19 Excited states: Single particles ( Phonons ) wave function: (r 1,...,r N ) / " N X n=1 cos(k l n)r n # GS Two phonons:

20 Quantum Field Theory Recall that H = NX l=1 p 2 l 2m + apple 2 X (r l r l+1 ) N 1 l=1 p l l

21 Quantum Field Theory Define Hamiltonian becomes H = a 0 N X l=1 m (la 0 ) 2 + apple 2 h i 2 ([l + 1]a 0 ) (la 0 ) Now consider the limit N!1, a 0! 0, apple!1 L = Na 0 and apple = applea 2 0 fixed volume vibrations remain a 0

22 H! Z L 0 dx " # corresponding Lagrangian density L(t, x) = " m (t, 2 apple 2 # (t, Massless (relativistic) Scalar Field

23 Which part of the physics does QFT describe? Large distances compared to lattice spacing a 0 Small frequencies ω(k), i.e. low energies i.e. normal modes in this region

24 Which part of the physics does QFT describe? length a 0 atomic scale emergent collective wave-like excitations non-relativistic Physics relativistic QFT

25 Which part of the physics does QFT describe? length a 0 atomic scale emergent collective wave-like excitations non-relativistic Physics relativistic QFT Emergent Physical Law

26 Measuring the collective modes: inelastic neutron scattering neutron gives energy ω and momentum k to the crystal excites a single normal mode if ω and k match Fig 4. g. 2 Neutron diffraction patterns from polycrystalline manganese oxide, MnO, at temperatures (a) belo Phonon spectrum of Cu

27 Fig 4.

28 Main Points so far Quantum mechanics of atoms in solids can give rise to collective excitations, which are described by a QFT The lowest energy state ( vacuum ) has a non-trivial wave function.

29 Main Points so far Quantum mechanics of atoms in solids can give rise to collective excitations, which are described by a QFT The lowest energy state ( vacuum ) has a non-trivial wave function. Much more exotic physics and QFTs can arise in this way!

30 Splitting the Electron Consider electronic degrees of freedom in quasi-1d crystals Sr 2 CuO 3 low energy electronic physics due to outer electrons on Cu atoms (black) anisotropy e - move essentially only along 1D chains

31 Basic model: Lattice: on each site either 0, 1 or 2 electrons (spin!)... Electrons can hop to neighbouring sites... Electrons repel through Coulomb interaction...

32 ( ) ( ) Field Theory Limit: description in terms of Ψ 1 (t,x) = ( ) R (t,x) L (t,x), Ψ 2 (t,x) = ( R (t,x) ) L (t,x) fermionic quantum fields of these spinor fields the Hamiltonian (10.24) without t 2 species of 2-dimensional Dirac spinors L(t, x) = 2X a=1 a(t, x) apple a(t, x)+g 3X =1 J 1 (t, x)j 1 (t, x) J 0 (t, x)j 0 (t, x) (t, x) = (t, x) 0 J µ = 1 2 a µ ab b, µ =0, 1 SU(2) Thirring Model 0 = 0 = i y 0 1 = = 1 = x 0 1 = 1 0

33 Measure excitations by scattering photons (Angle Resolved Photo Emission Spectroscopy) Count emitted electrons with given energy and momentum ( angles)

34 Measure excitations by scattering photons (Angle Resolved Photo Emission Spectroscopy) If the collective excitations are electrons, we expect to see something like this

35 Spectral Function : ARPES on SrCuO 2 (Kim et al 06) Instead, for quasi-1d systems one observes The electron has fallen apart! ( spin-charge separation )

36 How to understand this? In Sr 2 CuO 3 we have 1 electron per site Ground State (vacuum) e - emitted e - hopping spin flip holon spinon

37 The origin for this splitting of the electron is the highly non-trivial nature of the ground state (vacuum)!!

38 Summary Quantum Field Theories often describe collective properties of solids at large distances/low energies. Lorentz covariance can be an emergent feature in this regime. Low-energy physics can be very exotic because the vacuum is highly non-trivial. In the Cond. Mat. context it is possible to vary the spatial dimensionality D=1,2,3 (anisotropy!).