Assignments to Solid State Theory I/II Problem Set: Quantized Chains And

University of Regensburg, Physics Department WS 2004-05

Assignments to Solid State Theory I/II

Prof. Dr. M. Grifoni (Phy 3.1.29) Room Phy 5.1.03, 3pm Dr. G. Cuniberti (Phy 4.1.29) Tue 02.11.2004 (Sheet 2)

Problem set: Quantized chains and second quantization

2.1. The quantized elastic chain as a bosonic Hamiltonian

The quantized elastic chain Hamiltonian

L 2 2 1 2 ka Hˆ [φ,ˆ πˆ] = πˆ (x) + ∂xφˆ(x) dx. 0 2m 2 Z is the Hamiltonian of the continuum limit of the quantum harmonic chain of length L (done in class). The quantum fields φˆ and πˆ, which are canonically conjugated [πˆ(x), φˆ(x0)] = i~δ(x x0), can be expressed in Fourier series in the interval 0 x L: − − ≤ ≤ 1 +ikx φˆ(x) = e φˆk, √L Xk 1 ikx πˆ(x) = e− πˆk, √L Xk where the momentum represented fields are functions of the quantized momen- ta k = 2πm/L, m Z (not to be confused with the “quantum operator field momentum” πˆ) and∈are thus written as

L 1 ikx φˆk = e− φˆ(x) dx, √L 0 Z L 1 +ikx πˆk = e πˆ(x) dx. √L Z0 (a) Show that the Fourier representation of the canonical commutation relations reads [πˆ , φˆ 0 ] = i~δ 0 . k k − kk Note: The real classical ﬁelds φ(x) are quantized to an Hermitian quantum ﬁeld ˆ ˆ ˆ φ(x). As a consequence φk = φ† k and similarly for πˆk. − (b) Show that the Fourier representation of the Hamiltonian operator takes the form

2 1 ka 2 Hˆ = πˆkπˆ k + k φˆkφˆ k . 2m − 2 − Xk (c) Deﬁning

mωk 1 ak φˆk + i πˆ k , ≡ 2~ mω − r k where ω = a k/m k = v k show that operators a ’s obey the canonical k | | | | k commutation relations [a , a† 0 ] = δ 0 , and [a , a 0 ] = 0. p k k kk k k (d) Finally, with this deﬁnition, show that the Hamiltonian can be expressed in the form 1 Hˆ = ~ω a† a + . k k k 2 Xk Ergo, show that even the elastic quantum chain (as it is for the discrete quantum harmonic chain) is a diagonal “bosonic hamiltonian”. Note: We have dropped out, in the boson ﬁelds a’s, the “operator hat”: sooner or later one has to do that!

2.2. Speciﬁc heat of solids (Debye model)

Recall the “specific heat dilemma” and the solution provided by Einstein (done in class3). Note: The “specific heat dilemma” is due to the fact that classical mechanics cannot explain the low temperature vanishing behavior of the specific heat. Einstein’s solution consisted in modelling lattice dynamics as N normal modes vibrating at the same frequency ω0. Einstein’s model predicts an exponential vanishing low temperature behavior of the specific heat but experiments with first refrigerators showed T 3 dependences at low temperatures. This further “power law vs. exponential” dilemma was solved by Debye in a model which is the core of this exercise. The following figure shows the plot of silver specific heat (the experimental data points fit very nice- ly Debye theory solid curve at both low and high temperatures.4). Let us thus

calculate the correct Debye phonon curve stepwise:

3D. L. Goodstein, States of Matter (Dover Publications, Inc., 1985). 4 Taken from H. P. Myers, Introductory Solid State Physics, 2nd ed. (Taylor & Francis, 1997).

2 (a) Evaluate the phonon density of states (ω) for a 3d solid assuming a linear dispersion relation ω(k) = vk for the threeD acoustic branches (here v is an average sound speed) (b) Prove that the speciﬁc heat C of the solid then goes as C T 3 (for low T ) using the expression C = ∂U/∂T , where ≈

∞ U = ~ω n (~ω) (ω) dω, BE D Z0 where 1 nBE(E) = eE/(kBT ) 1 − is the Bose-Einstein distribution. What is the large T limit in this approximation? (c) Show that by cutting oﬀ the dispersion relation up to a maximum wave vec- tor kDebye a new density of states Debye(ω) is obtained. The latter is usually normalized by including the eﬀectDof the optical branches upon requiring

vkDebye N = (ω) dω. DDebye Z0

Show that Debye (which still leads to the correct low temperature behavior for the specificD heat) reproduces the expected classical result for the high temperature specific heat C Nk (also known as Dulong and Petit law). ≈ B Note: At even lower temperatures, the specific heats of metals show linear dependences. The reason for that is the linear electron contribution to specific heat for very low temperatures in metals which becomes larger than the T 3 dependent phonon contribution. The fact that this linear contribution does not emerge at higher temperatures was another mystery of early part of the 20th century: How could electrons contribute to electrical conduction and heat conduction and not to the specific heat? The electrons in the metal which contribute to conduction are very close to the Fer- mi level, “ripples on the Fermi sea”. But to contribute to bulk specific heat, all the valence electrons would have to receive energy from the nominal thermal energy kBT . The Fermi energy, however, is much greater than kBT and the overwhelming majority of the electrons cannot receive such energy since there are no available energy levels within kBT of their energy. The small fraction of electrons which are within kBT of the Fermi level does contribute a small specific heat, and this electron specific heat becomes significant only at very low temperatures.

2.3. Cubic correction to the q-harmonic oscillator

Calculate the correction to the frequency of an oscillator in its ground state due to a cubic anharmonicity. It arises by expanding the adiabatic potential beyond the harmonic terms. This corresponds to an oscillator problem with the hamiltonian

3 H = ~ω0 a†a + ∆(a† + a) .

3 Treat the anharmonicity by bringing the third order terms in the phonon operators ﬁrst into normal order and reduce them by replacing the number operator whenever possible by the thermal expectation value n(T ). The self-consistent harmonic phonon approximation consists in considering the anharmonic correction in Brillouin-Wigner perturbation theory. Find the lowest eigenvalue and discuss its dependence on T to understand the meaning of a soft mode.