Quantum Field Theory in Condensed Matter Physics Luca DellAnna and - PowerPoint PPT Presentation

Quantum Field Theory in Condensed Matter Physics Luca Dell’Anna and Luca Salasnich Dipartimento di Fisica e Astronomia “Galileo Galilei”, Universit` a di Padova PhD School of Physics, February 2016 Description: an introduction to quantum field theory and quantum entanglement for macroscopic quantum phenomena: superfluidity in ultracold atomic gases and liquid helium and superconductivity in metals. Theoretical framework: finite-temperature functional integration of bosonic and fermionic fields with broken symmetry.

Program of the course (I) First part (Salasnich, 12 hours) Partition function of bosons. Bosonic fields. Bosonic coherent states. Functional integration of bosonic fields. Broken symmetry and ideal Bose gas. Interacting Bose gas. Saddle-point approximation and gaussian fluctuations. Bogoliubov spectrum and dimensional regularization. Gross-Pitaevskii equation of the Bose-Einstein condensate. Solitons and quantized vortices. Critical velocity and Landau criterion for ultracold atoms and liquid helium. Von Neuman entropy and entanglement for bipartite systems. Interacting bosons in a double-well potential: coherence visibility, Fisher information and entanglement entropy.

Program of the course (II) Second part (Dell’Anna, 12 hours) Partition function of fermions. Fermionic coherent states. Gaussian integrals with Grassman variables. Functional integration of fermionic fields. BCS theory of metals by functional integration. Hubbard-Stratonovich transformation and the bosonic field of pairing. Saddle point approximation: gap equation and critical temperature. Ginzburg-Landau theory from the BCS effective action. Gaussian fluctuations: Goldstone mode, Meissner effect and Higgs mechanism.

Suggested Books N. Nagaosa, Quantum Field Theory in Condensed Matter Physics (Springer, 1999). A. Altland and B. Simons, Condensed Matter Field Theory (Cambridge Univ. Press, 2006). M. Le Bellac, A Short Introduction to Quantum Information and Quantum Computation (Cambridge Univ. Press, 2006). H.T.C. Stoof, K.B. Gubbels, and D.B.M. Dickerscheid, Ultracold Quantum Fields (Springer, Dordrecht, 2009).

1.1 Partition function of bosons (I) Let us consider a gas of non-interacting bosons in thermal equilibrium with a bath at the temperature T . The relevant quantity to calculate all thermodynamical properties of the system is the grand-canonical partition function Z , given by Z = Tr [ e − β ( ˆ H − µ ˆ N ) ] (1) where β = 1 / ( k B T ) with k B the Boltzmann constant, ˆ � ǫ α ˆ H = N α , (2) α is the quantum Hamiltonian, ˆ � ˆ N = (3) N α α is total number operator, and µ is the chemical potential, fixed by the conservation of the average particle number. Here α represents the set of single-particle quantum numbers.

1.1 Partition function of bosons (II) Figure: Ground-state of non-interacting bosons (a) and fermions (b) is a harmonic trap. Horizontal lines describe single-particle energies ǫ α . The single-particle-state number operator ˆ N α is given by ˆ c + N α = ˆ α ˆ c α , (4) c + where the operators ˆ c α and ˆ α act in the Fock space of the identical bosons. A generic state of this Fock space is given by |{ n α }� = | ... n α ... n β ... � , (5) meaning that there are n α bosons in the single-particle state | α � , n β bosons in the single-particle state | β � , et cetera.

1.1 Partition function of bosons (III) c + The operators ˆ c α and ˆ α are called annihilation and creation operators because they respectively destroy and create one boson in the single-particle state | α � , namely √ n α | ... n α − 1 ... � , ˆ c α | ... n α ... � = (6) √ c + ˆ α | ... n α ... � = n α + 1 | ... n α + 1 ... � . (7) Note that these properties follow directly from the commutation relations c + c + c + [ˆ c α , ˆ β ] = δ α,β , [ˆ c α , ˆ c β ] = [ˆ α , ˆ β ] = 0 , (8) where [ˆ A , ˆ B ] = ˆ A ˆ B − ˆ B ˆ A .

1.1 Partition function of bosons (IV) The vacuum state, where there are no particles, can be written as | 0 � = |{ 0 }� = | ... 0 ... 0 ... � , (9) and c + ˆ c α | 0 � = 0 , ˆ α | 0 � = | 1 α � = | α � , (10) where | α � is such that � r | α � = φ α ( r ) . (11) From Eqs. (6) and (7) it follows immediately that ˆ N α counts the number of bosons in the single-particle state | α � , i.e. ˆ N α | ... n α ... � = n α | ... n α ... � . (12)

1.1 Partition function of bosons (V) Taking into account the properties of Fock state, from the definition Z = Tr [ e − β ( ˆ H − µ ˆ N ) ] (13) of the grand-canonical partition function we find �{ n α }| e − β ( ˆ H − µ ˆ α ( ǫ α − µ ) ˆ � � �{ n α }| e − β P N ) |{ n α }� = N α |{ n α }� Z = { n α } { n α } � α ( ǫ α − µ ) n α = � � e − β P e − β ( ǫ α − µ ) n α = { n α } { n α } α ∞ e − β ( ǫ α − µ ) � n � e − β ( ǫ α − µ ) n α = � � � � = (14) α n α α n =0 and finally 1 � Z = 1 − e − β ( ǫ α − µ ) . (15) α

1.1 Partition function of bosons (VI) Quantum statistical mechanics dictates that the thermal average of any operator ˆ A is obtained as A � = 1 A e − β ( ˆ H − µ ˆ � ˆ Z Tr [ˆ N ) ] . (16) It is then quite easy to show that H ′ � = 1 N ) ] = − ∂ = − ∂ N ) e − β ( ˆ H − µ ˆ � Tr [ e − β ( ˆ H − µ ˆ � � ˆ Z Tr [(ˆ H − µ ˆ N ] ∂β ln ∂β ln( Z ) . (17) By using Eq. (15) we immediately obtain � 1 − e − β ( ǫ α − µ ) � � ln( Z ) = ln , (18) α and finally from Eq. (17) we get ǫ α 1 � ˆ � � ˆ � H � = and also N � = e β ( ǫ α − µ ) − 1 . (19) e β ( ǫ α − µ ) − 1 α α

1.2 Bosonic fields (I) Introducting the bosonic field operator ˆ � ψ ( r ) = ˆ c α φ α ( r ) , (20) α one immediately finds − � 2 � � 2 m ∇ 2 + U ( r ) � ˆ d 3 r ˆ ˆ � ψ + ( r ) c + H = ψ ( r ) = ǫ α ˆ α ˆ c α , (21) α where U ( r ) is the external trapping potential of bosons and φ α ( r ) are the orthonormalized single-particle eigenfunctions of the single-particle stationary Schr¨ odinger equation − � 2 � 2 m ∇ 2 + U ( r ) � φ α ( r ) = ǫ α φ α ( r ) (22) with single-particle eigenvalues ǫ α . Again α represents the set of single-particle quantum numbers.

1.2 Bosonic fields (II) A remarkable property of the adjoint field operator ˆ � ψ + ( r ) = c + α φ ∗ ˆ α ( r ) , (23) α is the following: ψ + ( r ) | 0 � = | r � . ˆ (24) That is the operator ˆ ψ + ( r ) creates a particle in the state | r � from the vacuum state | 0 � . In fact, ˆ � � ψ + ( r ) | 0 � c + α φ ∗ c + = ˆ α ( r ) | 0 � = ˆ α � α | r �| 0 � α α � � c + = ˆ α | 0 �� α | r � = | α �� α | r � = | r � , α α because of the completeness relation (closure) � | α �� α | = 1 . (25) α

1.2 Bosonic fields (III) It is straightforward to show that the bosonic field operators satisfies the following commutation rules [ ˆ ψ ( r ) , ˆ ψ + ( r ′ )] = δ ( r − r ′ ) . (26) and also [ ˆ ψ ( r ) , ˆ ψ ( r ′ )] = [ ˆ ψ + ( r ) , ˆ ψ + ( r ′ )] = 0 . (27) Let us prove Eq. (26). By using the expansion of the field operators one finds [ ˆ ψ ( r ) , ˆ ψ + ( r ′ )] � φ α ( r ) φ ∗ β ( r ′ ) [ˆ c + = c α , ˆ β ] α,β � φ α ( r ) φ ∗ β ( r ′ ) δ α,β = � φ α ( r ) φ ∗ α ( r ′ ) = α,β α � � � r | α � � α | r ′ � = � r | | α � � α | r ′ � = α α � r | r ′ � = δ ( r − r ′ ) . =

1.3 Bosonic coherent states (I) It is interesting to observe that c + � n α | ˆ c α | n α � = � n α | ˆ α | n α � = 0 . (28) This result of is due to the fact that the expectation value is performed with the Fock state | n α � , which simply means that the number of bosons in the single-particle state α is fixed because ˆ N α | n α � = n α | n α � , (29) with ˆ c + N α = ˆ α ˆ c α . In some case the number of bosons is not fixed, in other words the system is not in a pure Fock state. For example, the radiation field of a well-stabilized laser device operating in a single mode | α � is described by a coherent state | c α � , such that ˆ c α | c α � = c α | c α � . (30) The coherent state | c α � , introduced in 1963 by Roy Glauber, is thus the eigenstate of the annihilation operator ˆ c α with complex eigenvalue c α = | c α | e i θ α .

1.3 Bosonic coherent states (II) The coherent state | c α � does not have a fixed number of photons, i.e. it is not an eigenstate of the number operator ˆ N α , and it is not difficult to show that | c α � can be expanded in terms of number (Fock) states | n α � as follows ∞ c n α √ n α ! | n α � = e −| c α | 2 / 2 e c α ˆ | c α � = e −| c α | 2 / 2 c + � α α | 0 � , (31) n α =0 taking into account the normalization to one, i.e. � c α | c α � = 1 . (32) From Eq. (30) one immediately finds N α | c α � = | c α | 2 , N α = � c α | ˆ ¯ (33) and it is natural to set � N α e i θ α , ¯ c α = (34) where ¯ N α is the average number of photons in the coherent state, while θ α is the phase of the coherent state.