Tutorial class 1: Quantization of the harmonic chain 1

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1 M Quantum Physics Academic year 3-4 Broen symmetries and quantum phase transitions Tutorial class : Quantization of the harmonic chain We consider a one-dimensional crystal made of pointlie masses m = ρa separated by a distance a at rest. Neighboring masses are lined by springs with constant s = κ/a. We want to study the quantum dynamics of this system first by following the canonical quantization rules and then by writing the partition function as a functional integral over a displacement field. Classical analysis. Write the Lagrangian L and determine the eigenmodes of the system from the Euler-Lagrange equations. Give the expression of the sound velocity c in the crystal.. In the low-energy or long-distance limit, we can approximate the crystal by a continuous elastic chain. We note φx, t the displacement of the infinitesimal mass ρdx located between x and x + dx at rest. The Lagrangian L[φ] = dx L x φ, φ becomes a functional of the field φx, t. What is the Lagrangian density L? What is the equation of motion satisfied by φ? Compare with the result of question.. Canonical quantization. What is the momentum Πx, t conjugated to the displacement field? Give the expression of the classical Hamiltonian of the system.. In the quantum description, the classical fields φx, t and Πx, t become operators ˆφx and ˆΠx time independent in the Schrödinger picture. What are the commutation rules satisfied by these operators? Using the operators ˆφ = L dx e ix ˆφx = ˆφ, L ˆΠ = L dx e ix ˆΠx = ˆΠ, L show that the quantum Hamiltonian Ĥ corresponds to a set of decoupled harmonic oscillators with frequencies ω = c..3 Show that the Hamiltonian can be diagonalized by introducing the ladder operators ρω [ i ] â = ˆφ + ˆΠ, ρω â ρω [ = ˆφ i ] Π. ρω What is the physical meaning of the operators â and â? What are the eigenstates of Ĥ? 3 Students are encouraged to answer questions and before the tutorial class, only question 3 will be discussed in detail.

2 3 Functional integral 3. We introduce the states φ and Π defined by ˆφx φ = φx φ and ˆΠx Π = Πx Π. Show that these states satisfy the closure relations L/a a N lim N lim l= L/a a l= dφla φ φ = Î, dπla Π Π = Î, 4 Where Î is the identity operator. In order to correctly define the closure relations, we have discretized the chain the continuous variable x becoming a discrete variable la; the continuous chain is obtained in the limit a. N and N are normalization constants that will be ignored in the following. 3. Show that the partition function Z = φx,β=φx, D[φ] e S[φ] 5 can be written as a functional integral over a real field φx, τ τ [, β], with the Euclidean action S[φ] = 3.3 Write the action S[φ] as a function of the Fourier transformed field β dτ φ, iω n = βl β What is the expression of the frequency ω n. L dτ [ dx ρ φ + κ x φ ]. 6 L dx e ix ωnτ φx, τ Using properties of Gaussian integrals, compute the partition function Z from equation What can we deduce from the equation of motion δs/δφx, τ =? Show that at sufficiently high temperature, only the field φ, iω n= contribute to the functional integral. What is then the expression of the action? Show that at zero temperature we recover the action of a classical two-dimensional system. 3.6 Using properties of Gaussian integrals, calculate the propagator of the field φ G, iω n = φ, iω n φ, iω n = D[φ] φ, iω n φ, iω n e S[φ]. 8 Z Show that G, τ τ = φ, τφ, τ coincides with the correlation function T τ ˆφ, τ ˆφ, τ θτ τ ˆφ, τ ˆφ, τ + θ τ + τ ˆφ, τ ˆφ, τ, 9 where ˆφ, τ = eĥτ ˆφe Ĥτ is an operator in the Heisenberg picture in imaginary time. 3.7 Which information do we obtain from the poles of the retarded propagator G R, ω = G, iω n ω + iη η +? Show that the spectral function A, ω = I[G R, ω] contains information about the excitation spectrum. 3.8 We consider a more general case where the Lagrangian L[φ] includes the term dx V φ with V a polynomial? What is then the Euclidean action S[φ]? What is the relation with the quantization of a relativistic scalar field in field theory?

3 M Quantum Physics Academic year 3-4 Broen symmetries and quantum phase transitions Tutorial class : ϕ theory with ON symmetry From Ising model to ϕ 4 theory. We consider the Ising model, βh = σ i K ij σ j defined on a hypercubic lattice in d dimensions σ i = ±. K ij is equal to βj = J/T for two neighboring sites and vanishes otherwise.. Calculate the critical temperature T c in the mean-field approximation.. Using the Hubbard-Stratonovich transformation e i,j σikijσj i,j dϕ i e i i,j ϕik ij ϕj+ i ϕiσi, rewrite the partition function as an integral over real fields ϕ i varying between and. Show that the corresponding action or Hamiltonian can be written as { S[ϕ] = d d r ϕ + r ϕ + u } 4! ϕ4 3 in the continuum limit if terms of order ϕ 6, as well as higher-order derivative terms, are neglected. What is the expression of r and u? Can we generalize the derivation of Eq. 3 to the Heisenberg model or to a N-component spin model? ON model. We consider the ON model, defined by the action { S[ϕ] = d d r ϕ + r ϕ + u 4! ϕ }, 4 on a hypercubic lattice in d dimensions. ϕ = ϕ,, ϕ N T is a N-component vector, ϕ = N i= ϕ i and ϕ N i= ϕ i. We note r = r T T c, and assume r and u to be temperature independent. The model is regularized in the large momentum limit by a cutoff Λ.. Calculate m = ϕr in the mean-field approximation. Which symmetry is spontaneously broen in the low-temperature phase?. Expand the action S[ϕ] about its mean-field value to quadratic order in the fluctuations ϕ m Gaussian approximation. Deduce the expression of the propagator G i q = ϕ i qϕ i q ϕ i q ϕ i q 5 consider separately the high- and low-temperature phases. Do the results in the ordered phase agree with Goldstone s theorem?.3 What are the values of the critical exponents ν, β, γ and η in the Gaussian approximation?.4 Calculate the mean value ϕ i r m i in the ordered phase. What can we conclude regarding the existence of an ordered phase at nonzero temperatures in dimension d Mermin-Wagner theorem?

4 .5 Using dimensional analysis, determine the dimension [ϕ] of the field ϕr as well as [r ] and [u ]. We recall that X has dimension x = [X] if it is expressed in physical units of L x with L a length scale. Show that we can rewrite the action in the dimensionless form S[ ϕ] = d d r { r ϕ + ϕ + ũ ϕ }. 6 4! Deduce that the Gaussian approximation studied in., and more generally the perturbative approach in u, is correct for d > 4 but fails for d < 4. Show that in the latter case, one can nevertheless define a temperature T G the Ginzburg temperature above which the Gaussian approximation remains valid.. [ϕ] as defined in the text is called the naive scaling dimension or engineering dimension of the field ϕ.

5 M Quantum Physics Academic year 3-4 Broen symmetries and quantum phase transitions Tutorial class 3: Bogoliubov s theory of superfluidity We consider a three-dimensional boson gas at zero temperature β described by the action S[ψ, ψ] = β dτ { d 3 r ψ r, τ τ µ ψr, τ + g } m ψ r, τψ r, τψr, τψr, τ, with a ultraviolet momentum cutoff Λ Λ. Excitation spectrum.. What is the expression of the action in a saddle-point approximation where the field ψr, τ = ψ is assumed uniform and time independent. Under which condition do we have ψ? What is the symmetry which is spontaneously broen? Why can we choose ψ real with no loss of generality?. We now consider fluctuations ψ r, τ = ψr, τ ψ of the field about its saddle-point value ψ. What is the action S[ψ, ψ ] to quadratic order in ψ?.3 We introduce the two-component field ψ Ψ, iω n =, iω n ψ,, iω n Ψ, iω n = ψ, iω n, ψ, iω n, where ψ, iω n = β dτ d 3 r e i r ωnτ ψ r, τ, βv ψ, iω n = β dτ d 3 r e i r ωnτ ψ r, τ. βv Show that we can write the action in the form S = S + Ψ, iω n D, iω n Ψ, iω n, 4,ω n where S is the saddle-point value and D, iω n a matrix. What is the excitation spectrum in the superfluid phase ψ? Does it agree with Goldstone theorem? 3 Thermodynamic potential and equation of state.. Using equation, show that the thermodynamic potential reads,ω n Ω = Ω + β,ω n ln detd, iω n, 5 where Ω is the saddle-point contribution. The Matsubara frequency sum is divergent and the correct expression is Ω = Ω + [ D ln D, iω n e iωn+ + ln D, iω n e iωn+, iω n ] + ln. β D, iω n D, iω n 6

6 . Using the results, express the thermodynamic potential Ω as a function of µ do not carry out the sum over. Deduce the particle density n..3 For a contact interaction, the s-wave scattering length a is defined by the equation see page 3 m 4πa = g + V ɛ. 7 Using the results, express n as a function of µ and a tae the limit Λ. Deduce the equation of state µn in the limit na 3. 3 Condensate density. The particle density can be written as n = n + V where n is the particle density in the condensate. 3. Express n = ψ ψ as a function of D, iω n and deduce ψ ψ, 8 n = ɛ + µ E Using, determine the condensate density n in the limit na 3. 4 Derive equations and.

7 Supplementary material { D[ψ, ψ] exp where x = r, τ, dx = β dτ d 3 r, etc. β β ω n dxdy ψ ψy x, ψxax, y ψ y ln iω n + a e iωn + = pour a >, T =, ω ln n + a ωn + b = a b pour < a < b, T =. ω n } = det A /, where = E ɛ ɛ + µ E = π m3/ µ /, = 3π mµ3/, d 3 π 3, ɛ = /m and E = [ɛ ɛ + µ] /. s-wave scattering length When the interaction potential gr = gr r between particles is rotation invariant, collisions tae place only in the s-wave channel at low energy. In the center-of-mass frame, a scattering state reads where = ˆr and ψ r e i r e ir + f, r, 3 f, a + ia. 4 Low-energy interactions interactions are therefore entirely parameterized by the scattering length a. The latter can be obtained from the T matrix, and the equation T, /m = 4π m f,, 5 T ɛ = g + gg + ɛt ɛ, 6 where G + ɛ = ɛ + i+ H H is the inetic energy of the two particles in the center-of-mass frame. For a contact potential gr = gδr, T, ɛ depends only on ɛ and we obtain T ɛ = g + g ɛ + i + T ɛ 7 /m and in turn equation 7. 3

8 M Quantum Physics Academic year 3-4 Broen symmetries and quantum phase transitions Tutorial class 4: BCS theory of superconductivity Free electron gas. We consider non-interacting electrons. Write the Hamiltonian Ĥ in secondquantized form. What is the ground state of the system? Give the expression of the Euclidean action S [ψ, ψ] at finite temperatures, the one-particle propagator G, iω n and the spectral function A, ω = π I[GR, ω]. What is the physical meaning of A, ω? Superconducting metal. We now assume the electrons to interact via an attractive interaction. The Hamiltonian of the superconductor writes Ĥ + Ĥint, where Ĥ int = λ d 3 r ˆψ r ˆψ r ˆψ r ˆψ r and λ >. In conventional superconductors, the attractive interaction is due to the exchange of phonons between electrons and affects only particles in the vicinity of the Fermi surface, ξ = ɛ µ ω D ω D ɛ F is the phonon Debye frequency and ɛ F the Fermi energy. This last condition can be implemented explicitly by rewriting equation in terms of the operators ˆψ σ.. What is the action S int [ψ, ψ] corresponding to the Hamiltonian Ĥint?. Using the Hubbard-Stratonovich transformation e Sint[ψ,ψ] = D[, ] e β λ dτ d 3 r r,τ + β dτ d 3 r[ r,τ ψ r,τψ r,τ+c.c.] ψ σ r, τ is a Grassmann variable and r, τ a complex field, one can rewrite the partion function Z = D[ψ, ψ,, ] e S[ψ,ψ,, ] 3 as a functional integral over ψ and. What is the expression of the action S[ψ, ψ,, ]? 3 Mean-field approximation. We consider the action S[ψ, ψ,, ] within a saddle-point approximation or mean-field approximation where the fluctuations of the field r, τ about its mean value are neglected. We assume to be real. 3. Show that the action of the electrons in the presence of the mean field writes as a function of the Nambu spinor ψ, iω Ψ, iω n = n ψ, iω n S MF [ψ, ψ] = βv λ,ω n Ψ, iω n G, iω n Ψ, iω n 4, Ψ, iω n = ψ, iω n, ψ, iω n. 5 What is the expression of the matrix G, iω n? We shall introduce the energy E = ξ +. Show that the elements of G, iω n can be expressed as a function of G, iω n = ψ σ, iω n ψ σ, iω n F, iω n = ψ, iω n ψ, iω n F, iω n = ψ, iω n ψ, iω n normal Green function, anomalous Green function. 6

9 Give the expressions of G, iω n and F, iω n. 3. Which condition should we impose on the grand potential Ω or, equivalently, the partition function Z to obtain the value of? Deduce that = λ ψ r, τψ r, τ and obtain the equation satisfied by. Show that a non-trivial solution is possible below a temperature T c. Give the expression of T c in the wea-coupling limit λn. We shall assume that the density of states Nξ = V δξ ξ can be approximated by N when ξ ω D and use ωd dξ βξ ξ tanh γωd ln T ω D, 7 πt where γ is the exponential of the Euler constant. What is the physical meaning of T c? What is the broen symmetry in the low-temperature phase T < T c? By analogy with the bosonic superfluid where the order parameter is ˆψr, explain why we can interpret the low-temperature as a condensate of Cooper pairs, ;,. 3.3 Determine the value of at zero temperature in the limit λn. Deduce that the ratio /T c is universal. 3.4 Verify that the unitary transformation γ, iω n γ, iω = n where u = u v ψ, iω n v u ψ, iω n, 8 + ξ, v = ξ, 9 E E maes the action diagonal. What is the corresponding Hamiltonian in terms of the fermionic operators ˆγ σ and ˆγ σ? Show that the ground state satisfies ˆγ σ Ψ BCS = for all annihilation operators ˆγ σ and is given by the BCS state Ψ BCS = u + v ˆψ ˆψ vac originally introduced by Bardeen, Cooper and Schrieffer. What is then the physical meaning of u and v? Plot u, v and E / as a function of ξ /. How can we interpret the one-particle excitations ˆγ σ Ψ BCS? 3.5 Calculate the spectral function A, ω = π I[GR, ω] of the superconductor. Plot A, ω vs ω for ξ > and ξ <. 4 Collective modes. The collective modes of the superconductor correspond to the fluctuation eigenmodes of the complex field r, τ. 4. How many collective modes can we expect? 4. Show, without any calculation, that at least one of these modes must have a vanishing energy in the large wavelength limit. What low-energy effective action can we expect for this mode?

10 M Quantum Physics Academic year 3-4 Broen symmetries and quantum phase transitions Tutorial class 5: Superfluid Mott-insulator transition in a Bose gas We consider the Bose-Hubbard model, Ĥ = t r,r ˆb rˆbr + h.c. + r µˆb rˆb r + U ˆb rˆb rˆb rˆbr, describing bosons moving in a lattice t denotes the hopping amplitude between nearest-neighbor sites and subject to a local on-site interaction U. The chemical potential µ fixes the mean boson density i.e. the mean number of bosons per site. We assume an hypercubic lattice in d dimensions and consider only the zero-temperature limit. Superfluid Mott-insulator transition. What is the ground state in the absence of interaction U =? How does the local repulsion affect the boson dynamics? Deduce that, depending on the boson density and the strength of the interactions, the ground state is either a superfluid or a Mott insulator consider the limits t/u and t/u.. Does the Bogoliubov theory appear appropriate to study the transition between the superfluid state and the Mott insulator? Phase diagram. We first consider the limit of vanishing hopping amplitude t = local limit. What is the Hamiltonian Ĥloc of a single site? What are the eigenstates? What is the number n of bosons per site as a function of µ/u? Calculate the local Green function G loc τ = T τˆbτˆb and its Fourier transform G loc iω n.. Give the expression of the Euclidean action S[b, b] of the Bose-Hubbard model. Rewrite the partition function using the Hubbard-Stratonovich transformation e β dτ r,r b r t r,r b r = D[ψ, ψ] e β dτ r,r ψ r t r,r ψ r + β dτ r ψ r br+c.c. ψ is a complex field. t r,r equals t if the sites r and r are nearest neighbors and vanishes otherwise..3 It is possible to integrate out the field b in a cumulant expansion: D[b, b] e S loc[b,b] S [b,b] = Z loc e S + S S + where the mean value is to be taen with the local action S loc corresponding to the limit t =. What is the expression of the action to quadratic order in the auxiliary field ψ?.4 Compute the energy E = lim T T ln Z in a mean-field or saddle-point approximation where the fluctuations of the field ψ are neglected assuming that the saddle-point value of ψ r τ is uniform and time independent. Deduce the phase diagram in the plane dt, µ/u use the notations D = dt, δµ = µ Un / and x = n + /..5 We write the energy Eψ, ψ in the mean-field approximation as Eψ, ψ = a + a ψ + a 4 ψ 4 + O ψ 6, 4 3

11 where a and a were computed in question.4. What should be the sign of a 4? Calculate the mean boson density n in the Mott insulator and in the superfluid phase. Deduce that the compressibility κ = n dn/dµ of the Mott insulator vanishes. Show that there is a special point of the transition line where the superfluid Mott-insulator transition occurs at constant density. 3 Excitation spectrum in the Mott insulator 3. From the action S[ψ, ψ] obtained earlier question.3, deduce the propagator G, iω n of the field ψ. What is the condition on G =, iω n = for the Mott insulator to be stable? Show that one recovers the mean-field phase diagram question Show that the one-particle spectrum exhibits two branches, E + and E. Compute E ± =. How does the transition to the superfluid phase manifest itself? Why is the transition occurring at constant density special? 3.3 What is the behavior of E ± in the limit? Show that at the Mott transition E ± z where the exponent z taes the value or. 4 Low-energy effective action It is possible to write an effective action for the auxiliary field ψ by expanding G, iω n in powers of and ω n. Show that one obtains where S[ψ, ψ] = β dτ d d r ψ r, τ r + K τ K τ K 3 + ψr, τ + 5 r = G =, iω n =, K = r µ. 6 Deduce that K vanishes when the transition occurs at constant density. What can we deduce about the universality class of the superfluid Mott-insulator transition?

12 M Quantum Physics Academic year 3-4 Broen symmetries and quantum phase transitions Tutorial class 6: The two-dimensional quantum ON model in the large-n limit We consider the action S[ϕ] = ˆ β ˆ dτ { d r ϕ + c τ ϕ + r ϕ + u } 4!N ϕ of the two-dimensional quantum ON model, where ϕr, τ is a N-component real field satisfying ϕr, τ + β = ϕr, τ. r and u are temperature independent and c is a propagation velocity. The model is regularized by a ultraviolet momentum cutoff p < Λ. The large N limit. We use the identity = ˆ dρ δρ ϕ = N ˆ ˆ dρ dλ e i λϕ ρ we ignore the constant N in the following to rewrite the partition function as ˆ { ˆ β ˆ [ Z = D[ϕ, ρ, λ] exp dτ d r ϕ + c τ ϕ + r ρ + u 4!N ρ + i ]} λϕ ρ. 3. Integrate out the field ρ to obtain the action S[ϕ, λ].. We split the field ϕ = σ, π into a field σ and a N -component field π. Integrate out π to obtain the action ˆ β ˆ { [ S[σ, λ] = dτ d r σ + c τ σ + iλσ ] 3N } r iλ + N Tr ln g, 4 u where g r, τ; r, τ = [ r + c τ + iλr, τ ] δr r δτ τ 5 is the inverse propagator of the field π i in the presence of the fluctuating field λ..3 Show that the saddle-point approximation becomes exact in the limit N. Give the saddle-point equations for uniform and time-independent fields σ and λ; discuss qualitatively the solutions and their physical meaning. Zero-temperature phase diagram.. Show that there is a zero-temperature quantum phase transition when r = r c. What is the expression of r c?. Compute the excitation gap m = mt = m = iλc and the correlation length ξ = c/m in the disordered phase when m c G cλ with G cu. Deduce the value of the correlation-length exponent ν, the dynamical exponent z, and the anomalous dimension η..3 Compute the order parameter σ in the ordered phase. Is Goldstone s theorem satisfied? The transverse propagator can be written in the form gq, iω = σ c ρ s ω + c q, 6

13 where ρ s is the stiffness or rigidity. Show that ρ s has the dimension of an energy. 3 Finite-temperature phase diagram. 3. Show that at finite temperature the system is always disordered. Could have this result been anticipated? Show that { [ ]} π m = T argsinh exp r r c 7 T u in the vicinity of the quantum critical point. 3. We consider the case r r c where the system is disordered at zero temperature. Express m as a function of T and m. Show that at low temperatures the system is dominated by quantum fluctuations whereas at high temperatures both classical and quantum fluctuations matter. 3.3 We now consider the case r r c where the system is ordered at zero temperature. Express m as a function of ρ s and T and show that we can again distinguish two temperature regimes: a low-temperature regime where the system is disordered by classical fluctuations and the high-temperature regime studied in question Draw the phase diagram in the plane r, T with the crossover lines between the various finitetemperature regimes.