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Quantum Geometry, Non-Adiabatic Response and Emergent Macroscopic Dynamics

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Quantum Geometry, Non-Adiabatic Response and Emergent Macroscopic Dynamics Quantum geometry, non-adiabatic response and emergent macroscopic dynamics. Lecture notes. Karpacz Winter School, March 2014 Anatoli Polkovnikov1 1Department of Physics, Boston University, Boston, MA 02215 Contents I. Gauge potentials in classical and quantum Hamiltonian systems. 1 A. Classical Hamiltonian systems. 1 B. Quantum Hamiltonian systems. 4 C. Hamiltonian dynamics in the moving frame. Galilean transformation. 6 II. Geometry of the ground state manifold. Fubini-Study metric and the Berry curvature. 9 III. Geometric invariants: Geometry of the XY chain. 15 A. Basic definitions of the Euler characteristic and the first Chern number. 15 B. Geometric structure of the XY chain. 17 IV. Gauge potentials and non-adiabatic response 23 A. Dynamical Quantum Hall effect 23 B. General approach to non-adiabatic response. Emergent Newtonian dynamics. 31 Acknowledgments 42 References 42 I. GAUGE POTENTIALS IN CLASSICAL AND QUANTUM HAMILTONIAN SYSTEMS. A. Classical Hamiltonian systems. Hamiltonian systems, are generally defined by specifying a set of canonical variables pj; qj satisfying canonical relations pi; qj = δij; (1) f g where ::: denotes the Poisson bracket: f g @A @B @B @A A(~p;~q);B(~p;~q) = : (2) f g @p @q − @p @q j j j j j X 2 It is easy to check that any orthogonal transformation Q = R(λ)q; P = R(λ)p (3) preserves the Poisson bracket. A general class of transformations which preserve the Poisson brackets are known as canonical transformations and can be expressed through the generating functions (Landau and Lifshitz, 1982). It is easy to check that continuous canonical transformations can be generated by gauge potentials λ: A @ λ(λ, ~p;~q) qj(λ + δλ) = qj(λ) A δλ, (4) − @pj @ λ(λ, ~p;~q) pj(λ + δλ) = pj(λ) + A δλ, (5) @qj where λ parametrizes the canonical transformation and the gauge potential is an arbitrary function of canonical variables and the parameter. Then up to terms of the order of δλ2 the transformation above preserves the Poisson brackets 2 2 @ λ @ λ 2 2 pi(λ + δλ); qj(λ + δλ) = δij + δλ A A + O(δλ ) = δij + O(δλ ): (6) f g @p @q − @p @q j i j i Exercises. 1. Show that the generator of translations ~q(X) = ~q0 X~ is the momentum operator: − ~ ~ (~q; ~p) = ~p. You need to treat X~ as a three-component parameter ~λ. AX 2. Show that the generator of rotations around z-axis: qx(θ) = cos(θ)qx0 sin(θ)qy0; qy(θ) = cos(θ)qy0 + sin(θ)qx0; − px(θ) = cos(θ)px0 sin(θ)py0; py(θ) = cos(θ)py0 + sin(θ)px0; − is the angular momentum operator: θ = pxqy pyqx. A − 3. Find the gauge potential λ corresponding to the orthogonal transformation (3). A Hamiltonian dynamics is a particular canonical transformation parametrized by time dqj @ dpj @ = ; qj = H ; = ; pj = H (7) @t fH g @pj @t fH g −@qj Clearly these Hamiltonian equations are equivalent to Eqs. (5) with the convention t = . This A −H observation shows that the gauge potentials λ are generators of motion in the parameter space. A 3 One can extend canonical transformations to the complex variables. Instead of doing this in full generality we will focus on particular phase space variables which are complex wave ampli- tudes. For example let us consider a system of harmonic oscillators which can be always described by the normal modes parameterized by k (for translationally invariant system k represents the momentum). Each mode is an independent harmonic oscillator described by the Hamiltonian: 2 2 pk m!k 2 k = + q : (8) H 2m 2 k Let us define new linear combinations m!k ∗ 1 ∗ pk = i (a ak); qk = (ak + a ) (9) 2 k − 2m! k r r k or equivalently ∗ 1 i 1 i ak = qkpm!k pk ; ak = qkpm!k + pk : (10) p2 − pm!k p2 pm!k Next we compute the Poisson brackets of the complex wave amplitudes ∗ ∗ ∗ ak; ak = a ; a = 0; ak; a = i: (11) f g f k kg f kg To avoid dealing with the imaginary Poisson brackets it is convenient to introduce new coherent state Poisson brackets @A @B @B @A A; B c = ∗ ∗ : (12) f g @ak @a − @ak @a k k k X From this definition it is immediately clear that ∗ ak; a c = δkq: (13) f qg Comparing this relation with Eq. (11) we see that standard and coherent Poisson brackets differ by the factor of i: ::: = i ::: c: (14) f g f g Infinitesimal canonical transformations preserving the coherent state Poisson brackets can be also defined using the guage potentials: ∗ @ak @ λ @ak @ λ i = A∗ ; i = A : (15) @λ − @ak @λ @ak Exercise 4 0 Check that any unitary transformationa ~k = Uk;k0 a , where U is a unitary matrix, pre- • k ∗ serves the coherent state Poisson bracket, i.e. a~k; a~ c = δk;q. Verify that the Bogoliubov f qg transformation ∗ ∗ ∗ γk = cosh(θk)ak + sinh(θk)a−k; γk = cosh(θk)ak + sinh(θk)a−k; (16) with θk = θ−k also preserves the coherent state Poisson bracket, i.e. ∗ ∗ ∗ γk; γ−k c = γk; γ c = 0; γk; γ c = γ−k; γ c = 1: (17) f g f −kg f kg f −kg Assume that θk are known functions of some parameter λ, e.g. the interaction strength. Find the gauge potential λ = k, which generates such transformations. A k A P We can write the Hamiltonian equations of motion for the new coherent variables. Using that da @a @a = a; = i a; c (18) dt @t − f Hg @t − f Hg and using that our variables do not explicitly depend on time (such dependence would amount to going to a moving frame, which we will discuss later) we find ∗ dak @ dak ∗ @ i = ak; c = H∗ ; i = ak; c = H (19) dt f Hg @ak dt f Hg −@ak These equations are also known as Gross-Pitaevskii equations. Note that these equations are arbitrary for arbitrary Hamiltonians and not restricted to harmonic systems. And finally let us write down the Liouville equations of motion for the probability distribution ρ(q; p; t) or ρ(a; a∗; t). The latter just express incompressibility of the probability fllow, which directly follows conservation of the phase space volume dΓ = dqdp or dΓ = dada∗ for arbitrary canonical transformations including time evolution and from the conservation of the total proba- bility ρdΓ: dρ @ρ @ρ 0 = = ρ, H = i ρ, H c; (20) dt @t − f g @t − f g or equivalently @ρ @ρ = ρ, H ; i = ρ, H c (21) @t f g @t −{ g B. Quantum Hamiltonian systems. An analogue of canonical transformations in classical mechanics are unitary transformations in quantum mechanics. In classical systems these transformations reflect the freedom of choosing canonical variables while in quantum systems they reflect the freedom of choosing basis states. 5 The wave function representing some state can be always expanded in some basis: = n n 0; (22) j i n j i X where n 0 is some fixed, parameter independent basis. One can always make a unitary transfor- j i ∗ 1 mation to some other basis n(λ) = Unm(λ) n0 or equivalently n0 = U m(λ) . Then the j i j i j i nmj i same state can be written as j i ∗ ~ = nUnm m(λ) = m(λ) m(λ) ; (23) j i n j i m j i X X ~ ∗ y where m(λ) = Unm n = U . We can introduce gauge potentials by analogy with the classical systems as generators of continuous unitary transformations. Namely y y y i@λ ~(λ) = iU @λUU = iU @λU ~ = λ ~; (24) − − −A where we introduced the guage potential y λ = iU @λU: (25) A Note that up to the sign the gauge potential plays the role similar to the Hamiltoninan, i.e. it generates the motion in the parameter space. It is easy to check that the gauge potential is a Hermitian operator y y = i@λU U = λ; (26) Aλ A where we used that y y y @λU = U @λUU : (27) − The gauge potential can be also represented through the matrix elements: y n0 λ m0 = i n0 U @λU m0 = i n(λ) @λ m(λ) : (28) h jA j i h j j i h j j i or equivalently λ = i@λ: (29) A We work in units where ~ = 1, otherwise it is more appropriate to define λ = i~@λ. The diagonal A elements of the gauge potentials in the basis of some Hamiltonian (λ) are the Berry connections, H 1 Unless otherwise specified we always imply summation over repeated indices. 6 which we will discuss in detail later. So the Berry connections are direct analogues of energies representing the expectation values of the gauge potentials. With this definition one can extend the notion of the Berry connections to arbitrary states. Exercises 1. Verify that the gauge potential corresponding to the translations: ~(x) = (λ + x) is the momentum operator. Similarly verify that the gauge potential for rotations is the angular momentum operator. 2. Consider the quantum version of the Bogoliubov transformations discussed in the previous section (Eq. (16)). Show that the quantum and classical gauge potentials coincide if we iden- ∗ tify complex amplitudes ak and ak with the annihilation and creation operators respectively. C. Hamiltonian dynamics in the moving frame. Galilean transformation. Gauge potentials are closely integrated into the Hamiltonian dynamics. In particular, they naturally appear in gauge theories like electromagnetism to enforce the gauge invariance. We will come to this issue later. For now we simply note that the equations of motion should be invariant under the gauge transformations.
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