Computational Quantum Physics
Computational Quantum PhysicsPhilippe de Forcrand (forcrand@phys.ethz.ch)Philipp Werner (werner@phys.ethz.ch)ETH Zürich, FS 2009
Chapters 1-4 and parts of chapter 8 are taken from the lecture notes of Prof. M. Troyer.
Contents1 Introduction1.1 General . . . . . . .1.1.1 Lecture Notes1.1.2 Exercises . . .1.1.3 Prerequisites .1.1.4 References . .1.2 Overview . . . . . . .11112232 Quantum mechanics in one hour2.1 Introduction . . . . . . . . . . . . . . . . .2.2 Basis of quantum mechanics . . . . . . . .2.2.1 Wave functions and Hilbert spaces2.2.2 Mixed states and density matrices .2.2.3 Observables . . . . . . . . . . . . .2.2.4 The measurement process . . . . .2.2.5 The uncertainty relation . . . . . .2.2.6 The Schrödinger equation . . . . .2.2.7 The thermal density matrix . . . .2.3 The spin-S problem . . . . . . . . . . . . .2.4 A quantum particle in free space . . . . .2.4.1 The harmonic oscillator . . . . . .444455677899103 The quantum one-body problem3.1 The time-independent 1D Schrödinger equation . . . . . . . . . .3.1.1 The Numerov algorithm . . . . . . . . . . . . . . . . . . .3.1.2 The one-dimensional scattering problem . . . . . . . . . .3.1.3 Bound states and solution of the eigenvalue problem . . .3.2 The time-independent Schrödinger equation in higher dimensions3.2.1 Factorization along coordinate axis . . . . . . . . . . . . .3.2.2 Potential with spherical symmetry . . . . . . . . . . . . .3.2.3 Finite difference methods . . . . . . . . . . . . . . . . . . .3.2.4 Variational solutions using a finite basis set . . . . . . . .3.3 The time-dependent Schrödinger equation . . . . . . . . . . . . .3.3.1 Spectral methods . . . . . . . . . . . . . . . . . . . . . . .3.3.2 Direct numerical integration . . . . . . . . . . . . . . . . .3.3.3 The split operator method . . . . . . . . . . . . . . . . . .1212121314151516161718191920.i.
4 Introduction to many-body quantum mechanics4.1 The complexity of the quantum many-body problem4.2 Indistinguishable particles . . . . . . . . . . . . . . .4.2.1 Bosons and fermions . . . . . . . . . . . . . .4.2.2 The Fock space . . . . . . . . . . . . . . . . .4.2.3 Creation and annihilation operators . . . . . .4.2.4 Nonorthogonal basis sets . . . . . . . . . . . .222223232425275 Density Matrix Renormalization Group5.1 Introduction . . . . . . . . . . . . . . . . .5.2 Real-space RG for the “particle in a box” .5.3 Density matrix renormalization group . . .5.3.1 Infinite-size DMRG . . . . . . . . .5.3.2 Finite-size DMRG . . . . . . . . .5.4 Example: S 1/2 Heisenberg chain . . . .5.4.1 Measurements . . . . . . . . . . . .5.5 DMRG for the “particle in a box” . . . . .2828283030313132326 Path integrals and quantum Monte Carlo6.1 Introduction . . . . . . . . . . . . . . . . .6.2 Recall: Monte Carlo essentials . . . . . . .6.3 Notation and general idea . . . . . . . . .6.4 Expression for the Green’s function . . . .6.5 Diffusion Monte Carlo . . . . . . . . . . .6.5.1 Importance sampling . . . . . . . .6.5.2 Fermionic systems . . . . . . . . .6.5.3 Useful references . . . . . . . . . .6.6 Path integral Monte Carlo . . . . . . . . .6.6.1 Main idea . . . . . . . . . . . . . .6.6.2 Finite temperature . . . . . . . . .6.6.3 Observables . . . . . . . . . . . . .6.6.4 Monte Carlo simulation . . . . . .6.6.5 Useful references . . . . . . . . . 6060627 An7.17.27.37.4Introduction to Quantum Field TheoryIntroduction . . . . . . . . . . . . . . . . . . . . . . . .Path integrals: from classical mechanics to field theoryNumerical study of φ4 theory . . . . . . . . . . . . . .Monte Carlo algorithms for the Ising model . . . . . .7.4.1 Cluster algorithm . . . . . . . . . . . . . . . . .7.4.2 Worm algorithm . . . . . . . . . . . . . . . . .7.5 Gauge theories . . . . . . . . . . . . . . . . . . . . . .7.5.1 QED . . . . . . . . . . . . . . . . . . . . . . . .7.5.2 QCD . . . . . . . . . . . . . . . . . . . . . . . .7.6 Overview . . . . . . . . . . . . . . . . . . . . . . . . . .7.7 Useful references . . . . . . . . . . . . . . . . . . . . .ii.
8 Electronic structure of molecules and solids8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.2 The electronic structure problem . . . . . . . . . . . . . . . .8.3 Basis functions . . . . . . . . . . . . . . . . . . . . . . . . . .8.3.1 The electron gas . . . . . . . . . . . . . . . . . . . . .8.3.2 Atoms and molecules . . . . . . . . . . . . . . . . . . .8.3.3 Pseudo-potentials . . . . . . . . . . . . . . . . . . . . .8.4 The Hartree Fock method . . . . . . . . . . . . . . . . . . . .8.4.1 The Hartree-Fock approximation . . . . . . . . . . . .8.4.2 The Hartree-Fock equations in nonorthogonal basis sets8.4.3 Configuration-Interaction . . . . . . . . . . . . . . . . .8.5 Thomas-Fermi theory . . . . . . . . . . . . . . . . . . . . . . .8.6 Density functional theory . . . . . . . . . . . . . . . . . . . . .8.6.1 Hohenberg-Kohn theorem . . . . . . . . . . . . . . . .8.6.2 Hohenberg-Kohn variational principle . . . . . . . . . .8.6.3 Kohn-Sham equations . . . . . . . . . . . . . . . . . .8.6.4 Local density approximation (LDA) . . . . . . . . . . .8.6.5 Beyond LDA . . . . . . . . . . . . . . . . . . . . . . .8.7 Car-Parinello molecular dynamics . . . . . . . . . . . . . . . .646464656565666767676969707071727474759 Dynamical mean field theory9.1 Introduction . . . . . . . . . . . . . .9.2 Hubbard model . . . . . . . . . . . .9.3 Dynamical mean field approximation9.3.1 Single-site effective model . .9.3.2 Self-consistency condition . .9.3.3 DMFT self-consisteny loop . .9.4 LDA DMFT . . . . . . . . . . . . .7777787979818182.10 Diagrammatic Monte Carlo methods for impurity models10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .10.2 General recipe . . . . . . . . . . . . . . . . . . . . . . . . . .10.3 Weak-coupling approach . . . . . . . . . . . . . . . . . . . .10.3.1 Monte Carlo configurations . . . . . . . . . . . . . .10.3.2 Sampling procedure and detailed balance . . . . . . .10.3.3 Determinant ratios and fast matrix updates . . . . .10.3.4 Measurement of the Green’s function . . . . . . . . .10.3.5 Expansion order and role of the parameter K . . . .10.4 “Strong coupling” approach - expansion in the impurity-bathtion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.4.1 Monte Carlo configurations . . . . . . . . . . . . . .10.4.2 Sampling procedure and detailed balance . . . . . . .10.4.3 Measurement of the Green’s function . . . . . . . . .10.4.4 Generalization - Matrix formalism . . . . . . . . . .10.5 Comparison between the two approaches . . . . . . . . . . .iii. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .hybridiza. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .848486878788899091919194959596
A Numerical methodsA.1 Numerical root solvers . . . . . . . . . . . .A.1.1 The Newton and secant methods . .A.1.2 The bisection method and regula falsiA.1.3 Optimizing a function . . . . . . . .A.2 The Lanczos algorithm . . . . . . . . . . . .iv.aaabbc
Chapter 1Introduction1.1GeneralFor physics students the computational quantum physics courses is a recommendedprerequisite for any computationally oriented semester thesis, proseminar, diploma thesis or doctoral thesis.For computational science and engineering (RW) students the computational quantum physics courses is part of the “Vertiefung” in theoretical physics.1.1.1Lecture NotesAll the lecture notes, source codes, applets and supplementary material can be foundon our web page http://www.itp.phys.ethz.ch/education/lectures fs09/cqp.1.1.2ExercisesProgramming LanguagesExcept when a specific programming language or tool is explicitly requested you arefree to choose any programming language you like. Solutions will often be given eitheras C programs or Mathematica Notebooks.Computer AccessThe lecture rooms offer both Linux workstations, for which accounts can be requestedwith the computer support group of the physics department in the HPR building, aswell as connections for your notebook computers.1
1.1.3PrerequisitesAs a prerequisite for this course we expect knowledge of the following topics. Pleasecontact us if you have any doubts or questions.Computing Basic knowledge of UNIX At least one procedural programming language such as C, C , Pascal, Java orFORTRAN. C knowledge is preferred. Knowledge of a symbolic mathematics program such as Mathematica or Maple. Ability to produce graphical plots.Numerical Analysis Numerical integration and differentiation Linear solvers and eigensolvers Root solvers and optimization Statistical analysisQuantum MechanicsBasic knowledge of quantum mechanics, at the level of the quantum mechanics taughtto computational scientists, should be sufficient to follow the course. If you feel lost atany point, please ask the lecturer to explain whatever you do not understand. We wantyou to be able to follow this course without taking an advanced quantum mechanicsclass.1.1.4References1. J.M. Thijssen, Computational Physics, Cambridge University Press (1999) ISBN05215758852. Nicholas J. Giordano, Computational Physics, Pearson Education (1996) ISBN0133677230.3. Harvey Gould and Jan Tobochnik, An Introduction to Computer Simulation Methods, 2nd edition, Addison Wesley (1996), ISBN 002015060414. Tao Pang, An Introduction to Computational Physics, Cambridge University Press(1997) ISBN 05214859242
1.2OverviewIn this class we will learn how to simulate quantum systems, starting from the simpleone-dimensional Schrödinger equation to simulations of interacting quantum many bodyproblems in condensed matter physics and in quantum field theories. In particular wewill study The one-body Schrödinger equation and its numerical solution The many-body Schrödinger equation and second quantization Approximate solutions to the many body Schrödinger equation Path integrals and quantum Monte Carlo simulations Numerically exact solutions to (some) many body quantum problems Some simple quantum field theories3
Chapter 2Quantum mechanics in one hour2.1IntroductionThe purpose of this chapter is to refresh your knowledge of quantum mechanics andto establish notation. Depending on your background you might not be familiar withall the material presented here. If that is the case, please ask the lecturers and wewill expand the introduction. Those students who are familiar with advanced quantummechanics are asked to glance over some omissions and are encouraged to help usimprove this quick introduction.2.22.2.1Basis of quantum mechanicsWave functions and Hilbert spacesQuantum mechanics is nothing but simple linear algebra, albeit in huge Hilbert spaces,which makes the problem hard. The foundations are pretty simple though.A pure state of a quantum system is described by a “wave function” Ψi, which isan element of a Hilbert space H: Ψi H(2.1)Usually the wave functions are normalized:p Ψi hΨ Ψi 1.(2.2)Here the “bra-ket” notationhΦ Ψi(2.3)denotes the scalar product of the two wave functions Φi and Ψi.The simplest example is the spin-1/2 system, describing e.g. the two spin states of the electron (which can be visualized as anof an electron. Classically the spin Sinternal angular momentum), can point in any direction. In quantum mechanics it isdescribed by a two-dimensional complex Hilbert space H C2 . A common choice of4
basis vectors are the “up” and “down” spin states 1 i 0 0 i 1(2.4)(2.5)This is similar to the classical Ising model, but in contrast to a classical Ising spinthat can point only either up or down, the quantum spin can exist in any complexsuperposition Ψi α i β i(2.6)of the basis states, where the normalization condition (2.2) requires that α 2 β 2 1.For example, as we will see below the state1 i ( i i)2(2.7)is a superposition that describes the spin pointing in the positive x-direction.2.2.2Mixed states and density matricesUnless specifically prepared in a pure state in an experiment, quantum systems inNature rarely exist as pure states but instead as probabilistic superpositions. The mostgeneral state of a quantum system is then described as a density matrix ρ, with unittraceTrρ 1.(2.8)The density matrix of a pure state is just the projector onto that stateρpure ΨihΨ .(2.9)For example, the density matrix of a spin pointing in the positive x-direction is 1/2 1/2ρ ih .(2.10)1/2 1/2Instead of being in a coherent superposition of up and down, the system could alsobe in a probabilistic mixed state, with a 50% probability of pointing up and a 50%probability of pointing down, which would be described by the density matrix 1/2 0ρmixed .(2.11)0 1/22.2.3ObservablesAny physical observable is represented by a self-adjoint linear operator acting on theHilbert space, which in a final dimensional Hilbert space can be represented by a Hermitian matrix. For our spin-1/2 system, using the basis introduced above, the components5
S x , S y and S z of the spin in the x-, y-, and z-directions are represented by the Paulimatrices 0 1xS (2.12)σx 22 1 0 0 i y(2.13)σy S 22 i 0 1 0 z(2.14)σz S 22 0 1The spin component along an arbitrary unit vector ê is the linear superposition ofthe components, i.e. zxy ee iexxyyzz e S e S e S (2.15)ê · S ez2 ex ieyThe fact that these observables do not commute but instead satisfy the non-trivialcommutation relations[S x , S y ] S x S y S y S x i S z ,[S y , S z ] i S x ,[S z , S x ] i S y ,(2.16)(2.17)(2.18)will have important consequences.2.2.4The measurement processThe outcome of a measurement in a quantum system is usually intrusive and not deterministic. After measuring an observable A, the new wave function of the system will bean eigenvector of A and the outcome of the measurement the corresponding eigenvalue.The state of the system is thus changed by the measurement process!For example, if we start with a spin pointing up with wave function 1(2.19) Ψi i 0or alternatively density matrix 1 0ρ 0 0(2.20)and we measure the x-component of the spin S x , the resulting measurement will beeither /2 or /2, depending on whether the spin after the measurement points inthe or x-direction, and the wave function after the measurement will be either of 11/2 i ( i i) (2.21)1/22 11/2(2.22) i ( i i) 1/226
Either of these states will be picked with a probability given by the overlap of the initialwave function by the individual eigenstates:p h Ψi 2 1/2p h Ψi 2 1/2(2.23)(2.24)The final state is a probabilistic superposition of these two outcomes, described by thedensity matrix 1/2 0.(2.25)ρ p ih p ih 0 1/2which differs from the initial density matrix ρ .If we are not interested in the result of a particular outcome, but just in the average,the expectation value of the measurement can easily be calculated from a wave function Ψi ashAi hΨ A Ψi(2.26)or from a density matrix ρ ashAi Tr(ρA).(2.27)It is easy to convince yourself that for pure states the two formulations are identical.2.2.5The uncertainty relationIf two observables A and B do not commute [A, B] 6 0, they cannot be measuredsimultaneously. If A is measured first, the wave function is changed to an eigenstate ofA, which changes the result of a subsequent measurement of B. As a consequence thevalues of A and B in a state Ψ cannot be simultaneously known, which is quantified bythe famous Heisenberg uncertainty relation which states that if two observables A andB do not commute but satisfy[A, B] i (2.28)then the product of the root-mean-square deviations A and B of simultaneous measurements of A and B has to be larger than A B /2(2.29)For more details about the uncertainty relation, the measurement process or the interpretation of quantum mechanics we refer interested students to an advanced quantummechanics class or text book.2.2.6The Schrödinger equationThe time-dependent Schrödinger equationAfter so much introduction the Schrödinger equation is very easy to present. The wavefunction Ψi of a quantum system evolves according to Ψ(t)i H Ψ(t)i,(2.30) twhere H is the Hamilton operator. This is just a first order linear differential equation.i 7
The time-independent Schrödinger equationFor a stationary time-independent problem the Schrödinger equation can be simplified.Using the ansatz Ψ(t)i exp( iEt/ ) Ψi,(2.31)where E is the energy of the system, the Schrödinger equation simplifies to a lineareigenvalue problemH Ψi E Ψi.(2.32)The rest of the semester will be spent solving just this simple eigenvalue problem!The Schrödinger equation for the density matrixThe time evolution of a density matrix ρ(t) can be derived from the time evolution ofpure states, and can be written as i ρ(t) [H, ρ(t)](2.33) tThe proof is left as a simple exercise.2.2.7The thermal density matrixFinally we want to describe a physical system not in the ground state but in thermalequilibrium at a given inverse temperature β 1/kB T . In a classical system each microstate i of energy Ei is occupied with a probability given by the Boltzman distribution1(2.34)pi exp( βEi ),Zwhere the partition functionXZ exp( βEi )(2.35)inormalizes the probabilities.In a quantum system, if we use a basis of eigenstates ii with energy Ei , the densitymatrix can be written analogously as1Xρβ exp( βEi ) iihi (2.36)Z iFor a general basis, which is not necessarily an eigenbasis of the Hamiltonian H, thedensity matrix can be obtained by diagonalizing the Hamiltonian, using above equation,and transforming back to the original basis. The resulting density matrix is1ρβ exp( βH)(2.37)Zwhere the partition function now isZ Tr exp( βH)(2.38)Calculating the thermal average of an observable A in a quantum system is hencevery easy:TrA exp( βH).(2.39)hAi Tr(Aρβ ) Tr exp( βH)8
2.3The spin-S problemBefore discussing solutions of the Schrödinger equation we will review two very simplesystems: a localized particle with general spin S and a free quantum particle.In section 2.2.1 we have already seen the Hilbert space and the spin operators for themost common case of a spin-1/2 particle. The algebra of the spin operators given by thecommutation relations (2.12)-(2.12) allows not only the two-dimensional representationshown there, but a series of 2S 1-dimensional representations in the Hilbert spaceC2S 1 for all integer and half-integer values S 0, 12 , 1, 23 , 2, . . . The basis states { si}are usually chosen as eigenstates of the S z operatorS z si s si,(2.40)where s can take any value in the range S, S 1, S 2, . . . , S 1, S. In this basisthe S
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