Second Quantization Jan Von Delft, 17.11.2020 Hopping .

Second QuantizationJan von Delft, 17.11.2020Motivation: to simplify treatment of exchange symmetry in many-particle systemson-site energyhopping between sites i and jinteraction between sites i andforbosonsfermionsAssumed background:elementary quantum mechanics, Dirac bra-ket notation, Bose and Fermi statisticsLiterature: numerous textbooks on many-body physics have an introductory chapter or anappendix on 2nd quantization. Examples (these notes follow Altland & Simons):- A. Altland & B. Simons, Condensed Matter Field Theory, Cambridge University Press, 2nd Ed. (2010),Sec.2.1-2- A. L. Fetter & J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill (1971), Chapter 1.- G. Rickayzen, Greens Functions and Condensed Matter Physics, Dover (2013), Appendix A- S. M. Girvin & K. Yang, Modern Condensed Matter Physics, Cambridge University Press (2019), Appendix J.Single-particle basisConsider a single-particle quantum system.all values ofSingle-particle Hilbert space:Wavefunction:It is often convenient (though not necessary) to choose the basis states to be eigenstates of asingle-particle Hamiltonian:Eigenvalue equation:Having this example in mind, we will assume that thelabel takes the valuesExample: harmonic oscillator:In general,Or,can also be a continuous index. E.g. for free particles,can enumerate sites in a lattice, then it is a discrete index,All we need (later) is some ordering convention for its values.

Exchange symmetry: 2 particlesConsider a system of 2 identical particles, described by2-particle Hilbert space::'first' particle in state, 'second' particle in state:'first' particle in state, 'second' particle in stateBut particles are indistinguishable, states (3a), (3b) don't have independent physical meaning.Physically meaningful states must be fully symmetric (bosons) or anti-symmetric (fermions):forMeaningful state:bosonsfermionsWavefunction:invariant up to a sign under'Exchange symmetry':Physical part of 2-particle Hilbert space contains only symmetrized/antisymmetrized states:2-particle 'Fock space':Exchange symmetry: N particlesN-particle Hamiltonian:N-particleHilbert space:N copiesPhysical part of this space contains only fully symmetrized/antisymmetrized states of the form:these states are occupied,all others emptySum: over all permutations of N indices. E.g.Sign:for bosons:if even/odd number of transpostions areneeded to converttofor fermionsNormalization:N-particle Fock space:chosen such thatall values of

N-particle wave functionsN-particle position eigenstate:position operator in i-thsingle-particle Hilbert spaceeigenvalueN-particlewavefunction:'Exchange symmetry':fermionsFor fermions, wavefunctionis a determinant:'Slater determinant'Antisymmetry of determinant underexchange of rows or columns implies:two particles in same stateiftwo particles at same position'Pauli exclusionprinciple'N-particle basis: occupation number representationDue to exchange symmetry, we can fully specify a basis stateby specifying how many particles,, populate eachwithFor 'bosons',: eachcan contain arbitrarily many bosons.For 'fermions',: eachcan contain at most one fermion ('Pauli principle').Examples: the states on the right are denoted asrepresentationrepresentation3 bosons3 fermions

Fock space, creation operatorsbosonsfermionsN-particle Fock space:It is often convenient to not impose the condition of fixed particle number N. Then consider(many-particle)Fock space:total particlenumber not fixed'vacuum state''Vacuum space':Define 'creation operators' connecting states which differ by 1 for specified occupation number:creates particle in state'fermionic sign' depends on how many 'earlier' states are occupied:For fermions, occupation numbers are defined modulo 2, i.e.so,[this encodes Pauli principle (6.4)]All states can be obtained fromvacuum state by repeated action of(Anti)-commutation relations commutator, for bosonsDef: anti-commutator, for fermionsClaim: creation operators satisfyProof:- Equal indices,(trivially true)For bosons,For fermions:This holds for all states in, hence- Unequal indices,Simplest example: action on vacuum state,, and also

General case: assume (without loss of generality)(5) holds for all basis ketsof, hence it is an operator identity:'boson creation operators commute, fermion creation operators anti-commute'(Anti)-commutation relationsRecall definition of creation ) holds for all basis brasi.e.of'annihilates' or 'destroys' a particle in stateAction in Fock space:vacuumspaceHermitian conjugate of (9.6):, hence we conclude that

(Anti)-commutation relationsConsider equal indices:Bosons:holds for all basis kets!Fermions:(4,5) hold for all basis kets!Compact formulation of (3) & (6):(Anti)-commutation relationsGeneral case: assume (without loss of generality)(analogous to p. 9):ifholds for all basis kets!Summary:'boson operators commute, fermion creation anti-commute', except forGiven complex structure of Fock space, these relations are remarkably simple!

Change of basisThe single-particle states used above must form a basis offor discrete index, satisfyingfor continuous index, e.g.- orthogonality:- completeness:We make identifcation:Consider change of basis:Correspondingly:(anti)-commutation relations preserve their form:IfIf, then,then similarly,Representation of one-body operatorsDef: 'occupation number operator':operatoreigenvaluewithDiagonalone-body operator:When acting in:acts in i-th of N single-particle spaces: is the single particle?there found in the single-particle stateMany-body matrix elements:total number of particles found in single-particle state(5) holds for all basis kets ofoperator identity:Simply count the number of particles in single-particle stateTransformed to a general(non-diagonal) basis:and multiply by eigenvalue!

Examples of one-body operatorsVarious single-particle bases:energy eigenbasisposition basismomentum basisTotal particle number: density operatorKinetic energy:PotentialLattice Hamiltonian:on-site energyhopping between sites i and jZeeman field couplingto electron spin:spin operatorRepresentation of two-body operatorsPauli matricessymmetricInteraction potential between particles at positionsWe seek many-body operatorsuch thatAnsatz:Interpretation: 'take out two particles at x and x', let them feel the interaction, and put them back in'.Note:Check (3): act withon N-particle position eigenstate:commuteto the right:vacuum state is empty:each commutation operation yields an extradensity operator 'finds' the particles, yields

commuteNow multiply byto the right until it sits betweenand integrate over x and x', as in (16.3):Hence, Ansatz (16.3) does yield the result (16.2), as required.A general two-body operator with matrix elementsmnemonic:can be expressed as

- S. M. Girvin & K. Yang, Modern Condensed Matter Physics, Cambridge University Press (2019), Appendix J. bosons fermions for on-site energy hopping between sites i and j interaction between sites i and Single-particle basis Single-particle Hilbert space: Example: harmonic oscillator: Wavefunction: all values of Consider a single-particle .