Harmonic Crystals

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Lecture 1: Harmonic crystals

Christopher Mudry

Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland.

(Dated: September 6, 2001)

Abstract

Classical equations of motion for a finite chain of atoms are solved within the harmonic approximation.

The thermodynamic limit is constructed in terms of a one-dimensional (classical) field theory. Quantization

of the finite harmonic chain is undertaken. The thermodynamic limit is constructed in terms of a one-

dimensional quantum field theory describing phonons in a one-dimensional lattice.

Electronic address: [email protected]; URL: http://people.web.psi.ch/mudry

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I. CLASSICAL ONE-DIMENSIONAL CRYSTAL

A good reference for this section is chapter 12 of [1].

A. Discrete limit

For simplicity, I will consider a one dimensional world made ofN point-like objects (atoms) of

mass m and interacting through a potential V. I assume first that the potential V depends only on

the coordinates n , n 1 N, of the N atoms:

V V 1 N (1.1)

Furthermore, I assume that V has a non-degenerate minimum at

n na n 1 N (1.2)

where a is the lattice constant. For example, one could imagine that

V 1 Na

2

2

N 1

n 1

1 cos2

an 1 n

a

2

2

m2N

n 1

1 cos2

an

boundary terms (1.3)

For small deviations n about minimum (1.2), it is natural to expand the potential energy accord-ing to

V 1 1 N N V 1 N

N 1

n 1

2n 1 n

2 1

2m2

N

n 1

n2

boundary terms (1.4)

The dimensionfull constant is the elastic or spring constant. It measures the strength of the

linear restoring force between nearest neighbor atoms. The characteristic frequency measures

the strength of an external force that pins atoms to their equilibrium positions (1.2). To put it

differently, m2 is the curvature of the potential well that pins an atom to its equilibrium position.

Terms that have been neglected in are of several kinds. Only terms of quadratic order in the

nearest neighbor relative displacement n 1 n have been accounted for, and all interactions

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beyond the nearest neighbor range have been dropped. I have also omitted to specify boundary

terms. They are specified once boundary conditions have been imposed. In the limit N ,

the choice of boundary conditions should be immaterial since the bulk potential energy should

be of order L Na, whereas the energy contribution arising from boundary terms should be of

order L0 1. To minimize boundary effects in a finite system, one imposes periodic boundary

conditions

n N n n 1 N (1.5)

An open chain of atoms turns into a ring after imposing periodic boundary conditions. Further-

more, imposing periodic boundary conditions endows the potential with new symmetries within

the harmonic approximation defined by1

Vharmonic 1 1 N N :N

n 1

2n 1 n

2 1

2m2

N

n 1

n2

(1.7)

First, the shift of labelling

n n m n 1 N m (1.8)

leaves Eq. (1.7) invariant. Second, translational invariance is recovered in the absence of the

pinning potential:

0 Vharmonic 1 N Vharmonic 1 xa N xa x (1.9)

The kinetic energy of an open chain of atoms is simply given by

T 1 1 N N1

2m

N

n 1

dndt

21

2m

N

n 1

n2

(1.10)

As for the potential energy, the choice of boundary conditions only affects the kinetic energy by

terms of order L0. It is again natural to choose periodic boundary conditions if one is interested in

extensive properties of the system.

1

Without loss of generality, I have set the classical minimum of the potential energy to zero:

V 1 N 0 (1.6)

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The classical Lagrangian in the harmonic approximation and with periodic boundary condi-

tions is defined by subtracting from the kinetic energy (1.10) the potential energy (1.7)

:N

n 1

1

2m n

2

n 1 n2

m2 n2

(1.11)

The classical equations of motion follow from Lagrange equations

d

dt

n

n n 1 N (1.12)

They are

m n n 1 n 1 2n m2n n 1 N (1.13)

with the complex traveling wave solutions

n t ei kn t 2 2

m1 cos k 2 (1.14)

Imposing periodic boundary conditions allows to identify the normal modes. These are countable

many traveling waves with the frequency-wave number relation

l 2

m1 cos kl 2 kl

2

Nl l 1 N (1.15)

The most general real solution of Lagrange equations (1.13) obeying periodic boundary conditions

is

n t

N

l 1 Al e i kl n l t Al e i kl n lt n 1 N (1.16)

Here, the complex valued expansion coefficient Al is arbitrary.

To revert to the Hamilton-Jacobi formalism of classical mechanics, one introduces the canonical

momentum n conjugate to n through

n t :

n

imN

l 1

lA

le i kl n lt A

le i kl n l t n 1 N (1.17)

and construct the Hamiltonian

N

n 1

1

2

n2

m n 1 n

2m2 n

2 (1.18)

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from the Lagrangian (1.11) through a Legendre transformation. Hamilton-Jacobi equations of

motion are then

n

nn n

nn n 1 N (1.19)

where stands for the Poisson brackets.2

In the long wave number limit kl 1, the dispersion relation reduces to

2l

m k2l

2

k4l (1.21)

The pinning potential characterized by the potential wall curvature has opened up a gap in the

spectrum of normal modes. No solutions to Lagrange equations (1.13) can be found below the

characteristic frequency . By switching off the pinning potential, 0, the dispersion relation

simplifies to

2l

mk2l k

4l (1.22)

The proportionality constant m between frequency and wave number is interpreted as the

velocity of propagation of a sound wave in the one-dimensional harmonic chain.

B. Thermodynamic limit

The thermodynamic limit N emerges naturally if one is interested in the response of solids

to external perturbations as can be induced, say, by compressions. Of course, the characteristic

wave lengths of typical perturbations in daily life are much larger than the atomic separation.

Hence, the elastic response from a solid to a macroscopic perturbation is dominated by normalmodes with arbitrarily small wave numbers k 0. It is then much more economical not to account

for the discrete nature of the solid as is done in the Lagrangian (1.11). To this end, Eq. (1.11) is

first rewritten as

N

n 1

a1

2

m

an

2

an 1 n

a

2m

a2 n

2

:N

n 1

aLn (1.23)

2 The Poisson bracket f g of two functions f and g of the canonical variables n and n is defined by

f g :N

n 1

f

n

g

n

f

n

g

n(1.20)

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Interpret

:m

a :

n 1 na

Y : a and Ln (1.24)

as the mass per unit length, the elongation per unit length, the Youngs modulus,3 and the local

Lagrangian per unit length, respectively. Now write

L

0 dx

1

2

t

2

Y

x

2

2

2

:L

0dx (1.26)

whereby the following substitutions have been performed:

The discrete sum n has been replaced by the integral dx a over the semi-open interval

0 L .

The relative displacement n at time t has been replaced by the value of the real function at space-time coordinates x t obeying periodic boundary conditions in space:

x L t x t x 0 L t (1.27)

The time derivative of the relative displacement n at time t has been replaced by the value

of the time derivative t at space-time coordinate x t .

The discrete difference n 1 n at time t has been replaced by the lattice constant times

the value of the space derivative x at space-time coordinate x t .

The integrand in Eq. (1.26) is called the Lagrangian density. It is a real valued function of

space-time. From it, one obtains the continuum limit of Lagrange equations (1.12) according

to

t x t

t y tx

x t

x y t

x t

y t(1.28)

3 For an elastic rode obeying Hookes law, the extension of the rode per unit length is proportional to the exerted

force F with the Youngs modulus Y as the proportionality constant:

F Y (1.25)

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Here, the symbol x t is to be interpreted as the infinitesimal functional change of at

the given space-time coordinates x t induced by Taylor expansion,

x x t t x t

x x

t t

2 x2 t

2

(1.29)

One must keep in mind that , x , and t , are independent variables. Moreover,

one must use the rule

x t

y t x y

L

0dx x t

y t1 y 0 L (1.30)

that extends the rule

mn

m nN

m 1

mn

1 n 1 N (1.31)

to the continuum. Otherwise, all the usual rules of differentiation apply to .

Equations of motion (1.13) become the one-dimensional sound wave equation

2t v22x

2 0 v :Y

(1.32)

after replacing the finite difference

n 1 n 1 2n n 1 n n n 1 (1.33)

by a2 times the value of the second order space derivative 2x at space-time coordinates

x t .

The Hamiltonian in the continuum limit follows from Eq. (1.26) with the help of a (functional)

Legendre transform or directly from the continuum limit of Eq. (1.18),

L

0dx

1

2

2

Y

x

2

22

:

L

0 dx (1.34)

where the field is the canonically conjugate to :

x t :L

0dy

y t

t x t t x t (1.35)

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C. From sums to integrals

As we have seen, probing the one dimensional crystal on length scales much larger than the

lattice spacing a blurs our vision to the point where the crystal appears as an elastic continuum.

Viewed without an atomic microscope, the relative displacements n, n 1 N, become a

field x where x can be any real number provided N is sufficiently large.

The basic mathematical rules for this blurring or coarse graining is that for functions f thatvary slowly on the lattice scale,

n

f nadx

af x (1.36)

In particular,

f ma n

m nf na n

am n

af na f y dx x y f x (1.37)

justifies the identification

m na

x y (1.38)

Equation (1.38) tells us that the divergent quantity x 0 in real space should be thought of as

the reciprocal of the lattice spacing, i.e., the number of normal modes in reciprocal space per unit

volume 2 N in wave number space:

1

a

kl 1 kl

a

1

2 Nkl :

2

Nl (1.39)

How does one go from a discrete Fourier sum to a Fourier integral? Start from

N

l 1

eikl m n Nm n kl :2

Nl (1.40)

Multiply both sides of this equation by the reciprocal of the system size L Na:

1

L

N

l 1

eikl m nm n

a(1.41)

Since the right hand side should be identified with x y in the thermodynamic limit N ,

the left hand side should be identified with

1

L

N

l 1

eikla

m n a2 a

0

dk

2eik x y

dk

2eik x y (1.42)

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whereby

kl

ak m n a x y (1.43)

To see this, recall first that the periodic boundary conditions tell us that l 1 N could have

equally well be chosen to run between N 2 1 and N 2 if N is even or N 1 2 and

N 1 2 if N is odd. Hence, it is permissible to adopt the more symmetrical rule

1

L

N

l 1

f kl a

a

dk

2f k (1.44)

to convert a finite summation over wave numbers into an integral over the Brillouin zone (recip-

rocal space) a a as the thermodynamic limit N L a is taken. Now, if f x is a

slowly varying function on the lattice scale a, its Fourier transform f k will be essentially van-

ishing for k 1 a. In this case, the limits a can safely be replaced by the limits on

the right hand side of Eq. (1.44). We then arrive to the desired integral representation of the delta

function in real space,

x y

dk

2eik x y (1.45)

Observe that factors of 2 appear in an assymetrical way in integrals over real x and reciprocal

k spaces. Although this is purely a matter of convention when defining the Fourier transform,

there is a physical reasoning behind this choice. Indeed, Eq. (1.44) implies that dk 2 has the

physical meaning of the number of normal modes in reciprocal space with wave number between

k and k dk per unit volume L in real space. Correspondingly, the divergent quantity 2 k 0

in reciprocal space has the physical meaning of being the divergent volume L of the system

as is inferred from

x

dk

2eikx 2 k

dx eikx (1.46)

II. QUANTUM MECHANICAL ONE-DIMENSIONAL CRYSTAL

A. Reminiscences about the harmonic oscillator

I now turn to the task of giving a quantum mechanical description for a non-dissipative one-

dimensional harmonic crystal. One possible route consists in the construction of a Hilbert space

and of operators acting on it whose expectation values can be related to measurable properties of

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the crystal.4 In this setting, the time evolution of physical quantities can be calculated either in the

Schrodinger or in the Heisenberg picture. I will begin by reviewing these two approaches in the

context of a single harmonic oscillator. The extension to the harmonic crystal will then follow in

a very natural way.

The classical Hamiltonian that describes a single particle of unit mass m 1 confined to a

quadratic well with curvature 2 is

:1

2p2 2x2 (2.1)

Hamilton-Jacobi equations of motion are

dx t

dtx

pp t

d p t

dtp

x2x t (2.2)

Solutions to the classical equations of motion are

x t A cos t B sin t

p t A sin t B cos t (2.3)

The energy E of the particle is a constant of the motion that depends on the choice of initial

conditions through the two real valued constants A and B:

E1

2A2 B2 2 (2.4)

In the Schrodinger picture of quantum mechanics, the position x of the particle and its canonical

conjugate p become operators x and p that act on the Hilbert space of twice differentiable and

square integrable functions : and obey the commutation relation

x p : x p p x i (2.5)

The time evolution (or dynamics in short) of the system is encoded by Schrodinger equation

i t x t H x t (2.6)

4 Another route to quantization is by means of the path integral representation of quantum mechanics as is shown in

appendix B.

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where the quantum Hamiltonian H is given by

H1

2p2 2 x2 (2.7)

The time evolution of the wave function x t is unique once initial conditions x t 0 are

given. Solving the time-independent eigenvalue problem

Hn x nn x (2.8)

is tantamount to solving the time-dependent Schrodinger equation through the Ansatz

x t n

cnn x eint (2.9)

The expansion coefficients cn are time-independent and uniquely determined by the initial

condition, say x t 0 . As is well known, the energy eigenvalues n are given by

n n

1

2 n (2.10)

The energy eigenfunctions n x are Hermite polynomials multiplying a Gaussian:

0 x

1 4

e12x2

1 x4

3 1 4xe

12x2

2 x

4

1 4

2

x2 1 e12x2

...

n x1

2nn!

n 1 2

1 4 x

d

dx

n

e12x2 (2.11)

The Heisenberg picture of quantum mechanics is better suited than the Schrodinger picture to a

generalization to quantum field theory. In the Heisenberg picture and contrary to the Schrodinger

picture, operators are explicitly time-dependent. For any operator O, the solution to the operator

equation of motion5

i d

O tdt

O t H (2.12)

5 The assumption that the system is non-dissipative has been used here in that H does not depend explicitly on time:

t H 0.

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that replaces Schrodinger equation is

O t e iHt

O t 0 e iHt

(2.13)

By definition, the algebra obeyed by operators in the Schrodinger picture holds true in the Heisen-

berg picture provided operators are taken at equal-time. For example,

x t : e iHt

x t 0 e iHt

p t : e iHt

p t 0 e iHt

(2.14)

obey by construction the equal-time commutator

x t p t i t (2.15)

Finding the commutator of x t and p t at unequal times t t requires solving the dynamics of

the system, i.e., Eq. (2.12) with O substituted for x and p, respectively:

dx t

dtp t

d p t

dt 2

x t (2.16)

In other words, the Heisenberg operators x t and p t satisfy the same equation of motion as the

classical variables they replace:

d2 x t

dt22 x t 0 p t

dx t

dt(2.17)

The solution (2.3) can thus be borrowed with the caveat that A and B should be replaced by time-

independent operators A and B.

At this stage, it is more productive to depart from following a strategy dictated by the real valued

classical solution (2.3). The key observation is that the quantum Hamiltonian for the harmonic

oscillator takes the very simple form6

H a t a t1

2(2.18)

if the pair of canonically conjugate Hermitean operators x t and p t is traded for the pair a t

and a t of operators defined by

x t 2 a t a

t

p t2

ia t ia t (2.19)

6 Observe that I am anticipating that H does not depend explicitly on time.

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Once the equal-time commutator a t a t is known the Heisenberg equations are easily derived

from

ida t

dta t H

ida t

dta t H (2.20)

With the help of

a t

2x t i

p t

a t

2x t i

p t

(2.21)

one verifies that

x t p t i x t x t p t p t 0 a t a t 1 a t a t a t a t 0

(2.22)

The change of Hermitean operator valued variables to non-Hermitean operator valued variables

now proves advantageous in that the equations of motion for a t and a t decouple according to

da t

dti a t a t a t 0 e it

da t

dti a t a t a t 0 e it (2.23)

Below, I will write a for a t 0 and similarly for a. The time evolution of x t , p t , and H is

now explicitly given by

x t2

a e it a e it

p t i

2a e it a e it

H aa1

2(2.24)

As must be by the absence of dissipation, H is explicitly time-independent: t H 0. The Hilbert

space can now be constructed explicitly with purely algebraic methods. The Hilbert space is

defined by all possible linear combinations of the eigenstates

n :a

n

n!0 H n n n n (2.25)

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Here, the ground state or vacuum 0 is defined by the condition

a 0 0 (2.26)

One verifies that 0 x in Eq. (2.11) uniquely satisfies Eq. (2.26) by using the real space represen-

tation of the operator a.

B. Discrete limit

In the spirit of the Heisenberg picture for the harmonic oscillator and guided by the Fourier

expansions in Eqs. (1.16) and (1.17), I begin by defining the operators

n t :1

N

N

l 1

2lal e

i kl n l t al e

i kl n lt

n t : i1

N

N

l 1

l

2

al ei kl n l t a

l e

i kl n lt n 1 N (2.27)

where the frequency l and the integer label l are related by Eq. (1.15), i.e., (remember that m 1)

l 2 1 cos kl 2 kl :2

Nl l 1 N (2.28)

and the operator valued expansion coefficients al and al obey the harmonic oscillator algebra

al

al

l l al al al

al

0 l l 1 N (2.29)

The normalization factor 1 N is needed to cancel the factor of N present in the Fourier series

N

l 1

eikl m n Nm n (2.30)

that shows up when one verifies that the equal-time commutators

m t n t i m n m t n t m t n t 0 m n 1 N (2.31)

hold for all times. I am now ready to consistently define the Hamiltonian H for the quantum

one-dimensional harmonic crystal [compare with Eq. (1.18)]

H :N

n 1

1

2n t

2 n 1 t n t2 2 n t

2(2.32)

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With the help of the algebra (2.29), one verifies that H is explicitly time-independent and given by

HN

l 1

l al al

1

2(2.33)

Next task is to construct the Hilbert space for the one-dimensional quantum crystal by algebraic

methods. Assume that there exists a unique state 0 , the ground state or vacuum, defined by

al

0 0 l 1 N (2.34)

If so, the state

n1 n2 nN :N

l 1

1

nl!a

l

nl0 (2.35)

is normalized to one and is an eigenstate of H with energy eigenvalue

n1 nN :N

l 1

l nl1

2(2.36)

The ground state energy is of order N and given by

0 0 :1

2

N

l 1

l (2.37)

Excited states have at least one nl 0. They are called phonons. The eigenstate n1 n2 nN is

said to have n1 phonons in the first mode, n2 phonons in the second mode, and so on. Phonons can

be thought of as identical elementary particles since they possess a definite energy and momentum.

Because the phonon occupation number

nl n1 nl nN al al n1 nl nN (2.38)

is an arbitrary positive integer, phonons obey Bose-Einstein statistics. Upon switching on a suit-

able interaction [say by including cubic and quartic terms in the expansion (1.4)], phonons scatter

off one other just as other -ons (mesons, photons, gluons, and so on) known to physics. Although

we are en route towards constructing quantum fields x t out ofn t we have encountered par-

ticles. The duality between field and particle is the essence of quantum field theory.

The vector space spanned by the states labelled by the phonon occupation numbers

n1 nNN in Eq. (2.35) is the Hilbert space of the one-dimensional quantum crystal. The

mathematical structure of this Hilbert space is a symmetric tensor product ofN copies of the har-

monic oscillator Hilbert space. In physics, this symmetric tensor product is called a Fock space

when the emphasis is on the phonon as an elementary particle.

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C. Thermodynamic limit

Taking the thermodynamic limit N is a direct application to subsection II B of the rules

established in the context of the classical description of subsections I B and I C. Hence, with the

identifications7

N

n 1

a dx

1

Na

N

l 1

dk

2

l k 2v2

a21 cos ka 2 v2k2 2 ifka 1,

kln kx

al1

Naa k

n t a x t

n t a x t (2.39)

the canonically conjugate pairs of operators n t and n t are replaced by the quantum fields

x t :dk

2 2 ka k e i kx k t a k e i kx k t

x t : idk

2

k

2a k e i kx k t a k e i kx k t (2.40)

respectively.

8

Their equal-time commutators follow from the harmonic oscillator algebra

a k a k 2 k k a k a k a k a k 0 (2.41)

They are

x t y t i x y x t y t x t y t 0 (2.42)

7 Limits of integrations in real and reciprocal spaces are left unspecified in order to distinguish whether L Na is

held fixed or not in the thermodynamic limit N , i.e., whether the continuum limit a 0 is simultaneously

taken or not.8 The substitution rules al

1

Naa k , n t a x t , and n t a x t , are needed to cancel the

volume factor Na in Nl 1 Nadk2 .

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The Hamiltonian is

H dx1

2t x t

2v2 x x t

2 2 x t 2

dk

2

1

2 k a k a k a k a k (2.43)

The excitation spectrum is obtained by making use of the commutator between a k and a k .

It is given by

H E0 :dk

2 k a k a k (2.44)

and is observed to vanish for the vacuum 0 . The operation of subtracting from the Hamiltoni-

an the ground state energy E0 is called normal ordering. It amounts to placing all annihilation

operators a k to the right of the creation operators a k . The ground state energy

E0 : 0 H 0

dk

2

1

2 k 2 k 0

Volume in real spacedk

2

1

2 k

modes

1

2modes (2.45)

can be ill-defined for two distinct reasons. First, if N with a held fixed, there exists an upper

cut-off to the integral over reciprocal space at the Brillouin zone boundaries a and E0 is only

infrared divergent due to the fact that 2 k 0 is the diverging volume L Na in real space.

Second, even ifL Na is kept finite while both the infrared N and ultraviolet a 0 limits

are taken, the absence of an upper cut-off in the k integral can cause the zero point energy density

E0 L to diverge as well. Divergences of E0 or E0 L are only of practical relevance if one can

control experimentally k or the density of states modes and thereby measure changes in E0 or

E0 L. For example, this can be achieved in a resonant cavity whose size is variable. If so, changes

of E0 with the cavity size can be measured. These changes in the zero point energy are known

as the Casimir energy. Another possibility to measure E0 indirectly occurs when some internal

parameters entering the microscopic Hamiltonian can be tuned so as to lower E0 to the point

where E0 becomes negative thereby signalling an instability associated to spontaneous symmetry

breaking (the vacuum 0 is not non-degenerate anymore when E0 0). Finally, divergences of

E0 L matter greatly if the energy-momentum tensors of matter fields are dynamical variables as

is the case in cosmological models.

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D. Higher dimensional generalizations

Generalizations to higher dimensions are straightforward. The coordinates x 1 and k 1

in real and reciprocal one-dimensional spaces need only be replaced by the vectors r d and

k d, in real and reciprocal d-dimensional spaces, respectively.

APPENDIX A: THE HARMONIC OSCILLATOR ALGEBRA AND ITS COHERENT STATES

1. Boson algebra

The quantum Hamiltonian for the harmonic oscillator is

H aa1

2(A1)

when represented in terms of the lowering (annihilation) and raising (creation) operators a and a,

respectively, that obey the boson algebra

a a 1 a a a a 0 (A2)

A complete, orthogonal, and normalized basis of H is given by

na n

n!0 H n n

1

2n (A3)

where the ground state (vacuum) 0 is annihilated by a:

a 0 0 (A4)

As it should be ifH is to be Hermitean, annihilation a and creation a operators are thus adjoint to

each other and represented by

a n n n 1 a n n 1 n 1

m a n nm 1 n m a n n 1m 1 n (A5)

The single-particle Hilbert space1

of twice differentiable and square integrable functions onthe real line for the harmonic oscillator can be reinterpretedas the Fock space for the annihila-

tion and creation operators a and a, respectively, since the number operator

N : aa (A6)

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commutes with the Hamiltonian and the Fock space is, by definition, the direct sum of the

energy eigenspaces:

1 :

n 0

n (A7)

One possible resolution of the identity 11 on 1 is

11

n 0

n n (A8)

More informations on the harmonic oscillator can be found in chapter V of Ref. [2].

2. Coherent states

Define the uncountable set of harmonic oscillator coherent states, in short bosonic coherent

states, by

cs : e a 0 :

n 0

n

n!n (A9)

The adjoint set is

cs : 0 ea :

n 0

n n

n! (A10)

Properties of boson coherent states are:

Coherent state cs is a right eigenstate with eigenvalue of the annihilation operator a,9

a cs a e a 0

n 0

n

n!a n

n 1

n

n!n n 1

n 1

n 1

n 1 !n 1

cs (A11)

9 Non-Hermitean operators need not have same left and right eigenstates.

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Coherent state cs is a left eigenstate with eigenvalue of the creation operator a,

a cs cs cs a

cs (A12)

The action of creation operator a on coherent state cs is differentiation with respect to

,

a cs a e a

0

n 0

n

n!a n

n 0

n

n!n 1 n 1

n 0

d

d

n 1

n 1 !n 1

d

d cs (A13)

The action of creation operator a on coherent state cs is differentiation with respect to ,

a csd

d cs cs a

d

dcs (A14)

The overlap cs cs between two coherent states is exp ,

cs cs

m n 0

m mm!

nn!

n

m n m n m n

n 0

n

n!

e (A15)

There exists a resolution of the identity in terms of boson coherent states,

11 dz dz2i

e z z z cscs z

:1

dRez

dImz e z z z cscs z (A16)

Proof:

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Write

O :dz dz

2ie z z z cs cs z (A17)

By construction, O belongs to the algebra of operators generated by a and a.

Step 1: With the help of Eqs. (A11) and (A14),

a z cs cs z a z cs cs z z cs cs z a

z z cs cs z z csd

dzcs z

z does not depend on z so that z can be factorized zd

dzz cs cs z (A18)

Hence, after making use of integration by parts,

a Odz dz

2ie z z z

d

dzz cs cs z

0 (A19)

Step 2: By taking the adjoint of Eq. (A19), a

O 0.

Step 3:

0 O 0dz dz

2ie z z 0 z cs cs z 0

m n m n m ndz dz

2ie z z

1 (A20)

Step 4: Any linear operator from to belongs to the algebra generated by a and a. Since

O commutes with both a and a

by steps 1 and 2, O commutes with all linear operators from

to . By Schurs lemma, O must be proportional to the identity operator. By Step 3, the

proportionality factor is 1.

For any operator A : ,

Tr A :

n 0

n A n

By the resolution of identity Eq. (A16)

dz dz

2i

e z z

n 0

n z cscs z A n

dz dz

2ie z z cs z A

n 0

n n z cs

By the resolution of identity Eq. (A8)

dz dz

2ie z z cs z A z cs (A21)

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Any operator A : is some linear combination of products of as and as. Normal

ordering of A, which is denoted : A :, is the operation of moving all creation operators to the

left of annihilation operators as if all operators were to commute. For example,

A aaaa aaa : A : aaaa aaa A 2aa a (A22)

The matrix element of any normal ordered operator : A a a : between any two coherent

states cs z and z cs follows from Eqs. (A11), (A12), and (A15),

cs z : A a a : z cs cs z : A z z : z cs e

z z : A z z : (A23)

Here, : A z z : is the complex valued function obtained from the normal ordered operator

: A a a : by substituting a for the complex number z and a for the complex number z .

Define the continuous family of unitary operators

D : ea

a (A24)

From Glauber formula10

D e 2

2 e a

e a (A26)

which implies that

D 0 e 2

2 e a

e a 0

e 2

2 e a 0

e 2

2 cs (A27)

Hence, D is the unitary transformation that rotates the vacuum 0 into the coherent state

cs, up to a proportionality constant.

More informations on bosonic coherent states can be found in complement GV of Ref. [2].

10

Let A and B be two operators that both commute with their commutator A B . Then,

eA eB eA B e12 A B (A25)

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APPENDIX B: PATH INTEGRAL REPRESENTATION OF THE ANHARMONIC OSCILLA-

TOR

Define the anharmonic oscillator of order n 2 3 4 by

HnH0

HnH0 : a

a1

2Hn :

2n

m 3

m a a

m

(B1)

Of the real parameters m, m 3 4 2n, it is only required that 2n 0. This insures that there

exists a vacuum 0 annihilated by a. With the help of the boson algebra (A2) it is possible to move

all annihilation operators to the right of the creation operators in the interaction Hn. Having done

that, Hn is normal ordered, i.e., Hn : Hn :. Evidently, Hn : Hn : cannot be written anymore as a

polynomial in x a a of degree 2n. For example, a a 3 aaa 3aaa 2a H c .

In the representation in which H is normal ordered, the canonical partition function on the

Hilbert space in Eq. (A7) is defined to be

Z : Tr exp H Tr exp : H :

n 0

n exp : H : n (B2)

I will now give an alternative representation of the canonical partition function that relies on the

use of coherent states. I begin with the trace formula (A21):

Zd0d0

2ie 00 cs 0 exp : H : 0 cs (B3)

For M a large positive integer, write

exp : H : exp

M

M 1

j 0

: H :

1

M

M 1

j 0

: H :

M

2

(B4)

Insert the resolution of identity (A8) M 1 -times,

e :H: e

M:

H:1

j M 1

dj dj

2ie jj j cs cs j e

M:

H: (B5)

Equation (A23) together with Eq. (B4) gives

cs 0 e

M: H: M 1 cs e

0M 1

M:H 0 M 1 :

M

2

(B6)

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and

cs j e

M:H: j 1 cs e

jj 1

M:H j j 1 :

M

2

j M 1 M 2 1

(B7)

The operator valued function : H : of a and a has been replaced by a complex valued function

:H: of and , respectively. Altogether, a M-dimensional integral representation of the partition

function has been found,

Z exp 0M 1

j 0

dj dj2i

expM

j 0

j j j 1

M: H j j 1 :

M

2

(B8)

whereby

N : 0 N : 0 (B9)

It is customary to write, in the limit M , the functional path integral representation of the

partition function

Z exp 0 eSE (B10)

where the so-called Euclidean action SE is given by

SE

0

d : H : (B11)

and the complex valued fields and obey the periodic boundary conditions

(B12)

Hence, their Fourier transform are

1

l

l eil

1

l l eil

l :

2

l (B13)

The frequencies l are the so-called boson Matsubara frequencies.

Convergence of the (functional) integral representing the partition function is guaranteed by

the contribution 2n 2n to the interaction : Hn :. Thus, convergence of an integral

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is the counterpart in a path integral representation to the existence of a ground state in operator

language.

Quantum mechanics at zero temperature is recovered from the partition function after perform-

ing the analytical continuation (also called a Wick rotation)

it d idt it (B14)

under which

SE iS

i

dt t it t : H t t : (B15)

The path integral representation of the anharmonic oscillator relies solely on two properties of

bosonic coherent states: Equations (A8) and (A23). Raising, a, and lowering, a, operators are

not unique to bosons. As we shall see, one can also associate raising and lowering operators to

fermions. Raising and lowering operators are also well known to be involved in the theory of the

angular momentum. In general, raising and lowering operators appear whenever a finite (infinite)

set of operators obey a finite (infinite) dimensional Lie algebra. Coherent states are those states

that are eigenstates of lowering operators in the Lie algebra and they obey extensions of Eqs. (A8)

and (A23). Hence, it is possible to generalize the path integral representation of the partition

function for the anharmonic oscillator to Hamiltonians expressed in terms of operator obeying a

fermion, spin, or any type of Lie algebra. Due to the non-vanishing overlap of coherent states, a

first order imaginary-time derivative term always appears in the action. This term is called a Berry

phase when it yields a pure phase in an otherwise real valued Euclidean action as is the case,

say, when dealing with spin Hamiltonians.11 It is the first order time derivative term that encodes

11 By writing [compare with Eq. (2.21)]

1

2x ip

1

2x ip (B16)

we can derive the path integral representation of the (an)harmonic oscillator in terms of the coordinate and momen-

tum of the single particle of unit mass m 1, unit characteristic frequency 1, and with 1. The first order

partial derivative term becomes purely imaginary

0d i

0d xp (B17)

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quantum mechanics in the path integral representation of the partition function. A reference on

generalized coherent states is the book in Ref. [3].

APPENDIX C: HIGHER DIMENSIONAL GENERALIZATIONS

The path integral representation of the partition function for a single anharmonic oscillator is

a functional integral over the exponential of the Euclidean classical action in 0-dimensional space(B11). The path integral representation of the quantum field theory of a d-dimensional continuum

of coupled anharmonic oscillator is a functional integral over the exponential of the Euclidean

classical action in d-dimensional space of the form

SE

0dt ddr E

0d ddr r r : H r r : (C1)

The classical fields r , and r obey periodic boundary conditions in imaginary time ,

r r r r (C2)

At zero temperature, analytical continuation it of the action yields

S

dt ddr

dt ddr r it r : H r r : (C3)

The classical canonical field conjugate to r t is

r t :

t r t r t (C4)

Canonical quantization is obtained by replacing the classical fields r t and r t with quan-

tum fields r t and r t that obey the equal-time algebra

r t r t r r r t r t r t r t 0 (C5)

[1] H. Goldstein, Classical mechanics, (Addison-Wesley, New-York, 1980).

[2] C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum mechanics, (Hermann, Paris, 1977).

[3] A. M. Perelomov, Generalized coherent states and their applications, (Springer, Berlin, 1986).

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Harmonic Crystals