The Berry Phase by jau1990

The Berry Phase Amer Iqbal & Johar M. Ashfaque Center for theoretical physics, Lahore, Pakistan

We discuss the phase of the quantum mechanical states that arises as the Hamiltonian is changed adiabatically by varying the external parameters on which the Hamiltonian depends. In the standard approach to quantum mechanics, pure quantum states are represented by vectors in a complex Hilbert space H. Each vector ψ ∈ H describes a state by the collection of expectation values A→

hψ|A|ψi hψ|ψi

where A is a self-adjoint operator in H representing some physical quantity. For this reason, two vectors ψ and φ describe the same physical state if an only if they are linearly dependent that is to say ψ = λφ,

λ ∈ C.

If we normalize the state vector for example hψ|ψi = 1 there is still a freedom to choose an overall phase factor eiα . Two normalized state vectors ψ and φ are physically equivalent ψ ∼ φ ⇐⇒ ψ = eiα φ. Hence, one usually says that the above phase factor, eiα , has no physical meaning. For this reason we may equivalently represent pure quantum states as one-dimensional projectors in H ψ =⇒ Pψ = |ψihψ|. Clearly ψ ∼ φ ⇐⇒ Pψ = Pφ . However, we know that it is the phase that controls the key effect of quantum mechanics - quantum interference. This effect is governed by the relative phase. If we have two normalized state vectors ψ and φ such that ψ = eiα φ

2 then one calls α a relative phase between ψ and φ. The relative phase or equivalently the phase difference does have a physical meaning and hence can be measured. Superposition of two states ψ ∼ φ differing in phase by α leads to the following interference formula I ∝ |1 + eiα |2 = 2(1 + cos α) = 4 cos2 (α/2) which enables one to measure α. In the interference experiment the overall phases of ψ and φ are still unknown and are not important. It is evident that eiλ ψ and eiλ φ will produce the same interference as ψ and φ. Only the relative phase counts.

THE STANDARD DERIVATION

Consider a curve C on a manifold of external parameters M t −→ xt ∈ M and the adiabatic evolution of the quantum system described by the parameter dependent Hamiltonian H = H(x) along the curve C. Then the Hamiltonian depends on time solely via the time-dependence of the external parameters: H(t) = H(xt ). Suppose that for any x ∈ M the Hamiltonian H(x) has a purely discrete spectrum that is H(x)|n(x)i = En (x)|n(x)i with hn(x)|m(x)i = δnm . Moreover, let us assume that the eigenvectors |n(x)i are single-values (as functions of x ∈ M ) that is we assume the existence of the map x ∈ M −→ |n(x)i ∈ H with H being the system’s Hilbert space. Obviously, this map need not be defined globally on M . Therefore, we assume its existence only locally. Let us assume that the n-th eigenvalue En (x) is non-degenerate and let Pn (x) = |n(x)ihn(x)| be the corresponding one-dimensional projector onto the n-th eigenspace Hn (x) which we write as Hn (x) = {α|n(x)i|α ∈ C}.

3 The eigenvectors |n(x)i are not uniquely defined and so one can arbitrarily change the phase of |n(x)i. Due to the adiabatic theorem ψ(t) stays in the n-th eigenspace of H(xt ) during the adiabatic evolution that is ψ(t) ∈ Hn (xt ). Therefore, if the evolution is cyclic that is a curve C is closed then ψ(0) and ψ(T ) both belong to Hn (x0 ) and hence they may differ only by a phase factor ψ(T ) = eiγ ψ(0). The obvious guess for the phase γ would be 1 γ=− ~

En (t)dt 0

but Berry showed this guess to be wrong. There is an additional component that has a purely geometric origin. It depends upon the geometry of the manifold M and the circuit C itself. To find this, let us begin by noting that ψ(t) and |n(xt )i differ by a time-dependent phase factor ψ(t) = exp

i − ~

En (τ )dτ eiφn (t) |n(xt )i

where the Schrödinger equation implies the following equation for the function φn φ̇n = ihn|ṅi where for simplicity the argument for |ni has been omitted. This equation defines the following one-form on M A(n) = ihn|dni. Note that hn|dni is purely imaginary. We can solve φ̇n = ihn|ṅi by a simple integration t

Z hn(τ )|ṅ(τ )idτ =

φn (t) = i 0

A(n)

where one integrates the one-form A(n) along the curve C between x0 and xt . This shows that the guess for γ needs to be supplemented by the following geometric quantity γn (C) = φn (T )

4 that is the total phase shift γ splits into two parts namely the dynamical phase and the geometric phase. The geometric phase defines the much celebrated Berry phase corresponding to the cyclic adiabatic evolution along C. Using Stoke’s theorem one may rewrite Berry’s phase as Z γn (C) =

F (n)

where Σ is an arbitrary two-dimensional submanifold in M such that ∂Σ = C and F (n) = dA(n) . The quantity A(n) is usually called the Berry vector potential or Berry’s potential one-form. The quantity F (n) plays the role of a magnetic field for the potential A(n) and the equation Z γn (C) = F (n) Σ

shows that the Berry phase is an analog of the magnetic flux in the electromagnetic theory.

II.

SPIN-HALF IN A MAGNETIC FIELD

Consider an adiabatic evolution of the spin-half particle in a slowly-varying magnetic field B. It turns out that this system naturally leads to the Hopf fibration S 3 → S 2 . To show this let us consider the Hamiltonian which is given by 1 H(B) = µσ · B 2 where σ = (σ1 , σ2 , σ3 ) is a three-vector of Pauli matrices. The magnetic field B ∈ R3 plays the role of an external parameter. To find the eigenvalues and eigenvectors of the instantaneous Hamiltonian, express the magnetic field in spherical coordinates as B = B(sin θ cos φ, sin θ sin φ, cos θ). Clearly, if B = Be3 , then the corresponding eigenvalue is solved by H(Be3 )| ± (e3 )i = E± (B)| ± (e3 )i with 1 E± (B) = ± µB 2

5 and

1 , | + (e3 )i =

0 . | − (e3 )i =

Now, if an arbitrary B is parametrized by spherical angles (θ, φ) then it may be obtained from e3 by the following SO(3) rotation R(θ, φ) = R3 (φ) · R2 (θ) · R3 (−φ) where Rk (α) denoted the rotation about the k-axis by an angle α. The corresponding unitary operator acting in C2 is given by U (θ, φ) = U3 (φ) · U2 (θ) · U3 (−φ) where Uk (α) = exp(−iαJk ) with Jk = 21 σk being the so(3) ∼ = su(2) generators. Let | ± (B)i ≡ | ± (θ, φ)i denote the eigenvectors of the original Hamiltonian. Using the spherical parametrization | ± (B)i ≡ | ± (θ, φ)i = U (θ, φ)| ± (e3 )i. Now making use of the well-known formula exp(iαn · σ) = cos α + in · σ sin α with n being the unit vector in R3 , it can be easily found that

| + (θ, φ)i =

cos 2θ

eiφ sin θ2

− sin θ2

| − (θ, φ)i =

. eiφ cos θ2

Note that the eigenvectors |±i do not depend on |B| and therefore, it is natural to take as the parameter space a two-dimensional sphere M = S 2 . Obviously, the parametrization of |±i is not global. Note that the point B = 0 corresponds to the degeneracy of the spectrum and does not belong the parameter space M = S 2 . Inserting

| + (θ, φ)i =

cos 2θ

eiφ sin θ2

− sin θ2

| − (θ, φ)i =

. eiφ cos θ2

into A(n) = ihn|dni

6 one obtains the following

A+ = ih+(θ, φ)|d| + (θ, φ)i 1 = − (1 − cos θ)dφ 2

and similarly 1 A− = (1 + cos θ)dφ. 2 Obviously, A+ and A− correspond to the potential one-forms of the magnetic monopole of strength g = ∓ 12 placed at the origin B = 0. The corresponding Berrry curvature reads 1 F + = − sin θdθ ∧ dφ 2 and F− =

1 sin θdθ ∧ dφ. 2

Clearly F + + F − = 0. Finally, the formula for the Berry phase for a spin-half particle reads Z γ± (C) =

1 F (±) = ∓ Ω(C) 2 Σ

where Ω(C) is the solid angle subtended by C on S 2 . In particular, for θ =

θ 2

that is if the

circuit C stays in the xy-plane and encloses the degeneracy point B = 0 then γ± (C) = ∓π and therefore eiγ± (C) = −1 that is the wave-function of a spin-half particle changes sign after coming back to the initial point of the the parameter manifold.

7 III.

SPIN 1

Consider a system with 3 states and Hamiltonian given by: 

     0 1 0 B 0 −i 0  1 0 0  B      1 2 ~ · J~ = √ 1 0 1 + √  i 0 −i + B3  H=B 0 0 0      2 2 0 1 0 0 i 0 0 0 −1   B1√ −iB2 0 B3 2    B1 +iB2 B1√ −iB2  = √  0 2   2 B1√ +iB2 0 −B3 2

(3.1)

Define ~ = B sin(θ)cos(φ), sin(θ)sin(φ), cos(θ) B

(3.2)

Then the Hamiltonian is given by 



sin (θ) −iφ √ e

cos(θ) 0 2    sin(θ) iφ sin (θ) −iφ  √ H =B √ e 0 e  2  2  sin (θ) iφ √ 0 e −cos(θ) 2

(3.3)

The eigenvalues of the above Hamiltonian are 0, ±B. We label the eigenvectors with the corresponding eigenvalues:

H|0; ni = 0 ,

H|−; ni = −B|−; ni

H|+; ni = B|+; ni ,

(3.4)

The eigenvectors are given by:  √  −iφ −sin(θ) e / 2     |0; ni =   , cos(θ)   √ sin(θ) eiφ / 2



−iφ



sin (θ/2) e  √    |−; ni =  −sin(θ)/ 2  ,   cos2 (θ/2) eiφ



−iφ

cos (θ/2) e  √  |+; ni =  sin(θ)/ 2  sin2 (θ/2)eiφ

    

A0 = ih0; n|d |0; ni

(3.5)    √ √ −cos(θ) e−iφ / 2 isin(θ) e−iφ / 2     i h √ √     = i −sin(θ) eiφ / 2 cos(θ) sin(θ) e−iφ / 2  dθ +    dφ −sin(θ) 0     √ √ iφ iφ cos(θ) e / 2 isin(θ) e / 2 

A− = ih−; n|d |−; ni

(3.6) 

= i sin2 (θ/2) eiφ

h √  2 −iφ  −sin(θ)/ 2 cos (θ/2) e 



sin(θ/2)cos(θ/2) e−iφ  √   dθ −cos(θ)/ 2  −cos(θ/2) sin(θ/2)eiφ   2 −iφ −isin (θ/2) e  i    +  dφ 0   icos2 (θ/2) eiφ

= − cos4 (θ/2) − sin4 (θ/2) dφ = −cos(θ) dφ

A+ = ih+; n|d |+; ni

(3.7) 

= i cos2 (θ/2) eiφ

h √  2 −iφ  sin(θ)/ 2 sin (θ/2) e 



−sin(θ/2)cos(θ/2) e−iφ  √   dθ cos(θ)/ 2  cos(θ/2) sin(θ/2)eiφ   2 −iφ −icos (θ/2) e   i   +  dφ 0   isin2 (θ/2) eiφ

= cos (θ/2) − sin (θ/2) dφ 4

= −A−

1 2π

Z S2

F0 = 0

(3.8)

1 2π

F S2

1 =± 2π

Z sin(θ) dθ ∧ dφ = ∓2

(3.9)

• In general the state |j, mi gives rise to the bundle O(−2m) over CP1 . Aside remark: degree of a map Since π2 (S 2 ) = Z therefore each ψ : S 2 7→ S 2 falls in an equivalence class labelled by the degree of the map: Z

ψ · ∂µψ × ∂ν ψ dxµ ∧ dxν ∈ Z

(3.10)

An example of a degree k map is: ψ = sin(θ) cos(kφ), sin(θ) sin(kφ), cos(θ)

(3.11)

IV.

1 8π

THE CASE OF ARBITRARY SPIN

Let us start with a state of a particle with the total spin s and its z-projection m, denoted |s, mi. If we now take a state oriented in z-direction with total spin s, the z-projection is necessary of values {−s, ..., 1, 0, 1, ...s}. If we rotate |s, mi about the z-axis by angle ϕ and then about new (rotated) y’ axis by angle ϑ , we get a quantum state with total spin s, its z-projection m in the direction of a general unit vector ~n = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ). Recall now the generators of such rotations, that are well-known spin operators ŝi . In other words, if we want to rotate a |ψi about an arbitrary axis ~n by angle α, we need to act on this state with an exponential operator e−iα~n·~s . Now we deal with the rotations R̂z (ϕ) = e−iϕŝz R̂y0 (ϑ) = e−iϕŝy0 = e−iϕŝz e−iϑŝy0 eiϕŝz and finally the operator rotating from z-axis to axis in the general position is R̂~n (ϕ, ϑ) = R̂y0 (ϑ)R̂z (ϕ).

10 It is important to remind us, that matrix of this operator is unitary. Formally, at this stage, we could write the Berry’s connection in terms of these matrices: A = ihs, m|R−1 (ϑ, ϕ)dR(ϑ, ϕ)|s, mi but as can be seen, this is not very useful. To make progress, let us deal with the eigenstate in the general direction via |s, mi~n = R̂~n (ϕ, ϑ)|s, mi~z . Notice first that the matrix of the z-operator is diagonal having (2s + 1) values on its diagonal. Then the operator of rotation about z-axis is diagonal too having exponentials of the eigenvalues of the diagonal  e−iϕs . . . 0    . ..  .. =  .. . .    iϕs 0 ... e 

e−iϕŝz

and the last thing we need to do is to evaluate, in a similar fashion, the elements of the second rotation operator (about y-axis). However, this is a serious problem since matrix of this operator is not diagonal. Recall the lowering and raising operators s± , that act on eigenstates of sz and s2 in a well-known way s± |s, mi =

p s(s + 1) − m(m ± 1)|s, m ± 1i

where s± = sx ± isy . Recall that any arbitrary rotation can be composed from up to 3 particular rotations defined by the Euler angles: • about z axis by α • about y 0 axis by β, • about z 00 axis by γ. It is of great importance that the operator R(α, β, γ) = e−iαŝz e−iβŝy0 e−iγŝz00

11 does not change the s-value of the state it acts on. This leads to R(α, β, γ)|s, mi =

s X

s 0 Dm 0 ,m (α, β, γ)|s, m i

m0 =−s

which acts as the definition of Wigner matrices. The only non-trivial part is associated with the y-rotation 0

s −im α −imγ s Dm e dm0 ,m (β) 0 ,m (α, β, γ) = e

which defines the so-called Wigner small d-matrix.

SU (2): LIE GROUPS, LIE ALGEBRAS AND REPRESENTATIONS

We will start with the expression for the Berrys connection in terms of rotation matrices R A = ihs, m|R−1 (ϑ, ϕ)dR(ϑ, ϕ)|s, mi. We have to admit, that in this formula is only partially true. The rotation matrices are generally matrices from SU (2) group. That means the multiplication by them is only possible for 2-component columns. For other spins we need to represent these rotation matrices from SU (2) by elements of the group GL(2s + 1, C) in order rescue the validity of A = ihs, m|R−1 (ϑ, ϕ)dR(ϑ, ϕ)|s, mi. The rows and columns of Wigner D-matrices span the irreducible representations of the Lie algebras generated by operators of the total angular momentum j. Therefore it is wise to correct the expression in the following way: A = ihs, m|ρ(R−1 (ϑ, ϕ))dρ(R(ϑ, ϕ))|s, mi where ρ : SU (2) → GL(2s + 1, C) is the aforementioned irreducible representation. Let us compute the left-invariant 1-form(s) on SU (2). The most general matrix A ∈ SU (2) is of the form  A=

z −w w

 ,

|z|2 + |w|2 = 1,

z, w ∈ mathbbC,

12 that can be fulfilled by 3 Euler angles ϑ ∈ h0, πi, ψ ∈ h0, 2πi and ϕ ∈ h0, 4πi as follows: ϑ z = cos e−i/2(ψ+ϕ) , 2 ϑ w = sin e−i/2(ψ−ϕ) . 2 Then the canonical 1-form is     cos ϑ2 ei/2(ψ+ϕ) sin ϑ2 ei/2(ψ−ϕ) cos ϑ2 e−i/2(ψ+ϕ) − sin ϑ2 ei/2(ψ−ϕ)  · d  θ= ϑ −i/2(ψ−ϕ) ϑ −i/2(ψ+ϕ) ϑ −i/2(ψ−ϕ) ϑ i/2(ψ+ϕ) − sin 2 e cos 2 e sin 2 e cos 2 e which yields 

i θ=−  2 e1 + ie2

e1 − ie2 −e

 

where e1 = sin ψdϑ − sin ϑ cos ψdϕ, e2 = cos ψdϑ + sin ϑ sin ψdϕ, e3 = dψ + cos ϑdϕ. In our case, the rotation is given by ϑ, ϕ with ψ = 0. The Berry connection is then given by A = e3 hs, m|ŝz |s, mi = me3 = m cos ϑdϕ as only the ŝz operator leaves the|s, mi state unchanged contributing the factor m - its eigenvalue.