Disorder expansion - Free lattice bosons with weak disorder

Free lattice bosons with weak disorder

8.1 Disorder expansion

We begin by considering the functional-integral formulation of the Bose-Hubbard model with an action given by (4.1). Furthermore, we introduce to this action the source term expressed in (4.4). We are interested in the case where the hopping energy is much larger then the other energy scales in the system. Assuming thatU/J→0 and ∆^/J¿1, we can treat the system as composed of non-interacting particles and consider the local potential as a perturbation. Thus, in this limit, we define the unperturbed partition function to have the following form

Z0[j^∗, j]≡

HDψ^∗H

Dψe⁽^ψ|G⁰⁻¹^ψ^)+(j|ψ⁾⁺⁽^ψ|j) HDψ^∗H

Dψe⁽^ψ|G⁰⁻¹^ψ⁾ , (8.1) such that_Z₀[0, 0]=1. Moreover, we use the definition

(ψ|G0−1ψ)= Z _β

dτX

i j

ψ^∗_i(τ)

J_{i j}−δi j

µ ∂

∂τ−µ

¶¸

ψj(τ). (8.2)

124

By completing the square in the argument of the exponential function in the numerator of (8.1), we obtain

Z0[j^∗, j] = e⁻^(j^|G⁰^j)

HDψ^∗H

Dψe⁽^ψ+jG⁰^|G⁰⁻¹^[^ψ+G⁰^j]) HDψ^∗H

Dψe⁽^ψ|G⁰⁻¹^ψ⁾

= e⁻^(j^|G⁰^j),

(8.3)

where the second line is obtained by realizing that the integral in the numerator of the first line is equivalent to the integral in the denominator up to a change o variables.

Introducing the Matsubara-Fourier transform ψi(τ)=1

³ a 2π

´dX

dke^ik^·^xⁱ⁻ⁱ^ω^l^τψ(k, iωl), (8.4)

we can write the unperturbed Green’s function as G0(k, iωl)= 1

iωl+µ+J(k), (8.5)

where J(k)is the lattice dispersion for a^d-dimensional hypercubic lattice defined in (4.19).

Note that the above unperturbed Green’s function has the same form as the zero-temperature limit shown in equation (5.2) for the case ofn₀=0. Similarly to what we have done in Sec- tion 4.1 to find the hopping expansion to the partition function (4.10), we can consider the local potential as a perturbation, and calculate corrections with respect to it via

Z[j^∗, j]=exp Ã

−X

d^dk Z

d^dk⁰²(k,k⁰) δ²

δj^∗(k, iωl)δj(k⁰, iωl)

e⁻^(j^|G⁰^j), (8.6) where_²(k,k⁰)has the same form of (3.65). Therefore the full Green’s function can be written as

G(k,k⁰; iω)= − δ²Z[j^∗, j]

δj^∗(k, iωl)δj(k⁰, iωl)

¯j^∗,j=0

. (8.7)

We point out that analogously to the hopping expansion of (4.10), where for each term the tunneling matrix coupled different local Green’s functions (3.82) between different neighbor- ing sites, in the present case of (8.6), the local energy shift couples free propagators (8.5) of states with different wavevectors.

Considering order by order in the perturbative expansion, it is possible to show that the terms which contribute to the full Green’s function have the following form

G(k,k⁰; iωl)=δ(k−k⁰)G0(k, iωl)+G0(k, iωl)²(k,k⁰)G0(k⁰, iωl) +

d^dk₁G0(k, iωl)²(k,k₁)G0(k₁, iωl)²(k₁,k)G0(k⁰, iωl)+ · · ·. (8.8) To study the global properties of disorder, we must take the average over all realizations of

the random potential

G(k,k⁰; iωl)=δ(k−k⁰)G0(k, iωl)+G0(k, iωl)²(k,k⁰)G0(k⁰, iωl) +

d^dk₁G0(k, iωl)²(k,k₁)G0(k₁, iωl)²(k₁,k)G0(k⁰, iωl)+ · · ·. (8.9) We can understand the physical interpretation of the expansion, by treating each term separately. The first term on the right-hand side of (8.9) is not affected by the disorder and thus represents the free propagation of an excitation with wavevector k and energy iωl. The second term has a correction of first order in the disorder expansion. The averaging process turns out to be given by

G0(k, iωl)²(k,k⁰)G0(k⁰, iωl) = ³ a 2π

´dX

e⁻^ixⁱ^·^(k⁻^k⁰⁾G0(k, iωl)²iG0(k⁰, iωl)

= δ(k−k⁰)G0(k, iωl)²iG0(k, iωl),

(8.10)

where we have used the explicit form of the random potential (3.65) and the representation of the sum over all sites of the exponential in the first line as a Dirac-delta distribution (3.88).

This term can therefore be interpreted as a free propagation followed by one scattering event at site i and another free propagation. The averaging of the second-order term is given by

d^dk₁²(k,k₁)G0(k₁, iωl)²(k₁,k)=

³ a 2π

´2dX

i j

²i²j

d^dk₁e^−ixⁱ^·(k⁻^k¹^)−ix^j^·(k¹⁻^k⁰⁾G0(k₁, iωl).

(8.11) The correlation function of the random potential can be decomposed as

²i²j=δi j²²_i +²i ²j. (8.12) For the first term on the right hand side of (8.12), equation (8.11) reduces to

d^dk₁²(k,k₁)G0(k₁, iωl)²(k₁,k) = ²²_i³ a 2π

´2dX

e⁻^ixⁱ^·^(k⁻^k⁰⁾ Z

d^dk₁G0(k₁, iωl)

= δ(k−k⁰)²²_iG0ii(iωl),

(8.13)

where we have definedG0ii as the local form of the inverse Fourier of (8.5) from reciprocal space to real space

G0i j(iωl)=

³ a 2π

´dZ

d^dke^ik^·(xⁱ⁻^x^j⁾G0(k, iωl), (8.14) that is,_G_0ii corresponds to the above equation when i=j,

G0ii(iωl)=

³ a 2π

´dZ

d^dkG0(k, iωl). (8.15)

The expression (8.13) can be understood as representing a double scattering process at siteifollowed by a coherent superposition all values of the wavevector over the first Brillouin zone defined by (8.15). For the second term on the right hand side of (8.12), equation (8.11) simplifies to

d^dk₁²(k,k₁)G0(k₁, iωl)²(k₁,k)=

³ a 2π

´2dZ

d^dk₁X

i j

e^−ixⁱ^·(k⁻^k¹^)−ix^j^·(k¹⁻^k⁰⁾²iG0(k₁, iωl)²j

=δ(k−k⁰)²iG0(k₁, iωl)²j,

(8.16) which can be interpreted as a single scattering event at siteifollowed by a free propagation and then by another single scattering event at a different site j.

With this interpretation, we introduce a diagrammatic representation to describe all the possible terms of the perturbation expansion (8.9). Free propagation can be represented by

G0(k, iωl) =

. (8.17)

We define the diagram for a single scattering at one site between two free propagations (8.10) as

G0(k, iωl)²iG0(k, iωl) =

k k

, (8.18)

while the diagram representing (8.13) is defined as

²²_iG0ii(iωl) = . (8.19)

Note that arrow between the two dots in this case has no wavevector index, meaning that it represents the sum of the amplitude over all values of the wavevector in the first Brilloin zone. With this notation equation (8.9) of the average Green’s function can be rewritten as

G(k,k⁰; iωl)=δ(k−k⁰)

k +

k k+

k k k +

k k

+ k k k k +

k k +

k k k

+ k k k +

k k + · · ·

¸ .

(8.20)

Note that all diagrams consist of free propagation followed by scattering once to infinitely

many times at the same site and then by another another free propagation. Therefore, we can choose the sum of all internal diagrams representing the process of scattering once to infinitely many times with the disordered potential as the self-energy

Σ^{(k, i}ωl)= + + + + · · ·. (8.21)

Therefore, the sum of the full disorder-averaged Green’s function can be reorganized in terms of the self-energy and represented diagrammatically as

G(k,k⁰; iωl)=δ(k−k⁰)

k +

k Σ k ⁺

k Σ k Σ k ^{+ · · ·}

. (8.22) Summing up an infinite amount of such diagrams we obtain

G(k,k⁰; iωl)=δ(k−k⁰) 1

G0(k, iωl)⁻¹−Σ^{(k, i}ωl), (8.23) which can be rewritten as

G(k,k⁰; iωl)=δ(k−k⁰) 1

iωm+µ+J(k)−Σ^{(k, i}ωl). (8.24) Note that, as discussed in previous chapters, the averaged Green’s function recovers trans- lational invariance and represents the propagation of excitations which conserve crystal momentum. However, as a result of the scattering against the random potential, such states acquire a finite lifetime.

In the case of (8.24), applying the definition (3.60) leads to the following averaged spectral function

A(k,ω)= −1 π

ImΣ^(k,ω)

[ω+µ+J(k)−ReΣ^(k,ω)]²+[ImΣ^(k,ω)]², (8.25) which has a form similar to equation (5.13). However, in the present case, we can see that the real part of the self-energy shifts the dispersion relation of the excitations

ω(k)= −µ−J(k)+ReΣ^(k,ω), (8.26) while its imaginary part is related to their lifetime. Considering that the peaks of the spectral function are still close to the unperturbed dispersion,_ω0(k)= −µ−J(k), we can estimate such a lifetime as

τ˜(k)≈ 1

|ImΣ^(k,ω0(k))|. (8.27)

Thus, in this calculation, the self-energy incorporates the effects of disorder in the propaga-

tion of the elementary excitations.

In the hopping expansion, developed in Section 4.1, the unperturbed Green’s functions corresponded to localized excitations calculated in the strongly interacting limit. This perturbation method is usually referred to as strong-coupling expansion, or even locator expansion in the case of free particles. Unlike the previous hopping expansion, the unperturbed Green’s function in the present calculation corresponds to delocalized excitations.

One can see this, by noticing that the inverse Fourier transform of (8.5) to real time and space represents a Bloch state. For this reason the present method is usualy called prop- agator expansion. From this point of view, the two methods are different. However, it was shown in Refs. [137, 138] that there is an equivalence relation between this two kinds of expansions in the non-interacting limit. The theory developed in this section with the disorder expansion could be applied together with the hopping expansion in chapter 4.1. For the results of the present work, this would amount to the substitution of (8.5) by the clean-case limit of the result (4.20) obtained by partial summation, which amounts to (5.1). In this way, fluctuations in the hoppping expansion could be treated separately from fluctuations in the disorder expansion. Hence, the present calculations provide an insight for future work.

What follows is an account of how to obtain the explicit form of the self energy via the single-site approximation.

No documento Ultracold bosons in random lattice potentials (páginas 161-166)