In this section we use the techniques of the previous section to study models of two species of bosons with statistical interactions.[47,48,49] Consider the following action:
S = 1 2
X
k
hv1(k)|J~1(k)|2+v2(k)|J~2(k)|2i +iX
k
θ(k)J~1(−k)·~aJ2(k) + iX
k
hJ~1(−k)·A~ext1 (k) +J~2(−k)·A~ext2 (k)i
. (2.25)
The first two terms of this action are just two copies of Eq. (2.21). BothJ~1andJ~2behave like theJ~of the previous section; however. in real space they are defined on different cubic lattices, as described previously.
We will index the sites of lattice ofJ~1with the indexr, and call it the direct lattice, whileJ~2’s lattice will be indexed byRand called the dual lattice. The sites of the dual lattice are in the center of the cubes of the direct lattice.
Eq. (2.25) also contains an integer-valued gauge field~aJ2, which is defined such that
J~2(R) = [∇ ×~ ~aJ2(r)]. (2.26)
This field lives on the links of direct (r) lattice, but taking the curl of something on the links of the direct lattice gives a variable defined on the links of the dual lattice. The third term in this action is defined such that whenθ(k)is a constant, this action has a phase ofeiθwhen loops of opposite species are linked. This linking in the(2 + 1)-dimensional spacetime corresponds to an exchange in the corresponding2dproblem, so the effect of this term is to encode mutual statistics betweenJ~1andJ~2. Whenθ(k)is not constant this term gives
some additional interactions in addition to the statistical interactions. TheAextare external Maxwell fields which will be used to compute linear reponses.
In this section we will show how to reformulate this action in terms of some new integer-valued currents G, which represent the worldlines of some other kind of boson. We will obtain an action for the~ G~ bosons that is of the same form as Eq. (2.25), but with modified parametersv(k)andθ(k). We will show how to write thev(k),θ(k)for theG~ variables in terms of those for theJ~variables. The point of doing this is that depending on the choice of parameters, theG~ variables can be much easier to study than theJ~variables.
We can also relate observables in theJ~variables (which can be hard to measure) to observables in theG~ variables (which can be easier to measure). These techniques were essential to solving the problems studied in Refs. [47,48,49].
We can use the duality transform from the previous section [see Eq. (2.24)] to go from theJ1variables to dualQ1variables as follows:
S =1 2
X
k
2π ~Q1(k) +θ(k)J~2(k) + [∇ ×~ A~ext1 ](k)
2
|f~k|2v1(k) + 1
2 X
k
v2(k)|J~2(k)|2+iX
k
J~2(−k)·A~ext2 (k). (2.27)
TheQ~1variables represent the vortices of the bosons defined byJ~1, and like them they are divergenceless and therefore form closed loops.
We can now make the following change of variables:[48,49]
F~1 = a ~Q1−b ~J2, (2.28)
G~2 = c ~Q1−d ~J2. (2.29)
This change of variables is valid if the matrix
a b c d
∈P SL(2,Z), (2.30)
i.e.,a, b, c, dare integers such thatad−bc= 1. Since the above matrix is an element of the modular group, we call this change of variables a modular transformation and will often refer to it simply(a, b, c, d). Here F1andG2are new integer-valued conserved currents, with all the same properties (divergenceless, zero total current) as theJandQvariables. We can therefore perform the duality transform to go from theF~1variables to dualG1variables, which gives us an action in terms of theG1andG2variables. This transformation, from J1,J2variables toG1,G2variables, is the generalization of the duality operation to modular transformations.
After performing this change we are left with the following action:
S = 1 2
X
k
vG1(k)
G~1(k) +c[∇ ×~ A~ext2 ](k) 2π
2
+1 2
X
k
vG2(k)
G~2(k) +c[∇ ×~ A~ext1 ](k) 2π
2
+ iX
k
θG(k)G~1(−k)·~aG2(k)−iX
k
c[2πa−θG(k)c]
(2π)2 [∇ ×~ A~ext1 ](−k)·A~ext2 (k)
− iX
k
a−θG~(k)c 2π
h
G~1(−k)·A~ext1 (k) +G~2(−k)·A~ext2 (k)i
, (2.31)
whereG~2=∇ ×~ ~aG2and
vG1/2(k) = (2π)2v1/2(k)
[2πd+θ(k)c]2+v1(k)v2(k)|f~k|2c2, (2.32) θG(k)
2π =[2πb+θ(k)a][2πd+θ(k)c] +v1(k)v2(k)|f~k|2ca
[2πd+θ(k)c]2+v1(k)v2(k)|f~k|2c2 . (2.33) Neglecting theAextterms, the above action has the same form as Eq. (2.25), but with differentv(k)andθ(k).
Though the expressions for may look intimidating, with a proper choice of(a, b, c, d)they can in fact be quite easy to work with.
Often we are interested in measuring physical properties of theJ~variables, but find that theG~ variables are much easier to work with. In particular, we want to measure linear responses to applied electromagnetic fields. These responses are defined by:
Cabµν(k) =hJaµ(k)Jbν(−k)i, (2.34)
whereµ, νare lattice directions anda, brepresent boson species.Caaµµ(k)≡ρaµ(k)is the superfluid stiffness for speciesa, whileC12xy(k)is related to the cross-species Hall response: the current induced in bosons of species1to an applied field which couples to bosons of species2. The following equations give theCabµν(k) of theJ~variables, in terms of theCabµν(k)of theG~ variables:
C11Jxx (k) = v(k)|fk|2c2
(θc+ 2πd)2+|fk|2v(k)2c2 (2.35)
+[(θc+ 2πd)2− |fk|2v(k)2c2]C11Gxx (k)−4 sink2zv(k)c(θc+ 2πd)C12Gxy (k)
[(θc+ 2πd)2+|fk|2v(k)2c2]2 ·(2π)2, C12Jxy (k) = −2 sink2zc(θc+ 2πd)
(θc+ 2πd)2+|fk|2v(k)2c2 (2.36)
+[(θc+ 2πd)2− |fk|2v(k)2c2]C12Gxy (k) + 4 sink2zv(k)c(θc+ 2πd)C11Gxx (k)
[(θc+ 2πd)2+|fk|2v(k)2c2]2 ·(2π)2.
The above expressions are especially easy to evaluate if the G~ variables are gapped, because in that
case in the thermodynamic limit theCabGµν → 0. 3 In the next chapter we will have an action of the form Eq. (2.25) which is difficult to study, and we will determine its Hall conductivity by finding the(a, b, c, d) which produced gappedG~ variables. We can then read off the Hall conductivity from Eq. (2.36). We also took a different approach in Ref. [48], which is not covered by this report. We studied a model wherev(k) had the formg/|fk|for both theJ~andG~ variables, with only the constantgchanging under the modular transformation. We were therefore able to use the above equations to produce the entire phase diagram, and find the Hall conductivity and superfluid stiffness in each phase.