Hawking Unruh Lowest Landau Level
Platform Spacetime (x, t) Non-commutative plane [Rx, Ry] =
−i`2
B in LLL
Invariant structure Spacetime metric ds2= dt2− d~x2 Commutation relation [Ri, Rj] =
−i`2 Bij
Symmetry transfor- mations
Metric preserving Lorentz transforma- tions so(2, 1): Boost (K1, K2) and ro-
tations K0
Area preserving potentials sl(2, R): Shears/Saddle (K1, K2) and rota-
tion/Harmonic K0
Rindler Hamilto- nian
Boost Inverted Harmonic oscillator
Algebra of transformations [K1, K2] = −iK0 , [K0, K1] = iK2, [K2, K0] = iK1
so(2, 1) ≈ sl(2, R)
Table 4.1: Table highlighting the parallels between the symmetry structures and platforms in the Hawking- Unruh effect and the lowest Landau level
Let us rewrite the forms of the LLL potentials identifying P = Rx/
√
`B , X = Ry/
√
V1, V2, V3 as K0, K1, K3 respectively. Then we have the Hamiltonian in the LLL generated by applied
potential to be of the form
K0= (P2+ X2), K1= (P X + XP ), K2= (P2− X2) (4.149)
In the basis of LLL wavefunctions (Bargmann space), these can be written as differential operators [237]
K0= 1 4 − ∂ 2 ∂z2 + z 2 , K1= i 2(z ∂ ∂z+ 1 2), K2= 1 4 − ∂ 2 ∂z2 − z 2 (4.150)
These are exactly the generators of sl(2, R) (in a given representation). The group SL(2, R) consists to 2 dimensional matrices of unit determinant. One could think of these are area preserving deformations in two dimensions and there are three generators of such a deformation. To get some intuition, one could think of a square(4.6): it can be rotated within its plane and the area does not change- this is done by the rotation generator K0. One can stretch it sideways increasing the length and decreasing the width preserving the area
or one could deform it to a parallelogram. These two are the shear transformations. The non-trivial part is that the order of successive transformations do not commute, but the non-commuting part will always be related to the third generators. This is expressed as the sl(2, R) ‘Lie-Algebra’:
[K1, K2] = −iK0, [K0, K1] = iK2, [K2, K0] = iK1 (4.151)
These generators are now the Hamiltonians acting on the LLL states. One can also think of them as the generators of canonical transformations that preserve the commutation relation [Rx, Ry] = −i`2B. Such an
algebra that is present in the context of the symmetries of the Minkowski spacetime is the so(2, 1) which was reviewed in one of the previous sections. In that context, the generators lead to transformation of the spacetime (t, ~x) such that the metric ds2= dt2− d~x2is preserved. The generators were those of one rotation
and two boosts. The exact mathematical relation between two Lie algebras is a Lie algebra isomorphism.
sl(2, R) ∼ so(2, 1) (4.152)
Roughly speaking, one can map from one set to the other such that the algebraic relation in Eq.(4.151) is always preserved. In this sense, the shear generators or the saddle- electrostatic potentials are equivalent to the boosts and the rotation generators in LLL to rotation generators in spacetime. One must be cautioned that this parallel atleast at the level of Lie algebra isomorphism should not be taken too literally, in the sense
of treating the two dimensional quantum Hall system as a spacetime. We are particularly interested in the action of the generator on the quantum mechanical states. In the quantum Hall system, these generators are the Hamiltonians in the LLL and generate the time evolution of states. A summary of the parallels in the structures between the two setting is given in the table 4.1.
This connection is extremely powerful as one can now think of generating quantum mechanical behaviour that is generated by relativistic transformations, particularly Lorentz Kinematics. In the following, we shall explore only a facet of this which manifests in the quantum hall effect as an equivalent to the Unruh effect.
4.10.1
Rindler Hamiltonian and the Hawking-Unruh effect in the LLL
We have seen that in the relativistic case, the boost acts as a generator of time translation or more precisely angular time translation in euclidean time τ = it. In the specific case of a Rindler spacetime, the generator of time translation was in fact the boost and was called the ‘Rindler Hamiltonian’. The immediate consequence of the Rindler Hamiltonian/boost acting the quantum mechanical states was the Unruh effect. Therefore, let us examine the shear generator/ saddle potential K3that is parallel to the boost in the sense as discussed
above.
K2=
1 4(P
2− X2). (4.153)
The above operator nothing but a quantum mechanical Hamiltonian for an inverted Harmonic oscilla- tor(IHO)() . This is also equivalent to the generator K1through a canonical transformation:
K1= i
1
2(XP + P X) (4.154)
This is called the dilatation generator and is an extremely important object in conformal field theory and generates scaling transformations. We will make use of both the representations in out study of the IHO.
This identification of the inverted oscillator with the Rindler hamiltonian and the ensuing thermality has been explored in the context of string theory in the name of c = 1 Matrix model. But more recently following ideas of t’Hooft [192] of treating black hole collapse and decay as a scattering problem (as discussed in the overview section.), Betzios et al [193] have shown that to an approximation the S-matrix of the black hole can be exactly obtained from an inverted Harmonic oscillator. They have made use of the method of projective light-cone construction to show the equivalence between the boost and the dilatation operator and finally recognizing it as the IHO. In this context let us note even though we have considered the Lorentz lie-algebra in 2+1- dimensions, the identification of the boost generator with the IHO is valid in 3+1-dimensions as well [193].
Now, let us consider the problem of quantum mechanical scattering off an inverted Harmonic oscillator and show the appearance of a thermal-like tunneling probability. We start with the form of the Hamiltonian:
H = 1 2(
p2
2m− x
2) (4.155)
The left and right sides of the barrier are denoted by ± and we take the mass m = 1/2 for convenience. The scattering matrix relates the incoming states from left and right directions to the outgoing states on the left and right, at energy E. The energy spectrum is a continuum in E : (−∞, ∞).
The S-matrix relates the out and in states as follows
|E, +iout |E, −iout
= ˆS |E, +iin |E, −iin (4.156) ˆ S = √1 2πΓ 1 2+ iE
eiπ/4e−πE/2 e−iπ/4e−πE/2
eiπ/4e−πE/2 e−iπ/4eπE/2
(4.157)
The derivation of the scattering properties and analysis of the IHO is given in detail in the next chapter as it deserves a detailed treatment. Here we just state the results. From this the tunneling probability can be immediately calculated
|t|2= 1
1 + e2πE (4.158)
The form of the tunnleing probability resembles a thermal distribution of degrees of freedom with energy E similar to the one we encountered in the Hawking-Unruh effect.
One can obtain the thermal density matrix interpretation to this result by expressing an in-coming state to the outgoing state [3]:
|0, ini = N exp
i Z +∞
−∞
e−Eπ(ˆbout,+E ˆbout,−E + ˆbout,−−E ˆbout,+−E )dE 2π
|0, outi (4.159)
The above ˆaE, ˆbE operators act on the vacua |0, ini , |0, outi respectively. A thermal density matrix can be
obtained by tracing out states on the −side:
ˆ
ρ = Σie−2πEi|Ei, +i ⊗ hEi, +| (4.160)
For completeness, let us also write down an effective metric in the quantum Hall system following[3] and relating to the discussion on Rindler spacetime in the previous section. The effective velocity of the
electron in the quantum Hall system under the application of the potential V (x, y) = λ(x2− y2) is given
vef f = eBλ ∂V /∂y = −(λe/B)y. This vanishes at y = 0 and that point can be interpreted as the event
horizon. The effective space-time metric with an ‘event horizon’ is then given by
ds2= −dt2+ 1 vef f(y)2
dy2 (4.161)
Using the vef f = −κy with κ = λ/eB, we can write the metric in Euclidean time τ = it as
ds2= Ω2(y)(dy2+ κ2y2dτ2) (4.162)
where Ω(y) = 1/(κ2y2) is a conformal factor. This is equivalent to the Rindler metric we had studied in the previous section.
Thus, we have shown that the Rindler Hamiltonian, the crucial element in the the phenomenon of Unruh effect can be captured as an exact mathematical isomorphism in the context of quantum Hall effect. The thermality arises in the tunneling probability of scattering off an inverted oscillator potential. One can also obtain a thermal density matrix and an effective Rindler space description in the quantum Hall system. This concludes the demostration of the parallels of Hawking-Unurh effect in the quantum Hall effect with a saddle potential.