U convergence
indicates the bottom of the spectrum.
3
Floquet operators in a quasiperiodically driven systems
For the two tone system, in the low energy states of the system, e↵ectively form an Aubry Andre model, and we see the familiar spectral properties of a localised system.
3.1
Ivar’s observation
Ivar noted that the eigenstates of the Floquet operator of the periodic system converge as the quasiperiodic limit limit is taken. To be more concrete, consider the two parameter periodic hamiltonian H(✓1 , ✓2 ) such that H(✓1 , ✓2 ) = H(✓1 +2⇡, ✓2 ) = H(✓1 , ✓2 +2⇡). Then define a quasiperiodic Hamiltonian Hq (t) = H(2⇡t, 2⇡ t), where =
p
5+1 2
is the Golden mean. This Hamiltonian is obtained in the quasiperiodic limit i ! 1 of the
series of periodic Hamiltonians Hi (t) = H(2⇡t, 2⇡ i t) where to
given by the ratio of consecutive Fibonacci numbers.
7
i
= Fi+1 /Fi is a rational approximation
The periodicity is given by Hi (t) = Hi (t + Fi ), allowing definition of a Floquet operator Ui = T exp
"
i
Z
Fi
Hi (t)dt 0
#
(11)
Ivar observed that the eigenstates |ui i of the operator Ui converged in the quasiperiodic limit.
3.2
A marginally stronger observation
In fact it seems the floquet operators inheret the algebra of the underlying Fibonacci numbers in the quasiperiodic limit, ie that the property Fi + Fi+1 = Fi+2 appears as both Ui Ui+1 ! Ui+2
and
Ui+1 Ui ! Ui+2
(12)
becoming exact in the quasiperiodic limit. This is verified by plotting |Ui Ui+1
Ui+2 | vs i (where | · | denotes the spectral norm) for arbitrary
driven systems. In this case we look at random GUE 4 ⇥ 4 periodic Hamiltonians
1
»Ui Ui+1 -Ui+2 »
0.01
10-4 10-6 10-8 0
5
10 i
15
20
. A corollary of this claim is that the commutator [Ui , Ui+1 ] ! 0 in the quasiperiodic limit, which
can also be numerically verified by plotting [Ui , Ui+1 ] vs i
8
1
»@Ui ,Ui+1 D»
0.01
10-4 10-6 10-8 0
5
10 i
. The exponential decay in both these plots follows ⇠ 3.2.1
15
⇣p
p5 1 5+1
⌘i
20
, as expected.
Numerical checks with other metallic ratios
Given a series of integers defined by the recursion relation pi+1 = api + bpi p boundary p1 = 1, p2 = 1, the ratio ri = pi+1 /pi ! (a + a2 + 4b)/2.
1
(for a, b 2 N), and
The Floquet operators of the hamiltonians Hi (t) = H(2⇡t, 2⇡ri t) are then found to inherit the new
recursion relation and, converge according to Uib 1 Uia ! Ui+1 . This is observed numerically for the silver ratio obtained from (a, b) = (2, 1)
Ui b Ui+1 a -Ui+2
0.1
0.001
10-5
10-7
0
2
4
6 i
and the copper ratio obtained from (a, b) = (3, 1)
9
8
10
1
Ui b Ui+1 a -Ui+2
0.01
10-4 10-6 10-8 0
2
4 i
6
8
and for (a, b) = (2, 2)
1
»Ui b Ui+1 a -Ui+2 »
0.1 0.01 0.001 10-4 10-5 10-6 0
2
4
6
8
10
i where the exponential decays (plotted in dashed red) follow the expected form ⇠
given by the ratio of the two roots of the recursion equation. 3.2.2
⇣p
2 pa +4b a a2 +4b+a
⌘i
A naive and unsuccessful attempt at proof
I show a naive, and unsuccessful, approach at proving this. I hope this might be useful as a starting point. We note that the definition Ui = T exp
"
i
10
Z
Fi
Hi (t)dt 0
#
(13)
where Hi = H(2⇡t, 2⇡ i t). We can also define a generator for the unitary Ui Ui
2 Ui 1
= T exp
"
i
Z
Fi
Ki (t)dt 0
2 Ui 1
via
#
(14)
where 8 > > H (t) = H (2⇡t, 2⇡ i 1 t) , > < i 1 Ki (t) = Hi 2 (t + Fi 1 ) = H (2⇡t, 2⇡ ( > > > :K (t ± F ) i
for t < Fi i 2 (t
Fi ) + Fi+1 ))
Fi
i i 1
1 ) = ( 1)
i 1
8 > > 2⇡( i 1 > < 2⇡( i 2 i (t) = > > > : (t ± F ) i i
= ( 1)i /(Fi Fi
1)
1
< t < Fi .
(15)
otherwise
i
For more direct comparison with Hi (t) we can write Ki (t) = H(2⇡t, 2⇡
Using that
for Fi
1
i )t,
for t < Fi
i )(t
Fi )
for Fi
1
i
+
i (t))
where
1
(16)
< t < Fi
otherwise.
we see that the maximum value of
i (t)
is given by
/Fi , which is exponentially small in i. So we try expanding perturbatively in (1)
Ki (t) = Hi (t) + Hi (t) i (t) + higher order terms.
i (t
=
i
(17)
Where we use H (1) to denote a the first derivative in the second argument of H. By expansion of the exponential Ui
2 Ui 1
= Ui +
Z
Fi 0
⇣ ds T e
i
Rs
Hi (t)dt
0
⌘⇣
(1)
i i (s)Hi
⌘⇣ Te
i
RF s
i
Hi (t)dt
⌘
.
(18)
⌘
(19)
Thus to leading order in |Ui
2 Ui
1
Ui | = =
Z
Z Z Z
Fi 0 Fi
⇣ ds T e
ds 0 Fi
⇣
Fi
ds 0
i
Te
ds T e
0
i
i
Rs
Hi (t)dt
Rs
Hi (t)dt
0
Rs 0
0
Hi (t)dt
⌘⇣ ⌘⇣
(1)
i (s)Hi (s) (1) i (s)Hi (s)
(1) i (s)Hi (s)
(1) i (s)Hi (s)
⇣ ⌘Z (1) max Hi (s) s
⌘⇣ Te ⌘⇣ Te
Te
i
RF s
i
i
i
RF s
RF s
i
Hi (t)dt
i
Hi (t)dt
Hi (t)dt
⌘
(20) (21) (22)
Fi 0
ds | i (s)|
(1)
= ⇡ max Hi (s)
(23) (24)
s
However, since this bound does not converge to zero, but a constant, it does not imply the observed
11
convergence. In particular we have used that Z
Fi 0
ds | i (s)| = 2⇡
"Z
Fi 0
1
t dt F i Fi
+ 1
Z
Fi
dt(Fi Fi
1
t)
✓
1 F i 1 Fi
2
1 F i Fi
1
◆#
=⇡
(25) (26)
(1)
In order to improve this I think it is necessary to use that H1 (s) is rapidly oscillating on the timescales which
i (t)
varies. However the possible cancellations that may be induced by this property are lost
when the norm-of-the-integral is split into an integral-of-norms as in (20). Anushya also pointed out that any procedure for attempting to prove this must necessarily fail in the case of an infinite Hilbert space where we know there exist open orbits in phase space which give rise to non normalisable eigenstates of the Floquet operator and unbounded Hamiltonians.
12
3
Floquet operators in a quasiperiodically driven systems
For the two tone system, in the low energy states of the system, e↵ectively form an Aubry Andre model, and we see the familiar spectral properties of a localised system.
3.1
Ivar’s observation
Ivar noted that the eigenstates of the Floquet operator of the periodic system converge as the quasiperiodic limit limit is taken. To be more concrete, consider the two parameter periodic hamiltonian H(✓1 , ✓2 ) such that H(✓1 , ✓2 ) = H(✓1 +2⇡, ✓2 ) = H(✓1 , ✓2 +2⇡). Then define a quasiperiodic Hamiltonian Hq (t) = H(2⇡t, 2⇡ t), where =
p
5+1 2
is the Golden mean. This Hamiltonian is obtained in the quasiperiodic limit i ! 1 of the
series of periodic Hamiltonians Hi (t) = H(2⇡t, 2⇡ i t) where to
given by the ratio of consecutive Fibonacci numbers.
7
i
= Fi+1 /Fi is a rational approximation
The periodicity is given by Hi (t) = Hi (t + Fi ), allowing definition of a Floquet operator Ui = T exp
"
i
Z
Fi
Hi (t)dt 0
#
(11)
Ivar observed that the eigenstates |ui i of the operator Ui converged in the quasiperiodic limit.
3.2
A marginally stronger observation
In fact it seems the floquet operators inheret the algebra of the underlying Fibonacci numbers in the quasiperiodic limit, ie that the property Fi + Fi+1 = Fi+2 appears as both Ui Ui+1 ! Ui+2
and
Ui+1 Ui ! Ui+2
(12)
becoming exact in the quasiperiodic limit. This is verified by plotting |Ui Ui+1
Ui+2 | vs i (where | · | denotes the spectral norm) for arbitrary
driven systems. In this case we look at random GUE 4 ⇥ 4 periodic Hamiltonians
1
»Ui Ui+1 -Ui+2 »
0.01
10-4 10-6 10-8 0
5
10 i
15
20
. A corollary of this claim is that the commutator [Ui , Ui+1 ] ! 0 in the quasiperiodic limit, which
can also be numerically verified by plotting [Ui , Ui+1 ] vs i
8
1
»@Ui ,Ui+1 D»
0.01
10-4 10-6 10-8 0
5
10 i
. The exponential decay in both these plots follows ⇠ 3.2.1
15
⇣p
p5 1 5+1
⌘i
20
, as expected.
Numerical checks with other metallic ratios
Given a series of integers defined by the recursion relation pi+1 = api + bpi p boundary p1 = 1, p2 = 1, the ratio ri = pi+1 /pi ! (a + a2 + 4b)/2.
1
(for a, b 2 N), and
The Floquet operators of the hamiltonians Hi (t) = H(2⇡t, 2⇡ri t) are then found to inherit the new
recursion relation and, converge according to Uib 1 Uia ! Ui+1 . This is observed numerically for the silver ratio obtained from (a, b) = (2, 1)
Ui b Ui+1 a -Ui+2
0.1
0.001
10-5
10-7
0
2
4
6 i
and the copper ratio obtained from (a, b) = (3, 1)
9
8
10
1
Ui b Ui+1 a -Ui+2
0.01
10-4 10-6 10-8 0
2
4 i
6
8
and for (a, b) = (2, 2)
1
»Ui b Ui+1 a -Ui+2 »
0.1 0.01 0.001 10-4 10-5 10-6 0
2
4
6
8
10
i where the exponential decays (plotted in dashed red) follow the expected form ⇠
given by the ratio of the two roots of the recursion equation. 3.2.2
⇣p
2 pa +4b a a2 +4b+a
⌘i
A naive and unsuccessful attempt at proof
I show a naive, and unsuccessful, approach at proving this. I hope this might be useful as a starting point. We note that the definition Ui = T exp
"
i
10
Z
Fi
Hi (t)dt 0
#
(13)
where Hi = H(2⇡t, 2⇡ i t). We can also define a generator for the unitary Ui Ui
2 Ui 1
= T exp
"
i
Z
Fi
Ki (t)dt 0
2 Ui 1
via
#
(14)
where 8 > > H (t) = H (2⇡t, 2⇡ i 1 t) , > < i 1 Ki (t) = Hi 2 (t + Fi 1 ) = H (2⇡t, 2⇡ ( > > > :K (t ± F ) i
for t < Fi i 2 (t
Fi ) + Fi+1 ))
Fi
i i 1
1 ) = ( 1)
i 1
8 > > 2⇡( i 1 > < 2⇡( i 2 i (t) = > > > : (t ± F ) i i
= ( 1)i /(Fi Fi
1)
1
< t < Fi .
(15)
otherwise
i
For more direct comparison with Hi (t) we can write Ki (t) = H(2⇡t, 2⇡
Using that
for Fi
1
i )t,
for t < Fi
i )(t
Fi )
for Fi
1
i
+
i (t))
where
1
(16)
< t < Fi
otherwise.
we see that the maximum value of
i (t)
is given by
/Fi , which is exponentially small in i. So we try expanding perturbatively in (1)
Ki (t) = Hi (t) + Hi (t) i (t) + higher order terms.
i (t
=
i
(17)
Where we use H (1) to denote a the first derivative in the second argument of H. By expansion of the exponential Ui
2 Ui 1
= Ui +
Z
Fi 0
⇣ ds T e
i
Rs
Hi (t)dt
0
⌘⇣
(1)
i i (s)Hi
⌘⇣ Te
i
RF s
i
Hi (t)dt
⌘
.
(18)
⌘
(19)
Thus to leading order in |Ui
2 Ui
1
Ui | = =
Z
Z Z Z
Fi 0 Fi
⇣ ds T e
ds 0 Fi
⇣
Fi
ds 0
i
Te
ds T e
0
i
i
Rs
Hi (t)dt
Rs
Hi (t)dt
0
Rs 0
0
Hi (t)dt
⌘⇣ ⌘⇣
(1)
i (s)Hi (s) (1) i (s)Hi (s)
(1) i (s)Hi (s)
(1) i (s)Hi (s)
⇣ ⌘Z (1) max Hi (s) s
⌘⇣ Te ⌘⇣ Te
Te
i
RF s
i
i
i
RF s
RF s
i
Hi (t)dt
i
Hi (t)dt
Hi (t)dt
⌘
(20) (21) (22)
Fi 0
ds | i (s)|
(1)
= ⇡ max Hi (s)
(23) (24)
s
However, since this bound does not converge to zero, but a constant, it does not imply the observed
11
convergence. In particular we have used that Z
Fi 0
ds | i (s)| = 2⇡
"Z
Fi 0
1
t dt F i Fi
+ 1
Z
Fi
dt(Fi Fi
1
t)
✓
1 F i 1 Fi
2
1 F i Fi
1
◆#
=⇡
(25) (26)
(1)
In order to improve this I think it is necessary to use that H1 (s) is rapidly oscillating on the timescales which
i (t)
varies. However the possible cancellations that may be induced by this property are lost
when the norm-of-the-integral is split into an integral-of-norms as in (20). Anushya also pointed out that any procedure for attempting to prove this must necessarily fail in the case of an infinite Hilbert space where we know there exist open orbits in phase space which give rise to non normalisable eigenstates of the Floquet operator and unbounded Hamiltonians.
12
