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Localisation in Permutation Symmetric Fermionic Quantum Walks
A. P. Balachandran,a,b Anjali Kundalpady,bPramod Padmanabhan,cAkash Sinhac
aPhysics Department, Syracuse University,
Syracuse, NY, 13244-1130, USA
bInstitute of Mathematical Sciences,
CIT Campus, Taramani, Chennai 600113, India
cSchool of Basic Sciences,
Indian Institute of Technology, Bhubaneswar, 752050, India
apbal1938, anjali.kundalpady, pramod23phys, akash.sinha@gmail.com
Abstract
We investigate localisation in a quantum system with a global permutation symmetry
and a superselected symmetry. We start with a systematic construction of many-fermion
Hamiltonians with a global permutation symmetry using the conjugacy classes of the per-
mutation group SN, with Nbeing the total number of fermions. The resulting Hamilto-
nians are interpreted as generators of continuous-time quantum walk of indistinguishable
fermions. In this setup we analytically solve the simplest example and show that all the
states are localised without the introduction of any disorder coefficients. Furthermore,
we show that the localisation is stable to interactions that preserve the global SNsym-
metry making these systems candidates for a quantum memory. The models we propose
can be realised on superconducting quantum circuits and trapped ion systems.
arXiv:2307.01963v1 [quant-ph] 5 Jul 2023
1 Introduction
A quantum theory with superselection sectors is tightly constrained, restricting the possible
representations that the algebra of observables can take. In such theories, typically the oper-
ators are not charged under some symmetries, meaning they transform trivially under their
action. The Hilbert space of such a system splits into different superselection sectors that
are labelled by the irreducible representations of these symmetries. A physically important
example of such a symmetry is the permutation symmetry generated by the statistics opera-
tor, sij that effects the exchange of particles located at iand j. This occurs while studying a
system of identical and indistinguishable bosons (fermions) which correspond to the Abelian
representations of the permutation group where the exchange operator takes the eigenvalues
+1 (−1) respectively. In such theories the states are symmetrised (antisymmetrised) and the
operators are invariant under this exchange symmetry.
In this work we explore localisation in such constrained systems. We find that with an
additional global permutation symmetry SN, the fermionic system completely localises with-
out any disorder. The requirement of the global SNsymmetry enforces the interaction of
each fermion with every other fermion making the Hamiltonian similar in appearance to the
SYK model [1–4]. Furthermore these systems are also stable under interaction terms that
preserve the global SNsymmetry. This is contrasted with fermionic Hamiltonians without an
explicit global SNsymmetry such as the tight-binding Hamiltonians, where we show that such
localisation does not occur.
The models we write down can be interpreted as quantum walk Hamiltonians that have
received a lot of attention in the past [5–10] in the context of search algorithms [11–13], as
quantum simulators [14–17] and for universal quantum computation [18–21]. We are concerned
with continuous-time quantum walks (CTQW) of identical particles [22–28] and especially
those with an additional global symmetry. Symmetric quantum walks have been considered
in the case of discrete-time quantum walks (DTQW) [29–31] and have been shown to feature
topological phases [32–35] and localisation [36–40].
The contents are laid out as follows. We begin Sec. 2 with the space describing the
fermions and the operators acting on them. The Hamiltonians with global SNsymmetry are
constructed using simple ideas from the theory of permutation groups. Among the many
possibilities, we consider the simplest such Hamiltonian in Sec. 2.1. The solution of the
Hamiltonian is provided in Sec. 3 and the resulting features are compared with the models
1
that lack a global SNsymmetry. We conclude with a few remarks and future directions in
Sec. 4.
2 Construction
We begin with a brief description of the Fock space (spanned by the states diagonalising the
number operator) describing Nidentical and indistinguishable fermions. The vacuum,|Ω⟩
denotes the state with no fermions. The state |i⟩, for i∈ {1,2,··· , N}, describes the presence
of a fermion on site i. These are the 1-fermion states and they span a Ndimensional space
henceforth denoted H ≃ CNwith the canonical inner product. In this notation, multiparticle
states such as, |i⟩⊗|j⟩live in H⊗H. No relation between |i⟩⊗|j⟩and |j⟩⊗|i⟩is assumed a priori.
However in the case of indistinguishable fermions, we work with normalised antisymmetrised
states, 1
√2[|i⟩⊗|j⟩−|j⟩⊗|i⟩] that live in H ∧ H, with the ∧denoting antisymmetrisation.
Thus the full Hilbert space becomes the antisymmetrised Fock space,
N
M
n=0 H∧n.
This space is finite and its dimension is seen from
N
X
k=0 N
k!= 2N,
with N
k!being the dimension of H ∧ H ∧ · ·· ∧ H
| {z }
ktimes
.
The creation (a†
j) and annihilation (aj) operators satisfying the fermionic (CAR) algebra,
{aj, a†
k}=δjk ,
{aj, ak}={a†
j, a†
k}= 0,(2.1)
are realized on this space. The index jtake values in, {1,··· , N}. More generally we could
add an extra index µto each oscillator to denote an internal degree of freedom like a colour or
spin index. For simplicity we will stick to just the indices j∈ {1,2,··· , N }for the fermions,
allowing their interpretation as lattice sites.
An arbitrary k-fermion state expressed as
a†
i1a†
i2···a†
ik|Ω⟩,(2.2)
2
satisfies all the necessary antisymmetry properties as can be directly verified by using (2.1).
With the help of these aand a†, we can mutate between different particle sectors. For example,
the action of a†
ion an arbitrary state from H∧kyields
a†
ia†
i1a†
i2···a†
ik|Ω⟩=a†
ia†
i1a†
i2···a†
ik|Ω⟩ ∈ H∧k+1.(2.3)
The Fock space description ensures that the fermionic creation and annihilation operators
commute with the superselected exchange symmetry of this system.
Next we move on to the action of the global permutation symmetry SNon the site indices
{1,2,··· , N}. The action of these operators on the states and operators of the theory are
obtained as follows. The vacuum is invariant under permutations, sij |Ω⟩=|Ω⟩and the
transformation rule of operators under conjugation by permutation generators is,
sjk O···j···k···s−1
jk =O···k···j···,
where Ois an operator with several indices including jand k(Note that s−1
jk =sj k). Using
these properties we can deduce the action of the permutation group on arbitrary states of this
system.
Permutation invariant operators acting on these states satisfy
siOi1···ipsi=Oi1···ip,∀i∈ {1,2,··· , N −1},(2.4)
where si≡si,i+1, are the transposition operators that generate the permutation group (SN)
and satisfy
sisi+1si=si+1sisi+1, s2
i= 1, sisj=sjsiwhen |i−j| ≥ 2.(2.5)
For the particular value of N= 2, let us consider the two operators
a†
1a1+a†
2a2and a†
1a2+a†
2a1.(2.6)
They clearly are invariant under the action of S2and the objective is to construct such per-
mutation invariant operators for arbitrary SN. A natural place to look for such objects are in
the conjugacy classes of SNwhich are left invariant as a set under the action of the group by
definition. For the permutation group, the elements of a conjugacy class have the same cycle
structure and their order is given by
N!
N
Q
k=1
(kνk)νk!
3
where νkis the number of k-cycles. A clear invariant is the sum of the elements of a given
conjugacy class with a particular cycle structure1. These statements are independent of the
particular realisation of the transposition operators. For our current problem we will show a
realisation using the fermionic creation and annihilation operators. The operators we use are
such that the resulting Hamiltonians are hermitian and number-preserving as the permutation
operators do not change the number of fermions.
We will obtain the fermionic realisation of SNby showing the existence of the permutation
operators for each particle sector. The fermionic realisation for the transposition permutation
σ∈ SNin the k-fermion sector is given by,
σ=X
i1<i2<···ik
a†
σ(i1)a†
σ(i2)···a†
σ(ik)aik···ai2ai1.(2.7)
The permutation σhas a particular cycle structure. For example the fermionic realisations of
the transpositions (ij) in the 1-fermion and 2-fermion sectors are given by
(ij)1=a†
jai+a†
iaj+
N
X
k=1
k=i,j
a†
kak,(2.8)
and
(ij)2=a†
ja†
iajai+X
k=j
a†
ja†
kakai+X
k=i
a†
ka†
iajak+
N
X
k,l=1
k<l={i,j}
a†
ka†
lalak,(2.9)
respectively. The suffix α, on (ij)αdenotes fermion number sector on which this transposition
acts. Thus on the full Fock space the transposition is given by,
(ij) =
N
M
α=1
(ij)α.(2.10)
A more non-trivial example is that of a 3-cycle permutation in the 1-fermion sector,
(ijk)1=a†
jai+a†
kaj+a†
iak+
N
X
l=1
l={i,j,k}
a†
lal.(2.11)
Note that the hermitian conjugate of this term is (ikj)1. Indeed the Hamiltonian identified as
a sum of the elements of the conjugacy class will turn out hermitian. We can now write down
an SNinvariant Hamiltonian for a given conjugacy class made of p-cycles as
H(p)=
N
M
α=1
H(p)
α,(2.12)
1These are precisely the generators of the center of the permutation group algebra C(SN).
4
where
H(p)
α=X
i1<i2<···<ip
(i1i2···ip)α,(2.13)
acts on the α-fermion sector. Such SNinvariant Hamiltonians (2.12) are true for any realisation
of the permutation group. The fermionic realisation in (2.7) introduces simplifications to the
Hamiltonian due to the CAR algebra (2.1).
Before going into these we first note that the operators corresponding to cycles of length
larger than βannihilate the vectors in the β-fermionic sector. Thus the bilinear expression
acting on the 1-fermion sector affects all possible fermion number sectors in a system of N
fermions. It acts as an exchange operator on the 1-fermion states and has a non-trivial action
on the remaining sectors.
In what follows we will restrict ourselves to the Hamiltonians constructed out of the 2-cycle
conjugacy class. We will comment on models obtained from other conjugacy classes in Sec. 4
and carry out a more detailed investigation in a future work.
2.1 2-cycle Hamiltonian
Consider the conjugacy class made out of purely 2-cycles which are just the transpositions.
They include the exchange of any two of the Nindices and there are precisely N(N−1)
2of them.
The 2-cycle Hamiltonian in the 1-fermion sector is obtained using (2.8) and is bilinear in the
fermion creation and annihilation operators,
H(2)
1=X
i<j ha†
iaj+a†
jaii+(N−1)(N−2)
2ˆ
N. (2.14)
The factor accompanying the number operator ˆ
N=
N
P
k=1
a†
kakis a result of the substitution
(2.8) for the 2-cycles. Clearly the term in the [ ] commutes with ˆ
Nand represents a fermion
on a given site hopping to any other site. As noted earlier this Hamiltonian has a non-trivial
action on every fermion number sector except on the 1-fermion sector where it acts as a
permutation operator. It is clearly SNinvariant in its site indices2. The second term in (2.14)
dominates for large N. Our goal is to study the localisation features of this system for large N
and so the explicit presence of Nin the Hamiltonian can lead to incorrect conclusions about
2A more rigorous proof is shown in App. A.
5
the origin of the localisation. To avoid this we will choose the term in [ ] as our Hamiltonian,
H=X
i<j ha†
iaj+a†
jaii,(2.15)
where all the fermions interact with each other in a symmetric manner. This model can be
solved exactly by a simple change of basis as we shall see in Sec. 3.
In addition to the above bilinear Hamiltonian, we consider the operators acting on the
2-fermion states which are quartic in the fermion creation and annihilation operators. This
Hamiltonian can be thought of as an interaction term when added to the bilinear Hamiltonian
in (2.15). However a crucial point is that this 2-cycle Hamiltonian can be simplified3using
the CAR algebra in (2.1) resulting in,
H(2)
2=(N−2)(N−3)
2−1ˆ
N2−ˆ
N+ 2 X
i<j ha†
iaj+a†
jaiiˆ
N−1.(2.16)
This Hamiltonian continues to remain SNinvariant and acts on 2-fermion and higher states.
These terms represent interactions but reduce to the product of bilinears due to the CAR
algebra. As a consequence they commute with the Hamiltonian in (2.14) and thus merely
shift their eigenvalues while sharing the eigenstates. This further implies that localised states
of (2.15) are stable to such SNpreserving perturbations. This trend continues to hold for
higher order perturbations obtained using the 2-cycle Hamiltonians acting on 3- and higher-
fermion sectors (See App. B).
3 Solution
The bilinear Hamiltonian in (2.15) is solved with a simple change of variables in the space of
creation and annihilation operators. Consider a new set of annihilation (creation) operators,
Aα(A†
α) defined as,
Aα=1
√N
N
X
j=1
ωjα aj, A†
α=1
√N
N
X
j=1
ω−jα a†
j,(3.1)
with ω=e2πi
Nbeing a Nth-root of unity and α∈ {1,2,·· · , N }. These operators satisfy the
CAR relations required of fermionic operators,
{Aα, A†
β}=δαβ,
{Aα, Aβ}={A†
α, A†
β}= 0.(3.2)
3The proof for this is shown in App. B.
6
In these variables, the 2-cycle Hamiltonian in (2.15) reduces to
H=N A†
NAN−ˆ
N, (3.3)
where ˆ
N=
N
P
i=1
a†
iai=
N
P
α=1
A†
αAαcommutes with the Hamiltonian. A number of fermionic
symmetries for the Hamiltonian in (3.3) become apparent in this basis. We find that all
bilinears A†
αAβ4commute with the Hamiltonian when α, β =N. In fact the permutation
operators of (2.7) can be written as linear combinations of such bilinears and thus these are
the operators that map the states of a given eigenspace into each other.
The spectrum can be found by labelling the eigenspaces with the set {1,2,· ·· N}. The
dimension of the k-fermion sector is N!
k!(N−k) ! and these are spanned by two kinds of eigenstates
of the form
A†
α1A†
α2···A†
αk|Ω⟩,(3.4)
with no two α’s equal to each other. The first set of eigenstates are those where at least one
of the α’s is N. There are a total of (N−1) !
(k−1) !(N−k) ! such states and they share the eigenvalue,
(N−k). The second set are those where none of the α’s take the value N. These account
for the remaining (N−1) !
k!(N−k−1) ! states and they come with the eigenvalue −k. In evaluating the
spectrum we have used the identities,
hˆ
N, A†
αi=A†
α,hˆ
N, Aαi=−Aα.(3.5)
Having obtained the spectrum, we are in a position to compute the probability distribu-
tions. We will consider 1-fermion and 2-fermion walks which sufficiently illustrate the features
of the permutation invariant systems considered here. Following this we will also discuss the
general k-fermion sector. An important point to keep in mind is the role played by the global
SNsymmetry in determining the structure of the spectrum. For instance, it is enough to find
the time evolution of any single state in a particular number sector. The remaining states can
be computed by the action of the appropriate SNoperators on this state. Furthermore, an-
other crucial feature arises as a consequence of the global SNsymmetry, namely the restriction
on the subspace that a given state is allowed to evolve into. For example the 2-fermion state
|1,2(t)⟩only evolves into the |1, j⟩and |2, j ⟩states. There is no overlap with the states |j, k⟩
when j, k /∈ {1,2}. In other words any state in this system does not explore the full Hilbert
space under time evolution. This is not apparent from the Aα(A†
α) basis but becomes more
4Removing the number operator from (3.3) will enhance the number of fermionic symmetries as now Aα
and A†
αwill also commute with the Hamiltonian when α=N.
7
transparent in a new basis. We will demonstrate this for each of the fermion number sectors
below.
1-fermion walks : The features we wish to illustrate are immediately seen in the following
eigenbasis of the 1-fermion sector : there is one state of the form,
N
X
j=1
a†
j|Ω⟩,(3.6)
with eigenvalue N−1 and there are N−1 eigenstates of the form,
a†
1−a†
j|Ω⟩;j∈ {2,3,··· , N},(3.7)
with eigenvalue −1. The non-degenerate state in (3.6) is symmetric under the action of SN,
whereas the degenerate states in (3.7) are mapped into each other under the action of SN.
More precisely the transposition operators in the 1-fermion sector (2.8) perform this mapping.
These operators can be written as linear combinations of the bilinears A†
αAβand as noted
earlier these commute with the Hamiltonian (3.3).
The non-zero probability distributions are found to be,
|⟨1|1(t)⟩|2=1
N21+(N−1)2+ 2 (N−1) cos N t,(3.8)
|⟨j|1(t)⟩|2=2
N2[1 −cos Nt],(3.9)
for j∈ {2,3,··· , N }and |j(t)⟩=e−iHt|j⟩are the time evolved states. The non-oscillating
terms of both these expressions highlight the localisation effect. For large Nthe term in (3.8)
goes to 1 whereas the term in (3.9) goes to 0. These features are illustrated in Fig. 1a. The
reason for the restricted evolution can also be seen from the explicit structure of the unitary
evolution operator in the 1-fermion sector and this is shown in App. E.
The phase of the oscillating term in (3.8) and (3.9) is the difference between the two energy
levels in the 1-fermion sector. We have seen earlier that the addition of higher order interaction
terms (2.10), will only shift these energy levels of the Hamiltonian (2.15) leaving the structure
of the eigenstates intact. This implies that the localisation seen here is stable to the inclusion
of such SN-symmetric interactions. This argument continues to hold even in the k-fermion
sector as there are just two energy levels in each fermion number sector.
8
(a) 1-fermion walk in the symmetric case.
positions(1 to 10) v/s time(0 to 20)
initial position = 5
(b) 2-fermion walk in the symmetric case.
Probability on positions(0 to 19)
initial position (10,15)
(c) 2-fermion walk in the non-symmetric case.
Probability(to the power 0.1) on positions(0 to 19)
initial position (10,15)
Figure 1: (Color Online) The probability distributions for the 1-, and 2-fermion walks for the
Hamiltonian (2.15) in (a) and (b) respectively. The 2-fermion walk for the tight-binding model
on the circle
N
P
j=1
a†
jaj+1 +h.c., is shown in (c). In all the plots the regions with shades of red
correspond to probabilities close to zero and the shades of green correspond to probabilities
close to one. In the non-symmetric case (c) the 2 fermions spread out to all other locations with
probabilities of the order of 10−2. To visualise this spread we have plotted the probabilities
to the power of 0.1 in (c).
9
2-fermion walks : Using the eigenstates in (3.4) the amplitude for an initial 2-fermion
state, |i, j⟩to end up in a state, |k, l⟩after a time tis found to be,
⟨k, l|i, j (t)⟩=1
N2"e−i(N−2)t
N−1
X
α=1 ωiα −ωjα ω−kα −ω−lα
+e2it
N−1
X
α,β=1
α<β ωiα+jβ −ωiβ+jα ω−kα−lβ −ω−lα−kβ
,(3.10)
where |i, j(t)⟩are the time evolved 2-fermion states. As mentioned earlier the restricted
evolution is not apparent from (3.10) but it becomes more transparent in a changed basis for
the 2-fermion states5. Consider the normalised eigenstates,
|[j]⟩2=1
√N−1a†
j
N
X
i=1
i=j
a†
i|Ω⟩;j∈ {1,2,··· , N −1},(3.11)
|[jkN ]⟩2=1
√3a†
ja†
k+a†
ka†
N+a†
Na†
j|Ω⟩;j < k ∈ {1,2,··· , N −1}.(3.12)
For these two sets of states, the notation |[ ]⟩2indicates that it is a linear combination of
2-fermion states. From these expressions we see that there are N−1 eigenstates of this form
in (3.11) and they come with the eigenvalue (N−2) and there are (N−1)(N−2)
2states of the
form (3.12) with eigenvalue -2. This is consistent with the previous solution. As in the 1-
fermion sector, the SNsymmetries, generated using (2.9), map the degenerate eigenstates into
each other. These operators can be written as products of the bilinears in A†
αAβand hence
commute with the Hamiltonian in (3.3).
These eigenstates are used to expand |1,2⟩=a†
1a†
2|Ω⟩as
|1,2⟩=√N−1
N|[1]⟩2−|[2]⟩2+√3(N−2)
N|[12N]⟩2−√3
N
N−1
X
j=3 |[1jN ]⟩2−|[2jN ]⟩2.(3.13)
The first three states in the above expression, |[1]⟩2,|[2]⟩2and |[12N]⟩2, are linear combinations
of |1, j⟩,|2, j ⟩with j∈ {1,··· , N }. The states under the summation |[1jN]⟩2,|[2jN ]⟩2, also
contains the |j, N⟩states but these cancel while taking the difference of these two states. Thus
from these arguments it is clear that the time evolved |1,2⟩state will not overlap with a state
|j, k⟩where neither of jor kis (1,2). This is verified in the plot for the probability distribution
shown in Fig. 1b. This is to be contrasted with a similar plot for a Hamiltonian that is not
5The orthogonality and completeness of these states is discussed in App. C.
10
permutation invariant (see Fig. 1c), where the two fermions can now be found in states that
do not follow the constraint for the symmetrised case 6. Subsequently we can also compute
the probability distributions
|⟨1,2|1,2(t)⟩|2=1
N24+(N−2)2+ 4 (N−2) cos N t,(3.14)
|⟨ψ|1,2(t)⟩|2=2
N2[1 −cos Nt].(3.15)
Here |ψ⟩denotes the allowed 2-fermion states that have an overlap with |1,2⟩. These expres-
sions present a clear indication of the localisation for large Nas the term 1
N24+(N−2)2
in (3.14) tends to 1 and the term 2
N2in (3.15) approaches 0.
k-fermion walks : This constraining feature continues to hold true for the amplitudes and
the corresponding probability distributions in a general k-fermion sector. We will see below
that a point in k-dimensional space occupied by kfermions moves to points where at least
k−1 of the coordinates coincide with the initial state. As in the 2-fermion case this becomes
apparent when we work with (N−1) !
(k−1) !(N−k) ! eigenstates of the form,
|[i1i2···ik−1]⟩k=1
√N−k+ 1a†
i1a†
i2···a†
ik−1
N
X
j=1
j={i1,i2,···ik−1}
a†
j|Ω⟩,(3.16)
with i1< i2<··· < ik−1∈ {1,··· N−1}. The second set of eigenstates7account for (N−1) !
k!(N−k−1) !
of them and take the form,
|[i1i2···ikN]⟩k=1
√k+ 1 ha†
i1···a†
ik+ (−1)ka†
i2···a†
N+· ·· + (−1)k2a†
N···a†
ik−1i|Ω⟩.
(3.17)
An initial state of the form |1,2,··· , k⟩can be expanded using the above eigenstates such
that each of them contain at least k−1 of {1,2,···k}. Finally the probability distribution
for the time evolved k-fermion state to overlap with its initial state is,
|⟨1,2,3,··· , k|1,2,3,··· , k(t)⟩|2=1
N2k2+ (N−k)2+ 2k(N−k) cos Nt,(3.18)
generalising the result in (3.14). It is clear from this expression that the localisation feature
continues to hold for the k-fermion states as well.
6A similar result for quantum walks on Cayley graphs of the symmetric group is in [41].
7The proof for this is in App. D.
11
4 Discussion
We have explored the question of localisation in a system with superselection sectors and a
global discrete symmetry. It is important to understand the source of the localisation and the
role played by these two symmetries in obtaining this feature. To this end we consider two
possibilities. In the first situation we continue working with fermionic systems and reduce the
explicit global SNsymmetry. This is done by ‘marking’ a single site, say 1, to modify the
Hamiltonian in (2.15) to,
H=β
N
X
j=2 ha†
1aj+a†
ja1i+
N
X
j,k=2
j<k ha†
kaj+a†
jaki.(4.1)
This model has a global SN−1symmetry among the sites {2,3,··· , N }. To analyse the con-
sequences we rewrite the Hamiltonian in a different fashion. In the 1-fermion sector spanned
by
a†
i|Ω⟩→|i⟩:= (0,··· ,1
|{z}
i-th position
,··· ,0); i={1,2,··· , N}(4.2)
the Hamiltonian becomes an N×Nmatrix. However owing to the residual SN−1symmetry,
we further can reduce the dimension of the matrix. For example, we can work in the space
spanned by
|1⟩,|2⟩,1
√N−2(|3⟩+· ·· +|N⟩)(4.3)
The resulting Hamiltonian is
H=
0β β√N−2
β0√N−2
β√N−2√N−2N−3
(4.4)
Using this the probability that after evolving |1⟩in time, we still find it at |1⟩comes to be
|⟨1|1(t)⟩|2= 1 −
2(N−1)β21−cos [tp((N−2)2+ 4(N−1)β2)]
(N−2)2+ 4(N−1)β2(4.5)
Similarly we can compute |⟨2|2(t)⟩|2, and we show the plots for both of these in Fig.[2, 3]. The
essential difference between these two is that, we can localize |1⟩even for small Nby tuning
12
0.0 0.5 1.0 1.5 2.0
t
0.2
0.4
0.6
0.8
1.0
|〈1|1(t)〉2
Figure 2: |⟨1|1(t)⟩|2vs t
0.0 0.5 1.0 1.5 2.0
t
0.2
0.4
0.6
0.8
1.0
|〈2|2(t)〉2
β=1,n=10
β=10,n=10
β=1
10
,n=10
Figure 3: |⟨2|2(t)⟩|2vs t
β, which is not possible for the other states that continue to localise for large Nvalues as in
the case of the full global SNsymmetry.
To examine the role of the superselected symmetry, consider a spin chain system with no
superselected symmetry and just a global SNsymmetry. We obtain such an example by using
another realisation of the transposition operators, (ij),
(ij) = 1 + XiXj+YiYj+ZiZj
2,(4.6)
where X,Yand Zare the Pauli matrices. This system acts on Nsites with each site occupied
by a two-level system, {|↑⟩,|↓⟩} ≃ C2. In this case the 2-cycle Hamiltonian generalizes the
XXX spin chain,
H=1
2
N
X
j,k=1
j<k
[XjXk+YjYk+ZjZk] + N(N−1)
2.(4.7)
The 2-cycle Hamiltonian is just the sum of the N(N−1)
2transpositions acting on ⊗N
j=1C2
j. The
Hilbert space of this system splits into sectors labelled by the number of |↓⟩’s as the Hamilto-
nian, being the sum of permutation operators, cannot mix states containing different number
of |↓⟩’s. Thus the problem we have now is similar to the one encountered in the fermionic
realisation. For example consider the sector where there is a single |↓⟩ with the remaining
sites filled by a |↑⟩. There are Nsuch states and the Hamiltonian in this sector reduces to the
form,
(H1)ij =(N−1)(N−2)
2−1δij + 1 (4.8)
which is similar to the one obtained in the fermion case, (Appendix E.1) modulo the function
of Nappearing with the δij . We expect to see localisation here as well suggesting that this
13
property is true regardless of the realisation chosen for the global permutation symmetry. It
is not hard to see that this pattern continues for the other sectors of this Hilbert space and
so we conclude that the symmetric XX X chain will contain localised states for large N.
To summarise we see from these considerations that the localisation observed here is a
property of the global permutation symmetry and does not depend on its realisation. This
suggests that, in group theoretic Hamiltonians that preserve some symmetry group G, the
states will localise without any disorder providing a stable quantum memory protected by the
global symmetry. This would be an interesting and systematic way to obtain localised phases.
We also note that these setups are not merely a theoretical exercise and can in fact be seen in
experimental systems such as superconducting quantum circuits [42] and trapped ions [43].
We end with a few remarks and possible future works.
1. The out-of-time-order correlators (OTOC’s) indicate chaotic behavior in thermal systems
[44]. The Lyapunov exponent can be read off from such expressions [3]. The saturation of
this quantity implies behavior similar to that predicted by the AdS/CFT correspondence
and is like an SYK model [1–4]. The systems discussed in this work describe the opposite
behavior and thus the results on the OTOC’s do not hold for them.
2. An extension worthy of mention is adding an internal symmetry index α= 1,2 trans-
forming by the spin 1
2representation of SU (2) to the operators ai. Then since this
representation is pseudoreal, aα
i(σ2)α,α′aα′
jis SU (2) invariant and so is its adjoint (Here
σ2is the second Pauli matrix.). So we can add such SU(2) or colour singlet Majorana
terms which are also permutation invariant and incorporate features of the SYK model.
3. Furthermore it would be interesting to consider the algebra of observables that are per-
mutation invariant and study the corresponding Hilbert spaces built using the GNS
construction [45–50]. These can then be used to explore the entanglement entropy and
thermalisation properties of these systems and those derived from them.
Acknowledgments
Certain ideas in this paper were initiated in discussions of A.P.B and Fabio Di Cosmo which
we gratefully acknowledge.
A.P.B enjoyed the hospitality of The Institute of Mathematical Sciences while this work was
done. For that, he is grateful to the Director, Ravindran, and his colleague and friend Sanatan
14
Digal. A.S and P.P thank Tapan Mishra and Abhishek Chowdhury for useful discussions.
AH(2)
1in different particle sectors
Let us consider the operator
H(2)
1=X
ρ∈2-cycle
conjugacy class
N
X
i=1
a†
ρ(i)ai(A.1)
This is invariant in every particle sector under global permutations among the site indices. In
one-particle sector this can be seen very easily :
σ1H(2)
1σ−1
1a†
m|Ω⟩=H(2)
1a†
m|Ω⟩(A.2)
Here σ1is the realization of σ∈ SNin the one-particle sector. However before proceeding
further let us consider the action of H(2)
1on an arbitrary k-particle state, a†
i1···a†
ik|Ω⟩. We
can write
H(2)
1a†
i1···a†
ik|Ω⟩=X
ρ∈2-cycle
conjugacy class a†
ρ(i1)ai1+···a†
ρ(m)am+· ·· +a†
ρ(ik)aika†
i1···a†
ik|Ω⟩(rests give 0)
=X
ρ∈2-cycle
conjugacy class a†
ρ(i1)···a†
ik+a†
i1···a†
ρ(m)···a†
ik+a†
i1···a†
ρ(ik)|Ω⟩(A.3)
We will now demonstrate the invariance of H(2)
1in the k-particle sector.
σkH(2)
1σ−1
k|i1,··· , ik⟩=σkH(2)
1|σ−1(i1),··· , σ−1(ik)⟩
=σkX
ρ∈2-cycle
conjugacy class |ρσ−1(i1),··· , σ−1(ik)⟩+· ·· +|σ−1(i1),·· · , ρσ−1(ik)⟩
=X
ρ∈2-cycle
conjugacy class |σρσ−1(i1),··· , ik⟩+· ·· +|i1,·· · , σρσ−1(ik)⟩
=X
ρ′∈2-cycle
conjugacy class
(|ρ′(i1),··· , ik⟩+· ·· +|i1,··· , ρ′(ik)⟩)
=H(2)
1|i1,··· , ik⟩(A.4)
where σρσ−1=ρ′∈2-cycle conjugacy class also and σkis the realization of σ∈ SNin the
k-particle sector.
15
B Proof of (2.16)
The quartic Hamiltonian originating from the two-cycle conjugacy class is
H(2)
2=X
σX
i<j
a†
σ(i)a†
σ(j)ajai=1
2X
i,j X
σ
a†
σ(i)a†
σ(j)ajai(B.1)
We notice that all σ’s belonging to the two-cycle conjugacy class can be grouped as
•i→i j →jthere are N−2C2such elements.
•i→j j →ithere is one such element.
•i→i j →m(=i, j) there are N−2 such elements.
•i→m(=i, j)j→jthere are N−2 such elements.
Considering this, we can simplify (B.1) as
H(2)
2=1
2X
i,j "N−2C2a†
ia†
jajai+a†
ja†
iajai+X
m=i,j a†
ia†
m+a†
ma†
jajai#(B.2)
With A=Piai,ˆ
N=Pia†
iaibeing the number operator and using the relations [ ˆ
N, A/a] =
−A/a, we obtain
H(2)
2=1
2N−2C2−3ˆ
N2−ˆ
N+A†Aˆ
N−A†A
=1
2(N−2)(N−3)
2−1ˆ
N2−ˆ
N+ 2 X
i<j ha†
iaj+a†
jaiiˆ
N−1(B.3)
So, we can write H(2)
2in terms of H(2)
1and ˆ
Nand we expect this trend to continue for higher
order Hamiltonians like hextic Hamiltonian and so on.
C Orthogonality and Completeness of (3.11),(3.12)
The inner product between 2-fermion eigenstates ⟨u|v⟩is zero when |u⟩ ∈ (3.11) and |v⟩ ∈
(3.12). However when both |u⟩and |v⟩belong to any particular eigenvalue then they are not
orthogonal to each other. Nevertheless we can show that these states are complete using the
action of the permutation operators from SN. To do this we first note that the states in (3.11)
are mapped into each other,
sjk |[j]⟩2=|[k]⟩2;j, k ∈ {1,2,··· , N −1},(C.1)
16
using the transpositions sjk and the states in (3.12) are mapped into each other,
sj1j2sk1k2|[j1k1N]⟩2=|[j2k2N]⟩2(C.2)
using the permutations sj1j2sk1k2. Combining these identities with the expression of the 2-
fermion state |1,2⟩in (3.13) we see that any other 2-fermion state |j, k⟩, with j, k ={1,2},
can be obtained as
|j, k⟩=s1js2k|1,2⟩
=√N−1
N|[j]⟩2− |[k]⟩2+√3(N−2)
N|[jkN ]⟩2−√3
N
N−1
X
m=3 |[jmN ]⟩2− |[kmN ]⟩2.
(C.3)
Notice that we have used s1j|[2]⟩2=|[2]⟩2and s2k|[1]⟩2=|[1]⟩2. On the other hand the states
of the form |1, j⟩(|2, j ⟩), for j∈ {3,···N}, are obtained by applying s2j(s1j) on ±|1,2⟩
respectively. Thus any 2-fermion state can be written as a linear combination of the 2-fermion
eigenstates in (3.11) and (3.12) showing their completeness. These arguments can be extended
to a general k-fermion sector as well.
D Proof of (3.17)
We have the action of quadratic all-to-all on a general k-particle state as
H(2)
1a†
i1···a†
ik|Ω⟩=X
σa†
σ(i1)···a†
ik+a†
i1···a†
σ(ij)···a†
ik+· ·· +a†
i1···a†
σ(ik)|Ω⟩(D.1)
We concentrate on a single term
X
σ
a†
i1···a†
σ(ij)···a†
ik(D.2)
As done previously, we can group the σ’s as
•ij→ijthere are N−1C2such elements.
•ij→m=ijthere are N−1 such elements.
17
Then we finally have
X
σ
a†
i1···a†
σ(ij)···a†
ik=N−1C2a†
i1···a†
ij···a†
ik+a†
i1···
X
m=ij
a†
m
···a†
ik
=N−1C2−1a†
i1···a†
ij···a†
ik+a†
i1··· X
m
a†
m!···a†
ik
=NC2−Na†
i1···a†
ik+a†
i1··· A†
|{z}
j-th position · ·· a†
ik(D.3)
Therefore we have
H(2)
1a†
i1···a†
ik|Ω⟩=kNC2−Na†
i1···a†
ik|Ω⟩+A†a†
i2···a†
ik|Ω⟩+· ·· +a†
i1··· A†
|{z}
j-th position · ·· a†
ik|Ω⟩
+· ·· +a†
i1···a†
ik−1A†|Ω⟩(D.4)
Now let us consider the following situation :
H(2)
1a†
i1···X
ik
a†
ik|Ω⟩=kNC2−Na†
i1···X
ik
a†
ik|Ω⟩+A†a†
i2···X
ik
a†
ik|Ω⟩+
· ·· +a†
i1··· A†
|{z}
j-th position · ·· X
ik
a†
ik|Ω⟩+· ·· +a†
i1···a†
ik−1X
ik
A†|Ω⟩
=kNC2−Na†
i1···A†|Ω⟩+N a†
i1···A†|Ω⟩(D.5)
Thus we have the following eigenstates of H(2)
1
H(2)
1a†
i1···A†|Ω⟩=kNC2−N(k−1)a†
i1···A†|Ω⟩(D.6)
Now let us consider another kind of states
|i1,··· , ik, N;k⟩=a†
i1···a†
ik|Ω⟩+· ·· + (−1)j ka†
j+1 ···a†
ika†
N
|{z}
k−j+1-th position
a†
i1···a†
j−1|Ω⟩+
· ·· + (−1)k2a†
Na†
i1···a†
ik−1|Ω⟩(D.7)
Let us focus on two particular expressions
H(2)
1a†
i1···a†
ik|Ω⟩and H(2)
1(−1)jk a†
j+1 ···a†
ika†
N
|{z}
k−j+1-th position
a†
i1···a†
j−1|Ω⟩(D.8)
We should have two terms coming from the above
a†
i1··· A†
|{z}
j-th position · ·· a†
ik|Ω⟩and (−1)jk a†
j+1 ···a†
ikA†
|{z}
k−j+1-th position
a†
i1···a†
j−1|Ω⟩(D.9)
18
They have same index content. Rearrangement of the indices in the second of these yields
(−1)(k−j+1)(j−1)+(k−j)+jk a†
i1··· A†
|{z}
j-th position · ·· a†
ik|Ω⟩
= (−1)−j2+2jk+j−1a†
i1··· A†
|{z}
j-th position · ·· a†
ik|Ω⟩
= (−1)−j2+j(−1)2jk−1a†
i1··· A†
|{z}
j-th position · ·· a†
ik|Ω⟩
=−a†
i1··· A†
|{z}
j-th position · ·· a†
ik|Ω⟩(D.10)
Thus these terms always cancel and the states
|i1,··· , ik, N;k⟩(D.11)
are eigenstates of H(2)
1
H(2)
1|i1,··· , ik, N;k⟩=kNC2−N|i1,··· , ik, N ;k⟩(D.12)
E Time evolution operator in 1-particle sector
The one-particle Hilbert space spanned by {a†
i|Ω⟩;i= 1,2,··· , N}can equivalently be de-
scribed by the space spanned by {|i⟩= (0,··· ,1
|{z}
i-th position
,··· ,0)T}. In this framework, the
bilinear Hamiltonian in (2.15) is given by
H: (H)ij = 1 −δij =⇒H=I − I(E.1)
where the matrix Ihas entries Iij = 1 and the matrix Iis the N×Nidentity matrix with
entries Iij =δij. They satisfy the following properties
Im=Nm−1Iand Im=I(E.2)
Then one can simplify :
e−iHt =I−I
N+I
Ne−iNteit (E.3)
19
Further, the probability can be calculated as
|⟨j|j(t)⟩|2=1−1
N+1
Ne−iNteit
2
=1
N21+(N−1)2+ 2(N−1) cos (Nt)(E.4)
|⟨j′|j(t)⟩|2
ȷ′=j=−1
N+1
Ne−iNteit
2
=2
N2[1 −cos (Nt)] (E.5)
References
[1] A. Kitaev, “A simple model of quantum holography,” KITP strings seminar and Entan-
glement 2015 program, vol. http://online.kitp.ucsb.edu/online/entangled15/., Feb. 12,
April 7, and May 27, 2015.
[2] Sachdev and Ye, “Gapless spin-fluid ground state in a random quantum heisenberg mag-
net.,” Physical review letters, vol. 70 21, pp. 3339–3342, 1992.
[3] J. Maldacena, S. H. Shenker, and D. Stanford, “A bound on chaos,” Journal of High
Energy Physics, vol. 2016, pp. 1–17, 2015.
[4] V. Rosenhaus, “An introduction to the syk model,” Journal of Physics A: Mathematical
and Theoretical, vol. 52, 2018.
[5] J. Kempe, “Quantum random walks: An introductory overview,” Contemporary Physics,
vol. 44, pp. 307 – 327, 2003.
[6] S. E. Venegas-Andraca, “Quantum walks: a comprehensive review,” Quantum Informa-
tion Processing, vol. 11, pp. 1015–1106, 2012.
[7] D. Reitzner, D. Nagaj, and V. R. Buzek, “Quantum walks,” 2012.
[8] A. M. Childs, E. Farhi, and S. Gutmann, “An example of the difference between quantum
and classical random walks,” Quantum Information Processing, vol. 1, pp. 35–43, 2001.
[9] A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous, “One-dimensional
quantum walks,” in Symposium on the Theory of Computing, 2001.
[10] Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Physical Review
A, vol. 48, no. 2, p. 1687, 1993.
20
[11] M. Santha, “Quantum walk based search algorithms,” in Theory and Applications of
Models of Computation, 2008.
[12] N. Shenvi, J. Kempe, and K. B. Whaley, “Quantum random-walk search algorithm,”
Physical Review A, vol. 67, p. 052307, 2002.
[13] R. Portugal, Quantum Walks and Search Algorithms. Springer Publishing Company,
Incorporated, 2013.
[14] K. Bepari, S. Malik, M. Spannowsky, and S. Williams, “Quantum walk approach to
simulating parton showers,” Physical Review D, 2021.
[15] B. P. Nachman, D. Provasoli, W. A. de Jong, and C. W. Bauer, “Quantum algorithm for
high energy physics simulations.,” Physical review letters, vol. 126 6, p. 062001, 2019.
[16] R. D. Somma, S. Boixo, H. Barnum, and E. Knill, “Quantum simulations of classical
annealing processes.,” Physical review letters, vol. 101 13, p. 130504, 2008.
[17] F. W. Strauch, “Relativistic quantum walks,” Physical Review A, vol. 73, p. 054302, 2005.
[18] A. M. Childs, D. Gosset, and Z. Webb, “Universal computation by multiparticle quantum
walk,” Science, vol. 339, pp. 791 – 794, 2012.
[19] M. S. Underwood and D. L. Feder, “Universal quantum computation by discontinuous
quantum walk,” Physical Review A, vol. 82, p. 042304, 2010.
[20] R. Asaka, K. Sakai, and R. Yahagi, “Two-level quantum walkers on directed graphs. i.
universal quantum computing,” Physical Review A, 2021.
[21] A. M. Childs, “Universal computation by quantum walk.,” Physical review letters, vol. 102
18, p. 180501, 2008.
[22] L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osel-
lame, “Two-particle bosonic-fermionic quantum walk via integrated photonics.,” Physical
review letters, vol. 108 1, p. 010502, 2011.
[23] X. Qin, Y. Ke, X.-W. Guan, Z. Li, N. Andrei, and C. Lee, “Statistics-dependent quan-
tum co-walking of two particles in one-dimensional lattices with nearest-neighbor inter-
actions,” Physical Review A, vol. 90, p. 062301, 2014.
[24] M. K. Giri, S. Mondal, B. P. Das, and T. Mishra, “Two component quantum walk in
one-dimensional lattice with hopping imbalance,” Scientific Reports, vol. 11, 2020.
[25] A. A. Melnikov and L. Fedichkin, “Quantum walks of interacting fermions on a cycle
graph,” Scientific Reports, vol. 6, 2013.
21
[26] A. A. Melnikov, A. P. Alodjants, and L. Fedichkin, “Hitting time for quantum walks of
identical particles,” in International Conference on Micro- and Nano-Electronics, 2018.
[27] Y. Lahini, M. Verbin, S. D. Huber, Y. Bromberg, R. Pugatch, and Y. R. Silberberg,
“Quantum walk of two interacting bosons,” Physical Review A, vol. 86, p. 011603, 2011.
[28] D. Wiater, T. Sowi’nski, and J. J. Zakrzewski, “Two bosonic quantum walkers in one-
dimensional optical lattices,” Physical Review A, vol. 96, p. 043629, 2017.
[29] H. Krovi, “Symmetry in quantum walks,” 2007.
[30] J. Janmark, D. A. Meyer, and T. G. Wong, “Global symmetry is unnecessary for fast
quantum search,” Physical Review Letters, vol. 112, p. 210502, 2014.
[31] C. M. Chandrashekar, R. Srikanth, and S. Banerjee, “Symmetries and noise in quantum
walk,” Phys. Rev. A, vol. 76, p. 022316, Aug 2007.
[32] C. Cedzich, T. Geib, F. A. Gr¨unbaum, C. Stahl, L. Vel´azquez, A. H. Werner, and R. F.
Werner, “The topological classification of one-dimensional symmetric quantum walks,”
Annales Henri Poincar´e, vol. 19, pp. 325–383, 2016.
[33] C. Cedzich, T. Geib, F. A. Gr¨unbaum, L. Vel’azquez, A. H. Werner, and R. F. Werner,
“Quantum walks: Schur functions meet symmetry protected topological phases,” Com-
munications in Mathematical Physics, vol. 389, pp. 31 – 74, 2019.
[34] T. Geib, C. Cedzich, A. H. Werner, and R. F. Werner, “Topological aspects of discrete
and continuous time quantum walks on one dimensional lattices,” 2019.
[35] C. Cedzich, T. Geib, C. Stahl, L. Vel´azquez, A. H. Werner, and R. F. Werner, “Complete
homotopy invariants for translation invariant symmetric quantum walks on a chain,”
arXiv: Quantum Physics, 2018.
[36] B. Danacı, I. Yal¸cinkaya, B. C¸ akmak, G. Karpat, S. P. Kelly, and A. L. Suba¸sı, “Disorder-
free localization in quantum walks,” arXiv: Quantum Physics, 2020.
[37] A. Mandal, R. S. Sarkar, and B. Adhikari, “Localization of two dimensional quantum
walks defined by generalized grover coins,” Journal of Physics A: Mathematical and The-
oretical, vol. 56, 2021.
[38] S. Singh and C. M. Chandrashekar, “Interference and correlated coherence in disordered
and localized quantum walk,” arXiv: Quantum Physics, 2017.
[39] A. Joye, “Dynamical localization for d-dimensional random quantum walks,” Quantum
Information Processing, vol. 11, pp. 1251–1269, 2012.
22
[40] C. Cedzich and A. H. Werner, “Anderson localization for electric quantum walks and
skew-shift cmv matrices,” Communications in Mathematical Physics, vol. 387, pp. 1257
– 1279, 2019.
[41] H. Gerhardt and J. Watrous, “Continuous-time quantum walks on the symmetric group,”
in RANDOM-APPROX, 2003.
[42] S. Hazra, A. Bhattacharjee, M. Chand, K. V. Salunkhe, S. Gopalakrishnan, M. P.
Patankar, and R. Vijay, “Long-range connectivity in a superconducting quantum pro-
cessor using a ring resonator.,” arXiv: Quantum Physics, 2020.
[43] B. P. Lanyon, C. Hempel, D. Nigg, M. M¨uller, R. Gerritsma, F. Z¨ahringer, P. Schindler,
J. T. Barreiro, M. Rambach, G. Kirchmair, M. Hennrich, P. Zoller, R. Blatt, and C. F.
Roos, “Universal digital quantum simulation with trapped ions,” Science, vol. 334, pp. 57
– 61, 2011.
[44] K. Hashimoto, K. Murata, and R. Yoshii, “Out-of-time-order correlators in quantum
mechanics,” Journal of High Energy Physics, vol. 2017, pp. 1–31, 2017.
[45] A. P. Balachandran, T. R. Govindarajan, A. R. de Queiroz, and A. F. Reyes-Lega, “En-
tanglement and particle identity: a unifying approach.,” Physical review letters, vol. 110
8, p. 080503, 2013.
[46] A. P. Balachandran, T. R. Govindarajan, A. R. Queiroz, and A. F. Reyes-Lega, “Algebraic
approach to entanglement and entropy,” Physical Review A, vol. 88, p. 022301, 2013.
[47] A. F. Reyes-Lega, “Entanglement Entropy in Quantum Mechanics: An Algebraic Ap-
proach,” 12 2022.
[48] A. P. Balachandran, A. R. de Queiroz, and S. Vaidya, “Entropy of Quantum States:
Ambiguities,” Eur. Phys. J. Plus, vol. 128, p. 112, 2013.
[49] A. P. Balachandran, A. R. de Queiroz, and S. Vaidya, “Quantum Entropic Ambiguities:
Ethylene,” Phys. Rev. D, vol. 88, no. 2, p. 025001, 2013.
[50] F. Benatti, R. Floreanini, F. Franchini, and U. Marzolino, “Entanglement in indistin-
guishable particle systems,” Physics Reports, 2020.
23