Quantum Alchemy and Universal Orthogonality Catastrophe in One-Dimensional Anyons

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Quantum Alchemy and Universal Orthogonality Catastrophe

in One-Dimensional Anyons

Naim E. Mackel 1, Jing Yang 1,2, and Adolfo del Campo 1,3

1Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg

2Nordita, KTH Royal Institute of Technology and Stockholm University, Hannes Alfvéns vag 12, 106 91 Stockholm, Sweden

3Donostia International Physics Center, E-20018 San Sebastián, Spain

Many-particle quantum systems with in-

termediate anyonic exchange statistics are

supported in one spatial dimension. In this

context, the anyon-anyon mapping is re-

cast as a continuous transformation that

generates shifts of the statistical param-

eter κ. We characterize the geometry of

quantum states associated with diﬀerent

values of κ, i.e., diﬀerent quantum statis-

tics. While states in the bosonic and

fermionic subspaces are always orthogonal,

overlaps between anyonic states are gen-

erally ﬁnite and exhibit a universal form

of the orthogonality catastrophe governed

by a fundamental statistical factor, inde-

pendent of the microscopic Hamiltonian.

We characterize this decay using quantum

speed limits on the ﬂow of κ, illustrate our

results with a model of hard-core anyons,

and discuss possible experiments in quan-

tum simulation.

1 Introduction

The spin-statistics theorem dictates that in the

familiar three-dimensional world particles are ei-

ther bosons or fermions. In lower spatial dimen-

sions, however, intermediate exchange statistics

are allowed, giving rise to the existence of anyons.

Anyons are characterized by many-body wave-

functions that are not necessarily fully symmetric

or antisymmetric, but can pick up an arbitrary

phase factor under particle exchange.

A model of two-dimensional Abelian anyons

was ﬁrst introduced by Leinaas and Myrheim [1]

Naim E. Mackel : naimmackel@outlook.de

Jing Yang : jingyangyzby@gmail.com

Adolfo del Campo : adolfo.delcampo@uni.lu

and further elaborated by Wilczek [2,3]. The

study of two-dimensional anyons has grown into

a substantial body of literature [4,5]. This any-

onic behavior should not be confused with the

notion of generalized exclusion statistics, as de-

scribed by Haldane and Wu, possible in arbi-

trary spatial dimensions [6–8]. Decades later

it was appreciated that intermediate exchange

statistics is also possible in one spatial dimen-

sion [9,10]. Several models of interacting one-

dimensional anyons have been characterized in-

cluding contact interactions as well as hardcore

[10–14] and ﬁnite-range potentials. While the in-

clusion of spin degrees of freedom is possible, we

shall focus on spinless (or spin-polarized) quan-

tum states of one-dimensional anyons. The re-

sulting families of anyons are labeled by the sta-

tistical parameter κthat governs the statistical

phase factor arising from particle exchange. For

κ= 0 one recovers fully symmetric wavefunctions

while the case κ=πcorresponds to antisymmet-

ric fermionic wavefunctions. Proposals to real-

ize models of one-dimensional anyons have been

put forward using optical and resonator lattices

as quantum simulators [15–21]. An experimental

realization of one-dimensional anyons has been

reported using ultracold atoms in an optical lat-

tice [22]. In these scenarios, the statistical param-

eter κis not ﬁxed and it is possible to conceive

experiments in which its value is tuned dynami-

cally. Such prospects pave the way for quantum

alchemy, i.e., the transmutation of particles of one

kind into another, such as bosons into anyons [4].

The fact that the permutation of particles in

one spatial dimension is necessarily interwoven

with interparticle interactions gives rise to the

existence of several dualities, generalizing the

celebrated Bose-Fermi mapping introduced by

Girardeau between strongly interacting bosons

Accepted in Quantum 2023-12-14, click title to verify. Published under CC-BY 4.0. 1

arXiv:2210.10776v3 [quant-ph] 18 Dec 2023

and free fermions [23]. The description of one-

dimensional hardcore anyons is possible using

the anyon-anyon mapping, which relates states

with diﬀerent values of the statistical parame-

ter κ[12]. This generalized duality has spurred

the investigation of hardcore anyons, making it

possible to characterize eﬃciently ground-state

correlations [24–27], ﬁnite-temperature behav-

ior [28,29], and their nonequilibrium dynam-

ics [14,30,31].

In this context, we associate the anyon-anyon

mapping with a continuous transformation de-

scribing shifts of the statistical parameter. We

show that under statistical transmutation, per-

mutation symmetry yields a universal form of the

orthogonality catastrophe governing the decay of

quantum state overlaps in a way that is inde-

pendent of the underlying system Hamiltonian.

This universal behavior further determines the

distinguishability of anyonic quantum states and

the quantum geometry of the space of physical

quantum states encompassing diﬀerent quantum

statistics.

2 Anyon-anyon mapping as a continu-

ous transformation

When the spin degrees of freedom can be ignored

(e.g., in a fully polarized state), the spatial wave-

functions of bosons and fermions are respectively

fully symmetric and antisymmetric with respect

to particle exchange. No permutation-symmetric

operator can couple them and thus exchange

statistics imposes a superselection rule in which

the Hilbert space of a physical system of identi-

cal particles is the direct sum of the bosonic and

fermionic subspaces. However, the importance of

mappings between diﬀerent sectors has been long

recognized in many-body physics. The celebrated

Bose-Fermi duality provides a prominent exam-

ple, relating wavefunctions of hard-core bosons

ΨHCB to that of spin-polarized fermions ΨFin

one spatial dimension: ψHCB =Qi<j sgn(xij )ΨF

where xij =xi−xj. The extension of the

Bose-Fermi mapping [23] to anyons was put for-

ward by Girardeau [12], building on earlier results

by Kundu [10] and applied to the construction

of anyonic wavefunctions from either bosonic or

fermionic states. For instance, given ΨFone ob-

tains the corresponding state of hard-core anyons

Ψκwith statistical parameter κusing the map-

ping Ψκ= exp−iκ

2Pi<j sgn(xij )ΨHCB [12].

Although models with softcore interactions are

possible (as in the case of the well-studied Lieb-

Liniger anyons), the hardcore condition by which

Ψκ= 0 when xij = 0 is rather ubiquitous. It

arises when the strength of contact interactions is

divergent (i.e., as a limit of Lieb-Liniger anyons

with repulsive interactions), for pairwise power-

law potentials (e.g., V=Pi<j λ/|xij |αwith

λ, α > 0, as in the case of Calogero-Sutherland

anyons with α= 2 [12]), and with other interac-

tion potentials satisfying V→+∞as xij →0.

Here, we consider a natural generalization

transforming anyonic wavefunctions Ψκ′with sta-

tistical parameter κ′into anyonic wavefunctions

Ψκwith statistical parameter κvia the linear

mapping

Ψκ=ˆ

A(κ, κ′)Ψκ′.(1)

In doing so, the anyon-anyon mapping ˆ

A(κ, κ′)is

associated with a continuous unitary transforma-

tion in which the generator

G=1

i<j

sgn(xij ),(2)

induces shifts of the statistical parameter κ. As

Gdoes not depend on it explicitly, the mapping

is given by the unitary, ˆ

A(κ, κ′) = ˆ

A(κ−κ′) =

exp h−i(κ−κ′)ˆ

Gi, and thus satisﬁes all the group

properties, including the existence of the identity

A(0) = I, inverse ˆ

A(κ)−1=ˆ

A(−κ) = ˆ

A(κ)†,

and group multiplication ˆ

A(κ)ˆ

A(κ′) = ˆ

A(κ+κ′),

when κtakes values on the real line. We note

however that it suﬃces to consider the domain

κ∈[0,2π)upon identifying κ+ 2nπ ∼κfor any

integer n∈Z.

3 Many-anyon state overlaps

Consider None-dimensional spinless hardcore

anyons in an arbitrary quantum state Ψκbelong-

ing to the Hilbert space Hκ, which is a subspace

with anyonic exchange symmetry of the Hilbert

space of square-integrable functions L2(RN).

Speciﬁcally, Ψκ(x1, . . . , xi, xi+1, . . . , xN) =

eiκsgn(xi−xi+1)Ψκ(x1, . . . , xi+1, xi, . . . , xN). The

application of the mapping ˆ

A(δ)on an any-

onic state |Φκ⟩introduces a unitary ﬂow of

the state, leading to a distinguishable state

Accepted in Quantum 2023-12-14, click title to verify. Published under CC-BY 4.0. 2

|Φκ+δ⟩ ≡ e−iˆ

Gδ|Φκ⟩with the statistical pa-

rameter shifted by δ. Anyons with statistical

parameter κare thus transmuted into anyons

with statistical parameter κ′, motivating the

term “quantum alchemy” for such transformation.

Note that whenever κ+δ=π,|Φκ+δ⟩describes

a fermionic wave function, which vanishes at the

contact points where at least two coordinates

coincide. Thus, |Φκ⟩must vanish at the contact

points, obeying a hard-core constraint. We

note that the family of hard-core anyons is

not restricted to contact interactions but can

accommodate, e.g., power-law interactions [12].

Having justiﬁed the ﬂow of the states in the

Hilbert space of identical particles, we aim at

characterizing the quantum geometry of state

space and ask what is the distance between the

state in Hκ+δand the original state in Hκ. To

this end, we consider Ψκ∈ Hκand compute the

survival amplitude deﬁned by the overlap

⟨Ψκ|Φκ+δ⟩=⟨Ψκ|e−iˆ

Gδ|Φκ⟩.(3)

On a given sector R:xR(1) > xR(2) >··· >

xR(N), the action of the anyon-anyon mapping

can be replaced by a phase factor ωδ(R). We

thus consider a generalized Heaviside step func-

tion 1R:

1R=1xR(1)>xR(2)>···>xR(N)≡(1if xR(1) > xR(2) >··· > xR(N)

0otherwise .(4)

Making use of it, we note that survival amplitude can be written as the sum over N!sectors, associated

with permutations over the symmetric group SN(see Appendix Cfor more details),

⟨Ψκ|Φκ+δ⟩=X

R∈SN

ωδ(R)Iκ(R),(5)

where

ωδ(R)≡e−iδ

2Pi<j sgn(xi−xj)xR(1)>xR(2)>···>xR(N),(6)

and Iκ(R)involves the N-dimensional integral

Iκ(R) = ZRN

i=1

dxiΨ∗

κ(x1,···xN)Φκ(x1,···xN)1R.(7)

Its explicit evaluation is challenging as it in-

volves N!N-dimensional integrals. However,

an evaluation in closed form is possible mak-

ing use of permutation symmetry. We note that

for arbitrary pairs of Ψκ,Φκand Ψκ′,Φκ′con-

nected by the anyon-anyon mapping, the corre-

sponding integrals are equal, i.e., ∀κ, κ′∈R:

Iκ(R) = Iκ′(R)≡I(R). In addition, inte-

grals evaluated at diﬀerent sectors are all equal,

i.e., ∀R,R′∈SN:I(R) = I(R′), thanks

to the permutation symmetry of the integrand

Ψ∗

κ(x1,···xN)Φκ(x1,···xN). From the resolu-

tion of the identity PR∈SN1R=IRN, we con-

clude that I(R) = ⟨Ψκ|Φκ⟩/N!, and thus,

⟨Ψκ|Φκ+δ⟩=⟨Ψκ|Φκ⟩1

N!X

R∈SN

ωδ(R).(8)

The overlap between anyonic states with diﬀer-

ent quantum statistics depends on the state over-

lap ⟨Ψκ|Φκ⟩between states with common κand

on an additional contribution from the shift δof

the statistical parameter

ΩN(δ) := 1

N!X

R∈SN

ωδ(R).(9)

As shown in Appendix D, this yields the recursion

relation

ΩN(δ) = 1

NPN−1

n=0 e−iδ

2(N−1−2n)ΩN−1(δ),(10)

making it possible to ﬁnd by iteration the exact

close-form expression

ΩN(δ) = 1

n=2

sin nδ

2

sin δ

2,(11)

Accepted in Quantum 2023-12-14, click title to verify. Published under CC-BY 4.0. 3

0.0 0.5 1.0 1.5 2.0

-1.0

-0.5

0.0

0.5

1.0

Figure 1: Statistical contribution to the survival

amplitude at diﬀerent system sizes. As the sys-

tem size Nincreases, the statistical contribution ΩNto

the overlap far from the δ= 0,2πdecays quickly, and

becomes progressively steeper within this neighborhood.

Note that δ∈[0,2π]is the relevant part to plot, since

ΩNis an even function and 4π-periodic.

shown in Fig. 1as a function of the statistical

shift for diﬀerent values of N. Under the sole

consideration of a statistical shift δ, i.e., choosing

|Φκ⟩=|Ψκ⟩, the survival amplitude of an initial

state under the ﬂow induced by the anyon-anyon

mapping collapses to ΩN(δ). This is a key funda-

mental result from which our subsequent analysis

follows.

4 Quantum speed limits on the ﬂow of

the statistical parameter

Quantum speed limits (QSLs) provide lower

bounds on the time required for a process to

unfold. While introduced in quantum dynam-

ics [32,33], QSLs apply to the ﬂow of quan-

tum states under other continuous transforma-

tions described as a one-parameter ﬂow [34,35].

Given that the anyon-anyon mapping can be de-

scribed by a unitary ﬂow, the decay of the quan-

tum state overlap under shifts of κis subject to

generalizations of QSLs. In quantum dynamics,

the Mandelstam-Tamm QSL and the Margolus-

Levitin QSL bound the minimum time for the

evolving state to become orthogonal to the ini-

tial state in terms of the energy dispersion and

the mean energy of the initial state, respectively

[32,33,36]. They can be generalized in the cur-

rent context as shown in Appendix F, to identify

a bound on the minimum κ-shift required for the

initial state |Ψκ⟩to be transmuted into an orthog-

onal state. Speciﬁcally, the generalized QSLs take

the form

κ≥κMT =π

2q⟨ˆ

G2⟩−⟨ˆ

G⟩2

,(12)

κ≥κML =π

2(⟨ˆ

G⟩ − G0),(13)

where G0=−N(N−1)

4is the lowest eigenvalue

of ˆ

G. At variance with the familiar case con-

cerning time evolution under a given Hamilto-

nian, the generator ˆ

Gis uniquely set by the

anyon-anyon mapping: there is no freedom in its

choice. The brackets in ⟨ˆ

Gn⟩can be used to de-

note the expectation value over the initial state

|Ψκ⟩, or equivalently, the average over the N!sec-

tors in conﬁguration space, given that the value

of ⟨ˆ

Gn⟩is the same for any wavefunction of indis-

tinguishable particles. As a result, and at vari-

ance with the conventional QSL, the character-

istic shifts of the statistical parameter are inde-

pendent of the quantum state, and are univer-

sal, being solely governed by permutation sym-

metry. The universal factor ΩN(δ)can be viewed

as the generating function of the moments of

the generator Gover the initial state |Ψκ⟩, i.e,

⟨ˆ

Gn⟩=indnΩN(δ)

dδn|δ=0, from which one obtains

⟨ˆ

G⟩= 0,⟨ˆ

G2⟩=−N(2N2+3N−5)

72 . Thus, we ﬁnd

the universal lower bounds

κ≥κMT =3√2π

pN(2N2+ 3N−5),(14)

κ≥κML =2π

N(N−1).(15)

For unitary ﬂows generated by time-independent

generators and pure initial states, the MT speed

limit can be expressed as κMT =π/qFQ(κ),

where FQ(κ)≡4(⟨ˆ

G2⟩ − ⟨ ˆ

G⟩2)is known as the

quantum Fisher information, which describes the

Riemannian geometry of the quantum state space

[37]. Speciﬁcally, it is tied to the Fubini-Study

metric on the state manifold parameterized by κ,

i.e., ds2=1

4FQ(κ)dκ2, where ds is the inﬁnites-

imal distance between |Ψκ⟩and |Ψκ+dκ⟩. The

quantum Fisher information also characterizes

the variance in practical estimation theory via

the quantum Cramér-Rao bound [38]. For gener-

ators with linear interactions between particles,

it has been shown that the quantum Fisher infor-

mation scales at most as (Nln N)2[39,40], where

Nis the number of particles. In our case, FQ(κ)

scales as N3, reminiscent of the super-Heisenberg

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0.0 0.1 0.2 0.3 0.4

0.0

0.2

0.4

0.6

0.8

1.0

Figure 2: Comparison between the exact statis-

tical contribution and its Gaussian approxima-

tion. As the system size Nincreases, ΩNis increasingly

more precisely approximated by GN.

scaling in the context of parameter estimation

quantum metrology with a nonlinear generator ˆ

[39]. In addition, Anandan and Aharanov showed

that the saturation of the MT bound occurs when

the state evolves along a shortest Fubini-Study

geodesic [41]. The fact that MT QSL is not satu-

rated indicates that the unitary ﬂow induced by

the anyon-anyon mapping does not trace a short-

est geodesic in the space of physical states. This

raises the question as to whether such ﬂow is op-

timal in any certain sense.

5 Universal Orthogonality Catastrophe

Many-body eigenstates are highly sensitive to lo-

cal perturbations. For fermions, this dependence

is extensive in the system size and is known as the

orthogonality catastrophe [42]. This phenomenon

has been analyzed in the case of particles obeying

generalized exclusion statistics [43,44]. Its oc-

currence has been further related to QSL in [45].

It is natural to explore analogs of it under the

transmutation of particles. For the case at hand,

we note that hard-core anyons are related to the

one-dimensional spin-polarized Fermi gas by the

anyon-anyon mapping. Further, the generator ˆ

of κshifts is two-body and spatially nonlocal. For

an inﬁnitesimal ﬂow of the statistical parameter,

we show in Appendix Fthat the overlap between

anyonic states decays as

ΩN(δ)≈exp "−δ2

N(2N2+ 3N−5)

72 #=GN(δ).

(16)

A comparison between this Gaussian approxima-

tion and the exact result (11) is shown in Fig

2. This overlap decays rapidly as the particle

2 3 4 5 6 7 8

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Figure 3: Comparison of the two QSL estimates

with the exact determined QSL. Minimum shift of

the statistical parameter κestimated by the generalized

QSL in comparison with the exact values determined

as zeroes of the overlap ΩN(δ)between anyonic many-

body states. The dependence with the system size Nof

the orthogonalization κ-shift is incorrectly predicted by

the MT and ML bounds, which are too conservative and

never saturated in the ﬂow of particle transmutation.

number Nis increased. Given that the vari-

ance σ2=⟨ˆ

G2⟩governs this decay, one may be

tempted to conclude that the MT bound gov-

erns the orthogonality catastrophe, as proposed

in [45]. Yet, from the explicit expression in Eq.

(11), it is found that ΩN(δ)vanishes identically

for δ∈ ZN, where

ZN=2πk

nn= 2, . . . , N;k= 1, . . . , n −1.

(17)

Note that the values δ= 0,2πare not zeroes

and are thus excluded. For a given N, the

interval where these zeros accumulate is given

by I=h2π

N,2π−2π

Ni. In the thermodynamic

limit N→ ∞, the values on this interval be-

come all zero, in addition the interval becomes

IN→∞ = (0,2π), while the values in 0,2πremain

unchanged. Hence, one ﬁnds

|ΩN→∞|=δδ,2πk , where k∈Z.(18)

The ﬁrst zero of ΩN(δ)for a given Ndescribes

the exact κ-shift for the anyonic state to be trans-

muted to an orthogonal state and is given by

κ⊥≡2π

N.(19)

Despite the inverse scaling with the number of

particles, neither the MT nor the ML bound pre-

dicts the correct scaling (19). Direct comparison

between the QSLs and κ⊥is displayed in Fig. 3.

As illustrated, for any N > 2, the chain of in-

equalities κML < κMT < κ⊥is fulﬁlled and there-

fore κMT is tighter than the ML bound. And yet,

Accepted in Quantum 2023-12-14, click title to verify. Published under CC-BY 4.0. 5

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摘要：

QuantumAlchemyandUniversalOrthogonalityCatastropheinOne-DimensionalAnyonsNaimE.Mackel1,JingYang1,2,andAdolfodelCampo1,31DepartmentofPhysicsandMaterialsScience,UniversityofLuxembourg,L-1511Luxembourg,Luxembourg2Nordita,KTHRoyalInstituteofTechnologyandStockholmUniversity,HannesAlfvénsvag12,10691Stockh...

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