Reconstruction of classical skyrmions from Anderson towers quantum Darwinism in action O. M. Sotnikov1 E. A. Stepanov2 M. I. Katsnelson3 F. Mila4 V. V. Mazurenko1

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Reconstruction of classical skyrmions from Anderson towers:

quantum Darwinism in action

O. M. Sotnikov1, E. A. Stepanov2, M. I. Katsnelson3, F. Mila4, V. V. Mazurenko1

1Theoretical Physics and Applied Mathematics Department,

Ural Federal University, Mira Str. 19, 620002 Ekaterinburg, Russia

2CPHT, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, F-91128 Palaiseau, France

3Radboud University, Institute for Molecules and Materials,

Heyendaalseweg 135, 6525AJ, Nijmegen, Netherlands

4Institute of Physics, ´

Ecole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

(Dated: October 11, 2022)

The development of the quantum skyrmion concept is aimed at expanding the scope of the fun-

damental research and practical applications for classical topologically-protected magnetic textures,

and potentially paves the way for creating new quantum technologies. Undoubtedly, this calls for

establishing a connection between a classical skyrmion and its quantum counterpart: a skyrmion

wave function is an intrinsically more complex object than a non-collinear conﬁguration of classical

spins representing the classical skyrmion. Up to date, such a quantum-classical relation was only

established on the level of diﬀerent physical observables, but not for classical and quantum states

per se. In this work, we show that the classical skyrmion spin order can be reconstructed using only

the low-energy part of the spectrum of the corresponding quantum spin Hamiltonian. This can be

done by means of a ﬂexible symmetry-free numerical realization of Anderson’s idea of the towers of

states (TOS) that allows one to study known, as well as unknown, classical spin conﬁgurations with

a proper choice of the loss function. We show that the existence of the TOS in the spectrum of the

quantum systems does not guarantee a priori that the classical skyrmion magnetization proﬁle can

be obtained as an outcome of the actual measurement. This procedure should be complemented

by a proper decoherence mechanism due to the interaction with the environment. The later selects

a speciﬁc combination of the TOS eigenfunctions before the measurement and, thus, ensures the

transition from a highly-entangled quantum skyrmionic state to a classical non-collinear magnetic

order that is measured in real experiments. The results obtained in the context of skyrmions allow

us to take a fresh look at the problem of quantum antiferromagnetism. In particular, we provide

a quantitative characterization of the TOS contributions to classical antiferromagnetic structures

including frustrated ones.

INTRODUCTION

Despite the huge success of quantum physics, which is

one of the pillars of our science and technology, its foun-

dations still remain a subject of hot debates, and many

conceptual issues still require further investigations1–7.

In particular, searching for the connection between quan-

tum and classical descriptions of the same phenomenon

or object has a long history in physics starting from the

foundation of quantum mechanics. In this sense, the de-

velopment of the path integral concept8–10 is a bright

example showing that the classical trajectory of a parti-

cle is just one of numerous alternatives characterized by

diﬀerent probabilities. In these terms, classicality means

nothing but destruction of an interference between dif-

ferent alternatives, similar to a transition from wave to

geometric optics11.

However, in the case of the classical-quantum corre-

spondence the problem is more complicated. The tran-

sition between the classical and quantum regimes can-

not be determined only by the fact that the characteris-

tic size of the system, which can be related to the de

Broglie wavelength, becomes small compared to other

length scales of the problem. According to the popu-

lar decoherence program, the classical-quantum corre-

spondence is rather related to the openness of the quan-

tum system and to the destruction of quantum interfer-

ences by the interaction with the environment12–17. The

problem is closely connected to the measurement prob-

lem18. According to Bohr’s complementarity principle19,

a quantum measurement is nothing but the result of the

interaction of a quantum particle with a classical mea-

suring device. This picture is the basis of the formal

theory of measurements developed by von Neumann20.

This theory includes a mysterious collapse of the quan-

tum wave function after the measurement. Further devel-

opments have led to a more complicated picture, includ-

ing soft measurements21 and decoherence waves in dis-

tributed quantum systems22–24. There are also analyti-

cal25 and numerical26 attempts to derive von Neumann’s

postulate from a consequent quantum consideration of

the measurement process, including decoherence by the

environment. In general, the problem does not seem to

be completely solved, and further attempts at clarifying

these key issues are required.

These questions may look too general and too abstract

but, actually, they are very closely related to a very com-

mon and important phenomenon of physics around us.

Antiferromagnetism, a very usual property of condensed

matter27, is, probably, one of the best examples. The

classical N´eel picture of magnetic sublattices for the case

of an “antiferromagnetic” exchange interaction is in an

arXiv:2210.03922v1 [quant-ph] 8 Oct 2022

obvious contradiction with quantum mechanics predict-

ing a singlet ground state27. Actually, the antiferromag-

netic state can be described without introducing sublat-

tices28, but its diﬀerence with the singlet state remains

dramatic. A general way to establish the correspondence

between quantum and classical descriptions of antiferro-

magnets was open in the seminal work of P. W. Ander-

son29. It has been shown there that in some cases linear

combinations of eigenstates of a quantum Hamiltonian

that form a tower of low-energy states can be related

to an ordered state that would be the classical ground

state of the system in the thermodynamic limit. Such a

tower-of-states (TOS) approach may be of fundamental

importance because it treats the fundamental problem of

quantum-classical correspondence from a completely dif-

ferent perspective, without referring to measurements or

postulating decoherence due to the environment. More-

over, it has proven to be extremely helpful in detecting

broken symmetries with eigenspectrum of even small-size

supercells of quantum systems. Up to now, the Anderson

towers approach was mainly used for studying quantum

antiferromagnets30–32. However, it is worth mentioning

that the previous studies based on the group-theoretical

calculations were fully concentrated on the symmetry

identiﬁcation of the eigenfunctions contributing to the

TOS without attempting to quantify their partial con-

tributions. Such an approach is also not ﬂexible since it

requires to know the exact symmetry of the reconstructed

classical order, which prevents using the approach in the

case when the system is characterized by a transition to

an unknown classical state (a problem known as hidden

order).

In this paper, we report on a symmetry-free numeri-

cal technique based on gradient-descent optimization for

constructing TOS on the basis of a limited number of

calculated low-lying eigenstates of a quantum system. In

contrast to previous works, our approach provides quanti-

tative information on the TOS composition. By means of

the developed scheme we explore topologically-protected

classical magnetic skyrmions33 that attract a consider-

able attention due to their fundamental interest34–36 and

technological importance in design of atomic ultra-dense

memory37, probabilistic computing38, etc. Up to now,

most of the studies34 have been focused on a pure clas-

sical description of the skyrmionic structures, which as-

sumes that each spin in the system is a classical vec-

tor with three spatial projections. The experimental

discovery of nano-scale skyrmionic structures39 in sur-

face nanosystems signiﬁcantly heats up interest to search

for quantum analogues of classical skyrmions, for which

quantum eﬀects may play a crucial role40.

Recent theoretical studies41–47 suggested that the

ground state of some quantum spin Hamiltonians with

competing isotropic and anisotropic interactions can be

considered as analogs of classical skyrmions since the

magnetization, the susceptibility, and the scalar chirality

calculated for these quantum ground states agree with

those obtained for the corresponding classical models.

However, as was shown in a previous work [47], both pro-

jective measurements and site-resolved average magneti-

zation calculated in the ground state of periodic quantum

systems do not resemble the typical pattern of a classical

skyrmion as seen in experiments. For this reason, one

can formally deﬁne the quantum skyrmion as a quantum

state for which the spin-spin correlation functions repro-

duce the same quantities in the classical version of the

problem. The natural question is then what is the mech-

anism through which one can observe a classical skyrmion

in a system which is a priori quantum.

In this work, we show that the connection between the

classical and quantum skyrmion systems can be estab-

lished not only at the level of the observables, as done up

to now, but also at the more general level of a quantum

state and of macroscopic classical order per se. To this

aim we use the concept of Anderson’s tower of states and

explore both the towers of quantum wave functions to

reconstruct classical solutions, and the towers of classical

conﬁgurations needed to reproduce the quantum ground

state. The analysis of the composition of the TOS that

we have obtained for magnetic skyrmions and for antifer-

romagnets uncovers important details of Anderson’s the-

ory that were not addressed in previous group-theoretical

considerations. We argue that the environment plays a

crucial role in selecting speciﬁc combinations of the quan-

tum states that build the TOS that correspond to classi-

cal order and are known as pointer states in decoherence

theory (Quantum Darwinism). Thus, quantum decoher-

ence should be considered as an important part of the

TOS theory. Since the TOS analysis itself can be used as

a more eﬀective search for pointer states within the de-

coherence program, both theories will beneﬁt from this

integration.

RESULTS

Protocol for constructing the Anderson tower

Formally, a quantum wave function corresponding to

any classical spin texture can be deﬁned as a product of

coherent states of individual spins48–50

|ΨTi=Y

[cosθi

2eiφi

2|↑i + sinθi

2e−iφi

2|↓i],(1)

where the polar angles θjand φjset a local basis for

each spin. Below we will refer to this state as a coher-

ent state or a target wave function. The consequent

projective measurements20 of the state ΨTin σz,σx

and σybases result in a set of projections hˆ

ii,hˆ

ii,

and hˆ

iifor each spin. The latter can be associated

with the direction of the classical magnetic moment

miin magnetic structures measured in spin-polarized

scanning tunneling microscopy experiments39. More

speciﬁcally, hmx

ii= sin θicos φi,hmy

ii= sin θisin φi, and

hmz

ii= cos θi. Thus, one can establish a formal connec-

tion between parameters of the coherent state and the

A B

quantum

skyrmion

classical

skyrmion

N = 2 N = 3 N = 7

N = 20

Anderson towers

FIG. 1. Protocol for constructing Anderson towers of

states in the case of the skyrmionic problem. (A) (B) Evo-

lution of 19-site plaquette magnetization calculated for the

reconstructed wave function corresponding to the classical

skyrmion with di↵erent number of eigenstates of the parent

spin Hamiltonian.

As it can be seen from Fig.1, the ground state of the

quantum Hamiltonian Eq.2 is characterized by the uni-

form magnetization. Expectedly, the ﬁrst approximation

of the classical system shows initial trends toward tradi-

tional skyrmionic proﬁle and symmetry breaking. Such

an approximation is formed with the ground state and

6-fold degenerated ﬁrst excited state. Further extension

of the set of the excited states used to reconstruct | cl

ski

increase the ﬁdelity between target wave function and its

approximation. For N= 20 the ﬁdelity value reaches

95%.

We would like to stress that this magnetization analysis

is the ﬁrst stage on the way to a complete charaterization

of the transition from quantum skyrmion wave function

to symmetry-broken classical skyrmion state, which will

include the calculation of the scalar chirality operator

expectation values and scanning its distribution over the

system in question. Another quantity of our special in-

terest in this study is the entanglement entropy that is

undergoing drastical changes during the the reconstruc-

tion of the classical skyrmion state with zero entropy

from the wave functions with maximal entanglements.

Methodologically, the construction of the Anderson tower

of states is a kind of optimization problem aiming to de-

ﬁne a superposition of the quantum eigenstates that has

a maximal ﬁdelity with the coherent wave function rep-

resenting given classical state. Thus we will ﬁrst discuss

a gradient-descent method we developed to perform such

an optimization.

Random-phase gradient descent approach

Scalar Chirality. As it was shown in Ref.3 the only

physical quantity allows one to unambiguously identify

the quantum skyrmion state on the lattice is quantum

chirality, the following correlation function

ijk =1

⇡hˆ

Si·[ˆ

Sj⇥ˆ

Sk]i,(3)

i,j, and kdepict three di↵erent spins that form an el-

ementary plaquette. The total chirality is deﬁned as a

sum of local chiralities of individual plaquettes, Q =

PhijkiQ

ijk. In the case of the DMI Hamiltonian deﬁned

classical

skyrmion

scalar chirality density

quantum

skyrmion

FIG. 2. Triangle-resolved density of the scalar chirality

ijk calculated for real quantum skyrmion state (left plaque-

tte), coherent state with classical proﬁle of the magnetization

(right plaquette) and for quantum states being di↵erent ap-

proximations of the classical skyrmion within the Anderson

tower approach.

on the inﬁnite lattice that can be simulated with a super-

cell with periodic boundary conditions, all the triangular

plaquettes produce the same contribution to the total

chirality. In this sense the distribution of the chirality

is uniform and similar to the distribution of the mag-

netization over the system. In the case of the classical

skyrmionic structures the situation is opposite.

Entropy

H n=En n(4)

1M. A. Nielsen, I. L. Chuang, Quantum Computation and

Quantum Information, Cambridge University Press, 2010.

2P. W. Anderson, An Approximate Quantum Theory of the

Antiferromagnetic Ground State, Physical Review 86, 694

(1952).

3O. M. Sotnikov, V. V. Mazurenko , J. Colbois, F. Mila,

M. I. Katsnelson, and E. A. Stepanov, Phys. Rev. B 103,

L060404 (2021).

4R.A. Istomin, A.S. Moskvin, Overlap integral for quantum

skyrmions, Pis’ma v ZhETF 71, 487 (2000).

5A.M. Perelomov, Generalized coherent states and their ap-

plications, Springer-Verlag, Berlin, 1986.

6E.A. Stepanov, S.A. Nikolaev, C. Dutreix, M.I. Kat-

snelson and V.V. Mazurenko, Heisenberg-exchange-free

nanoskyrmion mosaic, J. Phys.: Condens. Matter 31

17LT01 (2019).

7D. I. Badrtdinov, S. A. Nikolaev, M. I. Katsnelson, and V.

V. Mazurenko, Spin-orbit coupling and magnetic interac-

tions in Si(111):C,Si,Sn,Pb, Phys. Rev. B 94, 224418 (2016).

8I. A. Iakovlev, O. M. Sotnikov, and V. V. Mazurenko,

Bimeron nanoconﬁned design, Phys. Rev. B 97, 184415

(2018).

quantum

skyrmion

classical

skyrmion

N = 2 N = 3 N = 7

N = 20

Anderson towers

FIG. 1. Protocol for constructing Anderson towers of

states in the case of the skyrmionic problem. (A) (B) Evo-

lution of 19-site plaquette magnetization calculated for the

reconstructed wave function corresponding to the classical

skyrmion with di↵erent number of eigenstates of the parent

spin Hamiltonian.

As it can be seen from Fig.1, the ground state of the

quantum Hamiltonian Eq.2 is characterized by the uni-

form magnetization. Expectedly, the ﬁrst approximation

of the classical system shows initial trends toward tradi-

tional skyrmionic proﬁle and symmetry breaking. Such

an approximation is formed with the ground state and

6-fold degenerated ﬁrst excited state. Further extension

of the set of the excited states used to reconstruct | cl

ski

increase the ﬁdelity between target wave function and its

approximation. For N= 20 the ﬁdelity value reaches

95%.

We would like to stress that this magnetization analysis

is the ﬁrst stage on the way to a complete charaterization

of the transition from quantum skyrmion wave function

to symmetry-broken classical skyrmion state, which will

include the calculation of the scalar chirality operator

expectation values and scanning its distribution over the

system in question. Another quantity of our special in-

terest in this study is the entanglement entropy that is

undergoing drastical changes during the the reconstruc-

tion of the classical skyrmion state with zero entropy

from the wave functions with maximal entanglements.

Methodologically, the construction of the Anderson tower

of states is a kind of optimization problem aiming to de-

ﬁne a superposition of the quantum eigenstates that has

a maximal ﬁdelity with the coherent wave function rep-

resenting given classical state. Thus we will ﬁrst discuss

a gradient-descent method we developed to perform such

an optimization.

Random-phase gradient descent approach

Scalar Chirality. As it was shown in Ref.3 the only

physical quantity allows one to unambiguously identify

the quantum skyrmion state on the lattice is quantum

chirality, the following correlation function

ijk =1

⇡hˆ

Si·[ˆ

Sj⇥ˆ

Sk]i,(3)

i,j, and kdepict three di↵erent spins that form an el-

ementary plaquette. The total chirality is deﬁned as a

sum of local chiralities of individual plaquettes, Q =

PhijkiQ

ijk. In the case of the DMI Hamiltonian deﬁned

classical

skyrmion

scalar chirality density

quantum

skyrmion

FIG. 2. Triangle-resolved density of the scalar chirality

ijk calculated for real quantum skyrmion state (left plaque-

tte), coherent state with classical proﬁle of the magnetization

(right plaquette) and for quantum states being di↵erent ap-

proximations of the classical skyrmion within the Anderson

tower approach.

on the inﬁnite lattice that can be simulated with a super-

cell with periodic boundary conditions, all the triangular

plaquettes produce the same contribution to the total

chirality. In this sense the distribution of the chirality

is uniform and similar to the distribution of the mag-

netization over the system. In the case of the classical

skyrmionic structures the situation is opposite.

Entropy

H n=En n(4)

1M. A. Nielsen, I. L. Chuang, Quantum Computation and

Quantum Information, Cambridge University Press, 2010.

2P. W. Anderson, An Approximate Quantum Theory of the

Antiferromagnetic Ground State, Physical Review 86, 694

(1952).

3O. M. Sotnikov, V. V. Mazurenko , J. Colbois, F. Mila,

M. I. Katsnelson, and E. A. Stepanov, Phys. Rev. B 103,

L060404 (2021).

4R.A. Istomin, A.S. Moskvin, Overlap integral for quantum

skyrmions, Pis’ma v ZhETF 71, 487 (2000).

5A.M. Perelomov, Generalized coherent states and their ap-

plications, Springer-Verlag, Berlin, 1986.

6E.A. Stepanov, S.A. Nikolaev, C. Dutreix, M.I. Kat-

snelson and V.V. Mazurenko, Heisenberg-exchange-free

nanoskyrmion mosaic, J. Phys.: Condens. Matter 31

17LT01 (2019).

7D. I. Badrtdinov, S. A. Nikolaev, M. I. Katsnelson, and V.

V. Mazurenko, Spin-orbit coupling and magnetic interac-

tions in Si(111):C,Si,Sn,Pb, Phys. Rev. B 94, 224418 (2016).

8I. A. Iakovlev, O. M. Sotnikov, and V. V. Mazurenko,

Bimeron nanoconﬁned design, Phys. Rev. B 97, 184415

(2018).

quantum

skyrmion

classical

skyrmion

N = 2 N = 3 N = 7

N = 20

Anderson towers

FIG. 1. Protocol for constructing Anderson towers of

states in the case of the skyrmionic problem. (A) (B) Evo-

lution of 19-site plaquette magnetization calculated for the

reconstructed wave function corresponding to the classical

skyrmion with di↵erent number of eigenstates of the parent

spin Hamiltonian.

As it can be seen from Fig.1, the ground state of the

quantum Hamiltonian Eq.2 is characterized by the uni-

form magnetization. Expectedly, the ﬁrst approximation

of the classical system shows initial trends toward tradi-

tional skyrmionic proﬁle and symmetry breaking. Such

an approximation is formed with the ground state and

6-fold degenerated ﬁrst excited state. Further extension

of the set of the excited states used to reconstruct | cl

ski

increase the ﬁdelity between target wave function and its

approximation. For N= 20 the ﬁdelity value reaches

95%.

We would like to stress that this magnetization analysis

is the ﬁrst stage on the way to a complete charaterization

of the transition from quantum skyrmion wave function

to symmetry-broken classical skyrmion state, which will

include the calculation of the scalar chirality operator

expectation values and scanning its distribution over the

system in question. Another quantity of our special in-

terest in this study is the entanglement entropy that is

undergoing drastical changes during the the reconstruc-

tion of the classical skyrmion state with zero entropy

from the wave functions with maximal entanglements.

Methodologically, the construction of the Anderson tower

of states is a kind of optimization problem aiming to de-

ﬁne a superposition of the quantum eigenstates that has

a maximal ﬁdelity with the coherent wave function rep-

resenting given classical state. Thus we will ﬁrst discuss

a gradient-descent method we developed to perform such

an optimization.

Random-phase gradient descent approach

Scalar Chirality. As it was shown in Ref.3 the only

physical quantity allows one to unambiguously identify

the quantum skyrmion state on the lattice is quantum

chirality, the following correlation function

ijk =1

⇡hˆ

Si·[ˆ

Sj⇥ˆ

Sk]i,(3)

i,j, and kdepict three di↵erent spins that form an el-

ementary plaquette. The total chirality is deﬁned as a

sum of local chiralities of individual plaquettes, Q =

PhijkiQ

ijk. In the case of the DMI Hamiltonian deﬁned

classical

skyrmion

scalar chirality density

quantum

skyrmion

FIG. 2. Triangle-resolved density of the scalar chirality

ijk calculated for real quantum skyrmion state (left plaque-

tte), coherent state with classical proﬁle of the magnetization

(right plaquette) and for quantum states being di↵erent ap-

proximations of the classical skyrmion within the Anderson

tower approach.

on the inﬁnite lattice that can be simulated with a super-

cell with periodic boundary conditions, all the triangular

plaquettes produce the same contribution to the total

chirality. In this sense the distribution of the chirality

is uniform and similar to the distribution of the mag-

netization over the system. In the case of the classical

skyrmionic structures the situation is opposite.

Entropy

H n=En n(4)

1M. A. Nielsen, I. L. Chuang, Quantum Computation and

Quantum Information, Cambridge University Press, 2010.

2P. W. Anderson, An Approximate Quantum Theory of the

Antiferromagnetic Ground State, Physical Review 86, 694

(1952).

3O. M. Sotnikov, V. V. Mazurenko , J. Colbois, F. Mila,

M. I. Katsnelson, and E. A. Stepanov, Phys. Rev. B 103,

L060404 (2021).

4R.A. Istomin, A.S. Moskvin, Overlap integral for quantum

skyrmions, Pis’ma v ZhETF 71, 487 (2000).

5A.M. Perelomov, Generalized coherent states and their ap-

plications, Springer-Verlag, Berlin, 1986.

6E.A. Stepanov, S.A. Nikolaev, C. Dutreix, M.I. Kat-

snelson and V.V. Mazurenko, Heisenberg-exchange-free

nanoskyrmion mosaic, J. Phys.: Condens. Matter 31

17LT01 (2019).

7D. I. Badrtdinov, S. A. Nikolaev, M. I. Katsnelson, and V.

V. Mazurenko, Spin-orbit coupling and magnetic interac-

tions in Si(111):C,Si,Sn,Pb, Phys. Rev. B 94, 224418 (2016).

8I. A. Iakovlev, O. M. Sotnikov, and V. V. Mazurenko,

Bimeron nanoconﬁned design, Phys. Rev. B 97, 184415

(2018).

Hamiltonian

problem

quantum

skyrmion

classical

skyrmion

N = 2 N = 3 N = 7

N = 20

Anderson towers

FIG. 1. Protocol for constructing Anderson towers of

states in the case of the skyrmionic problem. (A) (B) Evo-

lution of 19-site plaquette magnetization calculated for the

reconstructed wave function corresponding to the classical

skyrmion with di↵erent number of eigenstates of the parent

spin Hamiltonian.

As it can be seen from Fig.1, the ground state of the

quantum Hamiltonian Eq.2 is characterized by the uni-

form magnetization. Expectedly, the ﬁrst approximation

of the classical system shows initial trends toward tradi-

tional skyrmionic proﬁle and symmetry breaking. Such

an approximation is formed with the ground state and

6-fold degenerated ﬁrst excited state. Further extension

of the set of the excited states used to reconstruct | cl

ski

increase the ﬁdelity between target wave function and its

approximation. For N= 20 the ﬁdelity value reaches

95%.

We would like to stress that this magnetization analysis

is the ﬁrst stage on the way to a complete charaterization

of the transition from quantum skyrmion wave function

to symmetry-broken classical skyrmion state, which will

include the calculation of the scalar chirality operator

expectation values and scanning its distribution over the

system in question. Another quantity of our special in-

terest in this study is the entanglement entropy that is

undergoing drastical changes during the the reconstruc-

tion of the classical skyrmion state with zero entropy

from the wave functions with maximal entanglements.

Methodologically, the construction of the Anderson tower

of states is a kind of optimization problem aiming to de-

ﬁne a superposition of the quantum eigenstates that has

a maximal ﬁdelity with the coherent wave function rep-

resenting given classical state. Thus we will ﬁrst discuss

a gradient-descent method we developed to perform such

an optimization.

Random-phase gradient descent approach

Scalar Chirality. As it was shown in Ref.3 the only

physical quantity allows one to unambiguously identify

the quantum skyrmion state on the lattice is quantum

chirality, the following correlation function

ijk =1

⇡hˆ

Si·[ˆ

Sj⇥ˆ

Sk]i,(3)

i,j, and kdepict three di↵erent spins that form an el-

ementary plaquette. The total chirality is deﬁned as a

sum of local chiralities of individual plaquettes, Q =

PhijkiQ

ijk. In the case of the DMI Hamiltonian deﬁned

classical

skyrmion

scalar chirality density

quantum

skyrmion

FIG. 2. Triangle-resolved density of the scalar chirality

ijk calculated for real quantum skyrmion state (left plaque-

tte), coherent state with classical proﬁle of the magnetization

(right plaquette) and for quantum states being di↵erent ap-

proximations of the classical skyrmion within the Anderson

tower approach.

on the inﬁnite lattice that can be simulated with a super-

cell with periodic boundary conditions, all the triangular

plaquettes produce the same contribution to the total

chirality. In this sense the distribution of the chirality

is uniform and similar to the distribution of the mag-

netization over the system. In the case of the classical

skyrmionic structures the situation is opposite.

Entropy

H n=En n(4)

1M. A. Nielsen, I. L. Chuang, Quantum Computation and

Quantum Information, Cambridge University Press, 2010.

2P. W. Anderson, An Approximate Quantum Theory of the

Antiferromagnetic Ground State, Physical Review 86, 694

(1952).

3O. M. Sotnikov, V. V. Mazurenko , J. Colbois, F. Mila,

M. I. Katsnelson, and E. A. Stepanov, Phys. Rev. B 103,

L060404 (2021).

4R.A. Istomin, A.S. Moskvin, Overlap integral for quantum

skyrmions, Pis’ma v ZhETF 71, 487 (2000).

5A.M. Perelomov, Generalized coherent states and their ap-

plications, Springer-Verlag, Berlin, 1986.

6E.A. Stepanov, S.A. Nikolaev, C. Dutreix, M.I. Kat-

snelson and V.V. Mazurenko, Heisenberg-exchange-free

nanoskyrmion mosaic, J. Phys.: Condens. Matter 31

17LT01 (2019).

7D. I. Badrtdinov, S. A. Nikolaev, M. I. Katsnelson, and V.

V. Mazurenko, Spin-orbit coupling and magnetic interac-

tions in Si(111):C,Si,Sn,Pb, Phys. Rev. B 94, 224418 (2016).

8I. A. Iakovlev, O. M. Sotnikov, and V. V. Mazurenko,

Bimeron nanoconﬁned design, Phys. Rev. B 97, 184415

(2018).

quantum

skyrmion

classical

skyrmion

N = 2 N = 3 N = 7

N = 20

Anderson towers

FIG. 1. Protocol for constructing Anderson towers of

states in the case of the skyrmionic problem. (A) (B) Evo-

lution of 19-site plaquette magnetization calculated for the

reconstructed wave function corresponding to the classical

skyrmion with di↵erent number of eigenstates of the parent

spin Hamiltonian.

As it can be seen from Fig.1, the ground state of the

quantum Hamiltonian Eq.2 is characterized by the uni-

form magnetization. Expectedly, the ﬁrst approximation

of the classical system shows initial trends toward tradi-

tional skyrmionic proﬁle and symmetry breaking. Such

an approximation is formed with the ground state and

6-fold degenerated ﬁrst excited state. Further extension

of the set of the excited states used to reconstruct | cl

ski

increase the ﬁdelity between target wave function and its

approximation. For N= 20 the ﬁdelity value reaches

95%.

We would like to stress that this magnetization analysis

is the ﬁrst stage on the way to a complete charaterization

of the transition from quantum skyrmion wave function

to symmetry-broken classical skyrmion state, which will

include the calculation of the scalar chirality operator

expectation values and scanning its distribution over the

system in question. Another quantity of our special in-

terest in this study is the entanglement entropy that is

undergoing drastical changes during the the reconstruc-

tion of the classical skyrmion state with zero entropy

from the wave functions with maximal entanglements.

Methodologically, the construction of the Anderson tower

of states is a kind of optimization problem aiming to de-

ﬁne a superposition of the quantum eigenstates that has

a maximal ﬁdelity with the coherent wave function rep-

resenting given classical state. Thus we will ﬁrst discuss

a gradient-descent method we developed to perform such

an optimization.

Random-phase gradient descent approach

Scalar Chirality. As it was shown in Ref.3 the only

physical quantity allows one to unambiguously identify

the quantum skyrmion state on the lattice is quantum

chirality, the following correlation function

ijk =1

⇡hˆ

Si·[ˆ

Sj⇥ˆ

Sk]i,(3)

i,j, and kdepict three di↵erent spins that form an el-

ementary plaquette. The total chirality is deﬁned as a

sum of local chiralities of individual plaquettes, Q =

PhijkiQ

ijk. In the case of the DMI Hamiltonian deﬁned

classical

skyrmion

scalar chirality density

quantum

skyrmion

FIG. 2. Triangle-resolved density of the scalar chirality

ijk calculated for real quantum skyrmion state (left plaque-

tte), coherent state with classical proﬁle of the magnetization

(right plaquette) and for quantum states being di↵erent ap-

proximations of the classical skyrmion within the Anderson

tower approach.

on the inﬁnite lattice that can be simulated with a super-

cell with periodic boundary conditions, all the triangular

plaquettes produce the same contribution to the total

chirality. In this sense the distribution of the chirality

is uniform and similar to the distribution of the mag-

netization over the system. In the case of the classical

skyrmionic structures the situation is opposite.

Entropy

H n=En n(4)

1M. A. Nielsen, I. L. Chuang, Quantum Computation and

Quantum Information, Cambridge University Press, 2010.

2P. W. Anderson, An Approximate Quantum Theory of the

Antiferromagnetic Ground State, Physical Review 86, 694

(1952).

3O. M. Sotnikov, V. V. Mazurenko , J. Colbois, F. Mila,

M. I. Katsnelson, and E. A. Stepanov, Phys. Rev. B 103,

L060404 (2021).

4R.A. Istomin, A.S. Moskvin, Overlap integral for quantum

skyrmions, Pis’ma v ZhETF 71, 487 (2000).

5A.M. Perelomov, Generalized coherent states and their ap-

plications, Springer-Verlag, Berlin, 1986.

6E.A. Stepanov, S.A. Nikolaev, C. Dutreix, M.I. Kat-

snelson and V.V. Mazurenko, Heisenberg-exchange-free

nanoskyrmion mosaic, J. Phys.: Condens. Matter 31

17LT01 (2019).

7D. I. Badrtdinov, S. A. Nikolaev, M. I. Katsnelson, and V.

V. Mazurenko, Spin-orbit coupling and magnetic interac-

tions in Si(111):C,Si,Sn,Pb, Phys. Rev. B 94, 224418 (2016).

8I. A. Iakovlev, O. M. Sotnikov, and V. V. Mazurenko,

Bimeron nanoconﬁned design, Phys. Rev. B 97, 184415

(2018).

Eigenspectrum Approximation

quantum

skyrmion

classical

skyrmion

N = 2 N = 3 N = 7

N = 20

Anderson towers

FIG. 1. Protocol for constructing Anderson towers of

states in the case of the skyrmionic problem. (A) (B) Evo-

lution of 19-site plaquette magnetization calculated for the

reconstructed wave function corresponding to the classical

skyrmion with di↵erent number of eigenstates of the parent

spin Hamiltonian.

As it can be seen from Fig.1, the ground state of the

quantum Hamiltonian Eq.2 is characterized by the uni-

form magnetization. Expectedly, the ﬁrst approximation

of the classical system shows initial trends toward tradi-

tional skyrmionic proﬁle and symmetry breaking. Such

an approximation is formed with the ground state and

6-fold degenerated ﬁrst excited state. Further extension

of the set of the excited states used to reconstruct | cl

ski

increase the ﬁdelity between target wave function and its

approximation. For N= 20 the ﬁdelity value reaches

95%.

We would like to stress that this magnetization analysis

is the ﬁrst stage on the way to a complete charaterization

of the transition from quantum skyrmion wave function

to symmetry-broken classical skyrmion state, which will

include the calculation of the scalar chirality operator

expectation values and scanning its distribution over the

system in question. Another quantity of our special in-

terest in this study is the entanglement entropy that is

undergoing drastical changes during the the reconstruc-

tion of the classical skyrmion state with zero entropy

from the wave functions with maximal entanglements.

Methodologically, the construction of the Anderson tower

of states is a kind of optimization problem aiming to de-

ﬁne a superposition of the quantum eigenstates that has

a maximal ﬁdelity with the coherent wave function rep-

resenting given classical state. Thus we will ﬁrst discuss

a gradient-descent method we developed to perform such

an optimization.

Random-phase gradient descent approach

Scalar Chirality. As it was shown in Ref.3 the only

physical quantity allows one to unambiguously identify

the quantum skyrmion state on the lattice is quantum

chirality, the following correlation function

ijk =1

⇡hˆ

Si·[ˆ

Sj⇥ˆ

Sk]i,(3)

i,j, and kdepict three di↵erent spins that form an el-

ementary plaquette. The total chirality is deﬁned as a

sum of local chiralities of individual plaquettes, Q =

PhijkiQ

ijk. In the case of the DMI Hamiltonian deﬁned

classical

skyrmion

scalar chirality density

quantum

skyrmion

FIG. 2. Triangle-resolved density of the scalar chirality

ijk calculated for real quantum skyrmion state (left plaque-

tte), coherent state with classical proﬁle of the magnetization

(right plaquette) and for quantum states being di↵erent ap-

proximations of the classical skyrmion within the Anderson

tower approach.

on the inﬁnite lattice that can be simulated with a super-

cell with periodic boundary conditions, all the triangular

plaquettes produce the same contribution to the total

chirality. In this sense the distribution of the chirality

is uniform and similar to the distribution of the mag-

netization over the system. In the case of the classical

skyrmionic structures the situation is opposite.

Entropy

H n=En n(4)

n=0

↵n n(5)

1M. A. Nielsen, I. L. Chuang, Quantum Computation and

Quantum Information, Cambridge University Press, 2010.

2P. W. Anderson, An Approximate Quantum Theory of the

Antiferromagnetic Ground State, Physical Review 86, 694

(1952).

3O. M. Sotnikov, V. V. Mazurenko , J. Colbois, F. Mila,

M. I. Katsnelson, and E. A. Stepanov, Phys. Rev. B 103,

L060404 (2021).

4R.A. Istomin, A.S. Moskvin, Overlap integral for quantum

skyrmions, Pis’ma v ZhETF 71, 487 (2000).

5A.M. Perelomov, Generalized coherent states and their ap-

plications, Springer-Verlag, Berlin, 1986.

6E.A. Stepanov, S.A. Nikolaev, C. Dutreix, M.I. Kat-

snelson and V.V. Mazurenko, Heisenberg-exchange-free

nanoskyrmion mosaic, J. Phys.: Condens. Matter 31

17LT01 (2019).

7D. I. Badrtdinov, S. A. Nikolaev, M. I. Katsnelson, and V.

V. Mazurenko, Spin-orbit coupling and magnetic interac-

CGradient-descent optimization

quantum

skyrmion

classical

skyrmion

N = 2 N = 3 N = 7

N = 20

Anderson towers

FIG. 1. Protocol for constructing Anderson towers of

states in the case of the skyrmionic problem. (A) (B) Evo-

lution of 19-site plaquette magnetization calculated for the

reconstructed wave function corresponding to the classical

skyrmion with di↵erent number of eigenstates of the parent

spin Hamiltonian.

As it can be seen from Fig.1, the ground state of the

quantum Hamiltonian Eq.2 is characterized by the uni-

form magnetization. Expectedly, the ﬁrst approximation

of the classical system shows initial trends toward tradi-

tional skyrmionic proﬁle and symmetry breaking. Such

an approximation is formed with the ground state and

6-fold degenerated ﬁrst excited state. Further extension

of the set of the excited states used to reconstruct | cl

ski

increase the ﬁdelity between target wave function and its

approximation. For N= 20 the ﬁdelity value reaches

95%.

We would like to stress that this magnetization analysis

is the ﬁrst stage on the way to a complete charaterization

of the transition from quantum skyrmion wave function

to symmetry-broken classical skyrmion state, which will

include the calculation of the scalar chirality operator

expectation values and scanning its distribution over the

system in question. Another quantity of our special in-

terest in this study is the entanglement entropy that is

undergoing drastical changes during the the reconstruc-

tion of the classical skyrmion state with zero entropy

from the wave functions with maximal entanglements.

Methodologically, the construction of the Anderson tower

of states is a kind of optimization problem aiming to de-

ﬁne a superposition of the quantum eigenstates that has

a maximal ﬁdelity with the coherent wave function rep-

resenting given classical state. Thus we will ﬁrst discuss

a gradient-descent method we developed to perform such

an optimization.

Random-phase gradient descent approach

Scalar Chirality. As it was shown in Ref.3 the only

physical quantity allows one to unambiguously identify

the quantum skyrmion state on the lattice is quantum

chirality, the following correlation function

ijk =1

⇡hˆ

Si·[ˆ

Sj⇥ˆ

Sk]i,(3)

i,j, and kdepict three di↵erent spins that form an el-

ementary plaquette. The total chirality is deﬁned as a

sum of local chiralities of individual plaquettes, Q =

PhijkiQ

ijk. In the case of the DMI Hamiltonian deﬁned

classical

skyrmion

scalar chirality density

quantum

skyrmion

FIG. 2. Triangle-resolved density of the scalar chirality

ijk calculated for real quantum skyrmion state (left plaque-

tte), coherent state with classical proﬁle of the magnetization

(right plaquette) and for quantum states being di↵erent ap-

proximations of the classical skyrmion within the Anderson

tower approach.

on the inﬁnite lattice that can be simulated with a super-

cell with periodic boundary conditions, all the triangular

plaquettes produce the same contribution to the total

chirality. In this sense the distribution of the chirality

is uniform and similar to the distribution of the mag-

netization over the system. In the case of the classical

skyrmionic structures the situation is opposite.

Entropy

H n=En n(4)

n=0

↵n n(5)

h A| Ti=0.95 (6)

@ A

@↵n

(7)

1M. A. Nielsen, I. L. Chuang, Quantum Computation and

Quantum Information, Cambridge University Press, 2010.

2P. W. Anderson, An Approximate Quantum Theory of the

Antiferromagnetic Ground State, Physical Review 86, 694

(1952).

3O. M. Sotnikov, V. V. Mazurenko , J. Colbois, F. Mila,

M. I. Katsnelson, and E. A. Stepanov, Phys. Rev. B 103,

L060404 (2021).

4R.A. Istomin, A.S. Moskvin, Overlap integral for quantum

skyrmions, Pis’ma v ZhETF 71, 487 (2000).

Fidelity

k= 100

n=0

↵n n(5)

h A| Ti=0.95 (6)

@ A

@↵n

(7)

F=|h A| Ti|(8)

1M. A. Nielsen, I. L. Chuang, Quantum Computation and

Quantum Information, Cambridge University Press, 2010.

2P. W. Anderson, An Approximate Quantum Theory of the

Antiferromagnetic Ground State, Physical Review 86, 694

(1952).

3O. M. Sotnikov, V. V. Mazurenko , J. Colbois, F. Mila,

M. I. Katsnelson, and E. A. Stepanov, Phys. Rev. B 103,

L060404 (2021).

4R.A. Istomin, A.S. Moskvin, Overlap integral for quantum

skyrmions, Pis’ma v ZhETF 71, 487 (2000).

5A.M. Perelomov, Generalized coherent states and their ap-

plications, Springer-Verlag, Berlin, 1986.

6E.A. Stepanov, S.A. Nikolaev, C. Dutreix, M.I. Kat-

snelson and V.V. Mazurenko, Heisenberg-exchange-free

nanoskyrmion mosaic, J. Phys.: Condens. Matter 31

17LT01 (2019).

7D. I. Badrtdinov, S. A. Nikolaev, M. I. Katsnelson, and V.

V. Mazurenko, Spin-orbit coupling and magnetic interac-

tions in Si(111):C,Si,Sn,Pb, Phys. Rev. B 94, 224418 (2016).

8I. A. Iakovlev, O. M. Sotnikov, and V. V. Mazurenko,

Bimeron nanoconﬁned design, Phys. Rev. B 97, 184415

(2018).

k=1

n=0

↵n n(5)

h A| Ti=0.95 (6)

@ A

@↵n

(7)

F=|h A| Ti|(8)

F⇡0 (9)

F⇡1 (10)

1M. A. Nielsen, I. L. Chuang, Quantum Computation and

Quantum Information, Cambridge University Press, 2010.

2P. W. Anderson, An Approximate Quantum Theory of the

Antiferromagnetic Ground State, Physical Review 86, 694

(1952).

3O. M. Sotnikov, V. V. Mazurenko , J. Colbois, F. Mila,

M. I. Katsnelson, and E. A. Stepanov, Phys. Rev. B 103,

L060404 (2021).

4R.A. Istomin, A.S. Moskvin, Overlap integral for quantum

skyrmions, Pis’ma v ZhETF 71, 487 (2000).

5A.M. Perelomov, Generalized coherent states and their ap-

plications, Springer-Verlag, Berlin, 1986.

6E.A. Stepanov, S.A. Nikolaev, C. Dutreix, M.I. Kat-

snelson and V.V. Mazurenko, Heisenberg-exchange-free

nanoskyrmion mosaic, J. Phys.: Condens. Matter 31

17LT01 (2019).

7D. I. Badrtdinov, S. A. Nikolaev, M. I. Katsnelson, and V.

V. Mazurenko, Spin-orbit coupling and magnetic interac-

tions in Si(111):C,Si,Sn,Pb, Phys. Rev. B 94, 224418 (2016).

8I. A. Iakovlev, O. M. Sotnikov, and V. V. Mazurenko,

Bimeron nanoconﬁned design, Phys. Rev. B 97, 184415

(2018).

k=1

n=0

↵n n(5)

h A| Ti=0.95 (6)

@ A

@↵n

(7)

F=|h A| Ti|(8)

F⇡0 (9)

F⇡1 (10)

1M. A. Nielsen, I. L. Chuang, Quantum Computation and

Quantum Information, Cambridge University Press, 2010.

2P. W. Anderson, An Approximate Quantum Theory of the

Antiferromagnetic Ground State, Physical Review 86, 694

(1952).

3O. M. Sotnikov, V. V. Mazurenko , J. Colbois, F. Mila,

M. I. Katsnelson, and E. A. Stepanov, Phys. Rev. B 103,

L060404 (2021).

4R.A. Istomin, A.S. Moskvin, Overlap integral for quantum

skyrmions, Pis’ma v ZhETF 71, 487 (2000).

5A.M. Perelomov, Generalized coherent states and their ap-

plications, Springer-Verlag, Berlin, 1986.

6E.A. Stepanov, S.A. Nikolaev, C. Dutreix, M.I. Kat-

snelson and V.V. Mazurenko, Heisenberg-exchange-free

nanoskyrmion mosaic, J. Phys.: Condens. Matter 31

17LT01 (2019).

7D. I. Badrtdinov, S. A. Nikolaev, M. I. Katsnelson, and V.

V. Mazurenko, Spin-orbit coupling and magnetic interac-

tions in Si(111):C,Si,Sn,Pb, Phys. Rev. B 94, 224418 (2016).

8I. A. Iakovlev, O. M. Sotnikov, and V. V. Mazurenko,

Bimeron nanoconﬁned design, Phys. Rev. B 97, 184415

(2018).

FIG. 1. Protocol for constructing Anderson’s towers of states. (A) For the given Hamiltonian one calculates a set of low-lying

eigenstates. (B) On the basis of the calculated eigenstates, an initial approximation of the target state is prepared. The

complex coeﬃcients αnare chosen to be random. (C) The coeﬃcients are optimized within a gradient-descent approach aiming

to maximize the ﬁdelity between approximation and target wave functions. Nis the corresponding number of energy levels.

classical magnetic moments of a quantum system ob-

served in real or numerical experiments: θi= arccoshmz

and φi= arctan hmy

hmx

ii.

In order to construct the Anderson tower for a

quantum system we follow the key steps visualized in

Fig. 1 A-C. First, we perform the exact diagonalization

of a quantum Hamiltonian and determine its eigenstates

Ψn(Fig. 1 A). We consider only the low-lying part of the

eigenspectrum n∈[0, k] and introduce the initial approx-

imation for the target wave function ΨAwith random

complex coeﬃcients αn(Fig. 1 B). Further, these coef-

ﬁcients are varied using the gradient-descent method to

get the maximal ﬁdelity between ΨAand ΨT(Fig. 1 C).

Here, as for any optimization procedure, the choice of

the loss function that is responsible for the quality of the

resulting approximation and convergence speed plays a

central role. In this work it is given by the following

expression:

E(α)=1− |hΨT|ΨA(α)i|.(2)

The coeﬃcients are updated as

αnew =αold −γ∂E

∂αold

,(3)

where γis the gradient-descent step that is taken to be

1. This choice for the loss function can be justiﬁed by

the fact that the ﬁdelity is a standard metric to deﬁne

the distance between the two quantum states51.

An important property of the proposed numerical

scheme for constructing TOS is that we can quantita-

tively control the quality of the approximation for ΨA

with the parameter kthat truncates the eigenspectrum.

It is worth noting that in previous works the construction

of the Anderson TOS for antiferromagnets was mainly

based on a symmetry-based selection of the low-lying

states of the eigenspectrum in order to ﬁnd a signature

of a symmetry-broken state that can be the ground state

of the system in the thermodynamic limit52. Such a

symmetry-based realization of the Anderson idea limits

its possible applications and does not provide quantita-

tive information on the contribution of a particular eigen-

state to the TOS, which, as we show below, can be done

with our protocol. In addition, in our approach the loss

function (2) can be replaced by another form with milder

conditions, which might be useful in the case where the

precise form of the target state ΨTis unknown.

Skyrmionic tower of states

The implementation of the proposed protocol in the

case of magnetic skyrmions requires a speciﬁc choice of

spin Hamiltonian. While various microscopic mecha-

nisms for stabilizing skyrmion structures can be consid-

ered in the quantum case41–46, we focus on the most tra-

ditional one which is based on the competition between

the isotropic and the anisotropic exchange interactions

in the presence of an external magnetic ﬁeld. The cor-

responding quantum Hamiltonian on a two-dimensional

lattice is given by:

Hsk =X

Jij ˆ

Si·ˆ

Sj+X

Dij [ˆ

Si×ˆ

Sj] + X

Bˆ

i.(4)

Here, Jij is the isotropic Heisenberg exchange interac-

tion. Dij is an in-plane vector that points in the direction

perpendicular to the bond between neighboring iand j

lattice sites and describes the Dzyaloshinskii-Moriya in-

teraction (DMI). Bis an external uniform magnetic ﬁeld

applied along the zdirection. In the following, for con-

structing the skyrmionic towers, we use the parameters

J= 0.5, |D|= 1 and B= 0.44, which guarantee the

stabilization of the quantum skyrmion wave function as

the quantum ground state47.

In this work we are interested in exploring the proper-

ties of the eigenspectrum, thus the exact diagonalization

of the constructed Hamiltonian is performed. Following

Ref. [47], a 19-site supercell with periodic boundary con-

ditions on the triangular lattice is considered. The cor-

responding technical details are described in the Meth-

ods section. The low-energy part of the eigenspectrum

is presented in Fig. 2 A and is characterized by a size-

able gap between the ﬁrst 19 eigenstates and the rest of

the spectrum. The dependence of the gap on the value

of the magnetic ﬁeld is discussed in the Methods sec-

tion. As we will show below, the states at the bottom of

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ReconstructionofclassicalskyrmionsfromAndersontowers:quantumDarwinisminactionO.M.Sotnikov1,E.A.Stepanov2,M.I.Katsnelson3,F.Mila4,V.V.Mazurenko11TheoreticalPhysicsandAppliedMathematicsDepartment,UralFederalUniversity,MiraStr.19,620002Ekaterinburg,Russia2CPHT,CNRS,EcolePolytechnique,InstitutPolytechni...

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Reconstruction of classical skyrmions from Anderson towers quantum Darwinism in action O. M. Sotnikov1 E. A. Stepanov2 M. I. Katsnelson3 F. Mila4 V. V. Mazurenko1.pdf

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Reconstruction of classical skyrmions from Anderson towers quantum Darwinism in action O. M. Sotnikov1 E. A. Stepanov2 M. I. Katsnelson3 F. Mila4 V. V. Mazurenko1

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