Topological adiabatic dynamics in classical mass-spring chains with clamps

2025-05-06 0 0 388.76KB 6 页 10玖币

Atushi Tanaka

Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan

Abstract

The path dependence of adiabatic evolution in classical harmonic chains with clamps is examined. It is shown that cutting and

joining a chain may braid adiabatic normal mode frequencies. Accordingly, diﬀerent adiabatic paths with the same endpoints may

transport a normal mode to a diﬀerent one, and an adiabatic cycle pumps action variables, i.e., the adiabatic invariants of integrable

classical systems. Another adiabatic pump for artiﬁcial edge modes induced by clamps is shown as an application. Extensions to

completely integrable systems and quantum systems are outlined.

1. Introduction

The adiabatic theorems of classical and quantum mechani-

cal systems tell us the presence of conserved quantities, i.e.,

invariants, during the time evolution where system parameters

are slowly varied. In quantum theory, the populations of sta-

tionary states are adiabatic invariants. In classical mechanics,

the adiabatic invariants of integrable and ergodic systems are

actions [1] and phase-space volumes [2, 3], respectively. The

adiabaticity has been applied variously. To name a few, the adi-

abatic invariance of the classical action was a key to developing

the quantum theory, and the quantum adiabatic theorem oﬀers

bases for studying molecular physics [4, 5] and quantum com-

putation [6–8]. Note that we here discern the adiabaticity in

thermodynamics from the one in mechanics and focus on the

latter.

An adiabatic process induces changes in physical quantities,

and some are associated with the geometry of the adiabatic

path in the parameter space. Famous examples are the geo-

metric phase factors of quantum states and the angle variable

holonomy in classical integrable systems [9–14]. The geomet-

ric phase factor is crucial in understanding the quantum Hall

eﬀect [15] and topological pumps of quantum [16] as well as

classical [17–19] systems.

In quantum theory, the adiabatic invariants, i.e., the popu-

lations of stationary states, can also be changed by adiabatic

processes. Namely, an adiabatic cycle can transport a station-

ary state to another stationary state [20–22]. The result of the

adiabatic evolution of a stationary state is path-dependent, and

the topology of the path determines the ﬁnal state [23, 24]. Re-

cently, its experimental realization in a one-dimensional dipolar

gas has been dubbed a quantum energy pump [25].

On the other hand, in classical mechanics, it has been un-

known whether the adiabatic invariants can be path-dependent

or, equivalently, the pumping of the classical adiabatic invari-

ants is possible. Although the classical energy pump studied

URL: https://arxiv.org/a/tanaka_a_1.html (Atushi Tanaka)

by Lu, Jarzynski, and Ott [26] appears to be such an exam-

ple, their analysis proved that the pumping in their model is

nonadiabatic. The primary diﬃculty in ﬁnding a classical adia-

batic invariant pump would be the absence of quantum-classical

correspondence of the adiabatic theorem in generic systems,

whose phase-space involves both regular and chaotic compo-

nents [2, 27]. Also, in classically integrable systems, an adia-

batic path must avoid separatrix crossings to protect the adia-

batic invariance of action variables [28–30].

Our goal here is to show that the adiabatic evolution of clas-

sical adiabatic invariants is path-dependent in harmonic chains,

which are completely integrable systems with multiple degrees

of freedom. Crucial ingredients are adiabatic “cutting” and

“joining” operations to braid adiabatic normal mode frequen-

cies. We note that similar operations provide an example of

path-dependent quantum adiabatic invariants [31].

This paper is organized as follows. In section 2, we exam-

ine mass-spring chains with clamps to prove the path depen-

dence of the adiabatic invariants. An adiabatic cycle accord-

ingly pumps the normal modes of a mass-spring chain (Sec-

tion 3). These points reﬂect the topological nature of adiabatic

dynamics in the classical integrable systems (Section 4). As an

application, we introduce a topological pump for artiﬁcial edge

modes induced by clamps (Section 5). In section 6, we present

a summary of the present work. We also brieﬂy discuss the

extension of our results to completely integrable classical sys-

tems and quantum systems. Appendices provide details of the

adiabatic paths for our examples.

2. Mass-spring chains with adiabatic pinning

We examine classical mass-spring chains to explain the path-

dependence of adiabatic evolution. Assume that Nparticles of

identical mass mare connected by harmonic springs of iden-

tical spring constant k. We impose the ﬁxed boundary condi-

tion on the chain. Each particle is imposed pinning potential

adiabatically. We assume that the system is described by the

Preprint submitted to arXiv January 10, 2023

arXiv:2210.00663v2 [cond-mat.mes-hall] 9 Jan 2023

1 2 3 4

(a) K∅

(b)

(d) K{2}

(e) K{2,3}

(f) K{3}

Figure 1: The adiabatic time evolutions of a mass-spring system (N=4) along

two paths C1and C2, which share endpoints. (a) The mass-spring chain has

four particles (indicated by circles) and satisﬁes the ﬁxed boundary condition

(indicated by two crosses). The system is initially in the slowest normal mode.

(b) During C1, the third particle is adiabatically clamped (indicated by two

rectangles). (c) After completing C1, the oscillation localizes at the left part of

the system, and thus the right part (indicated by gray circles) ceases. During

C2, the second and third particles are clamped successively ((d) and (e)), and

the clamp for the second particle is released ﬁnally. The resultant oscillation

localizes at the right part of the system (f). The values of the adiabatic parameter

K(e.g., K∅and K{3}) are explained in the main text.

Hamiltonian

H=1

i=1

i+1

i,j=1

Ki j xixj,(1)

where xiand piare the displacement and momentum of the i-

th particle (1 ≤i≤N) and m=1 is assumed. The N×N

real symmetric matrix Kdescribes the harmonic springs and

pinnings and is adiabatically varied. According to the adiabatic

theorem, once the system is prepared to be in a normal mode,

speciﬁed by a normalized eigenvector ξaof K, the system stays

in a normal mode. Let ωadenote the normal mode frequency

corresponding to ξa.

Our adiabatic paths visit the points where the particles are

clamped to bring the braiding of adiabatic normal modes. The

clamps are made of inﬁnite pinning strength. For example, i-

th particle is clamped when Kii is inﬁnitely large. We denote

K=K{i}if i-th particle is clamped and no pinning is imposed

on other particles. We note that a mathematical analysis of the

parametric evolution along variable pining strength is shown in

Ref. [32] to examine classical oscillations with variable rigidity.

Also, similar operations have been utilized to study the role of

topology change in quantum systems [33, 34].

We introduce the endpoints of adiabatic operations. For sim-

plicity, N=4 is assumed. At the initial point of the paths,

no pinning is imposed on the mass-spring chain, denoted as

K=K∅(Fig. 1 (a)). On the other hand, we assume K=K{3},

i.e., inﬁnitely strong pinning is imposed to clamp the third par-

ticle, at the endpoint. The system at K{3}is accordingly divided

into two, where the “left” and “right” parts are N=2 and N=1

mass-spring chains under the ﬁxed boundary condition, respec-

tively (Fig. 1 (c) and (f)).

Both of two adiabatic paths C1and C2connect K∅and

K{3}. During the path C1, the pinning is imposed only on the

third particle (Fig. 1 (b)) to adiabatically cut the whole sys-

tem (Fig. 1 (c)). On the other hand, C2involves “cut” and

“splice”. Namely, C2connects K∅,K{2},K{2,3}and K{3}in or-

der, where i-th and j-th particles are simultaneously clamped at

(a)

(c)

(a)

(d)

(e)(f)

K∅K{3}K{2,3}K{2}K∅

ω1

ω2

ω3

ω4

ω1

ω2

ω3

ω4

Figure 2: Normal mode frequencies in N=4 mass-spring chain with clamps.

The horizontal axis represents the closed path C, where the conﬁgurations of

clamps are indicated. The adiabatic evolution of the ground normal mode ω1

at K∅along C1(thick line) and C2(thick dashed line) emanate from the left

and right ends, respectively, where the intermediate points shown in Fig. 1 are

depicted by circles. The evolutions of the ground mode from the left end (thick

line) to the right end (thin line) arrive at the ﬁrst excited mode ω2, where the

continuity of the normal mode vector is kept. Hence the closed path Cinduces

the mode holonomy.

K{i,j}(1 ≤i,j≤N) (Figs. 1 (a), (d), (e) and (f)). Examples of

the parametrization of the paths are shown in Appendix A.

We examine the adiabatic evolutions along C1and C2. We

assume that the initial system at K=K∅is in the ground normal

mode, with frequency ω1being the slowest.

The adiabatic evolution along C1delivers the ground mode

of K∅to the one of K{3}. This is due to the absence of spec-

tral degeneracy in C1(see, Fig. 2). Since the system at K{3}is

divided into two, the ﬁnal ground mode localizes at the larger,

left part. The rightmost particle accordingly ceases to oscillate

although it is not clamped (Fig. 1 (c)).

On the other hand, the adiabatic time evolution along C2pro-

ceeds as follows. The system at K{2}is in the ground mode and

localized at the right side of the clamp (Fig. 1 (d)) because of

the symmetry between K{2}and K{3}. The adiabatic clamp on

the third particle “compresses” the mode at K{2,3}to localize at

the rightmost particle (Fig. 1 (e)).

We must correctly handle the degeneracy point at K{2,3}. The

degenerate normal mode vectors localize at either end of the

mass-spring chain. Note that these normal modes are not hy-

bridized, as the complete clamps separate them. The system’s

state accordingly localizes at the right part around K{2,3}in C2.

Also, the state arrives at the ﬁrst excited normal mode after

K{2,3}and stays there. Hence the adiabatic evolution along C2

transports the ground mode at K∅to the ﬁrst excited mode at

K{3}.

We conclude that the result of the adiabatic evolution of a

normal mode from K∅to K{3}depends on the path. Further-

more, the adiabatic evolution is topological, as the possible can-

didates of the ﬁnal normal modes are discrete.

3. Normal mode pump

We utilize the path-dependent adiabatic time evolution to

construct a normal mode pump. Let Cdenote the closed path

made of C1and the inverse of C2. Namely, Cconnects K∅,

K{3},K{2,3},K{2}and K∅in order. The adiabatic evolution along

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摘要：

Topologicaladiabaticdynamicsinclassicalmass-springchainswithclampsAtushiTanakaDepartmentofPhysics,TokyoMetropolitanUniversity,Hachioji,Tokyo192-0397,JapanAbstractThepathdependenceofadiabaticevolutioninclassicalharmonicchainswithclampsisexamined.Itisshownthatcuttingandjoiningachainmaybraidadiabaticno...

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Topological adiabatic dynamics in classical mass-spring chains with clamps

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