Universit`
a degli Studi di Pisa
DIPARTIMENTO DI FISICA ENRICO FERMI
Corso di Laurea in Fisica - Curriculum di Fisica Teorica
Tesi di Laurea Magistrale
Localization of kinks in discrete classical models
Candidato:
Guglielmo Lami
Relatore:
Prof. Alessandro Silva
Correlatrice:
Prof.ssa. Maria Luisa Chiofalo
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La libert`a non `e star sopra un albero; non `e neanche il volo di un moscone. La libert`a non `e uno spazio libero...
G. Gaber
Libert`a e perline colorate, ecco quello che io ti dar`o.
E la sensualit`a delle vite disperate, ecco il dono che io ti far`o.
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Contents
1 Kinks in1 + 1 dimensions 19
1.1 Kinks in classical scalar field theories . . . 19
1.2 Kinks in the classical Ising model . . . 26
1.2.1 Spin systems . . . 26
1.2.2 The transverse field Ising model (TFIM) . . . 30
1.2.3 Continuum limit of the TFIM . . . 31
1.2.4 Kink solutions in the continuum limit . . . 32
1.2.5 Linear stability . . . 33
2 Kink dynamics in the discrete φ4 model 39 2.1 First remarks . . . 39
2.1.1 The discreteness parameter . . . 39
2.1.2 Central-Site (CS) and Central-Bond (CB) kinks . . . 41
2.2 Collective variables approach . . . 46
2.2.1 Hamiltonian formalism for the kink collective variables . . . 46
2.2.2 Motion of the kink in absence of radiation . . . 50
2.2.3 Effect of the radiation: numerical results . . . 53
2.2.4 Interpretation of the backwards emissions . . . 58
2.3 Final remarks . . . 61
3 Kinks in the discrete classical TFIM 65 3.1 Stationary CS and CB kinks . . . 65
3.2 Kink dynamics and localization . . . 71
Appendices 79 A Path-integral for spins 81 A.1 Spin coherent states . . . 81
A.2 Construction of the path integral . . . 83
B Constrained hamiltonian dynamics 87
C Newton’s equations for kink and phonons 91
D Peyrard-Kruskal’s arguments 95
Bibliography 97
Introduction
Localization was a different matter: very few believed it at the time, and even fewer saw its importance; among those who failed to fully understand it at first was certainly its author.
P. W. Anderson, Nobel Lecture
Thermodynamics is essentialy a description of the equilibrium states of systems with many degrees of freedom. A translationally invariant system can reach thermodynamic equilibrium only if any initial macroscopic inhomogeneity in local densities of conserved quantities is smoothed out by time evolution. This is what usually happens in a generic many-body system. The mechanism through which this happens is usually called transport process and is tipically governed by diffu-sion laws [1]. Examples of transport mechanisms include molecular diffudiffu-sion (mass transfer), heat conduction (energy transfer) and fluid flow (momentum transfer).
When transport mechanisms, for some reason, are suppressed, localization phenomena take place. For its conceptual importance, localization has become a fundamental paradigm of mod-ern condensed matter theory. Over the last decades, many efforts have been made to find and theoretically understand localization phenomena in many-body systems. Furthermore, recent ex-perimental progress in the context of cold atoms turned the study of out-of-equilibrium dynamics of closed systems from an academic debate into a concrete and extremely active research topic. We begin with an overview of localization.
Overview
The origin of this research branch dates back to 1958, when P. W. Anderson published his pio-neering theoretical work on the “Absence of Diffusion in Certain Random Lattices” [2]. In this work Anderson predicted the absence of diffusion for a quantum mechanical particle (an electron) in a sufficiently disordered medium. Today this phenomenon is known as Anderson localization or strong localization.
The simplest hamiltonian to understand this phenomenon is that of the Anderson model. This is a simple tight-binding model for a hopping particle in a discrete infinite lattice with a random potential in each site,
H = −gX
hi,ji
|ii hj| + |ji hi| +X
i
hi|ii hi| j ∈ Zd,
where the sum is over nearest neighbouring sites of the lattice Zd, g ≥ 0 is the nearest neighbor
hopping amplitude and hj are random independents variables distributed, for example, according
to an uniform distribution [−h, +h]. One can show that in every dimension d, for a sufficient strong yet finite disorder strength h > hc, all the eigenfunctions of H are exponentially localized [3]:
| hi|ψαi | ≈ e−
|i−iα|
ξα , for |i − iα| → ∞ ,
Here ξαis the wave-function localization length and iαis a certain point of the lattice
correspond-ing to the localization center. In particular, for d ≤ 2 it can be shown that hc= 0.
To appreciate, at least intuitively, why localization occours one can start with some wave-mechanics considerations. It is quite intuitive that the probability amplitude Ai→j to propagate
from site i to site j can be written as a sum over the amplitudes ACij contributed by each path C
connecting i and j. These heuristic considerations can be bolstered by a locator expansion of the single-particle Green’s function [3]. Thus, for the probability Pi→j = |Ai→j|2 we will have:
Pi→j = X C ACij 2 =X C |ACij| 2+X C,C0 A∗C ijAC0ij,
The first term of this equation corresponds to the classical contribution, the second one to the corrections due to quantum interferences between different paths. Because of the randomness of the on-site energies hi, the phase accumulated along different paths will be a random quantity as
well. Thus, in general quantum corrections to the classical probability will vanish upon a disorder average. If we consider the return probability Pi→i however, then for each path there exists a
time-reversed path. Since the system is invariant under time-reversal, both paths have the same amplitude and are phase coherent. Then we can write:
Pi→i = 2
X
C
|ACij|
2+ incoherent contributions ,
We conclude that the probability for a quantum mechanical particle to return to its initial site is twice as large than it would be for a classical particle. This effect is called coherent backscatter-ing. Furthermore, as the total probability is a conserved quantity and is normalized to one, the probability for the particle diffusing away into the infinite system must be reduced compared to the classical case. This is the origin of Anderson localization in a nutshell.
Naturally, Anderson localization has dramatic consequences on the transport properties of the system. For example, it implies that the diffusion coefficient D vanishes for h > hc. This is
unexpected from a semiclassical point of view. In fact dimensionally D is v2τ, with v a velocity
and τ a time. Treating the random term V =P
ihi|ii hi| perturbatively, v can be identified with
the group velocity ∂kEk in which Ek is the energy of the eigenfunctions of H in absence of V
(namely plane waves) and τ with the time between scattering over the impurities. Then τ can be computed as the inverse of the rate of scattering by using Fermi golden rule. Consequently τ and D will be proportional to h−2. This perturbative argument, valid for small h and for sufficently
large d (Figure 1), fails for larger h, as seen by the fact that the DC-conductivity σ at zero tem-perature vanishes for h > hc. Ultimately, we can see this phenomenology as the manifestation of
a disorder-driven quantum phase transition, i.e. a phase transition at zero temperature, driven by one of the parameters of H [5]. The phases in question are a metallic delocalized phase and an insulating localized one, where diffusive transport is completely suppressed. Decades of theoretical studies have clarified the nature of this transition. Furthermore, many experimental observations have confirmed the existence of Anderson localized phase, for example in cold atom systems [4].
CONTENTS 11
Figure 1: Diffusion coeffcient D of a particle in a random potential as a function of the disorder strenght h: semiclassical result (dotted line) and Anderson result (red line). Picture taken from Ref. [3].
The Anderson model, as remarked before, is a single-particle model but it can be trivially extended to describe a system of interacting particles in a random potential. One question arises naturally: what happens when we add interactions between the particles? Is the Anderson localized phase stable under generic two-body interactions? This has been a long-standing open question. Recent works have answered this question in the affermative: localization is indeed stable with respect to short-range and sufficiently weak interactions. Such a phase is called the “many-body localized” (MBL) phase.[6]
The MBL regime can be properly viewed as a phase of matter, because it is robust under sufficiently weak perturbations. The existence of this phase has been supported by extensive nu-merical simulations of one dimensional spin and fermionic models [7], [8], [9], and by a number of experiments [10], [11]. A crucial feature of many-body localized systems is that they are non-thermalizing, because the long-time state of system is determined by (and thus can “remember”) some local details of the system’s initial state. For example an initial macroscopic inhomogeneity in the energy density profile of the system can persist over arbitrarily long times. Thus, MBL systems cannot reach thermal equilibrium under their own dynamics. In this sense, MBL systems seem to form a very general class of exceptions to the common accepted thermalization paradigm for quantum closed systems, namely the Eigenstate Thermalization Hypothesis (ETH) [6].
So far we have been discussing disorder-generated localization. A long-standing and debated problem is rather the possible occurrence of localization phenomena in systems without disorder and their characterization [12], [13], [14]. In a recent work [15], Lerose et al. have shown that long-range ferromagnetic interactions in quantum spin chains can induce spatial quasi-localization of topological magnetic defects, i.e. domain-walls, even in the absence of disorder. The model considered is a quantum transverse field Ising chain with long-ranged ferromagnetic interactions:
H = − J Nα,L X 1≤i<j≤L σx iσjx |i − j|α− h L X i=1 σzi ,
where J is the ferromagnetic coupling and h the transverse field. The exponent α ≥ 0 characterizes the range of the ferromagnetic spin-spin interaction. Nα,L is a rescaling factor that ensures a
the system. This is: Nα,L= 1 L −1 X 1≤i<j≤L 1 |i − j|α
Open boundary conditions are assumed. The authors studied the non-equilibrium evolution gov-erned by H starting from the initial states |ψ(t = 0)i = |ni corresponding to a longitudinal domain-wall between sites n and n + 1:
|ni = |↑1... ↑n↓n+1... ↓Li ,
where |↑i , |↓i are the eigenstates of σx. By performing numerical computations with a
time-dependent variational principle on matrix product states (MPS-TDVP), they found that the be-havior of the local magnetization hσx
ii as function of t depend crucially on α being smaller or
larger than 2. In particular, for α > 2 they observed an unbounded light-cone spreading of the domain-wall. Instead, in presence of long-range interactions with 1 < α < 2, they observed that the inhomogeneity, initially spreading out from the center of the chain, bounces back at a certain characteristic distance and remains subsequently trapped within this finite region. The suppression of spatial diffusion of the domain wall during the accessible time scale shows that the excess energy density remains localized around the initial domain-wall position, hindering an efficient transport and then the thermalization.
Figure 2: Evolution of the longitudinal magnetization hσx
i(t)i in an open ferromagnetic quantum
Ising chain of L = 100 spins. If the exponent α, characterizing the range of the ferromagnetic interactions, is < 2 clearly appears a phenomenon of (quasi)localization. Picture taken from Ref. [15].
The theoretical analysis proposed in the paper aims to capture these features by projecting the complicated many-body hamiltonian in the Hilbert subspace spanned by the states {|ni}, namely the single-domain-wall states. In this way they obtain an effective hamiltonian for a single quantum particle corresponding to the domain wall, the following:
[Heff]nm= −h(δn+1,m+ δn−1,m) + Vα,L(n)δnm , where Vα,L(n) = 2 J Nα,L X 1≤i≤n X n<j≤L 1 |i − j|α
CONTENTS 13
is an effective potential equal to the ferromagnetic configurational excess energy respect to the state |n = Li = |↑1... ↑Li. This method is valid in the limit h J for the following reason.
The transverse field term hPL i=1σ
z
i gives rise to spin-flips during the time evolution (σz|↑i = |↓i
and σz|↓i = |↑i); however, a spin-flip far away from the domain-wall location has a very high
cost of configurational energy, compared with the perturbation strenght h. Then, at first order in perturbation theory we can completly ignore the probability amplitude of transition in a non-single-domain-wall state.
The qualitative behaviour of the dynamics generated by Heff is essentially determined by the
shape of the function Vα,L(n). In the L → ∞ limit one can evaluate the sums by using integrals.
In the case α > 2, one obtains that Vα,L(n) goes to a positive constant; contrary, in the case
1 < α < 2, Vα,L(n) has a non-trivial scaling limit for large L, described by a smooth function V:
Vα,L(n) → JL2−αcαVα(
n
L) Vα(x) = x
−α+2+ (1 − x)−α+2− 1 ,
in which cα is a positive constant. In the first case (α > 2) the eigenfunctions of Heff are plane
waves, since the potential Vα,L(n) is nearly flat. Contrary, in the second case (1 < α < 2) all
the eigenfunctions are spatially localized. To realize that, one can simply study the classical hamiltonian corresponding to Heff:
Hcl(q, p) = −2h cos p + JcαL2−α[q−α+2+ (1 − q)−α+2− 1] ,
with the phase space: (q, p) ∈ [0, L − 1] × [0, 2π). The classical trajectories, defined by Hamilton’s equations, are shown in Figure 3 for α = 1.25 and α = 3.0. The semiclassical eigenstates can be eventually obtained by quantizing the phase space area encircled in classical trajectories, following the Bohr-Sommerfeld quantization rule. In the case 1 < α < 2, the concurrence of the unbounded potential (∝ JL2−α) and the bounded kinetic energy (∝ h), together with the conservation of
total energy Hcl, implies that a particle initially placed in a given point can travel at most a finite
distance away from it. On the other hand, in the case α > 2 the trajectories are spatially-extended and a particle can travel along an extensive portion of the chain. This semiclassical analysis turns out to be valid in the limits L → ∞ and h/J → 0.
Figure 3: Trajectories defined by the classical Hamiltonian Hcl(q, p) in the phase space: (q, p) ∈
[0, L − 1] × [0, 2π), for α = 1.25 and α = 3.0. Picture taken from Ref. [15].
A similar scenario can be observed also in a model with short-range interactions. In another recent work [16], Mazza et al. analyze a simple quantum Ising chain with both a transverse and a longitudinal magnetic field and nearest neighbour ferromagnetic interactions:
H = −J X 1≤i≤L−1 σixσxi+1− hz L X i=1 σzi − hx L X i=1 σxi ,
open boundary conditions. As a consequence of the non-vanishing longitudinal field hx>0, the
chain acquires an initial macroscopic energy imbalance between the left (“cold”) half and the right (“hot”) half. One might expect the time evolution to give rise to an energy transfer between the two parts. However this does not happen. In fact, by performing numerical simulations, the authors clearly show that the energy transfer is suppressed even at late time. In Figure 4 it is reported the time evolution of the energy density hHi(t)i, where:
Hi= −Jσxiσ x i+1− hz σiz+ σz i+1 2 − hx σix+ σx i+1 2 .
Various values of the ratio hx/hz are explored. The oscillations of the profiles, shown in the
plots, may be interpreted as the quantum motion of the isolated domain wall initially localized at the junction between the two halves. In fact, the kinetic energy associated with this motion has a finite bandwidth ∝ hz on the lattice, and therefore, because of energy conservation, the kink
quasi-particle can travel at most a distance lconf∝ hz/hx, called confinement length scale, in the
linear confining potential V (l) = hxl, before bouncing back and oscillating.
Figure 4: Time evolution of the energy density profile hHi(t)i for hz= 0.2 (L = 50), 0.4 (L = 100)
and hx= 0.15, 0.3 varying as indicated by the axes. Picture taken from Ref. [16].
In this section we provided a brief overview of recent developments and the current state of the art about localization phenomena and suppression of transport. The purpose of this thesis is to show that localization phenomena are by no means a prerogative of quantum systems and that they can be observed in some classical discrete systems. In particular, we will analyze the discrete φ4 chain and the classical transverse field Ising chain. A key feature of these non-linear systems
is to admit a particular class of topological excitations, namely topological solitons, also known as kinks or domain walls according to the contexts [17]. Let us now summarize for the reader some basic aspects of topological solitons, before entering into details of this thesis.
CONTENTS 15
Topological solitons
In continuum models (i.e. field theories), solitons are non-linear localized waves, which can move undistorted with constant velocity, carrying a finite amount of energy. Textbook examples of soli-tons can be found in hydrodynamics, where they were first detected in 1834 by John Scott Russel. In this case the underlying non linear partial differential equation is the Korteweg-de Vries (KdV) equation [18]. Topological solitons basically are solitons with an additional property: they are “topologically protected” because intrinsically distinct from the vacuum solution. Let us clarify the exact mathematical reason behind this.
We consider a field theory, defined on the d−dimensional space Rd and, for simplicity, involving
only one scalar field φ(x). We are interested in static configurations of that field. The energy functional H[φ] will contain a certain (finite) number of terms involving spatial derivatives of φ, and a potential term V (φ) depending on the field but not on his spatial derivatives. A sufficient condition to ensure finiteness of energy is that V has at least one zero and that φ goes to one of this constant values for |x| → ∞. In this way the terms with spatial derivatives vanish. Let us define the “vacuum manifold”: M = {φ ∈ R : V (φ) = 0}. In general if we have n fields φ1... φn,
M will be a submanifold of Rn. We observe that M can also be defined as the coset space G/H,
where G is the group of simmetries of the considered field theory and H is the subgroup of G whose elements are the simmetries of the vacuum configurations [20]. Any finite energy configuration φ defines a map φ∞ from “the boundary of Rd” i.e. the sphere Sd−1 to the vacuum manifold:
φ∞: Sd−1 → M .
Two finite energy configurations are said to be topologically equivalent, or homotopic, if they can be deformed into each other without passing through “forbidden configurations”, i.e. configurations with infinite energy. This defines an equivalence relation on the maps φ∞. The corresponding
set of equivalence classes is called the (d − 1)th homotopy group of M, in symbols: πd−1(M).
Topological solitons are configurations of the field(s) belonging to a homotopy class distinct from the trivial uniform vacuum configuration (obiously necessary condition is to have a non-trivial ho-motopy group). For this reason topological solitons are extremly stable against local perturbations.
A very important advantage of this topological viewpoint is that it is easily generalizable to a wide variety of models and can be used to classify topological solitons [1], [17], [19]. In many cases the homotopy group involved in physical models is simply the group of integers Z (or the subgroup Z2). In these cases the topological character is captured by a single integer Q, called the
topological charge, that is a conserved quantity.
Elementary examples of topological solitons arise in one dimensional scalar field theories, like the sine-Gordon model. In these cases topological solitons are usually called kinks. Kinks interpo-late between distinct, degenerate vacua and they appear as particle-like objects, with finite mass and a smooth structure. Also the double-well φ4model admit kink solutions in one dimension, but
in this case few caveats are needed. In order to have solitons another property is frequently re-quired, namely that when two or more solitons (or anti-solitons) scatter, they simply pass through, except for a phase shift. In this strict mathematical sense, sine-Gordon kinks are solitons but φ4
kinks are not, because in this case kink scattering has a more complex character [20]. Nevertheless, we are not interested in scattering processes and hence we will use the word “topological solitons” in a broader sense.
Topological solitons are almost ubiquitous in condensed matter systems [22]. A non-exhaustive list of topics where play a role includes for example ferromagnetism [50], dislocation theory [1], charge density waves [25], [34]. Furthermore, topological structures, in this case usually called topological defects, play a key role in continuous phase transitions, via the Kibble–Zurek mecha-nism [26]. Other important applications can be found in elementary particle theory [17], cosmology [27], biophysics [28] [29], physical chemistry.
In this last context, a wonderful example of topological soliton arise in trans-polyacetylene, that is the thermodynamically stable isomer of polyacetylene, an organic polymer simply consisting of a linear chain of CH bonds. In the ground state, the carbon atom forms three σ bonds: one of them is to the H in the CH unit, one to the unit on the left and one to the right. In addition, there is one more electron orbital that can cause bonding. This is called the π electron, and the πbond can be formed to the left or to the right. Two sequences are possible: the first when the double (σ and π) bond is to the carbon on the right and the single to the left, the second when the double bond is to the left and the single to the right. Qualitatively, there is a sort of Z2 symmetry,
which is broken in the ground state. Kinks form if different ground states are chosen at different locations (see figure 5). A simple model, originally proposed by Su, Schrieffer and Heeger [31], can implement this intuition. The SSH Hamiltonian treats π electrons in tight-binding scheme, also approximating the hopping amplitude as a linear function of the distances between successive carbon atoms. Furthermore, the σ bonds are modelled like springs connecting the CH groups, giving rise to a quadratic term in the displacement variables un. Due to the so-called Peierls
instability [30], the ground state is dimerized (huni = (−1)nu0) and degenerate (u0and −u0 both
minimize the energy) . Using this model it has been rigorously proved the existence of kinks in polyacetylene. It turns out [32] that the kink width is approximately 14 lattice spacings and the mass is approximately six electron masses in good agreement with experiments.
Figure 5: Chemical structure of trans-polyacetylene showing the two possible topological phases joined by a (central) defect at the dashed red oval. Picture taken from Ref. [33].
Independently of the field, numerical simulation is a fundamental tool to study topological solitons. In performing simulations one is always forced to discretize space and this inevitably in-troduces discretization effects. In high energy physics this effects are fictitious and usually one tries to minimize them [23]. In this context, the standard discretization of a field equation is performed simply by replacing spatial derivatives with difference operators. This corresponds to replace the continuum field theory with a discrete system of oscillators, each oscillating in a certain substrate potential, with nearest neighbors coupled by springs. On the other hand, condensed matter theory concerns with systems with an intrinsic discrete nature. Therefore the effects of discreteness are meaningful and physically relevant, at least when the topological soliton characteristic width is close to the minimal length scale, i.e. lattice spacing. For this reason we are interested in discrete systems.
CONTENTS 17
Localized solitons in one dimensional discrete systems: summary of this
thesis
The study of the kinks dynamics in highly discrete one dimensional systems was originally per-formed by M. Peyrard and M.D. Kruskal [35], M. El-Batatouny, R. Boesch, C. Willis, P. Stancioff [38], [37] More recent are the reviews of J. M. Speight [23], O. M. Braun, Y. S. Kivshar [21], P. G. Kevrekidis [54]. Nowadays, the extraordinary advances in the study of cold atoms, ion traps and atomic physics has stimulated new interest. In fact, these technologies provide the concrete possibility to realize closed interacting quantum many-body systems, described with high degree of accuracy by simple known models. In particular, cold trapped ions offer a high degree of ex-perimental control, allowing the direct study of the non-linear physics of discrete solitons, even in the quantum regime [39]. Recently it has been shown that solitonic internal modes, physically describing “shape change” excitations of the kink, can preserve quantum coherence for long time in a ions trap [40]. It has been also suggested that this property can be used to implement quantum information processing in large systems [40]. Such applications and further studies of solitons in ion traps could benefit substantially from a deeper understanding of soliton dynamics in lattices. It is therefore important to understand in detail various aspects, such as for example, the effects of the discreteness.
In the continuum limit kinks can be rigidly translate without affecting their energy. In the discrete limit instead, kinks centered exactly on a lattice site or at the centre of a lattice cell have different energies. This fact is related to the presence of an effective potential, called Peierls-Nabarro (PN) potential or pinning potential, that is the potential effectively experienced by the kink. The PN potential is a crucial feature of such discrete systems and it has been discovered for the first time in studying the motion of dislocations in solids [18]. The PN potential is periodic, with periodicity equal to the lattice spacing and prevents the free propagation of the kink along the chain. It is usually well approximated by a simple cosine (or sine). There are two static configurations, corresponding to minima and maxima of the PN potential: one is stable under small perturbations, whereas the other is unstable. The energy difference between them is just the strength of the PN energy barrier, i.e. the barrier which a kink must surmount in order to propagate from one lattice cell to the next.
Some remarks are in order. In a continuum model the PN potential is trivially 0. This is be-cause the system is translationally invariant. Translations do not change the energy of the kink and then the spectrum of the possible kink’s excitations must contain a zero energy fluctuation mode, also called Goldstone mode. The existence of the PN potential is due to the lack of translational invariance in a discrete chain, that is explicitly broken by the presence of a finite lattice spacing. In this case the zero-frequency (zero-mass) translational Goldstone mode is replaced by a finite-frequency localized mode, known as the PN finite-frequency mode. One might expect the PN barrier, and its effects, to grow monotonically with the lattice spacing. However in general this is not true: there exist infinitely many substrate potentials with the property that at a particular (non-zero) lattice spacing the PN barrier vanishes exactly [36]. These are very particular cases: in general and in all the cases we will be interested in, the PN barrier is non-zero for any finite lattice spacing.
In the first instance, the kink can be viewed like a classical particle moving in the PN potential. Two things can happen, depending on the intial conditions: if the initial velocity is large enough the kink has enough energy to overcome the PN barrier and starts to move along the chain; on the contrary if the initial velocity is not large enough the kink will be trapped in a potential well. In this simple description, we can use only two colletive coordinates associated with the kinks to
describe the entire (and very complex) many-body dynamics, namely the kink position and the kink velocity (or momentum). The picture is made more interesting by including the other degrees of freedom in the description of the dynamics. Indeed, as the kink moves through the lattice, its motion in and out of the PN potential excites small amplitude traveling waves, i.e. phonons. The kink excites the phonon “field” and loses kinetic energy, which is converted into “radiation”, thus satisfying the conservation of the total energy of the system. This causes an effective phonon-induced friction on the kinks, producing a deceleration. Sometimes the kink slows down to the extent that at a certain instant in time it has insufficient energy to surmount the PN barrier, whereupon it becomes trapped into a lattice cell. Ultimately, this trapping is due to both the discreteness and the non-linearity of the underlying equations of motion and it take place at zero temperature.
In this work we will clarify the nature of this phenomenon, by using both theoretical considera-tions and numerical simulaconsidera-tions. The systems investigated are the classical φ4discrete model and
the classical transverse field Ising chain. The latter corresponds to the well-known quantum trans-verse field Ising model (TFIM), which is exactly solvable for S = 1
2 by means of a Jordan-Wigner
transformation and a Bogoliubov transformation [5], in the classical limit ~ → 0, equivalent to the large spin limit S 1. In the Ising chain, the above mentioned phonon excitations correspond to spin-waves or magnons, whereas topological excitations, i.e. kinks, are usually called domain walls.
The rest of this thesis is structured as follows.
• In Chapter 1 we will describe some of the crucial features of kink solutions in systems defined in 1 + 1 dimension (one spacial dimension plus the time). In particular, we will focus on standard scalar field theories, like φ4 theory and sine-Gordon theory, and on classical spin
chains. In this context, we will study the TFIM.
• In Chapter 2 we will analyze the discretized φ4model, reporting the results of our numerical
simulations illustrating the dynamical localization phenomenon described above. These are performed on systems of increasing size, showing that this kind of localization is a distinctive feature of the model in the thermodynamic limit. Furthermore, we will generalize an inter-esting collective variables approach originally proposed by Willis et al. [38] for the discrete sine-Gordon model and developed here in detail for the φ4model.
• In Chapter 3 we will report similar numerical results for classical Ising chain, defined on a discrete lattice.
The original part of this thesis consists of:
– the linear stability analysis of domain walls solutions in the classical Ising model (TFIM);
– the implementation of the collective variables approach to the φ4 model;
– the numerical study of the kink dynamics in the abovementioned models and the observations concerning the phenomenon of dynamal localization in the thermodynamic limit.
Chapter 1
Kinks in 1 + 1 dimensions
In this first Chapter we introduce some basic notions about kink solutions of classical field theories in (1 + 1) dimensions (one spatial dimension plus the time). We will focus on the Z2 kink in the
scalar φ4 theory and on domain-walls in one-dimensional ferromagnetic systems. In particular we
will consider the classical transverse field Ising model (TFIM).
1.1
Kinks in classical scalar field theories
We start with a prototype of kink, namely the so-called Z2kink in a theory with a quartic
double-well potential. Let us consider the 1 + 1 dimensional scalar field theory:
S = Z dt Z dx L= Z dt Z dx 1 2∂µφ ∂ µφ − U(φ) (1.1) U(φ) = λ 4(φ 2− η2)2, (1.2)
where ∂µ = (c−1∂t, ∂x) and λ > 0 (so that the energy is bounded from below). From now on,
we will set c = 1. The theory is invariant under translations in Minkowski space-time. Applying Noether’s theorem, it is straightforward to obtain the energy-momentum tensor [42]:
Tνµ = ∂L ∂(∂µφ)
∂νφ − δνµL (1.3)
Tνµ is such that ∂µTνµ= 0, i.e. there is a continuity equation for each component ν (for each
con-tinuus simmetry of the theory). The conserved charges associated with time translation symmetry and spatial translations are respectevely the total energy and the total momentum:
E= Z +∞ −∞ dx T00= Z +∞ −∞ dx 1 2φ˙ 2+1 2(φ 0)2+ U (φ) (1.4) P = − Z +∞ −∞ dx T10= − Z +∞ −∞ dx ˙φφ0 (1.5)
where ˙φ= ∂0φand φ0 = ∂1φ. Furthermore, the Lagrangian density L is clearly invariant under
the transformation φ → −φ, therefore it has a discrete Z2simmetry. The potential U (φ) has two
zeros in φ = ±η, that constitue the “vacuum manifold” M of the theory. By using the considera-tions made in the Introduction, we observe that our theory can support topological configuraconsidera-tions because the homotopy group π0(M) ' Z2is non-trivial. In fact we will obtain the exact expression
for the Z2-kink. On the contrary, the vacuum configurations φ(x) = ±η are topologically trivial.
By setting φ = η + ψ and then inserting in S[φ] = S[η + ψ], one can easily derive the action for the excitation field ψ around these configurations. The result is:
S = Z dt Z dx 1 2∂µψ ∂ µψ −1 2m˜ 2ψ2− r λ 2mψ˜ 3−λ 4ψ 4 ,
where ˜m= η√2λ is the effective mass of the excitation field ψ.
The equation of motion is obtained by applying the principle of least action on equation 1.1:
∂µ∂µφ= −
∂U
∂φ (1.6)
Let us start by looking for static solutions, so:
φ00= ∂U
∂φ . (1.7)
This equation strictly resembles the standard motion equation for a classical particle in a potential W: ¨ q= −1 m ∂W(q) ∂q
except for the substitutions t → x, q(t) → φ(x), W/m → −U . Invoking energy conservation we can obtain an equation of the first-order:
1 2m˙q
2+ W (q) = const.
and in our case:
1 2(φ
0)2− U (φ) = const. (1.8)
The constant is fixed by the boundary conditions at ±∞. We are interested in configurations topologically distinct from the trivial degenerate vacua φ(x) = ±η, so we set:
φ(±∞) = ±η or φ(±∞) = ∓η .
These choiches correspond to two types of solitons, called respectevely kinks and antikinks. In both cases we have:
1 2φ
02(±∞) − U (φ(±∞)) = 0
and therefore the constant must be fixed to be 0. By focusing on the kink, we find:
φ0= − r λ 2 (φ 2 − η2) (1.9) dφ η2− φ2 = r λ 2 dx ,
where the sign is chosen to be consistent with the choice of the boundary conditions. Then, by integration: 1 ηarctanh φ η = r λ 2(x − x0) .
1.1. KINKS IN CLASSICAL SCALAR FIELD THEORIES 21 Finally, we obtain: φK(x) = η tanh x − X l l= √ 2 η√λ = ˜m 2 −1 . (1.10)
X is the only constant of integration and labels the intersection point of the function of φK(x)
with the x-axis. Therefore, it respresents the position of the kink, whereas l is its characteristic width. The antikink solution can be obtained by solving φ0 = +pλ/2 (φ2− η2). The result is
obviously: φK¯(x) = −η tanh x − X l .
Since the lagrangian density L is a relativistic invariant, these solutions can be Lorentz-boosted to obtain a class of generalized solutions corresponding to a moving kink/antikink. The expression for a kink moving at velocity V = ˙X will be:
φK(x, t) = η tanh γx − V t l γ=√ 1 1 − V2 ≥ 1 (1.11)
The profile of the moving soliton explicitly depends on the velocity V , because of the relativistic factor γ wich leades to a contraction of the characteristic width of the kink.
Now we will enumerate a number of important features of the solutions just found.
• The total energy for the static case is:
E= Z +∞ −∞ dx E(x) E(x) = 1 2(φ 0 K) 2 + U (φK) . (1.12)
The energy density E(x) of the kink is plotted in figure 1.1, togheter with the profiles of the kink and of the antikink. One can clearly see that the energy is fully stored around the position X. Inserting equation 1.8 in the equation 1.12 we find:
E= Z +∞ −∞ dx(φ0K) 2 = Z +η −η dφ φ0K = r λ 2 Z +η −η dφ(η2− φ2K) = 2√2 3 η 3√λ , (1.13)
where we used equation 1.9. This value can be interpreted as the rest mass M of the kink. It is easy to check that the moving solution has energy E = M γ and total momentum P = M γv. The energy-momentum relationship
E=pM2+ P2
is also satisfied. Thus, kink (and antikink) solutions satisfy the usual relativistic kinematic re-lations. This fact suggests that these classical localized solutions may be effectively treated like particles.
• A way to characterize the topological nature of the Z2kink is to observe that:
jµ= 1 2η
µν∂
νφ µ, ν= 0, 1 (1.14)
corresponding conserved charge, usually called topological charge, is: Q= Z +∞ −∞ dx j0= 1 2η Z +∞ −∞ dx φ0(x) = 1 2η φ(+∞) − φ(−∞) . (1.15)
Obviously the following equalities hold:
kink −→ Q= +1 trivial vacua φ(x) = ±η −→ Q= 0 antikink −→ Q= −1 (1.16)
Qhas a different nature from energy or momentum: in fact, it is not a Noether’s charge and it is not the infinitesimal generator of any symmetry of the theory.
• We can study the linear stability of the kink splitting the field as: φ(x, t) = φK(x) + ψ(x, t), where
we denoted the kink by φK is the kink solution. Instead ψ(x, t) is the fluctuations field. Inserting
in the motion equation and linearizing, we obtain:
∂t2ψ − ∂x2ψ ' −ψ∂ 2U ∂2φ(φK) = −ψλ(3φ 2 K− η 2) .
To find the fluctuation eigenmodes we set ψ(x, t) = e−iωtψ(x), obtaining:
− ∂2 x+ 3λφ2K(x) − η2λ | {z } O ψ= ω2ψ . (1.17)
This equation has the form of a standard time-independent Schr¨odinger equation in one-dimension (~2/(2m) = 1). The corresponding potential is:
V(x) = 3λφ2 K(x) − λη2= 1 l2 4 − 6 cosh−2 x − X l , (1.18)
and it has the shape of a potential well, with width proportional to l. V is known as P¨oschl-Teller potential and the corresponding Schr¨odinger equation is exactly solvable. A priori, we can note that the operator O must have an eigenstate with zero-energy (ω = 0), since if φK(x; X) is a static
solution then also:
φK(x; X + dX) ' φK(x; X) +
∂φK
∂X (x; X) dX
is a static solution. Thus ∂XφK = −φ0K has to be a zero mode. This is the so-called Goldstone
mode. In our case, it is easy to check that
ψ0(x) = ∂XφK = − η l cosh −2 x − X l (1.19)
is a solution of equation 1.17 with ω = ω0 = 0. We note that the Goldstone mode must be the
eigenstate with the lowest value of ω2, since ∂
XφK is nodeless1. This observation proves that
the Z2 kink is stable under perturbations, i.e. the spectrum of the excitations field is positive
semidefinite. In addition, this kink has a second bound state, with oscillation frequency and
1This argument is highly generalizable, because usually kinks are represented by monotonic functions, hence
1.1. KINKS IN CLASSICAL SCALAR FIELD THEORIES 23
eigenfunction respectively equal to:
ω21= 3 4m˜ 2 ψ 1(x) = sinh x l cosh −2 x l (1.20)
for X = 0. This mode, usually called “shape mode”, corresponds to a modification of the profile of the kink localized around its position X. In addition to these bound states, the spectrum of the operator O has a continuum of eigenstates consisting of an incident wave, a transmitted wave and a reflected wave. These are scattering states and they are labelled by a single wave vector k. The corresponding oscillation frequencies and eigenfunctions are:
ω2k= k2+ ˜m2 ψk(x) = eikx 3 tanh2 x l − 1 − l 2k2 − 3ilk tanh x l (1.21)
From this expression it is easy to check a very special property of the P¨oschl-Teller potential: it is reflectionless, i.e. the reflection coefficient R vanishes for all the wave vectors k. This means that the interaction between an incident plane wave and the static kink only produces a phase shift. This feature can be rigorously proven, for example with the aid of supersimmetric quantum mechanics [43]. The potential V and the functions ψ1 and ψk are plotted in figure 1.2.
• Multi-kink configurations of the field can be formed by joining alternating kink and antikink in a smooth way. However in this way one cannot obtain exact static solutions of equation 1.6. In fact in the φ4 model kink-antikink pair initially at rest move towards each other and annihilate
4
2
0
2
4
x
1.00
0.75
0.50
0.25
0.00
0.25
0.50
0.75
1.00
kink
anti-kink
energy density
Figure 1.1: The field φ(x) of the Z2 kink/antikink (black curves) and their energy density profile
E(x) (blue curve). The parameters selected are: η = 1, λ = 2.
10
5
0
5
10
x
1.0
0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
effective potential V(x)
Re
kIm
kshape mode
1Figure 1.2: The effective potential V (x) experienced by the flucutuations ψ around the kink configuration (black curve), real and imaginary part of a radiative mode ψk for k = 1.5 (dotted
curves) and the localized shape mode (purple curve). The function ψk is normalized to 1 in the
1.1. KINKS IN CLASSICAL SCALAR FIELD THEORIES 25
We conclude this section with a brief overview on kinks in a different scalar field theory, namely the sine-Gordon theory. This model has additional characteristics extremely interesting.
The sine-Gordon model in (1 + 1) dimensions is defined by the action functional 1.1, with the potential: U(φ) = λη2 1 − cos φ η . (1.22)
Clearly, the energy-momentum tensor, the total energy and the total momentum still have the expression reported in equations 1.3, 1.4 and 1.5. The vacuum manifold is now constituted by the set of points:
φ= 2πη n n ∈ Z (1.23)
labeled by the integer n. √λhas the dimensions of a mass and is just the effective mass ˜m of the fluctuation field around a vacuum solution. The equation of motion is the well-known sine-Gordon equation: ∂µ∂µφ+ λη sin φ η = 0 .
Using the same trick as before, one can find exact static kink solutions by using the expression 1.8. The simplest static solution is:
φK(x) = 4η arctan e
x−X
l l=√1
λ, (1.24)
that interpolates between 0 and 2πη. It has topological charge Q equal to +1, provided one inserts an additional factor π−1 in the above definition of Q. The rest energy is:
E= 8η2√λ (1.25)
The dynamical kinks have the form:
φK(x, t) = 4η arctan γe
x−V t
l , (1.26)
and respect the usual relativistic kinematic relations. As usual, the antikink is expressed by the function 1.24 with an extra minus factor.
In contrast to the φ4 model, the sine-Gordon theory admits exact solutions involving
multi-ple, interacting kinks. These are obtainable by means of refined techniques, such as B¨acklund transformations. As an example, in the panel (a) of figure 1.3 we plotted an exact solution of the sine-Gordon equation in which a kink and an antikink, initally far apart, first approach each other, then collide and finally emerge with the same shapes and velocities. At both times t = ±∞ the field configuration is exactly the sum of φK and φK¯, with velocities ±V . The only effect of the
interaction is a time delay (a phase shift). For these reasons, the sine-Gordon kinks are genuine solitons according the stringent mathematical requirement given in the Introduction. The fact that the soliton scattering is “trivial” (i.e. no change of velocity or shape) is strictly correlated with the integrable nature of the sine-Gordon theory. Integrability means that it is possible to find infinitely many independent local conserved quantities in involution, namely with vanishing Poisson brackets between each other [44], [45], [42]. This is a far-reaching concept, which has im-portant consequences on the non-equilibrium behaviour of the system. However, a comprehensive discussion of the features of integrable systems is beyond our scope.
called breather, which consists in a bound state of a soliton and an anti-soliton, oscillating with respect to each other, with frequency ω. In this case the topological charge is Q = 0.
40 20 0 20 40
x
6 4 2 0 2 4 6(x
)
+2
2
t = 50t = 0 t = 30 (a) 20 15 10 5 0 5 10 15 20x
6 4 2 0 2 4 6(x
)
(b)Figure 1.3: (a) Kink-anti kink solution of the sine-Gordon model: φK ¯K(t, x) = 4 arctan cosh(γV t)/(V sinh(γx)) (η = 1, λ = 1). The velocity V is setted to 0.5. The differ-ent curves represdiffer-ent the solution evaluated at the times: t = −50, 0, 30. (b) Breather solution: φB(t, x) = 4 arctan p(1 − ω2)/ω sin(ωt)/ cosh(
√
1 − ω2x) (η = 1, λ = 1). ω is the oscillation
frequency and is set to 10−3.
The linear stability analysis around the static kink solution gives rise to the Schr¨odinger equa-tion: − ∂2 x+ λ cos( φK(x) λ ) | {z } O ψ= ω2ψ , (1.27)
where the effective potential experienced by the fluctuations is:
V(x) = λ cos(φK(x) λ ) = 1 l2 1 − 2 cosh−2 x l . (1.28)
Once again, V is a P¨oschl-Teller potential, hence it is reflectionless. The spectrum of the operator Oconsist of a single localized mode around the kink (i.e. the Goldstone mode), plus a continuous spectrum of scattering eigenstates, whose explicit form is given by:
ψk= eikx
k+ i tanh x l
i+ k (1.29)
for X = 0. There are no shape mode in this case.
1.2
Kinks in the classical Ising model
1.2.1
Spin systems
The majority of the literature about the dynamics of the classical Ising model is focused on the stochastic Glauber dynamics, i.e. the generation of a Markov chain of states corresponding to a given temperature. However, it is still interesting the dynamics properly associated with this model, namely the quantum dynamics in the classical limit ~ → 0. To better understand this limit, one can first write down the path-integral for a single quantum spin S subjected to a certain
1.2. KINKS IN THE CLASSICAL ISING MODEL 27
Hamiltonian H. The path-integral in imaginary time τ = it is nothing but the partition function:
Z = tr(e−βH) ,
commonly used in statistical mechanics. The idea is to split the exponential to get the well-known Trotter formula and then to insert the identity operator 1 expressed in terms of spin coherent states. These are known as the closest approximation of classical angular momentum states that it is possible to find in the Hilbert state of the spin S. Spin coherent states can be defined by applying an SU (2) trasformation to the maximum weight state |S, +Si of the irriducible represen-tation labelled by S (S = 1
2,1, 3
2...). Furthermore, they can be labelled as |ni, where n is a unit
vector living on a sphere S2of radius 1. A short summary on the main properties of these states is
given in Appendix A. Inserting the decomposition of the identity in spin coherent states, Z can be expressed as a path-integral over the functions n(τ ), in which the parameter 0 < τ < β identifies a curve on S2, i.e. a “cap”. Periodic boundary conditions are taken: n(0) = n(β), so the curve
is closed. In this way, one obtain an explicit formula for the euclidean action SE[n(τ )]. This
cal-culation is also fully shown in Appendix A, where we essentialy follow the discussion in [19] and [46].
Wick-rotating back to real time (t = −iτ , β = iT ), one can write the real-time action S, that takes the form:
S[n(t)] = Z T 0 dt − S ∂tn · A(n) − hn|H|ni . (1.30)
Here A(n) is a vector field defined on S2, with the property: ∇ × A = n. This means that the
“magnetic field” corresponding to this “vector potential” is radial and of constant strength. The first term in equation 1.30 is just the minimal coupling for a particle of charge S moving under the influence of A and it is known as Wess-Zumino action or Berry phase [19]. By using Stokes’ theorem, one can re-write:
Z T 0 dt ∂tn · A(n) = I ∂Ω dn · A(n) = Z Ω+ dˆσ · ∇ × A = Z Ω+ dˆσ · n ,
in which Ω+ is one of the two spherical caps bounded by the closed curve n(t), that we have
indicated as ∂Ω. From the last expression we see that this first part of the action is (proportional to) the geometrical area enclosed by the curve n(t). The definition of the value of this area has an intrinsic ambiguity of 4π. This because in the above equalities we could have used the other cap defined by the curve n(t), Ω−, and the two areas sum up to 4π:
AΩ++ AΩ− = 4π .
However, this ambiguity has no physical consequences. In fact, in the path-integral expression S is multiplied for the imaginary unity i and exponentieted, and: e4πiS is equal to 1, for any
inte-ger or half inteinte-ger value of S. In effect, this can be seen as a condition for the spin quantization [19].
In the second part of the action we have hn|H|ni, that is the expectation value of H on the spin coherent state labelled by n. In spherical coordinates: n = (sin θ cos ϕ, sin θ sin ϕ, cos θ) and:
∂tn= ˙θ eθ+ ˙ϕsin θ eϕ.
In addition,
A= 1 − cos θ
sin θ eϕ (1.31)
formulae of vector calculus in spherical coordinates. Thus, the particle moving on the sphere ex-periences the radial magnetic field generated by a “magnetic charge” placed in the center of the sphere. This is not in contrast with the Maxwell’s law: ∇ · B = ∇ · ∇ × A = 0. In fact, the vector potential 1.31 is clearly a non regular function of the position, because of the singularity along the line θ = π: thus the above equalities fail and there is no contradiction. These observations are strictly correlated with the notion of “Dirac monopole” [19], [17].
Defining H(θ, ϕ) = hn|H|ni, one obtains [47]:
L = −S(1 − cos θ) ˙ϕ − H(θ, ϕ) . (1.32)
For example, if H is the coupling with an external magnetic field B, aligned along the z axes, H= BSz, we have:
L = −S(1 − cos θ) ˙ϕ − SBcos θ (1.33)
The dynamics is that of a massless point on a two-dimensional sphere coupled with the fictitious magnetic field of the monopole and with the real magnetic field B.
Expression 1.33 obviously scales with S. Thus in the large spin limit S → ∞ the path-integral:
Z = Z
Dn e~iS[n(t)]
is dominated by the stationary points of the real time action S[n(t)] and Z can be evaluated by using a saddle point approximation. We see that the classical limit ~ → 0 is completely equivalent to the large spin limit S → ∞. In short: large spin values completely suppress the quantum fluctuations of the angular momentum. This result is generalizable to all the cases where H scales with S. Then, the classical mechanics of the spin is fully determined by the Euler-Lagrange equations derivable from the Lagrangian 1.32, that are:
Ssin θdϕdt = −∂H ∂θ Ssin θdθ dt = ∂H ∂ϕ (1.34)
These classical equations are correct up to the order 1/S2: corrections of higher order can be
ar-ranged in an expansion in powers of 1/S. The result is the so-called Holstein-Primakoff expansion [46].
Some remarks are in order. As a consequence of the gauge invariance, it is always possible to transform A adding the gradient of some scalar function χ:
A → A0= A + ∇χ ,
with no consequences for the “magnetic field” ∇ × A. This trasformation has also no effects on the equations of motion (and therefore on the classical physics), since the integrand ∂tn · ∇χ(n)
in the action is equal to the total derivative: dχ(n)/dt.
Ignoring the term ˙ϕ, which is a total derivative respect to t, we have: L/S = cos θ ˙ϕ−H(θ, ϕ)/S. The comparison with L = p ˙q − H allows us to identify cos θ as the conjugate momentum of the
1.2. KINKS IN THE CLASSICAL ISING MODEL 29
azimuthal angle ϕ. In fact it is easy to check that the Hamilton equations:
˙p = −1 S∂qH ˙q = +1 S∂pH (1.35)
are equivalent to the Euler-Lagrange equations 1.34. The canonical variables are q ∈ R and p ∈[−1, +1] and the phase space is a two dimensional sphere S2 (the Bloch sphere), coinciding
with the configuration space. However, we see that these variables are singular, because in any neighbourhood of the “north-pole” there is no one-to-one correspondance between Sx, Sy, Sz and
q, p: q is not well-defined for p = ±1 (θ = 0, π). When the curve n(t) contains the north or the south pole of the sphere the integral of 1.32 is not defined. Indeed, since we deal with the geometry of the sphere S2, it is not possible to assign a globally defined coordinate q together with a globally
defined conjugated momentum p. This contrasts with what happens in the standard problems of classical mechanics and it is a reason why the classical mechanics of spins is not discussed in introductory courses.
A way to proceed is to use the abstract (i.e. independent from the coordinates) formulation of hamiltonian mechanics in symplectic manifolds [48]. For our purposes it is sufficient to note that, by defining the Poisson brackets:
{f, g} = ∂f ∂q ∂g ∂p − ∂g ∂q ∂f ∂p ,
we can re-write the canonical relations:
{q, p} = 1 {q, q} = {p, p} = 0
in the equivalent form (~ = 1): {Sa, Sb} =
abcSc a, b, c ∈ {1, 2, 3} = {x, y, z} . (1.36)
The equations of motion are ˙p = {p, H}, ˙q = {q, H}, that we can re-write as:
˙
Sa = {Sa, H} .
In this case the Poisson bracket of two function A, B of the spin variables is:
{A, B} = abc ∂A ∂Sa ∂B ∂SbS c . (1.37) Thus we have: ˙S = S × ∂H ∂S , (1.38)
that is known as Landau-Lifshitz equation. From this formula we see that ∂H/∂S is the effective magnetic field experienced by the spin S.
Obviously only two of the cartesian coordinates Sa are effectively independent. Furthermore,
it is easy to check that consistently the dynamics preserve the total spin |S|2=P
a(Sa)2. In fact: d dt S aSa = 2 ˙SaSa= 2 abc ∂H ∂SbS cSa= 0 .
straight-forward. In fact, spin coherent states will be the factorized product of the coherent states on the Hilbert spaces of single spins. The total action will have a Berry phase term for each spin, in addition to the the hamiltonian terms. A number of classical field theories of this kind describing several spin systems has been studied. The best known of these theory is certainly the Heisenberg spin chain, which is exactly integrable in the continuum limit [53]. Below we will study another model: the classical transverse field Ising chain.
1.2.2
The transverse field Ising model (TFIM)
The model we consider is the classical version of the TFIM, i.e. the classical limit of the quantum hamiltonian: H= −J L−1 X i=1 SizSi+1z − B L X i=1 Six J, B >0 , (1.39)
defined on a one-dimensional lattice of L sites. J represents the ferromagnetic coupling between neighbouring lattice sites. As a consequence of this first term, it is energetically favourable for the spins to align parallel in the direction z. B is the transverse field and tends to align the spins along the direction x, orthogonal to the direction of the ferromagnetism. The interplay of these effects gives rise to a non trivial phenomenology. In the case S = 1/2 the quantum model can be exactly solved, by mapping the system in a free fermions model via a Jordan-Wigner and Bogoliubov transformations [5]. Now we are interested in the classical model, i.e. in the large spin limit S 1. By using the Poisson algebra of the spins, {Sa
i, Sjb} = δijabcSic, we find the equations
of motion: ˙ Si z = −BSiy ˙ Si y = −JSx i Si+1z + Si−1z + BSiz ˙ Si x = +JSiy Si+1z + Si−1z (1.40)
Alternatively we can express H in term of the canonical variables pi = cos θi = Siz/S and qi =
φi = arctan(S y i/S x i), getting: H S = −JS L−1 X i=1 pipi+1− B L X i=1 q 1 − p2 i cos qi (1.41)
The Hamilton’s equation are therefore:
˙qi = −JS pi+1+ pi−1 + B√pi 1−p2 i cos qi ˙pi= −Bp1 − p2i sin qi . (1.42)
As noted above, this last formulation of the dynamics is explicitly singular in the points pi = ±1
and it can not be used to describe a motion in which a spin is aligned along the z−axis at a certain moment in time. However, for our purpose, namely looking for kink solutions, we can use this formulation. By using 1/JS as unit of time, we find:
˙qi= − pi+1+ pi−1 + 2h√pi 1−p2 i cos qi ˙pi= −2hp1 − p2i sin qi (1.43)
1.2. KINKS IN THE CLASSICAL ISING MODEL 31
where h = B/(2JS). Let us start by searching for constant static solutions. We set: pi(t) = ¯pand
qi(t) = ¯q, finding: −2¯p+ 2h√p¯ 1− ¯p2 cos ¯q= 0 −2hp1 − ¯p2 sin ¯q= 0
The second line gives ¯q = 0 or ¯q = π, but only for ¯q = 0 the first line admit a solution, which is: ¯p= ±√1 − h2 if h ≤ 1. It is easy to check that for h ≤ 1, the Hamiltonian 1.41 is effectively
minimized by: (¯q= 0, ¯p= ±√1 − h2). For h > 1 these minima become: (¯q = 0, ¯p= 0), i.e. the
spins are aligned along the transverse field B = Bx. We see that depending on the strength of the transverse field, two “ground states” are possible for the system: if B ≤ 2JS the ground state is twofold degenerate and ferromagnetic, with magnetization ±√1 − h2along the z−axis; if B > 2JS
the ground state is non degenerate and paramagnetic, with zero magnetization along the z−axis. Bc= 2JS and hc = 1 are the critical values of the parameters. These observations are made in the
spirit of a mean-field approximation and are at the root of the zero-temperature quantum phase transition phenomenology in the quantum version of this model [5].
1.2.3
Continuum limit of the TFIM
We will now consider the continuum limit of our classical model. Setting the lattice spacing a to 1, we focus on spin configurations in which the lattice variables change slowly with i = 1... L, namely we consider only long-wavelengths modes. In such a case we can define the functions q(x), p(x), that coincide with the variables qi, pi on the lattice sites. We have:
pi+1+ pi−1= p(x + a) + p(x − a) ≈ 2p(x) + ∂2p ∂ x2 a 2 and then: ˙q(x) = −∂x∂2p2 − 2p(x) + 2h p(x) √ 1−p2(x)cos q(x) ˙p(x) = −2hp1 − p2(x) sin q(x) . (1.44)
These equation can be obtained from the continuum hamiltonian:
H = Z +∞ −∞ dx 1 2 ∂p(x) ∂x 2 − p2(x) − 2hp1 − p2(x) cos q(x) (1.45)
by using the Hamilton equations in their functional form: ˙q = δH/δp and ˙p = −δH/δq. In the above expression, we also sent L to infinity, considering the thermodynamic limit of the chain. One can alternatively write down the Hamiltonian in term of the spin field s(x):
H = Z +∞ −∞ dx 1 2 ∂sz ∂x 2 − (sz)2− 2hsx
The Landau-Lifshitz equations are: ˙s = s × (δH/δs(x)). These equations are manifestely in a non-relativistic form, hence the theory is not Lorentz-invariant.
The Hamiltonian 1.45 has obviously a continuum simmetry, namely translation invariance, and, according to Noether’s theorem, a corresponding quantity whose value is conserved by the time evolution. This quantity is the total momentum P . To get the correct expression of P we must
find a functional P of p(x), q(x) such that: −dq dx = {q(x), P } −dp dx = {p(x), P }
where the Poisson bracket is now defined as:
{F, G} = δF ∂q(x) δG ∂p(x) − δG ∂q(x) δF ∂p(x) .
It is immediate to verify that the functional:
P = − Z +∞ −∞ dxdq(x) dx p(x) = − Z +∞ −∞ dxdφ dxcos θ (1.46)
verify this relations. P has a simple geometric interpretation. If we fix the boundary conditions q(±∞), p(±∞), we have: P+q(+∞)−q(−∞) = − Z +∞ −∞ dx dq dx(1−p) = Z +∞ −∞ dxdφ dx(1−cos θ) = Z +∞ −∞ dφ(1−cos θ) (1.47)
Furthermore, we note that the area of a spherical triangle delimited by the “north-pole” corre-sponding to z, two meridians and one parallel is just: Rθ
0
Rφ 0 sin θ
0dθ0dφ0 = (1 − cos θ)φ. Thus, up
to a constant, the total momentum P associated with a certain spin configuration is the total area of the region of the sphere delimited by the north-pole and the curve s(x).
By using other arguments, one can obtain P =R ds·A(s), which is equivalent to our expression if the gauge 1.31 is chosen [49]. From this expression we clearly see that P is not gauge independent. However, if we fix the boundary conditions at ±∞ for the vector s, we see that the difference of the momenta of two different states is equal to the circulation of A, corresponding (via Stokes’ theorem) to the spherical area enclosed between the two curves s1(x) and s2(x). Thus, the difference
of momenta is in effect gauge invariant [51].
1.2.4
Kink solutions in the continuum limit
We are looking for moving kink solutions of the equation of motion C.5 for h < 1. In fact, in this case the two distinct vacua (q = 0, p = ±√1 − h2), gives rise to the non-trivial topology we need.
By using the traveling-wave ansatz: p(x, t) = p(ξ), q(x, t) = q(ξ) with ξ = x − V t, we find:
p00= V q0− 2p + 2h√p 1−p2 cos q p0= 2h V p1 − p 2sin q ,
in which0 represents the ξ−derivative. The topological solitons must interpolate between the two
vacua, therefore we must impose:
p(ξ = ±∞) = ±σp1 − h2 q(ξ = ±∞) = 0 .
σ = ±1 is the topological charge and distinguishes kink from antikink. By using the above equations, these boundary conditions and algebraic techniques similar to those used for the φ4
1.2. KINKS IN THE CLASSICAL ISING MODEL 33
scalar field theory, solitons can be found in the implicit form [50], [51], [52]:
σ√2(ξ − ξ0) = ψ − π 2 + tan α ln sin [1 2(ψ − α)] cos [1 2(ψ + α)] , (1.48) where: cos ψ = qcos θ 1 −V2 2 = q p 1 −V2 2 sin α = r c2− V2 2 − V2 (1.49)
and c =√2h. We omit a full derivation, because it is not particularly instructive. Details can be found in [52]. From the expression of sin α, we see that c represents the maximum value possible for the velocity V of the topological solitons. ξ0= X − V t is the position of the kink (i.e. the point
in which p = 0), whereas: tan α √ 2 = 1 γ c √ 2 − c2 = 1 γ h p2(1 − h2) = l γ (1.50)
is its characteristic width. The function p(x) represents the local magnetization along the z−axis, and it is a monotonic function of ξ. The azimuthal angle ϕ is obtainable from the relation:
cos q =
hβ2+q(1 − β2) 1 − p2− β2h2
p1 − p2 (1.51)
In figure 1.4 this kind of domain wall is plotted, by setting the parameters h = hc/2 and β = 0.9.
The energy and the momentum of the kink can be also computed. They evaluate to:
H − E0= √ 2(1 − h2β2) arcsin s 1 − h2 1 − h2β2 − cp(1 − h2)(1 − β2) P = 2 arcsin β s 1 − h2 1 − h2β2 − 2βh arcsin s 1 − h2 1 − h2β2
Here, E0 is the energy Egs = −
R+∞
−∞ dx(1 + h
2) of the ground-state, which we subtracted to
obtain a finite value. We note that P (β = 0) = 0, as result of the fact that in the static kink the rotation of the spins s(ξ) is restricted to the xz-plane (namely on the meridian φ = 0) as we see from equation 1.51. For β 1, the relations H − E0 ' const. + M V2/2, P ' M V and
H − E0 ' const. + P2/(2M ) are valid and one can define the effective mass of the domain wall,
which is a function of h: M = √ 2 h p 1 − h2− h arcsinp1 − h2 (1.52)
1.2.5
Linear stability
Expanding H up to quadratic order around a static solution q0(x), p0(x), one obtains:
H(2)= Z +∞ −∞ dx 1 2δp(x) − ∂ 2 ∂x2 − 2 + 2h cos q0(x) 1 − p2 0(x) 3/2 δp(x)+ −δp(x)2h sin q0(x) p0(x) p1 − p2 0(x) δq(x)+ +1 2δq(x)2h cos q0(x) q 1 − p2 0(x)δq(x)
S
x/S
1.0 0.5 0.0 0.5 1.0 5.0 2.5 0.0 2.5 5.0S
z/S
1.0 0.5 0.0 0.5 1.0S
x/S
1.0 0.5 0.0 0.5 1.0S
y/S
1.0 0.5 0.0 0.5 1.0S
z/S
1.0 0.5 0.0 0.5 1.0 3 2 1 0 1 2 3 0 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 sz sx syFigure 1.4: Top panels: plots of a ferromagnetic kink (i.e. a domain wall) in the Ising chain with velocity V = 0.9c and transverse field h = 0.5. On the top right we plot the curve s(ξ) parametrized by the position ξ ∈ (−∞, +∞). The total momentum associated with the configuration is just the area of the region between this curve and the meridian φ = 0 (dotted line). Bottom panel: the three components sa(ξ), a = x, y, z.
By means of the equations of motion 1.44 we see that a static solution never touching the poles p= ±1 satisfies sin q0(x) = 0. Then:
H(2)= Z +∞ −∞ dx 1 2δq2h cos q0 q 1 − p2 0 | {z } A δq+1 2δp − ∂ 2 ∂x2 − 2 + 2h cos q0 1 − p2 0 3/2 | {z } B δp (1.53)
where we defined the linear operators A and B, both Hermitian. Obviously, cos q0(x) is +1 or −1.
By using the vector δz(x) = (δq(x), δp(x)), we find the linearized Hamilton equations:
δ˙z(x) = ΣδH (2) δz(x) = 0 B −A 0 ! δz(x) = N δz(x) (1.54)
Here Σ is the standard Hamilton matrix 0 1 −1 0
!
. Taking the dime derivative d
dt we get:
δ¨z(x) = −O δz(x) O= −N2= BA 0
0 AB
!
1.2. KINKS IN THE CLASSICAL ISING MODEL 35
that in Fourier transform becomes:
O δz(x) = ω2δz(x) . (1.56)
Let us suppose now that cos q0(x) = +1, ∀x. O is not an hermitian operator in general, however
its spectrum is real. In fact, by noting that the matrix A is real, symmetric and strictly positive, we get M OM−1= A 1 2BA 1 2 0 0 A12BA 1 2 ! ,
where the positivity of A makes it possible to define its square root and the matrix
M = A 1 2 0 0 A−1 2 ! ,
which is symmetric and invertible. Furthermore, A1/2BA1/2 is obviously real and symmetric.
Thus, O is similar to a Hermitian operator and therefore its spectrum is real. In addition, the form of the right hand side in the above equation, with two identical blocks on the diagonal, ensures that: • if w is an eigenvector of A1/2BA1/2, then: w1= M−1 w 0 ! = A −1 2w 0 ! , w2= M−1 0 w ! = 0 A12w ! (1.57)
are two eigenvectors of O with identical eigenvalues;
• each eigenvalue of O is therefore (at least) double-degenerate.
The simplest case is that of the constant ground states q0(x) = ¯q= 0, p0(x) = ¯p=
√ 1 − h2. We get: A= 2h2 B= − ∂ 2 ∂x2 − 2 + 2 h2 and AB = BA = −2h2∂2
x+ 4(1 − h2). From 1.55, we obtain the equation:
2h2 ∂2 ∂x2 − 4(1 − h 2) δq= δ ¨q . (1.58)
By identifying c =√2h as the effective light speed, we see that this is the motion equation for the field δq which has effective mass ˜mc2= 2√1 − h2. The eigenfunctions are obviously δz = δz
keikx.
These mode are usually called spin-waves or magnons. In Fourier space we also get the relativistic expression:
ω2k= ˜m2c4+ k2c2 (1.59)
for the eigenvalues. The spectrum ω2 is strictly positive, i.e. the static solution is stable (as it
should be, since this is the ground state of the system).
For the static kink solutions the eigenproblem 1.56 cannot be fully solved in an analytical way, however few simple remarks are possible. The above considerations imply that the operators BA and AB have the same spectrum. Furthermore, it is straightforward to check that p0
0(x) belongs
to the kernel of B (and thus also to that of AB). The proof only make use of the staticity equation p000 = −2p0+ 2hp0/p1 − p20 (see the system 1.44, with q(x) = 0 and ˙p(x) = ˙q(x) = 0). Thus the
