本研究探討了SSH模型，它是可能發生非普通拓樸相變的最簡單模型，用來描
述在一維晶格中移動，具有交錯跳躍震幅( and ) 的極化費米子。我們探索了有
限以及無限模型的幾個情況，並取得了兩種情況的能譜。本研究強調在有限情
況下，晶格點數量L的宇稱的重要性。因為當晶格點數量為偶數時，有限系統
會出現拓樸特徵，所以本研究將會針對此情況進行探討。
利用IPR，我們對拓樸非普通的零能量邊緣態之有限一維SSH鏈進行了完整的
研究。有別於一般的認知，我們發現它存在一個遠大於穿透深度臨界長度，也
就是。當小於臨界長度時，傳播的訊息是可被拓樸保護的。因為在有限大小系
統的邊緣態之間的耦合能量不為零，所以邊緣態並非完全獨立於鏈的兩端。如
此一來，我們可以利用IPR在波的局域化中找到劇烈的變化，這支持了此特徵
傳播長度的存在。為了實驗上的驗證，我們結合了開口共振腔跟和其互補的共
振腔，與可以控制的跳躍震幅來實作SSH模型。我們利用脈衝激發，然後從邊
緣態的穿透光譜量測其群速度。我們觀察了由20個晶格點組成之晶格的兩邊緣
態之間的傳播速度，並由此結果得出能夠傳播邊緣態的最大系統大小。
另外，我們在晶格點上加上位能，使得有限的一維SSH鏈產生失序。我們發
現，藉由操作週期的倒數b，系統的能帶結構會遭到扭曲。即便如此，系統仍
保留一些局域態，這些態被強烈的限制在晶格的邊緣。由此可以得到，在晶格
點上加上位能，有利於邊緣態的局域化。

In this work we study the SSH model, the simplest model in which topological nontrivial
phase transitions may occur. It describes polarized fermions moving in a
one-dimensional lattice with staggered hopping amplitudes (v and w). We explore the
limiting cases of the model for an infinite and a finite chain, and we obtain the energy
spectrum for both cases, highlighting for the finite case, the importance of the parity of
the number of sites L. As the topological features of the finite system emerge when
the lattice has an even number of sites, we focus on this case for the rest of the work.
Using the IPR, we make a thorough study of the topologically non-trivial zero-energy
edge modes of a finite 1D SSH chain, obtaining that, contrary to the common belief,
there exists a critical length, that we call Lc, much larger than the penetration depth
i.e.,   << Lc, below which the transport of information with topological protection is
possible. Because of the non-zero coupling energy between the edge modes of a finite
size system, the edge states are not completely isolated at the two ends of the chain,
then, an abrupt change in the wave localization is found through the IPR, supporting the
existence of this characteristic transport length. In order to verify it experimentally, the
implementation of the SSH model is carried out by combining split ring resonators and
their complementary ones with controllable hopping amplitudes on a chain. By making
measurements on the group velocity from the transmission spectroscopy of the edge
modes with pulse excitations, we observe the transport velocity between the two edge
states for a lattice composed by 20 sites. With this result, we are able to give a guideline
on the maximum system size allowed to transport these edge states.
Also, we give some disorder to the finite 1D SSH chain by adding an on-site potential.
We find out that, by manipulating the inverse of the period b of the potential, the energy
band structure of the system is distorted. However, the system preserves a couple of
localized states. These states turn out to be strongly localized at the edges of the lattice,
resulting in the fact that the addition of an on-site potential like the one we add in this
work, benefits the localization of the edge modes.

Abstract 10
1 Introduction 15
2 The Su-Schrieffer-Heeger Model 17
2.1 Limiting cases of the SSH model . . . . . . . . . . . . . . . . . . . . 18
2.2 Energy spectrum of a finite SSH chain and its zero-energy edge modes 19
2.3 Localization and the inverse participation ratio . . . . . . . . . . . . . 26
3 Experimental realization of the SSH model 31
3.1 SSH model simulated by SRRs and CSRRs . . . . . . . . . . . . . . 32
3.1.1 Transmission spectroscopy . . . . . . . . . . . . . . . . . . . 34
4 Disorder and localization in lattices 41
4.1 Exploring the AAH model . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 The SSH model as a particular case of the AAH model . . . . . . . . 44
4.3 A finite SSH chain with on-site potential . . . . . . . . . . . . . . . . 45
5 Conclusions and perspectives 57
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
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