Exploring phases of the Su-Schrieffer-Heeger model with tSNE

Abstract

T-distributed stochastic neighborhood embedding (tSNE) is used as a tool to reveal the phase diagram of the Su-Schrieffer-Heeger model and some of its extended and non-Hermitian variants. Bloch vectors calculated at different points in the parameter space are mapped to a two-dimensional reduced space. The clusters in the reduced space are used to visualize different phase regions included in the input. The tSNE mapping is shown to be effective even in the challenging case of the non-Hermitian extended model where five different phases are present. An example of using wavefunction input, instead of Bloch vectors, is presented also. I. INTRODUCTIONMachine learning is being increasingly utilized in physics [1,2]. Examples of applications include event classification [3-5] and anomaly detection [6,7] in the analysis of particle physics experiments, aiding observations in astronomy [8], and the study of phases in condensed matter systems [9][10][11][12]. The focus here on the last topic, phases of matter.Many different machine learning methods have been applied to the exploration of phases and phase transitions. These include neural networks [13][14][15][16], principal component analysis [17,18], support vector machines [19] and diffusion maps [20][21][22]. In an interesting recent work, Yang et al. [23] suggested the use of t-distributed stochastic neighborhood embedding (tSNE) as an unsupervised learning method to obtain a visualization of phase diagrams and used this method to study a number of one-dimensional quantum spin systems.The idea of using unsupervised learning methods to reveal phases is particularly appealing. Specifically, topological phases [24] which can not be characterized by local order parameters can be studied without a priori input of domain knowledge. Ref. [21,22,[25][26][27][28][29][30] are examples of recent work on identifying topological phases and phase transitions using either neural networks or diffusion maps. The Ref. [22,[25][26][27][28][29] focus on the Su-Schrieffer-Heeger (SSH) model [31,32] which is also the subject of this work.The SSH model was introduced as a model for polyacetylene and is a simple extensively studied example of a model for a topological insulator [33,34]. The basic model consists of electrons (taken to be spinless) hopping on a one-dimensional lattice. The nearest-neighbor hopping amplitudes are taken to be staggered (see, for example, Fig. 1.1 in [33]) so the lattice can be divided into two-site units cells. Some details of the model are given in Sec. III.The SSH model can be extended by introducing longer range interactions. The extended SSH model considered here allows for the addition of next-next-nearest neighbor hopping terms [28,35]. A different type of modification of the model which has received considerable recent attention is to allow nonreciprocal intra-cell hoping [36,37]. This leads to a non-Hermitian Hamiltonian and the appearance of topological phases with fractional winding number [37].In this paper the tSNE algorithm, which has been used to s...