# Model-Independent Quantum Phases Classifier

F. Mahlow<EMAIL_ADDRESS>Faculty of Sciences, UNESP - São Paulo State
University, 17033-360 Bauru, São Paulo, Brazil  F. S. Luiz  Faculty of
Sciences, UNESP - São Paulo State University, 17033-360 Bauru, São Paulo,
Brazil  A. L. Malvezzi  Faculty of Sciences, UNESP - São Paulo State
University, 17033-360 Bauru, São Paulo, Brazil  F. F. Fanchini  Faculty of
Sciences, UNESP - São Paulo State University, 17033-360 Bauru, São Paulo,
Brazil

(August 27, 2024)

###### Abstract

Machine learning has revolutionized many fields of science and technology.
Through the $k$-Nearest Neighbors algorithm, we develop a model-independent
classifier, where the algorithm can classify phases of a model to which it has
never had access. For this, we study three distinct spin-$1$ models with some
common phases: the XXZ chains with uniaxial single-ion-type anisotropy, the
bound alternating XXZ chains, and the bilinear biquadratic chain. We show
that, with high probability, algorithms trained with two of these models can
determine common phases with the third. It is the first step toward a
universal classifier, where an algorithm is able to detect any phase with no
knowledge about the Hamiltonian, only knowing partial information about the
quantum state.

Introduction — The interest on quantum phases and phase transitions has been
recently renewed due to new physics uncovered by experiments on cuprate
superconductors, heavy fermion materials, organic conductors, and other
strongly correlated materials [1]. From a theoretical perspective, low-
dimensional quantum lattice models can capture many aspects of these new
phenomena [2, 3]. Therefore, determining the quantum (i.e., groundstate) phase
diagrams of these models is an important challenge in condensed matter and
statistical physics [4].

Concerning quantum phases classification, many alternatives have been
highlighted as promising [5, 6, 7, 8, 9] with special emphasis on machine
learning (ML). The main difference between ML and other statistical models is
the fact that an algorithm can improve its performance, that is, learn,
without the need for such explicit programming [10, 11]. ML is a form of
applied statistics, where computers use data (usually in large quantities) to
estimate functions with a high degree of complexity, which can be used to make
predictions and observe patterns in these data sets [12, 13]. ML has been
widely used in the physical sciences, including cosmology [14, 15, 16],
quantum information [17, 18, 19], many-body physics [8], and also to classify
quantum phases and detect their transitions [20, 21, 22, 23, 24, 25].

In this letter, we analyze the correlation between spins in a closed chain for
three distinct spin-1 models. We show that these correlations hold information
about the phases of these distinct models, and there is considerable overlap
between phase information of different models. This explain how ML algorithms,
training a known model, are capable to detect some phases of another unknown
model. To illustrate we use a machine learning classifier, fed with
correlation and phases labels of two known models, to detect the phases of a
third unknown model. We show that the prediction is succeed when the overlap
of the information about the different phases is minimal. Also, we show it is
possible to apply a transformation in the dataset of the correlations, which
allows minimizing the overlap of the information about the different phases,
making the ML predictor more accurate.

The Models — We use three well-known distinct spin-1 models, where the phase
diagram of these models and their central features are well established in the
literature [26, 27, 28]. The first mode, we present is the (i) XXZ chains with
uniaxial single-ion-type anisotropy, whose Hamiltonian is given by:

${\cal
H}_{1}=\sum_{l=1}^{N}[J({S_{l}^{x}}{S_{l+1}^{x}}+{S_{l}^{y}}{S_{l+1}^{y}})+J_{z}{S_{l}^{z}}{S_{l+1}^{z}}]+D\sum_{l=1}^{N}{S_{l}^{z}}^{2},$
(1)

where $S_{l}$ is a spin-$1$ operator acting on site $l$ of a one-dimensional
lattice (chain) with $N$ sites, $D$ represents uniaxial single ion anisotropy,
and $J(=1),J_{z}$ are spin couplings. For Hamiltonian Eq. 1, the phase diagram
consists of six distinct phases, namely, Haldane, Large D, XY1, XY2,
Ferromagnetic, and a Néel, and several different transitions can occur [27].
The next model is the (ii) bond-alternating XXZ chain, whose Hamiltonian is
given by:

${\cal
H}_{2}=\sum_{l=1}^{N}[1-\delta(-1)^{l}][{S_{l}^{x}}{S_{l+1}^{x}}+{S_{l}^{y}}{S_{l+1}^{y}}+\Delta{S_{l}^{z}}{S_{l+1}^{z}}],$
(2)

where $\Delta$ is the strength of the Ising-type anisotropy that originates
from the spin-orbit interaction in magnetic materials and $\delta$ is the
alternation of the bond that describes dimerization. The phase diagram of the
model Eq. 2 shows the Ferromagnetic phase, XY1, Néel, Haldane, and Dimerized
[26]. Finally, the last model analyzed was the (iii) bilinear biquadratic
chain, whose Hamiltonian is given by:

${\cal
H}_{3}=\sum_{l=1}^{N}[\cos\theta(S_{l}.S_{l+1})+\sin\theta(S_{l}.S_{l+1})^{2}],$
(3)

where $\theta\in[0,2\pi)$ quantifies the amount of coupling between the
nearest neighboring spins. The model Eq. 3 presents four phases, namely,
Haldane, Trimerized, Ferromagnetic, and Dimerized [28].

To illustrate and summarize the phases and the common phases contained among
these models, we prepare Table 1. As we can be noted, all phases of ${\cal
H}_{2}$ are contained in the combined phases of ${\cal H}_{1}$ and ${\cal
H}_{3}$, being the only phase diagram in which this occurs. The union of
${\cal H}_{1}$ and ${\cal H}_{2}$ contains three of four phases of ${\cal
H}_{3}$, and the union of ${\cal H}_{2}$ and ${\cal H}_{3}$ contains four of
six phases of ${\cal H}_{1}$. In total, we have five phases that are shared by
at least two models (Haldane, Néel, Ferromagnetic, XY1, and Dimer) and three
phases unique to a single model (Large-D, XY$2$, and Trimer).

We assume that even when different models appear on the phase diagram, a given
phase has a trademark that is model independent. We propose here that the spin
correlations can capture such a signature. To test this hypothesis, we will
analyze several correlations between the spins in the chain.

Quantum Phase |  ${\cal H}_{1}$ |  ${\cal H}_{2}$ |  ${\cal H}_{3}$
---|---|---|---
Haldane |  $\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\leavevmode\hbox to5.17pt{\vbox to5.17pt{\pgfpicture\makeatletter\hbox{\hskip 2.58333pt\lower-2.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.58333pt}{0.0pt}\pgfsys@curveto{2.58333pt}{1.42674pt}{1.42674pt}{2.58333pt}{0.0pt}{2.58333pt}\pgfsys@curveto{-1.42674pt}{2.58333pt}{-2.58333pt}{1.42674pt}{-2.58333pt}{0.0pt}\pgfsys@curveto{-2.58333pt}{-1.42674pt}{-1.42674pt}{-2.58333pt}{0.0pt}{-2.58333pt}\pgfsys@curveto{1.42674pt}{-2.58333pt}{2.58333pt}{-1.42674pt}{2.58333pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}_{(288)}$ |  $\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\leavevmode\hbox to5.17pt{\vbox to5.17pt{\pgfpicture\makeatletter\hbox{\hskip 2.58333pt\lower-2.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.58333pt}{0.0pt}\pgfsys@curveto{2.58333pt}{1.42674pt}{1.42674pt}{2.58333pt}{0.0pt}{2.58333pt}\pgfsys@curveto{-1.42674pt}{2.58333pt}{-2.58333pt}{1.42674pt}{-2.58333pt}{0.0pt}\pgfsys@curveto{-2.58333pt}{-1.42674pt}{-1.42674pt}{-2.58333pt}{0.0pt}{-2.58333pt}\pgfsys@curveto{1.42674pt}{-2.58333pt}{2.58333pt}{-1.42674pt}{2.58333pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}_{(473)}$ |  $\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\leavevmode\hbox to5.17pt{\vbox to5.17pt{\pgfpicture\makeatletter\hbox{\hskip 2.58333pt\lower-2.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.58333pt}{0.0pt}\pgfsys@curveto{2.58333pt}{1.42674pt}{1.42674pt}{2.58333pt}{0.0pt}{2.58333pt}\pgfsys@curveto{-1.42674pt}{2.58333pt}{-2.58333pt}{1.42674pt}{-2.58333pt}{0.0pt}\pgfsys@curveto{-2.58333pt}{-1.42674pt}{-1.42674pt}{-2.58333pt}{0.0pt}{-2.58333pt}\pgfsys@curveto{1.42674pt}{-2.58333pt}{2.58333pt}{-1.42674pt}{2.58333pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}_{(1150)}$
Néel |  $\color[rgb]{0.5,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0}\pgfsys@color@cmyk@stroke{0}{0}{1}{.5}\pgfsys@color<EMAIL_ADDRESS>to5.17pt{\vbox to5.17pt{\pgfpicture\makeatletter\hbox{\hskip 2.58333pt\lower-2.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0}\pgfsys@color@cmyk@stroke{0}{0}{1}{.5}\pgfsys@invoke{ }\pgfsys@color@cmyk@fill{0}{0}{1}{.5}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.58333pt}{0.0pt}\pgfsys@curveto{2.58333pt}{1.42674pt}{1.42674pt}{2.58333pt}{0.0pt}{2.58333pt}\pgfsys@curveto{-1.42674pt}{2.58333pt}{-2.58333pt}{1.42674pt}{-2.58333pt}{0.0pt}\pgfsys@curveto{-2.58333pt}{-1.42674pt}{-1.42674pt}{-2.58333pt}{0.0pt}{-2.58333pt}\pgfsys@curveto{1.42674pt}{-2.58333pt}{2.58333pt}{-1.42674pt}{2.58333pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}_{(2081)}$ |  $\color[rgb]{0.5,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0}\pgfsys@color@cmyk@stroke{0}{0}{1}{.5}\pgfsys@color<EMAIL_ADDRESS>to5.17pt{\vbox to5.17pt{\pgfpicture\makeatletter\hbox{\hskip 2.58333pt\lower-2.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0}\pgfsys@color@cmyk@stroke{0}{0}{1}{.5}\pgfsys@invoke{ }\pgfsys@color@cmyk@fill{0}{0}{1}{.5}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.58333pt}{0.0pt}\pgfsys@curveto{2.58333pt}{1.42674pt}{1.42674pt}{2.58333pt}{0.0pt}{2.58333pt}\pgfsys@curveto{-1.42674pt}{2.58333pt}{-2.58333pt}{1.42674pt}{-2.58333pt}{0.0pt}\pgfsys@curveto{-2.58333pt}{-1.42674pt}{-1.42674pt}{-2.58333pt}{0.0pt}{-2.58333pt}\pgfsys@curveto{1.42674pt}{-2.58333pt}{2.58333pt}{-1.42674pt}{2.58333pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}_{(1284)}$ |
Ferromagnetic |  $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\leavevmode\hbox to5.17pt{\vbox to5.17pt{\pgfpicture\makeatletter\hbox{\hskip 2.58333pt\lower-2.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.58333pt}{0.0pt}\pgfsys@curveto{2.58333pt}{1.42674pt}{1.42674pt}{2.58333pt}{0.0pt}{2.58333pt}\pgfsys@curveto{-1.42674pt}{2.58333pt}{-2.58333pt}{1.42674pt}{-2.58333pt}{0.0pt}\pgfsys@curveto{-2.58333pt}{-1.42674pt}{-1.42674pt}{-2.58333pt}{0.0pt}{-2.58333pt}\pgfsys@curveto{1.42674pt}{-2.58333pt}{2.58333pt}{-1.42674pt}{2.58333pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}_{(2168)}$ |  $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\leavevmode\hbox to5.17pt{\vbox to5.17pt{\pgfpicture\makeatletter\hbox{\hskip 2.58333pt\lower-2.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.58333pt}{0.0pt}\pgfsys@curveto{2.58333pt}{1.42674pt}{1.42674pt}{2.58333pt}{0.0pt}{2.58333pt}\pgfsys@curveto{-1.42674pt}{2.58333pt}{-2.58333pt}{1.42674pt}{-2.58333pt}{0.0pt}\pgfsys@curveto{-2.58333pt}{-1.42674pt}{-1.42674pt}{-2.58333pt}{0.0pt}{-2.58333pt}\pgfsys@curveto{1.42674pt}{-2.58333pt}{2.58333pt}{-1.42674pt}{2.58333pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}_{(800)}$ |  $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\leavevmode\hbox to5.17pt{\vbox to5.17pt{\pgfpicture\makeatletter\hbox{\hskip 2.58333pt\lower-2.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.58333pt}{0.0pt}\pgfsys@curveto{2.58333pt}{1.42674pt}{1.42674pt}{2.58333pt}{0.0pt}{2.58333pt}\pgfsys@curveto{-1.42674pt}{2.58333pt}{-2.58333pt}{1.42674pt}{-2.58333pt}{0.0pt}\pgfsys@curveto{-2.58333pt}{-1.42674pt}{-1.42674pt}{-2.58333pt}{0.0pt}{-2.58333pt}\pgfsys@curveto{1.42674pt}{-2.58333pt}{2.58333pt}{-1.42674pt}{2.58333pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}_{(1725)}$
Large-D |  $\color[rgb]{1,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,0}\pgfsys@color@cmyk@stroke{0}{0}{1}{0}\pgfsys@color@cmyk@fill{0}{0}{1}{0}{\leavevmode\hbox to5.17pt{\vbox to5.17pt{\pgfpicture\makeatletter\hbox{\hskip 2.58333pt\lower-2.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,0}\pgfsys@color@cmyk@stroke{0}{0}{1}{0}\pgfsys@invoke{ }\pgfsys@color@cmyk@fill{0}{0}{1}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.58333pt}{0.0pt}\pgfsys@curveto{2.58333pt}{1.42674pt}{1.42674pt}{2.58333pt}{0.0pt}{2.58333pt}\pgfsys@curveto{-1.42674pt}{2.58333pt}{-2.58333pt}{1.42674pt}{-2.58333pt}{0.0pt}\pgfsys@curveto{-2.58333pt}{-1.42674pt}{-1.42674pt}{-2.58333pt}{0.0pt}{-2.58333pt}\pgfsys@curveto{1.42674pt}{-2.58333pt}{2.58333pt}{-1.42674pt}{2.58333pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}_{(1623)}$ |  |
XY1 |  $\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}{\leavevmode\hbox to5.17pt{\vbox to5.17pt{\pgfpicture\makeatletter\hbox{\hskip 2.58333pt\lower-2.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@cmyk@fill{1}{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.58333pt}{0.0pt}\pgfsys@curveto{2.58333pt}{1.42674pt}{1.42674pt}{2.58333pt}{0.0pt}{2.58333pt}\pgfsys@curveto{-1.42674pt}{2.58333pt}{-2.58333pt}{1.42674pt}{-2.58333pt}{0.0pt}\pgfsys@curveto{-2.58333pt}{-1.42674pt}{-1.42674pt}{-2.58333pt}{0.0pt}{-2.58333pt}\pgfsys@curveto{1.42674pt}{-2.58333pt}{2.58333pt}{-1.42674pt}{2.58333pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}_{(207)}$ |  $\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}{\leavevmode\hbox to5.17pt{\vbox to5.17pt{\pgfpicture\makeatletter\hbox{\hskip 2.58333pt\lower-2.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@cmyk@fill{1}{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.58333pt}{0.0pt}\pgfsys@curveto{2.58333pt}{1.42674pt}{1.42674pt}{2.58333pt}{0.0pt}{2.58333pt}\pgfsys@curveto{-1.42674pt}{2.58333pt}{-2.58333pt}{1.42674pt}{-2.58333pt}{0.0pt}\pgfsys@curveto{-2.58333pt}{-1.42674pt}{-1.42674pt}{-2.58333pt}{0.0pt}{-2.58333pt}\pgfsys@curveto{1.42674pt}{-2.58333pt}{2.58333pt}{-1.42674pt}{2.58333pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}_{(1028)}$ |
XY2 |  $\leavevmode\hbox to5.17pt{\vbox to5.17pt{\pgfpicture\makeatletter\hbox{\hskip 2.58333pt\lower-2.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.58333pt}{0.0pt}\pgfsys@curveto{2.58333pt}{1.42674pt}{1.42674pt}{2.58333pt}{0.0pt}{2.58333pt}\pgfsys@curveto{-1.42674pt}{2.58333pt}{-2.58333pt}{1.42674pt}{-2.58333pt}{0.0pt}\pgfsys@curveto{-2.58333pt}{-1.42674pt}{-1.42674pt}{-2.58333pt}{0.0pt}{-2.58333pt}\pgfsys@curveto{1.42674pt}{-2.58333pt}{2.58333pt}{-1.42674pt}{2.58333pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{(33)}$ |  |
Dimer |  |  $\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}{\leavevmode\hbox to5.17pt{\vbox to5.17pt{\pgfpicture\makeatletter\hbox{\hskip 2.58333pt\lower-2.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@rgb@stroke{0}{1}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{1}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.58333pt}{0.0pt}\pgfsys@curveto{2.58333pt}{1.42674pt}{1.42674pt}{2.58333pt}{0.0pt}{2.58333pt}\pgfsys@curveto{-1.42674pt}{2.58333pt}{-2.58333pt}{1.42674pt}{-2.58333pt}{0.0pt}\pgfsys@curveto{-2.58333pt}{-1.42674pt}{-1.42674pt}{-2.58333pt}{0.0pt}{-2.58333pt}\pgfsys@curveto{1.42674pt}{-2.58333pt}{2.58333pt}{-1.42674pt}{2.58333pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}_{(2815)}$ |  $\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}{\leavevmode\hbox to5.17pt{\vbox to5.17pt{\pgfpicture\makeatletter\hbox{\hskip 2.58333pt\lower-2.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@rgb@stroke{0}{1}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{1}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.58333pt}{0.0pt}\pgfsys@curveto{2.58333pt}{1.42674pt}{1.42674pt}{2.58333pt}{0.0pt}{2.58333pt}\pgfsys@curveto{-1.42674pt}{2.58333pt}{-2.58333pt}{1.42674pt}{-2.58333pt}{0.0pt}\pgfsys@curveto{-2.58333pt}{-1.42674pt}{-1.42674pt}{-2.58333pt}{0.0pt}{-2.58333pt}\pgfsys@curveto{1.42674pt}{-2.58333pt}{2.58333pt}{-1.42674pt}{2.58333pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}_{(1150)}$
Trimer |  |  |  $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{\leavevmode\hbox to5.17pt{\vbox to5.17pt{\pgfpicture\makeatletter\hbox{\hskip 2.58333pt\lower-2.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\pgfsys@color@rgb@stroke{.75}{0}{.25}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{.75}{0}{.25}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.58333pt}{0.0pt}\pgfsys@curveto{2.58333pt}{1.42674pt}{1.42674pt}{2.58333pt}{0.0pt}{2.58333pt}\pgfsys@curveto{-1.42674pt}{2.58333pt}{-2.58333pt}{1.42674pt}{-2.58333pt}{0.0pt}\pgfsys@curveto{-2.58333pt}{-1.42674pt}{-1.42674pt}{-2.58333pt}{0.0pt}{-2.58333pt}\pgfsys@curveto{1.42674pt}{-2.58333pt}{2.58333pt}{-1.42674pt}{2.58333pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}_{(575)}$
Table 1: Phases contained in the diagrams corresponding to the three
Hamiltonians analyzed. The filled circles mean that the quantum phase is
present in the model. The numbers subscribed in the circles represent the
amount of data calculated for each phase of each model.

Data structure — The correlations between neighbors in the closed spin chain
are given by the expected values of the following observables: $\langle
S_{1}^{k},S_{i}^{k}\rangle$ and $\langle\prod_{j}S_{j}^{k}\rangle$, with
$k=\\{x,y,z\\}$, $i=[1,N/2+1]$, and $j=[1,N]$. Furthermore, $\langle
S_{1}^{k}S_{i}^{k}\rangle=\langle\lambda_{0}|S_{1}^{k},S_{i}^{k}|\lambda_{0}\rangle$,
are the expected values of the correlation for the lowest energy state of the
Hamiltonian and we take the number of spins $N=12$ [25]. Notice that since the
chain is closed and, consequently, the chain properties are cyclic, any non-
redundant correlation between two spins is obtained for $\langle
S_{1}^{k},S_{i}^{k}\rangle$ with $j=[1,N/2+1]$.

To generate the correlation dataset, we considered thousands of different
values for the parameters of the Hamiltonians ${\cal H}_{1}$, ${\cal H}_{2}$,
and ${\cal H}_{3}$. To ${\cal H}_{1}$, Eq. 1, we range the parameters $J_{z}$
and $D$ in the interval $[-4,4]$ with step size of $0.1$, this generates a
dataset with $6400$ data points. For the Hamiltonian ${\cal H}_{2}$, Eq. 2, we
range the parameters, $\Delta$ in an interval $[0,1]$, and the parameters
$\delta$ in an interval $[-1.5,2.5]$, with step sizes of 0.005 and 0.0125,
respectively, again this generate a dataset with $6400$ data points. Finally,
for the Hamiltonian ${\cal H}_{3}$, Eq. 3, we set the range of parameter
$\theta$ in the interval $[0,2\pi]$, with the step size of $4.35\times
10^{-4}$, which results in a dataset with $4600$ data points. The labels of
the phases for the Hamiltonians ${\cal H}_{1}$, ${\cal H}_{2}$, and ${\cal
H}_{3}$, are obtained from the literature [27], [26], and [28], respectively.

With the dataset of the three models, we could visualize the relation of the
correlations with the quantum phases. Since we intend to use a classifier
algorithm, the idea is to separate, in the multidimensional space, distinct
phases in distinct positions. For $12$ sites, for example, using the set of
correlations described above, we have $24$ distinct correlations (a space with
$24$ dimensions) and to visualize this amount of information in a 3D space is
complicated. Nevertheless, a clue as to what happens in the correlation space
can be obtained by considering only a pair of correlations. To illustrate
this, in Fig. (1) we choose a specific pair of correlations to plot, namely,
$\langle S^{x}_{1}S^{x}_{2}\rangle$ and $\langle S^{z}_{1}S^{z}_{6}\rangle$,
looking for the best graphical representation of what happens in the
correlation space. Note that $\langle S^{x}_{1}S^{x}_{2}\rangle$ and $\langle
S^{z}_{1}S^{z}_{6}\rangle$ are calculated for all dataset, where we consider
different values for the parameters of the Hamiltonians ${\cal H}_{1}$, ${\cal
H}_{2}$, and ${\cal H}_{3}$.

One important aspect to consider when working with machine learning is a
dataset transformation. In many cases, appropriate transformations can
separate the classes in the feature space, which facilitates the
classification process. In Fig. (1-a) to Fig. (1-d) we plot the raw data, and
in Fig. (1-e) to Fig. (1-h) we plot the raw data after scaling the dataset,
for each model, to have a unit norm. This is a well-known renormalization
procedure called spatial sign preprocessing [29]. Analyzing Fig. (1), we see
that even with the information contained in a single pair of correlations, we
are able to separate, with high distinctness, states with different phases.
Indeed, there is some overlap of this information for different phases,
especially when considering several different models, Fig. (1-d) and Fig.
(1-h). Moreover, for the normalized data, Fig. (1-e) to Fig. (1-h), the
overlap of information about the phases decreases for all models, almost
disappearing for ${\cal H}_{3}$, Fig. (1-g).

Figure 1: Correlation $\langle S^{x}_{1}S^{x}_{2}\rangle$ vs correlation
$\langle S^{z}_{1}S^{z}_{6}\rangle$, the color represents the phase of the
corresponding data point. And the figures (a) and (e) refer to the system
described by ${\cal H}_{1}$, (b) and (f) refers to ${\cal H}_{2}$, (c) and (g)
refers to ${\cal H}_{3}$, (d) and (h) are the data from the three systems. The
figures (a), (b), (c) and (d) are the raw data, and (e), (f), (g), (h) are the
data after performing the transformation.

Despite the improvement in reducing the overlap, Fig. (1-h) shows that there
are still overlaps of information from the Haldane phase with Large-D, from
the XY1, XY2, Néel, and Haldane phases with the Dimmer phase. However, it is
important to note that we are illustrating the procedure considering only two
features, which is insufficient to separate all phases given the complexity of
the problem. In fact, the use of the 24 features, as well as studying and
applying other transformations to separate the phases in the correlation
space, becomes crucial and is the focus of this work. Our main goal is to
classify the phases of an unknown model, given the distribution of these
phases in the correlation space of other known models. For this purpose, we
need an algorithm that, given a new quantum state and, consequently, a
position in the correlation space, labels it with the corresponding phase
based on the information from other known states, located in the vicinity of
the new one in the correlation space. As we show below, even a simple
algorithm is capable of performing this task with high accuracy.

Machine Learning — In this manuscript, we use an supervised algorithm, i.e.
where the learning process occurs through labeled data. To this task, one of
the the simplest machine learning algorithms is the $k$-Nearest Neighbors
classifier ($k$-NN) [30]. Despite its simplicity, it presented a good result
in our classification problem, which can be explained by the way the $k$-NN
works. When a data point of the unknown model is inserted, the algorithm
calculated the Euclidean distance (in this work, but the metric can be
changed) of this unknown model data point to the $k$-nearest neighbors. The
unknown model data point is classified by a plurality vote of its $k$
neighbors, with the unknown model data point being assigned to the class most
common among them. For example, if $k=1$, then the object is simply assigned
to the class of that single nearest neighbor [30, 31, 32].

In this work, we use $k=30$ and assume that all neighbors have the same voting
weight (it could be assumed that the closer, the greater the voting weight).
The values were obtained by performing the hypermetrization of the parameters
[31]. Analyzing Fig. (1) and understanding how $k$-NN works, it quite
plausible that a high accuracy universal classifier can be obtained.
Naturally, it is necessary to use an appropriate transformation that reduces
the overlap between the different phases in the correlation space. Once the
transformation decreases the overlap between different phases (such as that
made in Fig. (1-e) to Fig. (1-h)), it increases the accuracy of the $k$-NN
algorithm.

Results — To test the accuracy of the $k$-NN in transferring learning from the
known models to the unknown model, we begin by training the algorithm to
classify the data points of ${\cal H}_{1}$. For this case, we train the $k$-NN
with data from models ${\cal H}_{2}$ and ${\cal H}_{3}$ (it means that the
$k$-NN algorithm knows the dataset of ${\cal H}_{2}$ and ${\cal H}_{3}$ and
their respective phase labels), and predict the phases of ${\cal H}_{1}$. In
sequence, we do the same for ${\cal H}_{2}$ and ${\cal H}_{3}$ using the data
from models ${\cal H}_{1}$ and ${\cal H}_{3}$, and ${\cal H}_{1}$ and ${\cal
H}_{2}$, respectively. It is worth noting, however, that there are phases in
${\cal H}_{1}$ (Large-D and XY2) that do not exist in ${\cal H}_{2}$ and
${\cal H}_{3}$, and a phase in ${\cal H}_{2}$ (Trimer) that does not exist in
${\cal H}_{1}$ and ${\cal H}_{3}$. Clearly, there is no way to learn from the
data that were not provided [9, 33], for that reason, when we predict the
phase of ${\cal H}_{1}$, we remove the data for the Large-D and XY2 phases.
Furthermore, when we predict the phases of ${\cal H}_{2}$ we remove the Trimer
phase from the predicted dataset, which ensures that unknown phases do not
interfere with the accuracy of the $k$-NN.

Figure 2: Figure (a) is the phase prediction of model ${\cal H}_{1}$ using the
learning of models ${\cal H}_{2}$ and ${\cal H}_{3}$. Figure (b) is the phase
prediction of model ${\cal H}_{2}$ using the learning of models ${\cal H}_{1}$
and ${\cal H}_{3}$, and figure (c) is the phase prediction of model ${\cal
H}_{3}$ using the learning of models ${\cal H}_{1}$ and ${\cal H}_{2}$. The
white dots in panels (a) and (b), and light blue line in panel (c) are the
theoretical boundaries of phases given in the literature, [27, 26, 28], for
models ${\cal H}_{1}$, ${\cal H}_{2}$, and ${\cal H}_{3}$, respectively. The
name in black represents the real phase in that place of the model, and the
color represents the predicted phase.

In our analysis, we transform our dataset following a renormalization
procedure called spatial sign preprocessing, where the dataset is scaled to
have a unit norm [29]. As stated before, each ground state and its respective
labeled phase, are represented by the correlations between a pair of spins,
$\langle S_{1}^{k},S_{i}^{k}\rangle$, which provide 7 features for each
variable $k=\\{x,y,z\\}$, and global correlations
$\langle\prod_{j}S_{j}^{k}\rangle$, which provide 3 more features. As we note
in Fig. (1), where we use only 2 correlations to illustrate our strategy, even
after the dataset transformation, some overlap between phases is still
present. For example, in Fig. (1-f), it is easy to notice the overlap of the
XY1 phase with Haldane and Dimer phase, and the overlap between Néel and
Dimmer phases. So, the question emerges: is the inclusion of all features able
to separate the phases in the correlation space? To answer this question, Fig.
(2) shows the phase prediction of the $k$-NN algorithm, where in Fig. (2-a) we
estimate the phase diagram of ${\cal H}_{1}$ given the data from ${\cal
H}_{2}$ and ${\cal H}_{3}$, in Fig. (2-b) we estimate the phase diagram of
model ${\cal H}_{2}$ given the data from ${\cal H}_{1}$ and ${\cal H}_{3}$,
and analogous in Fig. (2-c) where we estimate the phases of the ${\cal H}_{3}$
given the data from ${\cal H}_{1}$ and ${\cal H}_{2}$.

The prediction of the phase diagram of ${\cal H}_{1}$, presented in Fig.
(2-a), was incredibly successful with an accuracy of $96.31\%$. All phases are
in the correct locations with few mistakes in the phase transitions. As in our
two-dimensional illustrative example, the Haldane phase invades the XY1 phase
space, and the confusion between the Néel and Dimmer phase persists.
Nevertheless, analyzing Fig. (1-f), it is important to note that the $k$-NN
correctly predicts the XY1 phase, even thought this phase is completely
overlapped with Dimer and Haldane phase in the two-dimensional case. When
considering ${\cal H}_{2}$, we observe that the algorithm commits mistakes in
the separations of the phases, which made its accuracy the lowest $72.72\%$.
Despite this, one aspect needs to be emphasized. The training dataset contains
information for ${\cal H}_{1}$ and ${\cal H}_{3}$, which includes all 8
distinct phases. Nevertheless, the Large-D, XY2, and Trimer phases were not
indicated by the $k$-NN algorithm for all data of ${\cal H}_{2}$, since the
algorithm detects all phases correctly, only making mistakes around the
boundaries.

Finally, the prediction for model ${\cal H}_{3}$ is presented in Fig. (1-c),
where we use the ${\cal H}_{1}$ and ${\cal H}_{2}$ models for learning. In one
hand, the confusion between the Haldane phase and the Dimer phase presented in
Fig. (1-f) was removed when all 24 features were included. However, it
incorrectly classifies the Haldane phase, mixing it with the Large-D, a phase
that is not even present in the model. Nevertheless, even making this mistake,
the algorithm achieve an accuracy of $89.17\%$. Although the good results,
different strategies can be used to increase the prediction accuracy. One is
to add new models to the predictor dataset, as adding new information to the
$k$-NN would help avoid incorrect phase prediction. The second is to find a
transformation that can separate the phase information in the correlation
space. Finally, different machine learning algorithms can certainly be
implemented to increase the accuracy. Nevertheless, these results demonstrate
that the correlations between the spins provide essential information about
the phases of the models. This shows that correlations obtained from known
models can be used to predict phases in other different models with high
accuracy.

Conclusion —

We have developed a method to study the phases of unknown magnetic systems
through spin correlations. We show that raw correlation data carry information
about the phases, which is independent of the model. With the spin correlation
information, we use a $k$-NN algorithm to predict the phases of an unknown
model with high accuracy. We present a proof of concept, showing that a ML
algorithm can classify unknown phases of a Hamiltonian through known phases of
another Hamiltonian, creating a model-independent quantum phase classifier. We
emphasize that no explicit use of the phases order parameters is made, so this
model-independent classifier opens up the possibility of universal classifier
creation, as more and more model-independent information is added to the
classifier database.

###### Acknowledgements.

F. M. and F.S.L. acknowledge support from Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior (CAPES), project number 88887.607339/2021-00 and
151435/2020-0, respectively; F.M. and F.F.F acknowledge support from Fundação
de Amparo à Pesquisa do Estado de São Paulo (FAPESP), project number
2019/00700-9 and 2021/04655-8, respectively.

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# Nanometric Turing Patterns: Morphogenesis of a Bismuth Monolayer

Yuki Fuseya∗ Department of Engineering Science, University of Electro-
Communications, Chofu, Tokyo 182-8585, Japan  Hiroyasu Katsuno  Institute of
Low Temperature Science, Hokkaido University, Kita-19, Nishi-8, Kita-ku,
Sapporo, Japan  Kamran Behnia  Laboratoire Physique et Etude de Matériaux
(CNRS-Sorbonne Université), ESPCI Paris, PSL Research University, 75005 Paris,
France  Aharon Kapitulnik∗ Departments of Physics and Applied Physics,
Stanford University, Stanford, CA 94305, United States of America

###### Abstract

Turing’s reaction-diffusion theory of morphogenesis has been very successful
in understanding macroscopic patterns within complex objects ranging from
biological systems to sand dunes. However, this mechanism was never tested
against patterns that emerge at the atomic scale, where the basic ingredients
are subject to constraints imposed by quantum mechanics. Here we report
evidence of a Turing pattern that appears in a strained atomic bismuth
monolayer assembling on the surface of NbSe2 subject to interatomic
interactions and respective kinetics. The narrow range of microscopic
parameters reflected in numerical analysis that observe stripe patterns and
domain walls with Y-shaped junctions is a direct consequence of the quantum-
mechanically allowed bond-lengths and bond-angles. This is therefore the first
demonstration of a dynamically formed Turing pattern at the atomic scale.

††preprint: APS/123-QED Figure 1:  Crystal structure of monolayer Bi on the
top Se layer. (a) Side view and (b) Top view. $\alpha_{i}$ and $\bm{u}_{i}$
are the positions corresponding to the adsorption potential minimum and the
displacement of the Bi atom at site $i$, respectively. (c) Schematic side view
of the equilibrium state. When only the elastic potential $V_{e}$ is
considered, the Bi atoms form staggered patterns with the period $2a$. The
adsorption potential $V_{a}$ reduces the amplitude of the staggered patterns.
When the bond angle potential $V_{b}$ is included, the Bi atoms form patterns
with wavelengths longer than $2a$.

Ordered patterns can arise out of randomness during morphogenesis. An
explanation for this puzzle was put forward by Turing, who explained the
emergence of stationary patterns by invoking interplay between an activator
and an inhibitor with different diffusion rates [1]. Turing’s theory of
diffusion-reaction has proven extremely influential across many disciplines
[2, 3, 4]. For example, the pigment patterns on sea shells [5], the stripes on
tropical fish [6, 7], and the purely chemical system of chlorite-iodide-
malonic acid [8, 9, 10] have been studied as Turing patterns. The typical
length scale for biological pigment-based patterns ranges from millimeters to
centimeters, and that for purely chemical systems is of the order of sub-
millimeters. However, it has never been employed to describe atomic scale
patterns. Because quantum mechanical considerations are expected to play a key
role at the scale of interatomic distances, the relevance of Turing’s theory
in such a context is yet to be demonstrated. Here, we show that atomic
monolayer bismuth grown by molecular beam epitaxy (MBE) provides such an
opportunity.

Strained MBE has become an important technique for its direct impact on
various functional properties for potential applications in electronic
devices, quantum sensors, or spintronics applications to name a few.
Traditionally, epitaxial growth of two-dimensional films has been described
within the framework of first-order phase transition, yielding three modes of
epitaxial growth [11]: island growth [12], layer growth [13] and their
combination, namely islands growing on one or two already-completed monolayers
[14]. When equilibrium conditions are met, the morphology of the newly grown
layer is obtained by balancing the elastic energy against the surface energy.
However, under far-from-equilibrium conditions, kinetic processes must
dominate, which arise from the relevant thermodynamic driving forces. This is
the regime where predictability is poor, since small changes in initial and/or
boundary conditions may have dramatic effects on the resulting morphology of
the film. In particular, where the growth of metallic (or semi-metallic) films
also involve quantum mechanics considerations associated with the itinerant
electronic states, constraints on allowed states of bonding, imposing specific
bond length or bond angle have to be met.

Our study was motivated by the recent work of Fang et al., who first found a
mysterious pattern of atomic-monolayer grown by MBE [15]. The Bi monolayer on
NbSe2 was originally fabricated in the interests of research in two-
dimensional topological physics, to which intensive efforts have been devoted
in recent years [16, 17, 18]. The patterns in Bi monolayer exhibit stripes
with a period of five atoms ($=1.7$ nm) and domain walls with Y-shaped
junctions (cf. Fig. 2b), which break the symmetry of the underlying lattice
[15]. This mysterious pattern strikingly resembles the stripes with a Y-shaped
pattern in angelfish Pomacanthus, where Turing’s mechanism has been studied
[6, 7], although their length scales differ by more than six orders of
magnitude.

The necessary conditions for the formation of Turing patterns are the
following: (i) auto-catalysis and cross-catalysis must exist between an
activator ($u_{a}$) and an inhibitor ($u_{h}$), and (ii) the diffusion of the
inhibitor ($D_{h}$) must be much faster than that of the activator ($D_{a}$),
i.e., $D_{a}\ll D_{h}$ [1, 2, 3, 4, 7]. Turing expressed this mechanism of
pattern formation in terms of a simple simultaneous differential equation of
the form [1]

$\displaystyle\frac{\partial u_{a,h}}{\partial
t}=D_{a,h}\nabla^{2}u_{a,h}+f_{a,h}(u_{a},u_{h}),$ (1)

which is called the reaction-diffusion equation. ($f_{a,h}(u_{a},u_{h})$ are
reaction functions.) In principle, no upper or lower limit of the length scale
is imposed by the reaction-diffusion equation. In other words, Turing patterns
possess an intrinsic wavelength, which depends only on the ratio of the
parameters in the equation. This scaleless property of Turing patterns
contrasts with the other well-known non-equilibrium structures, such as the
convective Bénard cells and the Taylor vortices in Couette flows, the length
scales of which are restricted by the geometric scales of the system [19, 8].
Therefore, one naively expects that Turing patterns can be realized at any
arbitrary length scale. However, the length scale of patterns hitherto
documented in biology and chemistry do not fall below a fraction of a
millimeter [3, 7, 8, 9, 10]. It is of fundamental interest to determine the
validity of Turing theory at the interatomic scale, where the solid state
remains stable thanks to quantum mechanics [20]. If Turing patterns can be
formed even at atomic scales, what are the activator and the inhibitor? Why
are their diffusion rates so different? What determines the wavelength? We
provide answers to each of these questions in the following discussion.

The present work aims to elucidate the origin and underlying mechanism of the
mysterious pattern exhibited by Bi monolayers on NbSe2. The lattice constant
of the Bi monolayer on NbSe2 matches that of the hexagonal lattice in the
NbSe2 ($a=3.45$ Å) [15]. This is substantially shorter than the lattice
constant of bulk Bi, $b=4.53$ Å , or of the Bi layer in Bi2Se3, $b=4.13$ Å.
Therefore, it can be expected that the NbSe2 substrate generates substantial
lateral strain in the Bi monolayer, inducing ripples to relieve the strain.
However, the ripples should maintain the rotational symmetry of the hexagonal
lattice because the direction of strain is three-fold. The expected ripple
pattern due to strain would be completely different from the mysterious
pattern observed in the Bi monolayer.

The crystal structure of Bi monolayer on NbSe2 is depicted in Fig. 1. Bi atoms
are deposited on the top Se layer of the substrate, and they form a hexagonal
lattice with a lattice constant equal to that of the Se layer. Let us consider
empirical two-body interatomic potentials between Bi–Bi atoms (elastic
potential $V_{e}$) and Bi–Se atoms (adsorption potential $V_{a}$). Although
the potentials are generally given by the Lennard-Jones or Morse potentials,
it is sufficient to consider the potential around its minimum in the present
case, namely, these are well approximated by the harmonic oscillator. The
elastic potential is given by $V_{e}=(K_{e}/2)\sum_{ij}(b-\ell_{ij})^{2}$,
where $K_{e}$ is the elastic constant, and $b$ ($>a$) is the natural lattice
constant of Bi. $\ell_{ij}=|\bm{u}_{i}-\bm{u}_{j}+\bm{\alpha}_{ij}|$ is the
distance between the $i$-th and $j$-th Bi atoms, $\bm{\alpha}_{ij}$ the vector
between the $i$-th and $j$-th sites, and $\bm{u}_{i}$ is the displacement of
Bi atoms at the $i$-th site. The Se atoms are tightly bound to the rest of the
NbSe2 substrate; consequently, the lattice distortion of Se is negligibly
small. Therefore, the adsorption potential can be expressed as a one-body
potential of the form $V_{a}=(K_{a}/2)\sum_{i}|\bm{u}_{i}|^{2}$.

If Bi atoms interact only through these two-body potentials, they would simply
form a staggered pattern to release the elastic energy, as shown in Fig. 1c.
This clearly fails to explain the observed patterns. This failure is due to
the lack of consideration of covalency. In particular, the covalency is known
to be crucial for understanding the crystal structure of pnictogens (Group
15), even if they are semimetals [21, 22]. To consider the covalency of the
crystal, it is necessary to include the three-body potential [23], which can
be expressed in terms of the bond angle in Bi–Bi–Bi. (The bond angle term
corresponding to Se–Bi–Se is renormalized to the one-body term $V_{a}$ because
the position of the Se atoms is fixed.) The bond angle potential is given by
$V_{b}=K_{b}a^{2}\sum_{i}(1+\cos\theta_{i})$, where $K_{b}$ is the coefficient
that has the same dimension as $K_{e,a}$. Further, $\theta_{i}$ is the bond
angle corresponding to Bi–Bi–Bi, and $i$ denotes the center Bi atom (Fig. 1a).
This bond angle potential is crucial in obtaining the Turing patterns.

Figure 2:  (a) Time evolution of the vertical displacement $u_{z}$ in the
$64\times 64$ hexagonal lattice with a periodic boundary condition for
$b/a=1.3$, $K_{a}/K_{e}=0.3$, and $K_{b}/K_{e}=0.12$. The unit of time is
given by $K_{e}^{-1}$. (b) Comparison of the numerical simulation at
$\tau=10000$ and the experiment by Fang et al. [15]. The wavelength obtained
via the simulation is $5a$ ($=1.7$ nm), which is in excellent agreement with
the experiment.

We investigated the dynamics of this model. Generally, a non-conserved
quantity will evolve such that the total energy decreases. In particular, we
aim to analyze the dynamics of the displacement $\bm{u}_{i}$ of Bi atoms. The
time evolution equation is given as $\partial\bm{u}_{i}/\partial
t=-\eta\partial V_{\rm tot}/\partial\bm{u}_{i}$, where $V_{\rm
tot}=V_{a}+V_{e}+V_{b}$ and $\eta>0$ is a prefactor. (Hereinafter, we use the
renormalized time $\tau=\eta t$, whose unit is $K_{e}^{-1}$.) This is a
simultaneous differential equation with respect to $u_{i}^{x,y,z}$. The
explicit form of the time evolution equation is provided in the Supplementary
Information [24]. Figure 2a shows the time evolution of the vertical
displacement $u_{i}^{z}$. The initial condition is given as a random
distribution of $\bm{u}_{i}$ with a small amplitude. Notably, the system
spontaneously breaks the symmetry and a pattern with a unique wavelength is
formed with time. We observed that stable stripe patterns with domains are
formed, with parameters $K_{a}/K_{e}=1.2$, $K_{b}/K_{e}=0.3$, and $b/a=1.3$.
The period (wavelength) of the stripes is $5a$, which is in exact agreement
with the experiment. The domains are oriented such that they form an angle of
120∘ with each other, and the aligned Y-junctions clearly appear at the domain
boundaries. All of these theoretical results are in excellent agreement with
the experimental results previously obtained with monolayer Bi on NbSe2 (Fig.
2b). The maximum vertical displacement is $\sim 0.5a$, which is larger than
that obtained in the experiment to some extent. If different initial
conditions are employed, stripe patterns with a different direction and domain
structure but the same wavelength are obtained [24].

In the following, we will demonstrate that our time evolution equation is
equivalent to Turing’s reaction-diffusion equation, although they may
initially seem entirely different. (See [24] for a detailed derivation.) The
vertical displacement $u_{i}^{z}$ corresponds to the activator, and the
horizontal displacement $u_{i}^{x,y}$ corresponds to the inhibitor. The bond
angle potential can be simplified as $-K_{b}\nabla^{4}u_{i}$, which entails
dominant (long-range) diffusion with regard to all $u_{i}^{x,y,z}$ [4]. By
contrast, the elastic potential gives $-K_{e}b/a\nabla^{2}u_{i}^{z}$, which
plays a role of auto-catalysis (a positive feedback) and reduces the diffusion
only in the $z$-direction, resulting in $D_{a}\ll D_{h}$. Further, the elastic
potential contributing to the reaction terms $f_{\mu}$ consists of
$\mathcal{O}(u_{i}^{2})$, which generates the cross-catalysis. The adsorption
term reduces the wavelength of the patterns, which is crucial for achieving
quantitative agreements with the experiments. Finally, our time evolution
equation can be simplified as

$\displaystyle\frac{\partial u_{i}^{\mu}}{\partial\tau}$
$\displaystyle=D_{\mu}\widetilde{\nabla}^{2}u_{i}^{\mu}-K_{a}u_{i}^{\mu}+f_{\mu}(u_{i}^{z},u_{i}^{x}),$
(2)

where $\widetilde{\nabla}=a\nabla$, $\mu=z,x$, $D_{z}=D_{x}-K_{e}b/a$, and
$D_{x}=-K_{b}\widetilde{\nabla}^{2}+K_{e}-2K_{b}$. (Here, we omitted the
$y$-components to make the argument as clear as possible. However, the essence
of our model is fully accounted for this one-dimensional array.) This is
essentially equivalent to the reaction-diffusion equation (1). Therefore, the
excellent agreement between the theory and the experiment strongly suggests
that the unique patterns appearing in monolayer Bi are realized by the Turing
mechanism. To the best of our knowledge, a wavelength of 2 nm is the shortest
length scale reported for Turing patterns thus far. At the same time, this
agreement is strong evidence that Turing’s theory is valid from centimeters to
nanometers, a surprisingly wide extent.

Figure 3:  (a) & (b) Fourier component $c_{k}$ of the numerically obtained
patterns in the model of equation (2) for 1024 sites and $b/a=1.1$ with (a)
$K_{b}/K_{e}=0.1$ and (b) $K_{a}/K_{e}=0.01$. The brightest region corresponds
to the most dominant $c_{k}$ of the patterns, which clearly exhibit
$(K_{a}/K_{b})^{1/4}$ dependencies. (c) Pattern diagram as a function of
$1/K_{a}$ and $1/K_{b}$. The Turing pattern appears in the region
corresponding to $K_{a}K_{b}<\beta K_{e}^{2}$.

The most unstable wavenumber of equation (2) is given by

$\displaystyle
k_{m}=a^{-1}\sqrt{k_{+}k_{-}}=a^{-1}\left(K_{a}/K_{b}\right)^{1/4},$ (3)

according to linear stability analysis [4, 24]. This form clearly indicates
that the wavelengths are determined only by intrinsic parameters. The bond
angle potential enhances the pattern wavelength, whereas the adsorption
potential reduces it. Figure 3a and b shows plots of the Fourier component
$c_{k}$ of the patterns obtained by our numerical simulation with the non-
linear terms $f_{\mu}(u_{i}^{z},u_{i}^{x})$, equation (2), as a function of
$K_{a}$ and $K_{b}$. The most dominant $c_{k}$ (the brightest region) exhibits
a clear $(K_{a}/K_{b})^{1/4}$ dependence, which agrees well with the analytic
result of equation (3). The $(K_{a}/K_{b})^{1/4}$ dependence never changes
even for the different system sizes [24], indicating the intrinsic nature of
the wavelength that is inherent in Turing patterns.

In addition to the wavelength, the pattern can be modified by tuning the
parameters. The changes in the pattern are schematically depicted in Fig. 3c.
(The precise pattern diagram is given in [24].) The patterns are formed only
for $K_{a}K_{b}<\beta K_{e}^{2}$, where the prefactor $\beta$ depends on
$b/a$. Otherwise, Bi atoms form a perfectly flat surface with a hexagonal
lattice even if the initial film is bumpy. The stripe patterns appear in the
narrow region close to the boundary. From these pattern diagrams, we infer the
reason for the formation of the Turing patterns in Bi monolayers as follows:
First, it is well known that bulk Bi exhibits substantial structural
instability owing to its specific electronic states [25, 26, 27]. In fact, Bi
under pressure has 10 different solid phases [28]. (Only sulfur has more
phases among the elemental solids [28].) This structural instability is
further enhanced in ultrathin Bi, resulting in a rich variety in the structure
[29, 30, 18, 15]. The structure instability reduces the bond angle potential,
i.e., reduces $K_{b}$. In typical materials, $K_{b}$ is substantially large,
and the system corresponds to the “no pattern” region. By contrast, in the
case of Bi, $K_{b}$ is relatively low; consequently, the system crosses the
boundary and enters the pattern formation region.

Figure 4:  (a) Simulation of wound healing. The patterns are wounded,
$\bm{u}_{i}=0$, at $\tau=5000$. (b) Simulation of uniaxial stress. The lattice
constant along the $x$-direction is extended by 3% at $\tau=5000$ to simulate
the application of uniaxial tensional strain. In both simulations, the same
parameters as in Fig. 2 are used.

Turing patterns are generated by interference between nonlinear waves. They
appear static; however, they are in dynamic equilibrium. One of the
significant consequences of the dynamic property of Turing patterns is wound
healing [6, 7, 4]. Particularly, wound healing is crucial for living creatures
to support their lives, and it can be explained using reaction-diffusion
equations [4]. Herein, we show that inorganic solids also possess the ability
of wound healing. Figure 4a shows the wound healing property of our model. We
inflicted a “wound” by imposing $\bm{u}_{i}=0$ in a circular area at
$\tau=5000$. The wound rapidly healed before $\tau=5100$. Surprisingly, the
domain structures and Y-junctions are regenerated exactly as before at
approximately $\tau=6000$ (shown in [24]). This clearly indicates the dynamic
nature of the Turing patterns generated by our model.

We show further dynamic properties of the patterns by applying uniaxial
strain. We simulated uniaxial strain by changing the lattice constant along
one particular direction. We applied tensional strain (increased the lattice
constant by 3%) along the $x$-direction after $\tau=5000$. Notably, as shown
in Fig. 4b, the two separated domains combined even though the tensional
strain was applied in the direction that would lead to their further
separation. Consequently, the domain structure of the patterns can be
controlled by applying strain.

Unnoticed Turing patterns can be ubiquitously detected in solid-state physics
publications. For example, magnetic patterns in ferrimagnetic garnet films
(BiGdY)3(FeGa)5O12 [31] may be reinterpreted as Turing patterns. The patterns
exhibited by type-I superconductors in the intermediate state [32, 33] or
those exhibited by ferromagnetic superconductors [34] may have a close
relationship to Turing patterns because the normal or ferromagnetic states
inhibit the superconducting states of the activator. The common concept of
spontaneous symmetry-breaking due to diffusion-driven instability hidden in
these patterns will bring about a new point of view in solid-state physics.

Our reaction-diffusion equation implies more than the fact that it can explain
the mysterious pattern of Bi on NbSe2. It proposes that the patterns can be
controlled by changing parameters, i.e., changing substate, overlayer, or
growth conditions. It can even remove undesirable patterns and make perfectly
flat thin films. Obtained different patterns can be building blocks to new
devices or new physics that was prior unexpected. It enables us to design
structures with long-periods, which create new band structures like Moiré
bands in twisted bilayer graphene [35, 36]. We believe that this approach,
together with recent advances in machine learning protocols, may pave the way
for the design of new materials that may otherwise not be predicted by
equilibrium thermodynamics.

###### Acknowledgements.

We thank A. Fang for providing the raw STM data discussed in this manuscript,
and N. Sasaki and K. Izawa for fruitful discussions. Funding: This work was
initiated through a “QuantEmX” Exchange Awards (AK) and the Promotion of Joint
International Research (YF) at ESPCI. Work at UEC (Japan) was supported by the
JSPS Grant No. 15KK0155 and 19K21844. Work at Stanford University was
supported by the U. S. Department of Energy (DOE) Office of Basic Energy
Science, Division of Materials Science and Engineering at Stanford under
contract No. DE-AC02- 76SF00515.

## Author contributions

Y.F., K.B., and A.K. initiated this work. Y.F. and H.K. constructed the theory
and carried out the calculations. Y.F., K.B., and A.K. wrote the paper.

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