# From Edge State Physics to Entanglement Spectrum: Studying Interactions and
Impurities in Two-Dimensional Topological Insulators

Marcela Derli  E. Novais<EMAIL_ADDRESS>Centro de Ciências
Naturais e Humanas, Federal University of ABC, Brazil.

(September 1, 2024)

###### Abstract

We present a novel theoretical approach to incorporate electronic interactions
in the study of two-dimensional topological insulators. By exploiting the
correspondence between edge state physics and entanglement spectrum in gapped
topological systems, we deconstruct the system into one-dimensional channels.
This framework enables a simple and elegant inclusion of fermionic
interactions into the discussion of topological insulators. We apply this
approach to the Kane-Mele model with interactions and magnetic impurities.

_Introduction_ : Two dimensional systems have captivated physicists with their
intriguing and often counter intuitive properties. Frequently, they demand the
development of new theoretical and numerical tools for investigation. A
notable historical example of this behavior is observed in the quantum Hall
effect. While the integer quantum Hall effect can be comprehensively explained
within the framework of non-interacting particles filling Landau bands, the
discovery of non-integer Hall plateaus due to electronic interactions and
disorder was a surprising revelation. Recently the better understanding of
topological properties in the band structure of two dimensional systems have
yielded new and exciting theoretical and experimental results. However, the
interplay of topology and correlations is still a relatively open area of
investigation.

In the study of strongly correlated two dimensional systems, one promising
approach is to leverage insights gained from one-dimensional models by
suitably identifying one dimensional structures in the higher-dimensional
system. Several methods can be employed to achieve this dimensional reduction.
One intuitively appealing approach is to first consider quantum wires and
subsequently introduce tunneling between them(Kane et al., 2002; Teo and Kane,
2014; Tam and Kane, 2021). The connection to the two dimensional system is
obtained using renormalization group arguments to the tunneling amplitude.
Another approach is to deconstruct the topological system into one dimensional
wires in order to classify the topological phases(Neupert et al., 2014;
Iadecola et al., 2016). In a complementary view to this anisotropic starting
point, we propose a novel approach that is particularly well-suited for
studying impurities in two-dimensional topological systems.

The manuscript revolves around a pivotal concept: the spectrum of entanglement
(Peschel, 2003) in gapped systems provides a valuable insights into the
dynamics of local operators(Qi et al., 2012). This fresh perspective
introduces a fictitious boundary, enabling us to represent the original wave
function as a superposition of _bulk states_ and _boundary states_. When
focusing on the behavior of local operators solely defined on the _boundary_ ,
it becomes evident that only the boundary states interact with these
operators. However, this introduction of the _boundary_ comes at the cost of
working within a formalism at a finite fictitious temperature, which is
determined by the problem’s characteristics, such as the presence of the
spectral gap.

By starting with a two-dimensional problem and transitioning to a one-
dimensional model at a finite temperature, we gain the ability to incorporate
interactions into the problem on much better footing than in a straightforward
approach within the original two-dimensional model. This avenue of
investigation presents a novel approach to studying electronic systems,
enabling a deeper comprehension of their dynamics and properties. Furthermore,
it presents a clear pathway towards two-dimensional bosonzation, and possibly
new numerical techniques, on these systems.

Though out this manuscript we consider natural unit, $\hbar=c=k_{B}=1$. The
manuscript is organized as follows: first the general formalism is introduced.
Subsequently, we delve into the analyses of the prototypical topological band
insulators, the _Kane-Mele model_. We add to this model local degrees of
freedoms and electronic interactions to investigate the interplay of
magnetism, interactions and topology. We finish with a conclusion and some
possible future perspectives.

_The physics of a hard-wall boundary:_ In a D-dimensional system, a hard-wall
boundary is a D-1 surface that prohibits the flow of particles and
information. These properties are implemented by specific boundary conditions
on the Schrödinger equation. For instance, let’s consider massive Dirac’s
fermions in two dimensions with a hard boundary located at $x=0$(Lieb et al.,
1961; Edward Witten, 2016; Fukui, 2020). We need to ensure that the single
particle wave function $\psi\left(x,y\right)=X\left(x\right)Y\left(y\right)$
follows the hard wall conditions

$\frac{\partial\psi\left(0,y\right)}{\partial
x}=0,\text{and}\,\psi\left(x<0,y\right)=0.$ (1)

These conditions lead to two very distinct type of $X\left(x\right)$
functions: _bulk_ and _boundary_ solutions. Bulk solutions correspond to
standing waves and must be zero at the boundary,

$\displaystyle X\left(x\right)$ $\displaystyle=x_{0}\sin\left[\varepsilon
x\right],$ (2)

with $x_{0}$ and $\varepsilon$ constants. Conversely, zero energy boundary
solutions can exist. They correspond to exponentially decaying functions

$\displaystyle X\left(x\right)$
$\displaystyle=x_{0}\left(1+\frac{x}{\xi}\right)e^{-\frac{x}{\xi}},$ (3)

where $\xi$ is the characteristic length of the state. The behavior of
$Y\left(y\right)$ can depend on the details of the microscopic theory. A well
known example is graphene(Lado et al., 2015). In graphene, an _armchair_ edge
in graphene gives rise to boundary states with energies away from the Fermi
level, while a _zigzag_ edge introduces states at the Fermi level with no
dispersion. The fact that we can classify solutions in the presence of a real
hard wall boundary as _bulk_ states and _boundary_ states is a powerful
insight into the spectrum of entanglement.

_Spectrum of entanglement and edge states_ : The concept of the entanglement
spectrum in topological systems was introduced by Li and Haldane in 2008 (Li
and Haldane, 2008). They investigated a Hall system on the surface of a sphere
and employed the Schmidt decomposition to to recast the problem in terms of
wave functions defined on each hemisphere of the sphere. The logarithm of the
Schmidt coefficients was then coined the _entanglement spectrum_. It encodes
the physical information on how the original wave function is weaved at the
fictitious boundary created by the choice of basis. Remarkably, through
numerical analysis, they observed that the entanglement spectrum accurately
reproduces the low energy spectrum of a genuine boundary in a Hall bar.

Following their pioneering work, subsequent contributions have further support
for these findings(Casini and Huerta, 2009; Giudici et al., 2018; Vidal et
al., 2003; Fidkowski, 2010; Qi et al., 2012; Swingle and Senthil, 2012; Chen
and Fradkin, 2013; Ejima et al., 2014; Buhr et al., 2011). These studies
consistently demonstrate that in topological systems the entanglement spectrum
faithfully reproduces the spectrum of genuine edge states, albeit at finite
fictitious _temperature of entanglement_ , $T_{E}$, and low energies.

In what follows, we assume that the initial wave function that we aim to
describe , denoted as$\left|g\right\rangle$, is the ground state wave function
of a gapped local Hamiltonian of a topological insulator. We proceed by
dividing the system into two partitions, $\left|A\right\rangle$ and
$\left|B\right\rangle$, with an arbitrary cut. The Hamiltonian can be written
as the sum of Hamiltonians pertaining to the individual partitions, namely
$H_{A/B}$ , along with the inter-partition Hamiltonian $H_{AB}$,

$H_{0}=H_{A}+H_{B}+H_{AB}.$ (4)

Figure 1: Partitioning a system using an arbitrary cut establishes a
fictitious boundary. The original ground state is re-written using the basis
of each partitions,
$\left\\{\left|A\right\rangle,\left|B\right\rangle\right\\}$. The expectation
value of operators defined at the boundary, $\left\\{{\cal O}_{i}\right\\}$,
can be done using only "boundary states" in
$\left\\{\left|A\right\rangle,\left|B\right\rangle\right\\}$. Many partitions
are possible and they correspond to different 1D channels that correlations
can propagate in the bulk.

The state $\left|g\right\rangle$ can always be expressed using the Schmidt
decomposition
$\left|g\right\rangle=\sum_{i}\lambda_{i}\left|A_{i}\right\rangle\left|B_{i}\right\rangle$,
where $\left|A_{i}/B_{i}\right\rangle$ form a complete basis of each
partition. These bases comprise both _bulk_ and _boundary_ states. The key
observation is that _bulk_ states must vanish at the boundary. As a
consequence, any set of operators, $\left\\{{\cal O}_{i}\right\\}$, that live
at the cut defining the partitions will couple only to the _boundary_ states
of the basis (Qi et al., 2012). When calculating expectation values of
$\left\\{{\cal O}_{i}\right\\}$, the state $\left|g\right\rangle$ can be
truncated to include only the _boundary_ states, leading to the definition of
two boundary theories,$H_{A}^{{\cal B}}$ and $H_{B}^{{\cal B}}$, which can be
used to label these states. In this context, these boundary theories can be
called as _entanglement models_. Notably, it has been demonstrated that if the
boundary theory is conformal(Qi et al., 2012), the maximally entangled
conformal state, known as the Ishibashi state
$\left|g^{*}\right\rangle=\sum_{i}\left|{\cal B}_{A,i}\right\rangle\left|{\cal
B}_{B,i}\right\rangle$, is projected into the desired wave function by the
_extrapolation length_ $\beta$,

$\left|g\right\rangle=e^{-\frac{\beta}{2}\left(H_{A}^{{\cal B}}+H_{B}^{{\cal
B}}\right)}\left|g^{*}\right\rangle.$ (5)

A similar discussion can also be found in (Chen and Fradkin, 2013; Swingle and
Senthil, 2012) and it was established that $\beta^{-1}=T_{E}$ is proportional
to the bulk gap of the topological insulator.

The evaluation of correlations perpendicular to the boundary direction
exhibits exponential decay due to the expansion of the ground state in terms
of boundary states. A direct consequence of Eq. (5) is that any two operators
situated on the fictitious boundary will also have exponentially decaying
correlations. These observations can be understood within the framework of the
Lieb-Robinson bound(Bravyi et al., 2006; Hastings, 2010; Nachtergaele and
Sims, 2010). In its more restricted form, the bound says that for a system
with a uniform gap and local interactions, the two point correlation function
of local operators ${\cal O}_{1}$ and ${\cal O}_{2}$ calculated over the
ground state should decay exponentially. Thus, the fictitious temperature on
Eq. (5) enforces the Lieb-Robinson bound along the direction of the cut.

The degree of arbitrariness in choosing the cut that creates the partitions is
an important question. When evaluating the expectation value of an operator at
the same spatial position, the particular cut is irrelevant. The main
advantage to use the spectrum of entanglement in this case is to leverage the
knowledge of the boundary theory and its symmetries. This approach enables us
to incorporate, for instance, fermionic interactions into the analysis,
expanding the scope of the discussion.

However, it is when evaluating operators $\left\\{{\cal O}_{i}\right\\}$ at
different positions we truly capture the underlying physics (see Fig. (1)).
All possible cuts are equally suitable to create the _entanglement model_ ,
hence each cut corresponds to an one-dimensional channel through which
information can propagate in the bulk. To obtain the expectation value, we
need to consider all possible one-dimensional channels between the operator
sites

$\left\langle{\cal O}_{1}{\cal
O}_{2}\right\rangle=\frac{1}{N}\sum_{\left\\{c\right\\}}\left\langle g|{\cal
O}_{1}{\cal O}_{2}|g\right\rangle,$

where $\left\\{c\right\\}$ are all possible cuts that passes through ${\cal
O}_{1}$ and ${\cal O}_{2}$ and $N$ is the number of these cuts. The shortest
path that connects ${\cal O}_{1}$ and ${\cal O}_{2}$ dominates the sum, since
Eq. (5) tell us that correlations will decay exponentially along the cut.

If we introduce additional terms to Eq. (5)_, $V_{A}^{{\cal B}}$_, the simple
description of the ground state in terms of Eq.(5) may no longer hold. A
common scenario is to add another Hilbert
space,$\left\\{\left|\vec{s}_{i}\right\rangle\right\\}$, at the boundary,
$V_{A}^{{\cal B}}=\sum_{i,\vec{s}}{\cal
O}_{i}\otimes\left|\vec{s}_{i}\right\rangle\left\langle\vec{s}_{i}\right|$.
The true ground state, $\left|G\right\rangle$, is not known _a priori_.
Nevertheless, the same reasoning that lead to Eq. (5) can be used to write

$\left|G\right\rangle=e^{-\frac{\beta}{2}\left(H_{A}^{{\cal B}}+H_{B}^{{\cal
B}}+V_{A}^{{\cal
B}}\right)}\left|g^{*}\right\rangle\otimes\left[\sum_{i}\left|\vec{s}_{i}\right\rangle\right],$
(6)

as an equivalent one dimensional problem at finite temperature. An entire
range of 1D tools can now be used to find $\left|G\right\rangle$. We now use
the Kane-Mele model to illustrate all these possibilities.

_Kane-Mele model with an IRLM:_ Arguably the simplest topological model that
preserves time reversal symmetry is the Kane-Mele model of free fermions on a
honeycomb lattice(Kane and Mele, 2005). Let us consider this model in its
standard form(Kane and Mele, 2005) and introduce a Hubbard interaction between
the fermions

$\displaystyle H_{0}$ $\displaystyle=\sum_{\left\langle
i,j\right\rangle,\sigma}c_{i\sigma}^{\dagger}c_{j\sigma}+i\Delta\sum_{\left\langle\left\langle
i,j\right\rangle\right\rangle,\sigma,\sigma^{\prime}}\nu_{ij}s_{\sigma\sigma^{\prime}}^{z}c_{i\sigma}^{\dagger}c_{j\sigma^{\prime}}$
$\displaystyle+u\sum_{i}n_{i\uparrow}n_{i\downarrow}$

where $n_{i\sigma}=c_{i\sigma}^{\dagger}c_{i\sigma}$, $\Delta$ in the next
nearest neighbor hopping strength, $\nu_{ij}=\pm 1$ depending on the hopping
orientation, $s^{z}$ in the $z$ Pauli matrix and $u$ is the Hubbard
interaction strength.

Figure 2: (color on-line) The Kane-Mele model with Coulomb interactions
between the electrons. A magnetic impurity is represented as a red dot. A
fictitious cut creates the two partitions,
$\left\\{\left|A\right\rangle,\left|B\right\rangle\right\\}$, and zip-zag
edges. The boundary theory on partition $\left|A\right\rangle$ is
$H_{A,K}^{{\cal B}}$, a Kondo model in a L.L. at finite temperature $T_{E}$,
represented by the dotted line.

In Fig (2), we introduce a cut that separates the two sublattices of the Kane-
Mele model, resulting in two zig-zag edges that can be described in terms of
Luttinger liquids(Kane and Mele, 2005). These edges exhibit fermionic
currents, namely $J_{A/B}^{R\uparrow}$ and $J_{A/B}^{L\downarrow}$. The fact
that the model preserves time-reserval symmetry implies that at low energies,
only marginal forward-scattering interactions need to be considered. These
interactions can be described by the boundary Hamiltonians

$H_{A/B,I}^{{\cal B}}=\frac{v_{f}}{4\pi}\int dx\sum_{a\neq
b\in\left\\{R\uparrow,L\downarrow\right\\}}\left[g_{1}J_{A/B}^{a}J_{A/B}^{b}+g_{4}\left(J_{A/B}^{a}\right)^{2}\right],$
(7)

where $v_{f}$ represents the Fermi velocity. By employing Abelian
bozonization(von Delft and Schoeller, 1998), the low energy theory of _each_
boundary shown in Figure 2) can be expressed as

$H_{A/B,0}^{{\cal B}}+H_{I}^{{\cal
B}}=\frac{\tilde{v}}{8\pi}\int_{-\infty}^{\infty}dx\frac{1}{g}\partial_{x}\phi_{A/B}^{2}+g\partial_{x}\theta_{A/B}^{2},$
(8)

where
$\left[\phi_{A/B}\left(a\right),\partial_{x}\theta_{A/B}\left(b\right)\right]=-4\pi
i\delta\left(a-b\right)$,
$g=\sqrt{\left(1+g_{4}-g_{1}\right)/\left(1+g_{4}+g_{1}\right)}$ is the
Luttinger parameter and
$\tilde{v}=v_{f}\sqrt{\left(1+g_{4}-g_{1}\right)\left(1+g_{4}+g_{1}\right)}$
is the bosonic velocity. The energy gap and the interactions in the bulk
system determine the Fermi velocity and the band width, which can be
approximated as $\Lambda\sim T_{E}$.

At a site on the edge of partition $A$ we introduce an interacting ressonant
level model with on-site repulsive interactions $H_{A,1}^{{\cal
B}}=H_{A,0}^{{\cal
B}}+V\sum_{\sigma}c_{0,\sigma}^{\dagger}d_{\sigma}+h.c.+UN_{\uparrow}N_{\downarrow},$
with $N_{\sigma}=d_{\sigma}^{\dagger}d_{\sigma}$. Using the Schrieffer-Wolff
transformation(Hewson, 1993) and taking the large $U$ limit, we obtain the
Kondo model in a L.L. as _one_ of the boundary theories(Lee and Toner, 1992;
Furusaki and Nagaosa, 1994; Gogolin et al., 2004)

$\displaystyle H_{A,K}^{{\cal B}}$ $\displaystyle=H_{A,0}^{{\cal
B}}+H_{A,I}^{{\cal B}}+J_{z}\partial_{x}\theta_{A}\left(0\right).S^{z}$
$\displaystyle+\left(J_{\perp}\Lambda\right)e^{-i\phi_{A}\left(0\right)}S^{-}+h.c.$
(9)

where $J_{k=z,\perp}\sim V^{2}/U$ and $\vec{S}$ the spin projection of the
fermion at the the $d$ level. The scaling equations for this Kondo problem
flow to strong coupling for repulsive interactions, $g<1$, and/or
antiferromagnetic Kondo coupling, $J_{z,\perp}>0$. The Kondo temperature can
be estimated using the usual argument as

$\frac{T_{K}}{T_{E}}=J_{\perp}{}^{\frac{-1}{1-g+\frac{4J_{z}}{\tilde{v}}}}.$
(10)

The physics of the ground wave function will depend on a comparison between
the Kondo temperature, $T_{k}$, and the temperature of entanglement, $T_{E}$,
which acts as an infrared cut-off to the renormalization process.

For repulsive interactions and when $T_{k}\gg T_{E}$ the Kondo singlet forms.
In the limiting case of $J_{z,\perp}\to\infty$, a $c$-fermion at the the
ressonat level site is bound to form the Kondo singlet. This effectively
removes the site from the lattice and introduces a zig-zag edge in the middle
of the bulk, see Fig. 3a. However, for a finite $J_{z,\perp}$ the many body
state is more complex. The fermion at the local level $d$ is entangled with
many electrons on the $c$-band, forming what is known as the Kondo cloud(V.
Borzenets et al., 2020). The Kondo cloud is not a well define entity in real
space, but it is usually assumed to have a characteristic length scale of
$\xi_{k}=\tilde{v}/T_{k}$. This region of the material, characterized by the
Kondo cloud, is an intriguing state of matter, that should be surrounded by
gapless real edge states. If a diluted set of these Kondo impurities is
separated by distances of the order of $\xi_{k}$, it could allow current to
pass through the bulk of the material. The tunneling of electrons in internal
edge states created by impurities have been proposed before as a mechanism to
destabilize the topological phase(Shen, 2017). The difference here is that the
region is not related to size of the impurities, but rather to the length of
the Kondo cloud.

(a)

(b)

Figure 3: (color on-line) (a) If $T_{E}\ll T_{k}$ a Kondo singlet is formed
(red region of length $\xi_{k}$). We expect that edge states exist at the end
of the Kondo cloud (depicted as dotted lines in the figure). (b) In the
perturbative regime, $T_{k}\lesssim T_{E}$, an effective $RKKY$ interaction
controls the dynamics of the magnetic impurities.

In the limit where $T_{k}\lesssim T_{E}$, we enter a perturbative regime where
the renormalized $J_{z,\perp}$ remains small due to an irrelevant flow or an
infrared cutoff. In this regime, multiple magnetic impurities can interact
through the $c$-band. To study this interaction, we need to evaluate the
Green’s functions of the $c$-electrons that mediate the interaction. For
simplicity, let us consider a pair of magnetic impurities,
$\left\\{\vec{S}_{1},\vec{S}_{2}\right\\}$, located on the same sublattice of
the honeycomb lattice, see Fig. 3b. We introduce a partition that connects the
two impurities through a minimum path of distance $\Delta x$ along the edge.
Since we are in the perturbative regime, it is straightforward to write a
Ruderman-Kittel-Kasuya-Yosida (RRKY) like interaction between the two
impurities using the finite temperature Green’s function of the L.L.(von Delft
and Schoeller, 1998),

$H_{RKKY}^{{\cal
B}}\propto\frac{J_{k}^{2}}{\left[\frac{\beta_{E}}{\pi}\sinh\left(\frac{\pi}{\beta_{E}}\Delta
x\right)\right]^{g}}\vec{S}_{1}.\vec{S}_{2}+\text{l.r.t.}\,.$ (11)

The ground state of the magnetic impurities is a singlet. Therefore, a
photoemission like experiment can then be used to directly determine the
Luttinger parameter in these types of systems by measuring several of these
singlet energies for different $\Delta x$.

The physics of a dense array of magnetic impurities will ultimately depend on
the interplay of the three energy scales in the problem: $T_{E}$, $T_{K}$,
$T_{RRKY}$. This interplay opens up a wide range of possibilities for
exploring the rich phenomenology of heavy fermions.

_Conclusions:_ In this manuscript we propose a new theoretical approach for
incorporating electronic interactions in the study of two dimensional
topological insulators. Building upon the established correspondence between
edge state physics and the entanglement spectrum in gapped topological
systems, we leverage this connection to deconstruct the two dimensional system
as one dimensional channels that connect any two points within in the system.
The effective theory of these channels is precisely the edge state theory at
the fictitious temperature of entanglement. The effective theory carries the
same topological protection that real edge states have. Consequently, we can
elegantly incorporate fermionic interactions into the framework. To analyze
these systems we can employ well-known techniques such as Abelian bosonazation
or te density matrix renormalization group. While considering one dimensional
channels in topological insulators is not a new idea(Kane et al., 2002; Teo
and Kane, 2014; Tam and Kane, 2021; Neupert et al., 2014; Iadecola et al.,
2016), our approach naturally emerges from the ground state wave function.

We follow the general discussion by considering the Kane-Mele model with
interactions and in the presence of magnetic impurities. It is well-
established that impurities can give rise to localized regions, or islands,
within a topological system, which harbor edge states(Shen, 2017). Our
findings reveal two intriguing scenarios. Firstly, in the Kondo regime we
identify a possible mechanism to destabilize the topological phase. Secondly,
in the perturbative regime we observe that the impurities can order
magnetically through a RRKY-like interaction. Interestingly, this magnetic
ordering in a dilute impurity system presents a unique opportunity to
experimentally measure the strength of fermionic interactions in the
topological system.

###### Acknowledgements.

M. Derli would like to thank CAPES for financial support. E. Novais would like
to thank useful discussions with T. C. Jordão and F. M. Hungria.

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