# Non-local corrections to the typical medium theory of Anderson localization

H. Terletska  A. Moilanen  K.-M. Tam  Y. Zhang  Y. Wang  M. Eisenbach  N. S.
Vidhyadhiraja  L. Chioncel  J. Moreno  Department of Physics & Astronomy,
Computational Science Program, Middle Tennessee State University,
Murfreesboro, Tennessee 37132, USA  Department of Physics & Astronomy,
Louisiana State University, Baton Rouge, Louisiana 70803, USA  Center for
Computation & Technology, Louisiana State University, Baton Rouge, LA 70803,
USA  Kavli Institute for Theoretical Sciences, University of the Chinese
Academy of Sciences, Beijing, 100190, China  Center for Nanophase Materials
Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA  Pittsburgh
Supercomputing Center, Carnegie Mellon University, PA 15213, USA  Center for
Computational Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37831,
USA  Theoretical Sciences Unit, Jawaharlal Nehru Centre for Advanced
Scientific Research, Bangalore, India.  Theoretical Physics III, Center for
Electronic Correlations and Magnetism, Institute of Physics, University of
Augsburg, and Augsburg Center for Innovative Technologies, University of
Augsburg, D-86135 Augsburg, Germany

###### Abstract

We use the recently developed finite cluster typical medium approach to study
the Anderson localization transition in three dimensions. Applying our method
to the box and binary alloy disorder distributions, we find a fast convergence
with the cluster size. We demonstrate the importance of the typical medium
environment and the non-local spatial correlations for the proper
characterization of the localization transition. As the cluster size
increases, our typical medium cluster method recovers the correct critical
disorder strength for the transition. Our findings highlight the importance of
the non-local cluster corrections for capturing the localization behavior of
the mobility edge trajectories. Our results demonstrate that the typical
medium cluster approach developed here provides a consistent and systematic
description of the Anderson localization transition in the framework of the
effective medium embedding schemes.

###### keywords:

Disorder , Anderson localization , cluster embedding , typical medium approach
, TMDCA

††journal: Localisation2020 - Special Issue in Annals of Physics

##  1 Introduction

Disorder as a ubiquitous feature of materials can cause profound effects on a
variety of their properties [1, 2]. Consequently, a careful control on the
concentration of defects in materials can be used to rationally change and
design new functionalities of modern quantum systems. One of the most
pronounced effects of disorder is the electron localization (Anderson
localization) and the associated metal-insulator transition [3]. The theory of
Anderson localization, where multiple scattering off impurities leads to the
spatial confinement of electrons, is well developed. It has been demonstrated
that in one and two dimensions, an arbitrarily small amount of disorder
localizes electrons, whereas in three dimensions states may be localized or
extended depending on the amount of disorder [4].

Numerical methods have played an important role in understanding the mechanism
of Anderson localization. Several standard computational tools have been
employed for finite-size lattice calculations, including exact
diagonalization, the transfer matrix method, the kernel-polynomial method, as
well as the multifractal approach [1, 5, 6]. While numerically robust, the
application of these methods to real materials often faces the challenge of
having to treat large localization lengths when being restricted to finite
system sizes. Moreover, often the techniques developed for detecting Anderson
localized states, in non-interacting systems, are not directly applicable to
interacting electron systems, as they are built on the knowledge of single and
not many-particle eigenstates.

Effective medium embedding methods have been developed over the past several
decades; presently, these aproaches constitute an alternative and
complementary way for treating disorder in materials. The most commonly used
approach for disordered systems is the coherent potential approximation (CPA)
[7, 8]. The CPA shares a similar conceptual construction with the dynamical
mean-field theory (DMFT), which has proven to be a very successful theory for
strongly correlated electron systems [9, 10]. Both CPA and DMFT are Green’s
function-based methods and can be easily combined to study the interplay of
disorder and electron localization. Conceptually, in these methods, the
original lattice is mapped to a single impurity embedded in a dynamical
effective medium determined self-consistently. The dynamics of the medium
allows for effectively capturing the disorder or interaction-induced
correlations effects at the impurity level. By construction, the CPA is a
local approximation, and to capture the multi-impurity scattering effects,
various cluster extensions have been developed. This includes the momentum-
space based Dynamical Cluster Approximation (DCA) and the real-space-cluster
molecular CPA [11, 12, 13].

Although these commonly used effective medium embedding methods allow studying
disorder effects, they, however, fail to capture the Anderson localization.
The main challenge here is that the linearly averaged (arithmetic average)
density of states (DOS), obtained from the corresponding disorder averaged
impurity (cluster) Green’s function calculated within the CPA (DCA), is not
critical at the Anderson transition. Hence, it cannot be used as an order
parameter to characterize the localized states due to disorder.

It is therefore of interest to explore the effective medium methods that
employ a proper order parameter capable of describing the Anderson
localization. There have been several proposals according to which the
Anderson transition might be detected by studying the statistical properties
of the local density of states (LDOS) and its distribution [3, 14, 15, 16, 17,
18, 19, 20]. Dobrosavljevic et al. [21] incorporated such ideas in the context
of the effective medium approach. They developed the typical medium theory
(TMT) and showed that the geometrically averaged LDOS is indeed an order
parameter for the Anderson transition. In the typical medium analysis, instead
of using the arithmetically averaged disorder Green’s function (as it is
implemented in the CPA and the DCA), the geometrical averaging is used in the
self-consistency loop. Such typical medium analysis has also been extended to
interacting disordered systems [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
33]. However, by construction, the TMT is a local single-site approximation
and, hence, it neglects the non-local spatial correlations. As a result, the
TMT underestimates the critical disorder strength for the Anderson transition
in a three dimensional (3D) model, and does not capture properly the mobility
edge trajectories for the uniform box distribution.

To overcome such limitations, recently, we have developed the typical medium
dynamical cluster approach (TMDCA) [34, 35, 36, 37, 38, 39, 40], which is a
cluster extension of the single-site typical medium method [21]. As we
demonstrated in ref. [35, 41], such typical medium analysis can properly
capture the non-self-averaging behavior of the Anderson localization
phenomena. In particular, it captures the dramatic changes in the distribution
of the local density of states (LDOS) through the transition. At small
disorder strength, the LDOS follows a Gaussian distribution, while it is a
skewed log-normal distribution at large disorder. The calculated typical-
medium DOS (TDOS) obtained from the geometrically averaged cluster Green’s
function can capture such behavior. As shown in [35, 41], the TDOS vanishes
for the localized states, while it is finite for the extended states. The
cluster TMDCA method when applied to the three dimensional Anderson model not
only captures accurately the critical strength of the disorder, but also the
reentrance behavior of the mobility edge. We have also extended the
application of the TMDCA method beyond the simple Anderson model, including
systems with electronic interactions [23, 40, 42], off-diagonal disorder [36,
43], multi-band [38], and phonon systems [44, 45, 46]. Recently, such
developments have been successfully applied in the context of ab-initio
calculations of Anderson localization in superconductors [38], dilute magnetic
semiconductors [39], photovoltaics [47], and binary alloy systems [17].

The goal of the present work is to further benchmark the TMDCA method for the
Anderson model in three dimensions. We perform a careful systematic cluster
size analysis of the electron localization for both box and binary disorder
distributions. Our results indicate that non-local cluster corrections are
significant in capturing electron localization, hence, the finite cluster
TMDCA analysis is necessary for the proper description of disorder effects in
the Anderson model.

The paper is organized as follows. In section 2, we provide a short overview
of the model and the TMDCA method. In section 3, we present the results of the
application of the TMDCA method for the 3D Anderson model with box and binary
alloy disorder distributions. We conclude, with a discussion of the prospects
of the method, in the last section.

##  2 Model and Method

We study the Anderson model of non-interacting electrons subjected to a
disordered random potential

$H=-t\sum_{<{i,j}>}(c^{\dagger}_{i}c_{j}+H.c.)+\sum_{i}V_{i}n_{i},$ (1)

here the operators $c^{\dagger}_{i}$ and $c_{i}$ are the creation and
annihilation operators, respectively, for an electron on site $i$,
$n_{i}=c_{i}^{\dagger}c_{i}$ is the number operator, and $t$ is the hopping
energy between nearest neighbors $i$ and $j$. The first term is the kinetic
energy operator due to the hopping of electrons on a lattice, and the second
term is the local on-site disorder potential. The disorder potential is a
random quantity distributed according to some specified probability
distributions $P(V_{i})$. We set $4t=1$ to serve as the energy scale.

We perform our analysis for two types of disorder distribution: the (uniform)
box disorder, which is given by the distribution function
$P(V_{i})=\frac{1}{W}\Theta(|W/2-V_{i}|)$ (the disorder strength is
characterized by $W$), and the binary alloy disorder distribution with
$P(V_{i})=c_{a}\delta(V_{i}-V_{A})+c_{b}\delta(V_{i}-V_{b})$. Here $c_{a}$ is
the concentration of the host $A$ ions, and $c_{b}=1-c_{a}$ stands for the
concentration of the impurity $B$ ions. We introduce a shorthand notation for
disorder averaged quantities: $<...>=\int dV_{i}P(V_{i})(...)$.

The TMDCA is a typical medium extension of the conventional DCA scheme [11,
12]. Just as in the DCA approach [12], we map the original lattice into a
cluster of size $N_{c}$ (constructed in momentum space) embedded in the
effective medium. The effective medium is determined self-consistently. The
TMDCA utilizes the geometric averaging over disorder for the cluster Green’s
function, while the DCA uses the arithmetic (linear) averaging. To construct a
$N_{c}$ cluster, the first Brillouin zone is divided into $N_{c}=L_{c}^{D}$
($L_{c}$ is the linear cluster size, $D$ is the dimension) coarse-grained
cells with the cluster centers $K$ surrounded by points $\tilde{k}$ within the
cell such that the lattice momentum $k=K+\tilde{k}$. Both DCA and TMDCA
systematically incorporate the non-local spatial fluctuations as the cluster
size $N_{c}$ increases, and becomes exact in the limit
$N_{c}\rightarrow\infty$. The non-local short-range spatial correlations are
treated explicitly within the range of the cluster $N_{c}$, while the long
length scale correlations are treated within the typical medium. As in the DCA
scheme, the TMDCA self-consistency loop is constructed for the momentum $K$
dependent quantities, while to solve the cluster problem, one employs the
Fourier transform to the real space of $N_{c}$ sites with $(I,J)$ the site
indices [11].

In the TMDCA scheme, the main quantity of interest is the cluster typical
Green’s function $G_{typ}^{c}(K,w)$ which is obtained from the Hilbert
transform of the corresponding cluster typical density of states
$\rho_{typ}^{c}(K,w)$. Here $\rho_{typ}^{c}(K,w)$ is obtained using the
geometrical averaging ansatz of the form [35, 41]:

$\rho_{typ}^{c}(K,w)=\overset{local-
TDOS}{{\overbrace{exp\left(\frac{1}{N_{c}}\sum_{I=1}^{N_{c}}<\ln(\rho_{I}^{c}(w,V)>\right)}}}\overset{non-
local}{\times{\overbrace{\left<\frac{\rho^{c}(K,w,V)}{1/N_{c}\sum_{I}\rho_{I}^{c}(w,V)}\right>}}}.$
(2)

Here $\rho_{I}^{c}(w,V)=-\frac{1}{\pi}ImG^{c}_{II}(w,V)$ is the local density
of states at site $I$ obtained from the cluster Green’s function
$G^{c}_{IJ}(w,V)$; $\rho^{c}(K,w,V)=-\frac{1}{\pi}G^{c}(K,w,V)$ is a non-local
density of states determined from the Fourier transform of the cluster Green’s
function $G_{IJ}^{c}$. In the ansatz of Eq. 2, to avoid self-averaging at
strong disorder, we separate the ”local-TDOS”, which utilizes the geometric
averaging over disorder, from the ”non-local” $K$-dependent contributions [35,
37, 34]. Later we will show that such ansatz indeed can capture effectively
the electron localization in the Anderson model. To understand better the
contribution to the TDOS coming from the local and non-local parts in the
above ansatz, we will also perform our calculations using the ”local” ansatz
only, with

$\rho^{local-
TDOS}_{typ}(K,w)={\exp\left(\frac{1}{N_{c}}\sum_{I=1}^{N_{c}}<\ln(\rho_{I}^{c}(w,V)>.\right)}$
(3)

Also notice that $\rho_{typ}^{c}(K,w)$ in Eq. 2 possesses the following
properties [35, 37, 34]: for the $N_{c}=1$ case, it reduces to the local TMT
with $\rho_{typ}^{c}(K,w)=\exp(<\ln\rho^{c}(w,V)>)$. And, at weak disorder
strength, the TMDCA reduces to the DCA with
$\rho_{typ}^{c}(K,w)\rightarrow<\rho^{c}(K,w,V)>$.

In the following we outline the TMDCA self-consistent iterative procedure that
we use in our calculations:

1\. Starting from a guess for the effective medium hybridization function
$\Delta(K,w)$, we first construct the cluster-excluded Green’s function

$\mathcal{G}(K,w)=\frac{1}{w-\Delta(K,w)-\bar{\epsilon}(K)},$ (4)

where $\bar{\epsilon}(K)$ is the coarse-grained bare dispersion. For the 3D
cubic lattice, the bare lattice dispersion is given as
$\varepsilon(k)=-2t(\cos(k_{x})+\cos(k_{y})+\cos(k_{z}))$.

2\. Since the cluster problem is solved numerically in real space [11, 12], we
then Fourier transform $\mathcal{G}(K,w)$ to real space with
$\mathcal{G}_{I,J}=\sum_{K}\mathcal{G}(K)e^{iK(R_{I}-R_{J})}$.

3\. Now we are ready to solve the cluster problem using, e.g., a random
sampling. For this, we stochastically generate a random configuration of
disorder potentials, $V_{i}$, and construct the cluster Green’s function
$G_{c}$ by inverting the matrix

$\hat{G}_{c}(V)=(\mathcal{\hat{G}}^{-1}-\hat{V})^{-1}.$ (5)

Then we calculate the disorder-averaged cluster typical density of states
$\rho_{typ}^{c}(K,w)$ using the ansatz of Eq. 2, and the Hilbert transform to
obtain the cluster typical (geometrically averaged over disorder) Green’s
function

$G_{typ}^{c}(K,w)=\int
dw^{\prime}\frac{\rho_{typ}^{c}(K,w^{\prime})}{w-w^{\prime}}$ (6)

4\. With the cluster problem solved, we close the self-consistency loop by
calculating the lattice coarse-grained Green’s function

$\bar{G}(K,w)=\int\frac{N_{o}^{c}(K,\epsilon)d\epsilon}{(G_{typ}^{c}(K,w))^{-1}+\Delta(K,w)-\epsilon+\bar{\epsilon}(K)}$
(7)

which is then used to obtain a new estimate for the cluster-excluded Green’s
function $\mathcal{G}(K,w)$. Such an iterative procedure is repeated, until
the self-consistency is reached, i.e., when the cluster typical Green’s
function $G_{typ}^{c}(K,w)$ and the coarse-grained lattice Green’s function
$\bar{G}(K,w)$ become equal.

##  3 Results and Discussion

###  3.1 Box disorder distribution

Figure 1: ADOS (solid line) and the TDOS (shaded area) as function of
frequency $\omega$ at different values of box disorder strength
$W=1.0,1.2,1.6,2.1$ calculated using the $N_{c}=64$ DCA and TMDCA methods,
respectively. The local TDOS (dashed line) is obtained using Eq. 2. Vertical
arrows indicate the approximate position of the mobility edge boundaries.

While the localization properties of the Anderson model with box disorder
distribution are well known from the literature [1], we consider it here to
demonstrate the validity of our numerical method. First, we start the
discussion of the results by comparing the disorder evolution of the $ADOS(w)$
(obtained using the conventional DCA scheme with the arithmetic averaging over
disorder in the self-consistency loop) and the typical $TDOS(w)$ (obtained
from our TMDCA procedure with the geometric averaging over disorder). Our
results for the cubic cluster $N_{c}=64$ are shown in Fig. 1. For the
$TDOS(w)$ , we also show the results obtained with the local ansatz of Eq. 3.
As expected, the disorder dependence of the $ADOS(w)$ and the $TDOS(w)$ are
very different: while the ADOS remains finite with increasing disorder
strength, the TDOS continuously gets narrower and eventually gets fully
suppressed. At weaker disorder strength $W$, the localization of electrons
starts at the band tails, and is detected by vanishing $TDOS(w)$ at higher
frequencies $w$. The mobility edge (shown by vertical arrows) separates the
extended states (with finite TDOS) from the localized (with zero TDOS) states.
As disorder strength $W$ increases, the TDOS gets suppressed at all
frequencies, indicating the localization of all electrons in the band. Such
suppression of the TDOS with disorder strength $W$ indicates that the TDOS
indeed can serve as an order parameter for Anderson localization.

To better understand the role of the non-local contribution (with the full
momentum $K$ dependence) in the ansatz for the geometrically averaged cluster
Green’s function in Eq. 2, in Fig. 1, we also show the results for the
$TDOS(w)$ obtained with the local ansatz of Eq. 3. Our data indicate that the
most contribution to the $TDOS(w)$ is actually coming from the geometrically
averaged local DOS part in the ansatz, i.e., it is well-captured by Eq. 3. The
critical behavior at the Fermi level is the same for both the local and non-
local ansatze. However, the non-local $K$\- dependent contribution of Eq. 2
seems to be important for capturing properly the mobility edge behavior
(marked by vertical arrows). Here, at edges, we observe the biggest difference
between the local $TDOS(w)$ and the $TDOS(w)$ obtained with full $K$ momentum
dependence. This indicates that while the critical behavior at the band center
is captured properly in the local ansatz with the geometric averaging over the
LDOS, the mobility edge trajectories for the local ansatz case will, however,
converge slower with the cluster size $N_{c}$.

Figure 2: (Left a) panel): The $TDOS(\omega=0)$ (obtained using ansatz of Eq.
2) vs disorder strength $W$ for different cluster sizes $N_{c}=1,8,27,64$. For
$N_{c}=64$ cluster, we also show the data for $TDOS(w=0)$ calculated using
local ansatz of Eq. 3 (shown by orange crosses). (Right b) panel: The critical
disorder strength of the Anderson transition $W_{c}$ as function of $N_{c}$.
The critical disorder strength $W_{c}$ is determined from $TDOS(w=0)=0$ data
of the left panel.

Next, we consider the evolution of the critical disorder strength $W_{c}$ of
the Anderson localization as a function of the cluster size $N_{c}$. The
critical disorder strength $W_{c}$ is obtained by calculating the TDOS at the
band center TDOS($w$=0) as a function of disorder strength $W$. The $W_{c}$ is
then defined by vanishing TDOS($w$=0)=0. We have done such analysis for
several cluster sizes $N_{c}=1,8,27,64$ on a cubic 3D Anderson model lattice.
Our results are shown in Fig. 2 (panel a)), where we plot the $TDOS(w=0)$ at
the band center as a function of disorder strength $W$. The TDOS($w$=0)
decreases with increasing disorder strength $W$, and approaches zero at
$W_{c}$. Performing a careful analysis for different clusters $N_{c}$, we
demonstrate that the critical disorder strength $W_{c}$ converges quickly with
the cluster size $N_{c}$ (panel b) of Fig. 2). These data also highlight the
importance of going beyond the single-site approximation when describing the
critical behavior of the Anderson transition. For $N_{c}=1$ (which corresponds
to the local TMT approximation case) the $W_{c}\approx 1.675$, and increases
gradually to the converged value of $W_{c}\approx 2.25$ as cluster size
$N_{c}$ gets larger. These results are in a good agreement with the $W_{c}$
reported in the literature. While TMDCA slightly overestimates the $W_{c}$ as
compared to exact results, the advantage of our method is that it can be
applied for the interacting and realistic systems. Finally, for $N_{c}=64$, we
also show the results obtained with the local ansatz for the TDOS of Eq. 3
(the corresponding data are shown by the orange crosses in Fig. 2). The local
TDOS data fall on the top of the $TDOS(w=0)$ obtained using full $K$
dependence. This indicates the critical disorder strength $W_{c}$ of Anderson
localization can be very well captured by simplified ansatz of Eq. 3, which
should be an important simplification when applying the TMDCA for more
realistic models.

###  3.2 Binary alloy disorder distribution

Figure 3: (Top panel): $N_{c}=1$ CPA and TMT results for the ADOS($w$) and
TDOS($w$) at different disorder strength values $V_{A}=0.2,0.5,0.65.,0.8$.
(Bottom panel): $N_{c}=64$ DCA and TMDCA results for the ADOS($w$) and
TDOS($w$) as function of increasing disorder $V_{A}=0.2,0.5,0.675,0.835.$ The
$N_{=}64$ TMDCA data for the TDOS($w$) (shaded region) are obtained using
ansatz of Eq. 2, the local TDOS($w$) curves (dashed lines) are obtained using
a simplified local ansatz of Eq. 3. Other parameters: $c_{a}=0.5$

To further explore the application of our TMDCA approach, we now consider its
implementation for the binary alloy systems. First, in Fig. 3, we show the
results for the ADOS($\omega$) and TDOS($\omega$) at different disorder
strength $V_{A}$. To highlight the significance of the non-local spatial
effects, we present the data for the local $N_{c}=1$ (TMT) and the non-local
$N_{c}=64$ TMDCA case. For binary alloy systems, when increasing the disorder
strength $V_{A}$, the system undergoes two phase transitions, i.e., the
Anderson transition which is detected by vanishing $TDOS(w=0$) at the band
center, and the band-gap opening insulator transition detected by vanishing
$ADOS(w=0$) at the Fermi level. For small disorder strength ($V_{A}=0,2$),
both the ADOS and the TDOS are practically the same. As disorder strength
increases, the band gap opens in both the ADOS and the TDOS at the Fermi
level. For the local $N_{c}=1$ case, the Anderson localization of the states
at the Fermi level and the band splitting transition occur at almost the same
disorder strength, i.e., the TDOS$(w=0)=0$ at $V_{c}^{typ}\approx 0.6275$, and
ADOS$(w=0)=0$ at $V_{c}^{ave}\approx 0.635$. As disorder strength $V_{A}$
increases, the TDOS becomes significantly smaller and narrower than the ADOS.
Regions where $ADOS(w)$ remains finite but $TDOS(w)$ is zero, indicate the
Anderson localized states, separated by the mobility edges (marked by vertical
arrows). Comparing the $N_{c}=1$ and finite cluster $N_{c}=64$ results, we
find that the non-local spatial correlations, which are included in the TMDCA
scheme, introduce a noticeable difference in the localization behavior. In
particular, the non-local effects are responsible for the finite structures in
the $ADOS(w)$, which are completely smoothed out in the local $N_{c}=1$ case.
Moreover, we find that the Anderson transition at the Fermi energy with
$TDOS(w=0)$ occurs much faster than the gap opening in the $ADOS(w)$.
$N_{c}=64$ results also indicate that the mobility edge trajectories are wider
than in $N_{c}=1$ case, i.e., the local TMT scheme underestimates the extended
states region [41, 35]. We also show the results (black dashed lines) for
$N_{c}=64$ local $TDOS(w)$ obtained using a simplified ansatz of Eq. 3. As in
the described above box distribution case, we see that the major contribution
to the $TDOS(w)$ is coming from the geometrically averaged local TDOS factor.
The full $K$\- dependence in the ansatz of Eq. 2 seems to be mostly relevant
for capturing the higher frequency mobility edge behavior.

Figure 4: (Right a) panel). Top: ADOS$(w=0)$ as function of disorder strength
$V_{A}$ for different cluster sizes $N_{c}=1,8,27,64,125$. Bottom: TDOS$(w=0)$
as function of disorder strength $V_{A}$ for different cluster sizes
$N_{c}=1,8,27,64,125$. (Left b) panel): The critical disorder strength of the
Anderson transition $V_{c}^{typ}$ and of the band-splitting transition
$V_{c}^{ave}$ as function of the cluster size $N_{c}$. $V_{c}$ are determined
from the corresponding TDOS$(\omega=0)=0$ and ADOS$(w=0)=0$ data of the left
panel. Other parameters: $c_{a}=0.5$

To explore the cluster size dependence of the critical disorder strength
$V_{c}$ for the corresponding transitions in the binary alloy model, in Fig. 4
we plot the ADOS$(w=0)$ (a) panel, top graph) and the TDOS$(w=0)$ (a) panel,
bottom graph) as a function of increasing disorder strength $V_{A}$ at
different cluster sizes $N_{c}=1,8,27,64,125$. The critical value for the
Anderson localization transition $V_{c}^{typ}$ is then extracted from the
vanishing $TDOS(w=0)$, and zero in $ADOS(w=0)$ determined the critical
disorder strength $V_{c}^{ave}$ for the band-gap opening transition. For the
$N_{c}=1$, the Anderson localization and the the band-splitting transition
occur almost simultaneously. However, at $N_{c}>1$, the Anderson localization
clearly proceeds the band-gap opening, and occurs at smaller values of
$V_{c}$. The difference in cluster size $N_{c}$ convergence is better observed
in Fig. 4 (b) panel), where we plot the critical disorder strength
$V_{c}^{typ}$ and $V_{c}^{ave}$ (extracted from the corresponding data on the
left panels) as function of the cluster size $N_{c}$. We observe that
$V_{c}^{typ}$ converges rather quickly with increasing cluster size $N_{c}$,
while there is significant $N_{c}$ dependence of $V_{c}^{ave}$ for the band-
splitting transition implying the importance of strong non-local spatial
correlations.

##  4 Conclusion

We use the developed finite cluster typical medium TMDCA approach to study the
Anderson localization in three dimensions for the box and binary alloy
disorder distributions. By performing a careful systematic cluster size
analysis, we demonstrate that TMDCA presents a successful and numerically
manageable effective medium embedding approach for the Anderson localization.
We show that non-local correlations are significant for the proper analysis of
the Anderson transition, and hence the importance of employing beyond the
local single site approximations. Using the typical density of states as the
order parameter for Anderson transition, we obtained the cluster size
converged critical disorder strengths for the transition $W_{c}\approx 2.25$
(box distribution) and $V_{c}\approx 0.675$ (binary alloy case), which is in
good agreement with the known results in the literature. Finite cluster
$N_{c}$ values for the critical disorder strength $W_{c}$ are of noticeable
improvement over the local single-site TMT results with $W_{c}\approx 1.68$,
and $V_{c}\approx 0.6275$ for the box and binary distributions, respectively.
We have also demonstrated the importance of the non-local correlations in
capturing the spectral properties of the disordered systems. The application
of the TMDCA to more realistic and complex models is often faced with the
challenge of constructing a proper ansatz for the calculation of the
geometrically averaging Green’s function in the self-consistency numerical
loop. Our results show that the geometrically averaged local density of state
factor (in the expression for the typical medium Green’s function ansatz of
Eq. 2) is important for capturing properly the localization at the band center
and the critical disorder strength. While the non-local momentum $K$\-
dependent part in the ansatz plays felicitates a faster cluster size
convergence of the localized states at the edges. The performed analysis
presented in this work will allow for more effective application of the
developed TMDCA method to more complex disorder models, and provide a better
understanding of the role of non-local spatial correlations in the disordered
systems.

##  5 Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

##  6 Acknowledgement

HT has been supported by NSF OAC-1931367 and NSF DMR-1944974 grants. KMT is
partially supported by NSF DMR-1728457 and NSF OAC-1931445. YW is partially
supported by NSF OAC-1931525. The work of ME has been supported by U.S.
Department of Energy, Office of Science, Basic Energy Sciences, Material
Sciences and Engineering Division and it used resources of the Oak Ridge
Leadership Computing Facility,which is a DOE Office of Science User Facility
supported under Contract DE-AC05-00OR22725. LC gratefully acknowledges the
financial support offered by the Augsburg Center for Innovative Technologies
and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -
Projektnummer 107745057 - TRR 80/F6. JM is supported by the U.S. Department of
Energy, Office of Science, Office of Basic Energy Sciences under Award Number
DE-SC0017861.

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# SO($N$) singlet-projection model on the pyrochlore lattice

Matthew S. Block  Department of Physics & Astronomy, California State
University, Sacramento, CA 95819  Jared Sutton  Unaffiliated

###### Abstract

We present an extensive quantum Monte Carlo study of a nearest-neighbor,
singlet-projection model on the pyrochlore lattice that exhibits SO($N$)
symmetry and is sign-problem-free. We find that in contrast to the previously
studied two-dimensional variations of this model that harbor critical points
between their ground state phases, the non-bipartite pyrochlore lattice in
three spatial dimensions appears to exhibit a first-order transition between a
magnetically-ordered phase and some, as yet uncharacterized, paramagnetic
phase. We also observe that the magnetically-ordered phase survives to a
relatively large value of $N=8$, and that it is gone for $N=9$.

##  I Introduction

The search for exotic quantum phase transitions in two dimensions has been a
fruitful endeavor from the perspective of numerical investigations. Of note is
the prediction, detection, and characterization of so-called deconfined
quantum critical points. These critical points defy traditional Landau-
Ginzburg-Wilson theory of phase transitions by allowing for a direct,
continuous transition between two phases that break fundamentally different
symmetries. Furthermore, numerical evidence corroborated the claim that at the
critical point, an emergent U(1) gauge field mediates interactions between
spinon degrees of freedom that are normally confined in the adjacent
phases.Senthil et al. (2004) The numerical linchpin to the success of these
studies was the development of SU($N$)-symmetric spin-singlet-projection
models deployed on several different bipartite two-dimensional (2D)
lattices,Sandvik (2007); Lou et al. (2009); Kaul and Sandvik (2012) which
allowed for comparison to the large-$N$ calculations on the descriptive gauge
field theory. A natural extension was to consider the same type of sign-
problem-free operator on a non-bipartite lattice, such as the triangular Kaul
(2015) or kagome, Block et al. (2020) where the symmetry is merely SO($N$).
Both of these studies showed evidence of exotic critical points separating the
phases as well as the presence of spin-liquid phases.

An obvious question is whether these critical transitions persist into three
dimensions where previous studies of a similar nature have seen them lost to
comparatively plain first-order transitions.Block and Kaul (2012) Here we take
the first step in these investigations by deploying the same SO($N$)-symmetric
nearest-neighbor singlet-projection model, augmented by a next-nearest-
neighbor permutation term, on the three-dimensional (3D), non-bipartite
pyrochlore lattice.

##  II Model

We consider the pyrochlore lattice where each site has a Hilbert space of $N$
states, denoted for site $j$ as $\Ket{\sigma}_{j}$ where $\sigma=1,\ldots,N$.
We will refer to this as the _color_ of the spin. By using the fundamental
representation of SO($N$) on each site it is possible to construct spin
singlets on any two sites:
$\Ket{S_{ij}}=\frac{1}{\sqrt{N}}\sum_{\sigma}\Ket{\sigma\sigma}_{ij}$. We can
then construct the singlet-projection operator for a pair of sites as
$\hat{\mathcal{P}}_{ij}=\Ket{S_{ij}}\Bra{S_{ij}}$. This follows closely the
previous numerical studies on the triangular lattice Kaul (2015) and kagome
lattice Block et al. (2020). We consider this operator acting on nearest
neighbors of the lattice and this is the first term in our model Hamiltonian:

$\hat{\mathcal{H}}_{J_{1}}=-J_{1}\sum_{\braket{ij}}\hat{\mathcal{P}}_{ij}.$
(1)

We can study this model on its own for integer values of $N$ to map out the
phase diagram as a function of the symmetry order (see Results below). To gain
a more detailed understanding of the phase transition between observed phases,
we can add a second term that acts on the shortest bonds joining sites on the
same sublattice (denoted $\\{ij\\}$; see Fig. 1):

Figure 1: (color online). A small cluster of the pyrochlore lattice ($L=4$).
Sites on the each of the four sublattices are shown in the same color. The
solid lines connect nearest-neighbor sites where the coupling is $J_{1}$. A
single representative dotted line is shown connecting nearest-neighbor sites
_on the same sublattice_ where the coupling is $J_{2}$. All such next-nearest-
neighbor sites are coupled in the same way.

$\hat{\mathcal{H}}_{J_{2}}=-J_{2}\sum_{\\{ij\\}}\hat{\Pi}_{ij},$ (2)

where
$\hat{\Pi}_{ij}=(1/N)\sum_{\sigma,\eta}\Ket{\sigma\eta}_{ij}\Bra{\eta\sigma}_{ij}$,
the so-called permutation operator, which encourages magnetic ordering. Our
full model is thus
$\hat{\mathcal{H}}=\hat{\mathcal{H}}_{J_{1}}+\hat{\mathcal{H}}_{J_{2}}$. In
the $J_{2}$-only model for any finite $N$, each sublattice would be perfectly,
but independently, magnetically ordered. By turning on a small $J_{1}$ at that
point, the sublattices couple together and all spins align. Therefore, if we
start in the paramagnetic phase for some large $N$, there must exist some
$g\equiv J_{2}/J_{1}$ beyond which magnetic order is restored. By varying $g$,
we can continuously tune from one phase to the other and perform a detailed
study of the properties of the phase transition.

##  III Measurements

In all cases, we study lattices with $N_{\text{spin}}=L^{3}/4$ sites where $L$
is the side length of the cubic lattice in which we could inscribe our section
of pyrochlore. Periodic boundary conditions are enforced along all three
standard axes, which preserves the rotational symmetry of the lattice. We
employ the stochastic series expansion (SSE) method for our quantum Monte
Carlo (QMC), which samples via local bond updates and global loop updates.
Aside from some proprietary measurement code and the generalization to an
arbitrary symmetry order $N$, the QMC algorithm was developed and described in
detail by Anders Sandvik. Sandvik (2010)

The loops built by the SSE algorithm live within a $(3+1)$-dimensional
spacetime the size of which scales as $L^{3}\beta$, where $\beta$ is the usual
reciprocal temperature. They connect spins of the same color and, as such, the
proliferation of these loops throughout the lattice speaks to the degree of
long-range magnetic ordering. More precisely, if the loops wrap around the
entire lattice, we call this a _winding_ and the total number of times this
occurs among all loops along one of the three standard axes constitutes the
_winding number_ for that direction:

$W_{\mu}=\sum_{i=1}^{\text{\\# of loops}}\frac{\text{displacement of loop $i$
along $\mu$-axis}}{L}$ (3)

for $\mu=x,y,z$. This turns out to be an extensive quantity since the
spacetime grows with lower temperature (larger $\beta$), but, in three
dimensions, we can define the intensive _spin stiffness_ along each direction:

$\rho_{s,\mu}\equiv\frac{\Braket{W_{\mu}^{2}}}{\beta\,L}$ (4)

and this is our primary order parameter. Given the symmetry of the lattice, we
expect to have $\rho_{s,x}=\rho_{s,y}=\rho_{s,z}$ and can therefore look at
any one component or average across all three components to improve the
quality of our statistical estimates. We note that $N=9$ appeared to be just
on the paramagnetic side of the transition, but very close to it, such that it
suffered from the ergodicity issue we will describe below. A similar
resolution to what is described therein involving a comparison of energies
allowed us to conclude the $N=9$ was indeed paramagnetic.

##  IV Results

###  IV.1 The $J_{1}$-only Model and the Critical Value of $N$

We began by studying the $J_{1}$-only model while varying $N$ to determine how
the symmetry order affected the realized phase. A modest investigation using
system sizes $L=4,8,12,16,24,48$ (Fig. 2) revealed that magnetic order
persists for $N\leq 8$ and vanishes for $N\geq 9$.

Figure 2: (color online). The magnetic order parameter, $\rho_{\mathrm{s},x}$,
as a function of reciprocal system size for various values of $N$ for the
$J_{1}$-only model. For sufficiently large $L$, we conclude that magnetic
order survives up through $N=8$ and is destroyed for larger $N$. We use
$\beta=10L$ for this part of the study. Error bars are too small to be
visible.

###  IV.2 The Ergodicity Problem

Our next goal is to characterize the nature of the transition between the two
known phases. To accomplish this, we start with a large value of $N$ on the
paramagnetic side of the critical value — in this case, we used $N=14$ — and
turn on the $J_{2}$ term. We used $\beta=6.25L$ for the remainder of this
study, relaxing a bit from the $\beta=10L$ used in the $J_{1}$-only model
above, which was perhaps a bit more than necessary.

To locate the transition, we considered a range of values of $g$ on system
sizes $L=8,12,16,24$. However, it became clear early in this investigation
that there was a breakdown of ergodicity in the Monte Carlo sampling. As $g$
was decreased, the stiffness appeared to have a finite value, but then,
occasionally, it would drop off to a significantly lower value, or even zero,
and then return to the trend of finite stiffness values for still lower values
of $g$. These jumps in the middle of otherwise smooth data are tell-tale signs
of inadequate equilibration and indeed a careful examination of the data
files, which were binned, showed the value of stiffness dropping as additional
bins were collected. However, some processor cores returned bins that didn’t
drop at all. Despite running millions of equilibration Monte Carlo sweeps, we
were forced to conclude that a report of zero stiffness was trustworthy — as
it exhibited lower energy and stiffness only ever _decreased_ with additional
bin collection — while a report of finite stiffness could not be believed.

Already, this behavior is compelling evidence of the presence of a first-order
transition. There appear to be two phases, close in energy, but quite distinct
in character with a magnetic order parameter that exhibits a discontinuity in
the neighborhood of the transition. But finding the location of the transition
precisely is very challenging if we cannot pinpoint where the order parameter
genuinely drops to zero due to the breakdown of ergodicity. It is a nightmare
scenario for a QMC practitioner because we are already performing millions of
equilibration sweeps at great computational cost for the larger systems sizes
and there is no way of knowing how many more millions would be necessary to
find the true ground state, or if that were even possible.

This breakdown of ergodicity is ultimately a consequence of the inadequacies
of our sampling algorithm, which relies on both local spin updates and global
loop updates within our spacetime. The loop updates are necessary for
ergodicity, but are maximally efficient when the loops are long such that they
update many spins at once. The function of the loop updates is therefore
analogous to the famous Swendsen-Wang Swendsen and Wang (1987) and Wolff Wolff
(1989) cluster updates for the Ising model (incidentally, our algorithm using
the Swendsen-Wang version in choosing to update every loop in the spacetime
with a random spin color each Monte Carlo sweep). The magnetically-ordered
phase, with its long-range order, has a small number of very long loops
joining spins of the same color, while the paramagnetic phase typically has a
huge number of tiny loops each with a random color. In the paramagnetic phase,
updating these loops is at once computationally costly and inefficient at
updating the states. The development of a new process to augment our sampling
algorithm that would more efficiently sample the paramagnetic phases has thus
far eluded us.

Ergodicity issues like this can often be addressed by merely raising the
temperature, but our analysis of energy versus temperature showed that we
required rather small temperatures to capture the ground state behavior. One
can try simply performing more equilibration sweeps, but, as mentioned above,
we are already doing quite a lot and, even if we did significantly more, we
could never be sure that we had indeed settled to the ground state near the
transition point. Another popular technique is thermal annealing Kirkpatrick
et al. (1983) where the temperature starts high and is systematically lowered
in an attempt to “trap” the system in the global minimum of energy while
avoiding any local minima of similar depth, but implementing this would have
required a significant overhaul of our code and there existed a much quicker
resolution that would also provide 100% confidence in the outcome: an ad hoc
form of variational Monte Carlo.

An unintended consequence of the metastability of the two states within the
QMC is that we are able to stay in one state or the other as we sweep past the
transition point while varying $g$. On the magnetically-ordered (MO) side, our
sampling algorithm will very efficiently find the lowest-energy state with
finite stiffness by starting with a random spin state and empty spacetime
(i.e., no loops initially); however, this may not be the true ground state as
$g$ is lowered and the transition is passed. Meanwhile, deep within the
paramagnetic (PM) phase, our simulation, which, to be clear, is fully ergodic
in principle, will find its way to a zero stiffness state (i.e., many tiny
loops in the spacetime) and record the QMC spacetime configuration to a file.
We can then copy this file for larger values of $g$ to use as a starting
point. By lightly equilibrating to adjust for the different value of $g$, we
can then find the lowest-energy state as we march to larger and larger values
of $g$. But, again, ergodicity breaks down near the transition and the
simulation will not easily find its way to the MO state until $g$ is raised
significantly beyond the transition point.

In summary, starting with a random state will incorrectly tell us that
magnetic order persists to a much lower value of $g$ than the transition value
while starting with a PM state will tell us that the paramagnetism persists to
a much higher value of $g$ than the transition value. But the true ground
state must have the lowest energy; the process we used here is reminiscent of
a variational Monte Carlo study Scherer (2017) in which an approximate ground
state is determined by minimization of energy as parameters are varied within
the proposed states. The famous shortcoming of this method is that one cannot
be sure that the true ground state looks anything like those being proposed.
Here, we do not have that problem; our ground state is either MO or PM, so
whichever has the lower energy must be the true ground state.

Figure 3: (color online). The energy per site as a function of $g=J_{2}/J_{1}$
calculated from two different metastable states within the QMC for $L=24$ and
$N=14$, the magnetically-ordered (MO) state and the paramagnetic (PM) state.
The states differ qualitatively in the value of the magnetic order parameter —
finite for MO and zero for PM — but yet they have very similar energies. The
energy data in the vicinity of the crossover is fitted with two separate best-
fit lines and the intersection point is calculated and reported as the
transition value of $g$, which we call $g_{c}$. Error bars are too small to be
visible.

Figure 4: (color online). The transition point drifts to larger values of $g$
as we increase system size suggesting that the magnetically-ordered phase is
less stable for larger $L$ with fixed $N$. We fit a power law to the values of
$g_{c}$ to extrapolate to the thermodynamic limit.

With this strategy, we compare the energy of the MO and PM states as a
function of $g$. The transition point, which we call $g_{c}$, occurs where the
energies cross over; see Fig. 3 for an example of this for $L=24$. We repeated
this energy crossover analysis for each of our system sizes and found the
value of $g_{c}$ in each case. We plot these values versus $1/L$ in Fig. 4 and
determine that the value of $g$ in the thermodynamic limit is
$g_{c,\infty}\approx 0.3993$ for $N=14$. Naturally, we expect this value to be
less for smaller $N$ (still above $N=8$) and greater for larger $N$, though we
did not investigate this in detail.

Figure 5: (color online). The magnetic order parameter, $\rho_{\mathrm{s},x}$,
as a function of $g$ for various system sizes. The sudden drop in the order
parameter provides strong evidence of a first-order transition. Error bars are
too small to be visible.

###  IV.3 The Character of the Transition

At last, we can return to the task of plotting the magnetic order parameter
versus $g$ to visualize the transition on various system sizes; see Fig. 5. No
longer are there artifacts of the inadequate equilibration due to the
breakdown of ergodicity. The data points show a smooth trend on either side of
the transition with invisible error bars and we can have certainty as to which
data to use. Once we have determined the transition value of $g$ from the
energy plots as described above, we can choose to use the PM stiffness data
when $g<g_{c}$ and the MO data when $g>g_{c}$ for each system size. This is
how the plot in Fig. 5 was created. The sudden drop of stiffness from a finite
value to zero across the transition, which persists on all system sizes
studied, provides additional and compelling evidence of a first-order
transition for this model on the pyrochlore lattice and that is our main
result.

##  V Conclusion

Our quantum Monte Carlo study of the SO($N$) singlet-projection model on the
pyrochlore lattice reveals the existence of a first-order transition
separating the magnetically-ordered phase from the paramagnetic phase. This
result is in contrast to the wide range of studies of the same model on
various two-dimensional lattices, which, with rare exception, all harbored
exotic critical points, though it is consistent with other 3D studies where
critical behavior is lost and replaced by mundane first-order transitions.

There exists ample opportunity for extensions of these investigations. For
example, we avoided the critical value of $N$ where magnetic order first
breaks down in the $J_{1}$-only model when studying the transition using the
$J_{1}$-$J_{2}$ model, but one could certainly lower $N$ to see if any
qualitative differences emerged, particularly for an odd value of $N$. Of
greater theoretical interest is to consider our same model on a non-trivial,
bipartite, three-dimensional lattice where we return to SU($N$) symmetry, such
as the diamond lattice. The diamond lattice project is already underway and
will serve as a logical capstone to this line of inquiry.

##  VI Acknowledgements

The authors wish to acknowledge and thank Ribhu Kaul (Penn State) who authored
much of the original QMC code and suggested the pyrochlore lattice as an
interesting application for the model studied herein. We gratefully
acknowledge the National Science Foundation and, in particular, the ACCESS
collaboration (formerly XSEDE) and the San Diego Supercomputer Center whose
Expanse cluster was instrumental to this study. The research reported here was
supported in part by NSF DMR-130040. M. Block was additionally supported by
Sacramento State’s Department of Physics & Astronomy Iloff Endowment Mentor
Support Grant. J. Sutton was supported by Sacramento State’s College of
Natural Sciences & Mathematics Summer Undergraduate Research Experience (SURE)
Award.

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