# Systematic Quantum Cluster Typical Medium Method for the Study of Localization in Strongly Disordered Electronic Systems

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## Abstract

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## 1. Introduction

## 2. Background: Electron Localization in Disordered Medium

#### 2.1. Anderson Localization

#### 2.2. Order Parameter of Anderson Localization

- We approximate the effect of coupling the block to the reminder of the lattice via Fermi’s golden rule—coupling $\Delta $ which is proportional to the density of accessible states.
- Since on average each cluster is equivalent to all the others, this density will also be proportional to some appropriate block DOS.

#### 2.3. On the Role of Interactions: Thomas-Fermi Screening

#### 2.4. The Mott Transition

#### 2.5. Interacting Disordered Systems: Beyond the Single-Particle Description

## 3. Direct Numerical Methods for Strongly Disordered Systems

#### 3.1. Transfer Matrix Method

#### 3.2. Kernel Polynomial Method

#### 3.3. Diagonalization Methods

## 4. Coarse-Grained Methods

#### 4.1. A Few Fundamentals

#### 4.2. The Laue Function and the Limit of Infinite Dimension

#### 4.3. The DCA

#### 4.3.1. Coarse-Graining

_{c}such samplings.

#### 4.3.2. DCA: A Diagrammatic Derivation

_{c}/2, where L

_{c}is now the linear size of the cluster, the Fourier transform of the Green’s function $\overline{G}(r)\approx G(r)+\mathcal{O}({(r\u2206k)}^{2})$, so that short-ranged correlations are reflected in the irreducible quantities constructed from $\overline{G}$; whereas, longer ranged correlations r > L

_{c}/2 are cut off by the finite size of the cluster [24]. Longer ranged interactions are also cut off when the transformation is applied to the interaction. To see this, consider an extended Hubbard model on a (hyper)cubic lattice with the addition of a near-neighbor interaction V∑

_{〈ij〉}n

_{i}n

_{j}where 〈ij〉 denotes near-neighbor pairs. When the point group of the cluster is the same as the lattice the coarse-grained interaction takes the form V sin (∆k/2)/(∆k/2)∑

_{〈ij〉}n

_{i}n

_{j}. It vanishes when N

_{c}= 1 so that ∆k = 2π. If N

_{c}is larger than one, then non-local corrections of length ≈ π/∆k to the DMFT/CPA are introduced.

#### 4.3.3. DCA: A Generating Functional Derivation

## 5. Typical Medium Theories of Anderson Localization: Model Studies

#### 5.1. Building Quantum Cluster Theories for the Study of Localization

- We approximate the coupling of the clusters to their lattice environment at the single-particle level (akin to the Fermi golden rule) neglecting two-particle and higher processes. This coupling is proportional to the square of a matrix element between the cluster and its host, times an appropriate DOS which describes the states available on the surrounding clusters.
- Since on average each cluster is equivalent to all the others, this DOS will also be proportional to some appropriate cluster density of states. In addition, since the distribution of the DOS is highly skewed, the typical DOS is quite different than the average DOS. The typical cluster DOS, which is clearly more representative of the local environment, will be used to define the effective medium.

- 3.
- Maintain the translational invariance of the impurity averaged cluster, i.e., there should be no distinction between, e.g., sites in the center and those at the boundary of the cluster.
- 4.
- The clusters should maintain the point group symmetries of the lattice.
- 5.
- The method should be fully causal, with positive definite spectra $A(\mathbf{K},\omega )=-1/\pi \Im G(\mathbf{K},\omega )>0$
- 6.
- It should recover the DCA when the disorder is weak.
- 7.
- it should recover the TMT when ${N}_{c}=1$
- 8.
- In lieu of interactions, the scatterings at different energies are completely independent of each other.
- 9.
- For large ${N}_{c}\to \infty $ it should become exact while avoiding self-averaging effects.
- 10.
- It should be extensible to multiple bands, and realistic models with longer ranged diagonal and off-diagonal disorder

#### 5.2. Typical Medium Dynamical Cluster Approximation (TMDCA)

- Ansatz 1$${\rho}_{typ}^{c}(\mathbf{K},\omega )=exp\frac{1}{{N}_{c}}\sum _{I}^{{N}_{c}}\langle ln{\rho}_{I}^{c}(\omega ,V)\rangle \u2329\frac{{\rho}^{c}(K,\omega ,V)}{\frac{1}{{N}_{c}}{\sum}_{I}{\rho}_{I}^{c}(\omega ,V)}\u232a\phantom{\rule{0.166667em}{0ex}}.$$$${G}_{typ}^{c}(\mathbf{K},\omega )=\int d{\omega}^{\prime}\frac{{\rho}_{typ}^{c}(\mathbf{K},{\omega}^{\prime})}{\omega -{\omega}^{\prime}}\phantom{\rule{0.166667em}{0ex}}.$$
- Ansatz 2While Ansatz 1 works rather well for simple single-band models with local and non-local disorder, we find that it can suffer from numerical instabilities when applied to complex first-principle effective Hamiltonians with many orbitals and non-local disorder potentials. Such numerical instabilities arise due to the Hilbert transformation which is used to calculate the Green’s function from the TDOS ${\rho}_{typ}^{c}(\mathbf{K},\omega )$. To avoid such numerical instabilities, we constructed the following Ansatz 2 [172] where we calculate ${G}_{typ}^{c}(\mathbf{K},\omega )$ directly as$${G}_{typ}^{c}(\mathbf{K},\omega )=exp\frac{1}{{N}_{c}}\sum _{I}^{{N}_{c}}\langle ln{\rho}_{I}^{c}(\omega ,V)\rangle \u2329\frac{{G}^{c}(\mathbf{K},\omega ,V)}{\frac{1}{{N}_{c}}{\sum}_{I}{\rho}_{I}^{c}(\omega ,V)}\u232a\phantom{\rule{0.166667em}{0ex}}.$$

- We start with a guess for the cluster self-energy $\mathsf{\Sigma}(\mathbf{K},\omega )$, usually set to zero.
- Then we calculate the coarse-grained cluster Green’s function $\overline{G}(\mathbf{K},\omega )$ as$$\overline{G}(\mathbf{K},\omega )=\frac{{N}_{c}}{N}\sum _{\tilde{k}}\frac{1}{\omega +\mu -\epsilon (\tilde{k}+\mathbf{K})-\mathsf{\Sigma}(\mathbf{K},\omega )}\phantom{\rule{0.166667em}{0ex}}.$$
- The cluster problem is now set up by calculating the cluster-excluded Green’s function $\mathcal{G}(\mathbf{K},\omega )$ as$$\mathcal{G}(\mathbf{K},\omega )=\frac{1}{\frac{1}{\overline{G}(\mathbf{K},\omega )}+\mathsf{\Sigma}(\mathbf{K},\omega )}\phantom{\rule{0.166667em}{0ex}}.$$
- Since the cluster problem is solved in real space, we then Fourier transform $\mathcal{G}$(
**K**,$\omega $) to real space: ${\mathcal{G}}_{I,J}={\sum}_{\mathbf{K}}\mathcal{G}\left(\mathbf{K}\right)exp(i\mathbf{K}\xb7({\mathbf{R}}_{I}-{\mathbf{R}}_{J}))$. - We solve the cluster problem using, e.g., a random sampling simulation. Here, we stochastically generate random configurations of the disorder potential V. For each disordered configuration, we construct the new fully dressed cluster Green’s function as$${G}^{c}\left(V\right)={({\mathcal{G}}^{-1}-V)}^{-1}.$$
- With the cluster problem solved, we use the obtained typical cluster Green’s function ${G}_{typ}^{c}(\mathbf{K},\omega )$ to obtain a new estimate for the cluster self-energy$$\mathsf{\Sigma}(\mathbf{K},\omega )={\mathcal{G}}^{-1}(\mathbf{K},\omega )-{({G}_{typ}^{c}(\mathbf{K},\omega ))}^{-1}$$
- We repeat this procedure starting from 2, until $\mathsf{\Sigma}(\mathbf{K},\omega )$ converges to the desired accuracy.

#### 5.3. Off-Diagonal Disorder

#### 5.3.1. DCA with Off-Diagonal Disorder

#### 5.3.2. TMDCA with Off-Diagonal Disorder

#### 5.4. TMDCA for Multi-Orbital Systems

#### 5.5. Disorder in Interacting Systems

#### 5.5.1. SOPT

#### 5.5.2. Stat DMFT Approach

#### 5.6. Two-Particle Calculations

## 6. Methodology for First-Principles Studies of Localization

#### 6.1. From DFT to the EDHM

- In the first step two DFT calculations are performed: a normal cell calculation of the pure host material and a supercell calculation of the host material with a single impurity in it. For example, for KFe${}_{2-y}$Se${}_{2}$, an iron-based superconductor that contains Fe vacancies, the normal cell of the host will be KFe${}_{2}$Se${}_{2}$. To capture the impurity potential of an Fe vacancy one can run a DFT calculation for a K${}_{8}$Fe${}_{15}$Se${}_{16}$ supercell containing a single Fe vacancy [199].
- The second step is to derive the low-energy Hamiltonians using a projected Wannier function transformation in which a set of atomic orbitals is projected on the bands close to the Fermi level [75,76,202]. For the case of KFe${}_{2-y}$Se${}_{2}$, one can project Fe-d and Se-p orbitals on the bands within [−6, 2] eV [199]. This results in two ordered tight-binding Hamiltonians. One for the normal cell ${H}^{0}$, and one for the single-impurity supercell ${H}^{\left({x}_{j}\right)}$.
- Finally, a superposition of these ordered Hamiltonians is used to build Hamiltonians of arbitrary impurity configurations. Specifically, the difference between the single impurity and pure Hamiltonian is taken to derive the single-impurity potential: ${V}^{\left({x}_{j}\right)}={H}^{\left({x}_{j}\right)}-{H}^{0}$. To remove the influence of the periodically repeated impurities in the single-impurity supercell calculation a partitioning procedure is necessary. A detailed account of this procedure is given in [202]. From single-impurity potential the effective Hamiltonian of a disordered impurity configuration with N impurities can be assembled as follows: ${H}_{\mathrm{eff}}^{({x}_{1},\dots ,{x}_{N})}={H}^{0}+{\sum}_{j=1}^{N}{V}^{\left({x}_{j}\right)}$.

#### 6.2. From the EDHM to TMDCA

## 7. Applications of the Typical Medium DCA to Systems with Disorder

#### 7.1. Results for the Anderson Model

#### 7.1.1. Typical DOS as an Order Parameter for Anderson Localization

#### 7.1.2. Cluster Size Convergence

#### 7.2. Results for Models with More Realistic Parameters

#### 7.2.1. Off-Diagonal Disorder

#### 7.2.2. Multiple Orbitals

#### 7.3. Results for Two-Particle Calculations

#### 7.4. Results for Interacting Models

#### 7.4.1. Results from SOPT

#### 7.4.2. Results from Stat-DMFT

#### 7.5. Results of First-Principles Studies of Localization

#### 7.5.1. Application to K${}_{y}$Fe${}_{2-x}$Se${}_{2}$

#### 7.5.2. Application to (Ga,Mn)N

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Acronym | Description |

ADOS | Average Density of States |

AL | Anderson Localization |

BEB | Blackman Esterling Berk |

CPA | Coherent Potential Approximation |

DCA | Dynamical Cluster Approximation |

DFT | Density-Functional Theory |

CDMFT | Cluster Dynamical Mean-Field Theory |

EDHM | Effective Disorder Hamiltonian Method |

JDM | Jacobi-Davidson Method |

KKR | Korringa-Kohn-Rostoker method |

KPM | Kernel Polynomial Method |

LAPW | Linear Augmented Plane Wave |

LDOS | Local Density of States |

LMA | Local Moment Approach |

MCPA | Molecular Coherent Potential Approximation |

MS | Multiple-Scattering |

NLCPA | Non-Local Coherent Potential Approximation |

ODD | Off-Diagonal Disorder |

QC | Quantum-Critical |

QMC | Quantum Monte Carlo |

SOPT | Second Order Perturbation Theory |

TDOS | Typical Density of States |

TMDCA | Typical Medium Dynamical Cluster Approximation |

TMM | Transfer Matrix Method |

TMT | Typical Medium Theory |

Symbol | Description |

$\mathbf{k}$ | wavenumber |

$\mathbf{K}$ | Cluster wavenumber |

$\mathbf{x}$ | lattice site coordinate |

$\mathbf{X}$ | Cluster site coordinate |

N | Number of lattice sites |

${N}_{c}$ | Number of cluster sites |

$\omega ,{\omega}_{n},z$ | Real and complex frequencies |

$M\left(\mathbf{k}\right)$ | DCA coarse-graining many to one map |

$\rho $ | Density of states |

V | Electronic potential |

$\u03f5$ | Electronic energy |

$\mu $ | Electronic chemical potential |

$\sigma $ | spin index |

t | Electronic Hopping matrix element (energy) |

m | Magnetization |

h | Magnetic Field |

$\chi $ | Two-particle Green’s function (tensor) |

F | Full vertex function (tensor) |

G | Single-particle Green’s function |

A | Single-particle spectral function |

$\Delta $ | Mean-field hybridization between cluster and host |

$\mathcal{G}$ | Host or cluster-excluded Green’s function |

$\mathsf{\Sigma}$ | Single-particle self-energy |

$\mathsf{\Gamma}$ | Irreducible vertex function |

$\mathsf{\Lambda}$ | Laue function |

${O}^{c}$ | A superscript “c” designates a cluster quantity |

${O}^{l}$ | A superscript “l” designates a lattice quantity |

${O}_{typ}$ | A subscript “typ” designates a cluster quantity |

$\overline{O}$ | denotes a coarse-grained quantity |

${O}_{I,J,\cdots}$ | uppercase subscripts indicate indices in cluster space |

${O}_{i,j,\cdots}$ | lowercase subscripts indicate indices in lattice space |

$\underline{O}$ | denotes a matrix in the Blackman formalism or in the multi-orbital system |

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**Figure 1.**Examples of various types of disorder, including substitution and interstitial impurities, and vacancies. In addition (not shown), disorder can originate from other ways of breaking the translational symmetry, including the external disorder potentials, amorphous systems, random arrangement of spins, etc.

**Figure 2.**Simultaneous treatment of the material-specific parameters, modeling disorder and electron-electron interactions present one of the major challenges for theoretical studies of electron localization in real materials.

**Figure 3.**The TMDCA may be used to study electron localization in both simple model Hamiltonians as well as those extracted from first-principles calculations.

**Figure 4.**To help understand localization, we divide the system into blocks. The average spacing of the energy levels of a block is $\delta E$ and the Fermi golden rule width of the levels is $\Delta $. If $\Delta \gg \delta E$ then we have a metal and if $\Delta \ll \delta E$, an insulator.

**Figure 5.**The global average (dashed lines) and the local (solid lines) DOS of the 3D Anderson model for small, moderate, and large disorder strength W with units $4t=1$ where t is the near-neighbor hopping (see text for details).

**Figure 6.**The evolution of the probability distribution function of the local DOS ${\rho}_{i}$ at the band center ($\omega =0$) with disorder strength W. The data is the same as in Figure 5.

**Figure 7.**The distribution of the local density of states at the band center (zero energy) in a single-band Anderson model with disorder strength $\gamma /t$ where $t=1$ is the near-neighbor hopping. Near the localization transition, $\gamma /t=16.5$ the distribution becomes log-normal (see also the inset) for over ten orders of magnitude, while for values well below the transition, $\gamma /3$ is shown, the distribution is normal [106].

**Figure 8.**The shift in the DOS parabola near a charged defect causes electrons to move away from the defect.

**Figure 9.**Screened defect potentials. The screening length increases with decreasing electron density n, causing states that were free to become bound.

**Figure 10.**Schematic of a transfer matrix method (TMM) calculation. Assuming the system has a width and height equal to M for each slice of a N-slice cuboid, forming a “bar” of length N, the amplitude of the wavefunction in the 0-th slice can be related to that in the N-th slice via the transfer matrix, Equation (10).

**Figure 11.**The mapping from the cluster to the lattice is accomplished by replacing the Green’s function and interaction by their coarse-grained analogs in the diagrams for the generating functional, self-energy and irreducible vertices. In the map back to the cluster, this self-energy is used to calculate a new cluster host Green’s function.

**Figure 12.**The first few graphs in the irreducible self-energy of a diagonally disordered system. Each ⚬ represents the scattering of a state $\mathbf{k}$ from sites (marked X) with a local disorder potential distributed according to some specified probability distribution $P\left(V\right)$. The numbers label the $\mathbf{k}$ states of the fully dressed Green’s functions, represented by solid lines with arrows.

**Figure 13.**The first few diagrams for the Hubbard model single-particle Green’s function. Here, the solid black line with an arrow represents the single-particle Green’s function and the wavy line the Hubbard U interaction.

**Figure 14.**The Laue function $\mathsf{\Lambda}$, which described momentum conservation at a vertex (left) with two Green’s function solid lines and a wiggly line denoting an interaction (perhaps mediated by a Boson). In the DMFT/CPA we take $\mathsf{\Lambda}=1$, so momentum conservation is neglected for irreducible graphs (right) so that we may freely sum over the momentum labels $\tilde{\mathbf{k}},{\tilde{\mathbf{k}}}^{\prime}\cdots $ leaving only local ($\mathbf{X}=0$) propagators and interactions.

**Figure 15.**The first few graphs of the CPA local self-energy of the Anderson model. Here the solid Green’s function line represents the average local propagator and the dashed lines the impurity scattering. These graphs may be obtained from the full set of graphs shown in Figure 12 by replacing each graphical element (Green’s function and impurity scattering lines) with its local analog coarse-grained through the entire first Brillouin zone.

**Figure 17.**Coarse-graining cells for ${N}_{c}=8$ (differentiated by alternating fill patterns) that partition the first Brillouin zone (dashed line). Each cell is centered on a cluster momentum $\mathbf{K}$ (filled circles). To construct the DCA cluster (e.g., for ${N}_{c}=8$) we map a generic $\mathbf{k}$ to the nearest cluster point $\mathbf{K}=\mathbf{M}\left(\mathbf{k}\right)$ (c.f. Figure 18) so that $\tilde{\mathbf{k}}=\mathbf{k}-\mathbf{K}$ remains in the cell around $\mathbf{K}$.

**Figure 18.**The DCA many-to-few mapping of an arbitrary point in the first Brillioun zone to one of ${N}_{c}=8$ cluster momenta $\mathbf{K}$.

**Figure 19.**Use of the DCA Laue function ${\mathsf{\Lambda}}_{DCA}$ leads to the replacement of the lattice propagators $G\left({\mathbf{k}}_{1}\right)$, $G\left({\mathbf{k}}_{2}\right)$, … by coarse-grained propagators $\overline{G}\left(\mathbf{K}\right)$, $\overline{G}\left({\mathbf{K}}^{\prime}\right)$, … The impurity scattering dashed lines and unchanged by coarse-graining since the scatterings are local.

**Figure 20.**Path-integral interpretation of the screening of a propagating particle. The single-particle lattice Green’s function, ${G}^{l}$, describes the quantum phase and amplitude the particle accumulates along its path as it propagates from space-time location 0 to x. It is poorly approximated by the cluster Green’s function from a small cluster calculation, ${G}^{l}\approx {G}^{c}$, especially when $x,r\le {L}_{c}$, the linear cluster size. Its self-energy, which describes generally short-ranged r screening processes, is well approximated ${\mathsf{\Sigma}}^{l}\approx {\mathsf{\Sigma}}^{c}$, by a small cluster calculation, especially when the cluster size ${L}_{c}$ is greater than the screening length. As discussed in Section 2 this screening length ${f}_{TF}\approx r$ which may be less than an Angstrom for a good metal. Therefore, rather than directly approximating the lattice Green’s function by the cluster Green’s function, the cluster self-energy is used to approximate the lattice self-energy in a Dyson equation for the lattice Green’s function ${G}^{l}={G}^{l}+{G}^{l0}+{G}^{l0}{\mathsf{\Sigma}}^{l}{G}^{l}$, where ${G}^{l0}$ is the bare lattice Green’s function.

**Figure 21.**A second-order term in the generating functional of the Hubbard model. Here the undulating line represents the interaction U, and on the LHS (RHS) the solid line the lattice (coarse-grained) single-particle Green’s functions. When the DCA Laue function is used to describe momentum conservation at the internal vertices, the momenta collapse onto the cluster momenta and each lattice Green’s function and interaction is replaced by the corresponding coarse-grained result.

**Figure 22.**A second-order term in the self-energy of the Hubbard model obtained from the first functional derivative of the corresponding term in the generating functional $\mathsf{\Phi}$ (Figure 21). When the DCA Laue function is used to describe momentum conservation at the internal vertices, the momenta collapse onto the cluster momenta and each lattice Green’s function and interaction is replaced by the corresponding coarse-grained result.

**Figure 24.**For off-diagonal disorder the hopping amplitude depends on the occupancy of the neighboring sites.

**Figure 25.**The diagrams for the first and second-order self-energy of the Hubbatr model labeled in real space. The indices $I,J$ indicate sites in the real-space cluster, while the lines are Hartree-corrected propagators $\tilde{\mathcal{G}}$.

**Figure 26.**The detailed algorithm implemented to solve the interacting disordered problem with a cluster solver built by combining statistical DMFT and a local impurity solver which could be, for example LMA or NRG. Note the self-consistency loop within the stat-DMFT cluster algorithm.

**Figure 27.**Bethe-Salpeter equation relating the two-particle Green’s function $\chi $ and the irreducible vertex $\mathsf{\Gamma}$. While $\mathbf{k}$, ${\mathbf{k}}^{\prime}$ and $\mathbf{q}$ represent momentum indices, $\omega $ and $\nu $ represent frequency indices (for fermionic and bosonic frequencies respectively) and the spin indices are suppressed. Please note that for the disordered systems considered here, the scatterings are elastic and thus the energy is conserved following any fermionic Green’s function line. Therefore, we only need two frequency indices to represent the frequency degree of freedom of the system.

**Figure 28.**Organization of the modular approach to first-principles calculations of localization. A DFT of the pure system and a DFT supercell calculation of a single impurity are performed as the first step. In the second step, the EDHM converts the DFT output into model parameters of the disordered system. In the third step, the TMDCA is used to study the materials-specific localization properties.

**Figure 29.**Spectral functions of the clean reference system KFe${}_{2}$Se${}_{2}$ (

**a**) and K${}_{4}$Fe${}_{8}$Se${}_{10}$ with one K vacancy and two Fe vacancies obtained from DFT (

**b**) and the effective Hamiltonian method (

**c**). Reprint from [199].

**Figure 30.**(Left) TMT (${N}_{c}=1$); (Right) TMDCA (${N}_{c}=38$). The ADOS (dash line) and the TDOS (solid lines) of the 3D Anderson model for different disorder strengths W in units where $4t=1$. At low disoder, the ADOS and TDOS coincide. This also indicates that the TMDCA and the DCA solutions are very similar. As disoder strength W increases the TDOS becomes suppressed and vanishes above the transition. In the local TMT (${N}_{c}=1$) scheme, the mobility edge (indicated by arrows) moves towards the band center $\omega =0$ monotonically, while in the TMDCA the mobility edge first moves to higher energy, and roughly around $W>1.75$ it starts moving towards the band center, indicating that TMDCA can successfully capture the re-entrance behavior missing in the TMT scheme. Reprint from [174].

**Figure 31.**Phase diagram of the Anderson localization transition in 3D obtained from TMDCA simulations. As ${N}_{c}$ increases, a systematic improvement of the trajectory of the mobility edge is achieved. At large enough ${N}_{c}$ and within computation error, our results converge to those determined by the TMM [209].

**Figure 32.**The TDOS at the band center $TDOS(\omega =0)$ vs. disorder strength W for the 3D Anderson model calculated with the TMDCA using Ansatz 1 for different cluster sizes ${N}_{c}=1,10,12,38,92$ with units where $4t=1$. The $TDOS(\omega =0)$ vanishes at the critical disorder strength ${W}_{c}$ when all states become localized. For ${N}_{c}=1$, which corresponds to the TMT method, the critical disorder strength ${W}_{c}({N}_{c}=1)\approx 1.65$. As cluster size ${N}_{c}$ increases, the critical disorder strength ${W}_{c}$ increases quickly to $\approx 2.25$, which is in very good agreement with the results from the TMM ${W}_{c}\approx 2.1$ [210].

**Figure 33.**ADOS and TDOS of the A-B binary alloy model with off-diagonal disorder. The left panel displays results for the TMT ${N}_{c}=1$ and the right panel for the TMDCA with ${N}_{c}>1$. The average DOS (dash-dotted line) and the TDOS (shaded regions) for ${N}_{c}=1$ (left panel), ${N}_{c}={4}^{3}$ (right panel) and blue dash lines for ${N}_{c}={5}^{3}$ (left panel) for various values of the local potential ${V}_{A}$ with off-diagonal disorder parameters: ${t}^{AA}=1.5$, ${t}^{BB}=0.5$, ${t}^{AB}=0.5({t}^{AA}+{t}^{BB})$, and ${c}_{A}=0.5$. We show the TDOS for several cluster sizes ${N}_{c}=1$, ${4}^{3}$, and $={5}^{3}$ to show its systematic convergence with increasing cluster size ${N}_{c}$. The ADOS converges within our numerical precision for cluster sizes beyond ${N}_{c}={4}^{3}$. The TDOS is finite for the extended states and zero for localized states. Reprint from [211].

**Figure 34.**Disorder-energy phase diagram of the A-B binary alloy model with off-diagonal and diagonal disorder. Parameters used are ${t}^{AA}=1.5$, ${t}^{BB}=0.5$, ${t}^{AB}=1.0$, and ${c}_{A}=0.5$. The mobility edges obtained from the TMT ${N}_{c}=1$ (black dashed line), TMDCA ${N}_{c}={3}^{3}$ (green dot-dashed line), ${N}_{c}={4}^{3}$ (purple double-dot-dashed line) and ${N}_{c}={5}^{3}$ (red solid line), and the transfer matrix method (TMM) (blue dotted line). The single-site ${N}_{c}=1$ strongly underestimates the extended states region especially for higher values of ${V}_{A}$. The mobility edges obtained from the finite cluster TMDCA (${N}_{c}>1$) converges gradually with increasing ${N}_{c}$ and shows good agreement with those obtained from the TMM, in contrast to the single-site TMT. See the text for parameters and details of the TMM implementation. Reprint from [211].

**Figure 35.**The TDOS at the band center ($\omega =0$) vs. ${V}^{aa}={V}^{bb}$ in the a-b two-orbital model with increasing cluster size, for ${t}^{aa}={t}^{bb}=1.0$, ${t}^{ab}=0.3$, ${V}^{ab}=0.0$. For ${N}_{c}=1$, the critical disorder strength is 0.65 and as ${N}_{c}$ increases, it increases and converges to 0.74 for ${N}_{c}=98$. Reprint from [12].

**Figure 36.**Evolution of the mobility edge of the a-b two-orbital model as ${t}^{ab}$ increases, while ${V}^{aa}$ and ${V}^{bb}$ are fixed. The results are calculated for ${N}_{c}=64$. A dome-like shape shows up around the band center, signaling the closing of the TDOS gap. Reprint from [12].

**Figure 37.**Comparison of the ADOS and TDOS of the a-b two-orbital model calculated with the DCA, TMDCA and KPM with fixed disorder strength ${V}^{aa}={V}^{bb}=0.8$ with impurity concentration $x=0.5$ and various values of the inter-band hopping ${t}^{ab}$. The KPM uses 2048 moments on a cubic lattice of size ${48}^{3}$ and 200 independent realizations generated with 32 sites randomly sampled from each realization. Reprint from [12].

**Figure 38.**The evolution of the ADOS, TDOS and DC conductivity of the single-band 3D Anderson model at various disorder strengths W for the single-site TMT and the TMDCA with cluster size ${N}_{c}=64$. Here, for the DC conductivity, $\omega $ corresponds to the chemical potential used in the calculation. Arrows indicate the position of the mobility edge, which separates the extended electronic states from the localized ones. Reprint from [190].

**Figure 39.**DC conductivity of the 3D Anderson model at $T=0$ and $\mu =0$ (band center) vs. disorder W for different cluster size ${N}_{c}=1,10,12,64,92$. The DC conductivity vanishes at ${W}_{c}$ where all states become localized. For ${N}_{c}=1$ (TMT), the critical disorder strength ${W}_{c}^{{N}_{c}=1}\approx 1.65$ (units $4t=1$). As the cluster size increases, ${W}_{c}$ systematically increases with ${W}_{c}^{{N}_{c}\gg 12}\approx 2.10\pm 0.10$ (in units of $4t=1$), showing a quick convergence with cluster size to the KPM result. Reprint from [190].

**Figure 40.**(Top) The typical DOS as a function of frequency, for the non-interacting case (${N}_{c}$ = 38, U = 0.0, units $4t=1$) and two weakly interacting cases ($U=0.1,0.2$) are shown for a disorder value W that is close to the critical disorder, i.e., $W/{W}_{c}\left(U\right)=0.86$ [11] of the 3D Anderson-Hubbard model. The $U=0$ TDOS shows a sharp band edge, while for $U>0$, exponential tails are seen, indicating the broadening of the mobility edge. (Bottom) The typical DOS as a function of frequency, for the interacting case (${N}_{c}=38,U=0.2$, units $4t=1$) at various chemical potentials ($\mu $). As the $\mu $ approaches the non-interacting mobility edge, the exponential tail seen in the top panel is replaced by a sharp edge.

**Figure 41.**Screening of disorder effects by weak interactions in the 3D Anderson-Hubbard model: The main panel shows the momentum integrated typical DOS, TDOS(

**R**= 0; $\omega =0$) for ${N}_{c}=38$ as a function of disorder, W for various U values (units $4t=1$). The inset shows that the critical disorder value, ${W}_{c}\left(U\right)$ increases with increasing U for three cluster sizes.

**Figure 42.**Distribution of Kondo scales vs. ${T}_{K}$ for various disorder values in the 3D Anderson-Hubbard model (units $4t=1$). For larger W values, the distribution develops a finite intercept. The inset shows the same data on a log-linear scale. Reprint from [189].

**Figure 43.**The negative of the imaginary part of the low-frequency ($\omega $) self-energy as a function of $\omega $ on a linear (a-panel) and log-log scale (b-panel) for various disorder values (legends), ${N}_{c}=38$ and $U=1.6$. The left panel shows that the $-\mathrm{Im}\mathsf{\Sigma}\left(\omega \right)$ is quadratic close to the Fermi level and crosses over to a power law form (see more clearly in the right panel) with an exponent $\alpha \left(W\right)<2$, that is disorder-dependent.

**Figure 44.**A schematic phase diagram in the disorder-energy plane of the Anderson-Hubbard model showing a disorder-driven QCP separating a Fermi liquid from an as yet unidentified Phase-2. Reprint from [189].

**Figure 45.**The average and typical density of states of KFe${}_{2}$Se${}_{2}$ with 12.5% Fe vacancy concentration calculated by multiband DCA and TMDCA with cluster size ${N}_{c}=1$ and ${N}_{c}=16$, compared with the average density of states of the clean (no vacancy) KFe${}_{2}$Se${}_{2}$. Reprint from [12].

**Figure 46.**DOS (blue) and typical DOS (red) of Ga${}_{1-x}$Mn${}_{x}$N for various Mn concentrations: x = 0.02, 0.03, 0.05, 0.1, with ${N}_{c}$ = 32, showing that the impurity band is completely localized for $x\le 0.03$. The chemical potential is set to be zero and denoted as the dash line. Inset: Zoom in of the DOS and TDOS around the chemical potential. Reprint from [172].

**Table 1.**A progression of TMDCA algorithms, with each one able to incorporate greater chemical detail as we go down the list. The first column lists systems that may be studied together with the label of the defining Ansatz. The second column lists some additional characteristics including a brief discussion of the desirable properties. The columns labeled VDP and ODP identify the desirable properties, discussed above, which are notably violated and observed.

System/Ansatz | Characteristics | ODP | VDP |
---|---|---|---|

Single Band Local (diagonal) Disorder Ansatz Equation (45) | Recovers TMT at ${N}_{c}=1$. Recovers DCA for $W<<{W}_{c}$ Calculate ${\rho}_{typ}$ Hilbert trans. for ${G}_{typ}^{c}$ | 8 | 7 |

Single Band Local (diagonal) Disorder Ansatz Equation (47) | Not TMT when $Nc=1$. Recovers DCA for $W<<{W}_{c}$ Calculate ${G}_{typ}^{c}$ directly | 7 | 8 |

Single Band Off-Diagonal Disorder Ansatz Equation (59) | $2\times 2$ matrix Calculate ${\rho}_{typ}$ matrix HT to get ${G}_{typ}^{c}$ matrix | 8 | 7 |

Multiband Systems Local Disorder Ansatz Equation (61) | Matrix in orbital space Calculate ${\rho}_{typ}$ matrix HT to get ${G}_{typ}^{c}$ matrix Recovers DCA for $W<<{W}_{c}$ | 8 | 7 |

Realistic Material Systems Complex Disorder Potentials with full DFT detail Ansatz Equation (47) | Matrix in orbital space ${G}_{typ}^{c}$ Recovers DCA for $W<<{W}_{c}$ | 7 | 8 |

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**MDPI and ACS Style**

Terletska, H.; Zhang, Y.; Tam, K.-M.; Berlijn, T.; Chioncel, L.; Vidhyadhiraja, N.S.; Jarrell, M. Systematic Quantum Cluster Typical Medium Method for the Study of Localization in Strongly Disordered Electronic Systems. *Appl. Sci.* **2018**, *8*, 2401.
https://doi.org/10.3390/app8122401

**AMA Style**

Terletska H, Zhang Y, Tam K-M, Berlijn T, Chioncel L, Vidhyadhiraja NS, Jarrell M. Systematic Quantum Cluster Typical Medium Method for the Study of Localization in Strongly Disordered Electronic Systems. *Applied Sciences*. 2018; 8(12):2401.
https://doi.org/10.3390/app8122401

**Chicago/Turabian Style**

Terletska, Hanna, Yi Zhang, Ka-Ming Tam, Tom Berlijn, Liviu Chioncel, N. S. Vidhyadhiraja, and Mark Jarrell. 2018. "Systematic Quantum Cluster Typical Medium Method for the Study of Localization in Strongly Disordered Electronic Systems" *Applied Sciences* 8, no. 12: 2401.
https://doi.org/10.3390/app8122401