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Tianran Chen

Publications and source records attributed to Tianran Chen.

At least 37 records · Page 2Linked to original sources

Temporally-decoherent and spatially-coherent vibrations in metal halide perovskite

The long carrier lifetime and defect tolerance in metal halide perovskites (MHPs) are major contributors to the superb performance of MHP optoelectronic devices. Large polarons were reported to be responsible for the long carrier lifetime. Yet microscopic mechanisms of the large polaron formation including the so-called phonon melting, are still under debate. Here, time-of-flight (TOF) inelastic neutron scattering (INS) experiments and first-principles density-functional theory (DFT) calculations were employed to investigate the lattice vibrations (or phonon dynamics) in methylammonium lead iodide ($\rm{MAPbI_3}$), a prototypical example of MHPs. Our findings are that optical phonons lose temporal coherence gradually with increasing temperature which vanishes at the orthorhombic-to-tetragonal structural phase transition. Surprisingly, however, we found that the spatial coherence is still retained throughout the decoherence process. We argue that the temporally decoherent and spatially coherent vibrations contribute to the formation of large polarons in this metal halide perovskite.

cond-mat.mtrl-sci↗

Graph edge contraction and subdivisions for adjacency polytopes

Adjacency polytopes, a.k.a. symmetric edge polytopes, associated with undirected graphs have been defined and studied in several seemingly independent areas including number theory, discrete geometry, and dynamical systems. In particular, the authors are motivated by the tropical intersections problem derived from the Kuramoto equations. Regular subdivisions of adjacency polytopes are instrumental in solving these problems. This paper explores connections between the regular subdivisions of an adjacency polytope and the contraction of the underlying graph along an edge. We construct a special regular subdivision whose cells are in one-to-one correspondence with facets of an adjacency polytope associated with an edge-contraction of the original graph. Moreover, this subdivision induces a decomposition of the original graph into ``cell subgraphs''. We explore the combinatorial, graph-theoretic, and matroidal aspects of this connection.

math.CO↗

On the root count of algebraic Kuramoto equations in cycle networks with uniform coupling

The Kuramoto model is a classical model used in the study of spontaneous synchronizations in networks of coupled oscillators. In this model, frequency synchronization configurations can be formulated as complex solutions to a system of algebraic equations. Recently, upper bounds to the number of frequency synchronization configurations in cycle networks of N oscillators were calculated under the assumption of generic non-uniform coupling. In this paper, we refine these results for the special cases of uniform coupling. In particular, we show that when, and only when, N is divisible by 4, the upper bound for the number of synchronization configurations in the uniform coupling cases is significantly less than the bound in the non-uniform coupling cases. This result also establishes an explicit formula for the gap between the birationally invariant intersection index and the Bernshtein-Kushnirenko-Khovanskii bound for the underlying algebraic equations.

math.AG↗

A toric deformation method for solving Kuramoto equations

The study of frequency synchronization configurations in Kuramoto models is a ubiquitous mathematical problem that has found applications in many seemingly independent fields. In this paper, we focus on networks whose underlying graph are cycle graphs. Based on the recent result on the upper bound of the frequency synchronization configurations in this context, we propose a toric deformation homotopy method for locating all frequency synchronization configurations with complexity that is linear in this upper bound. Loosely based on the polyhedral homotopy method, this homotopy induces a deformation of the set of the synchronization configurations into a series of toric varieties, yet our method has the distinct advantage of avoiding the costly step of computing mixed cells. We also explore the important consequences of this homotopy method in the context of direct acyclic decomposition of Kuramoto networks and tropical stable intersection points for Kuramoto equations.

math.AG↗

Directed acyclic decomposition of Kuramoto equations

The Kuramoto model is one of the most widely studied model for describing synchronization behaviors in a network of coupled oscillators, and it has found a wide range of applications. Finding all possible frequency synchronization configurations in a general non-uniform heterogeneous sparse network is an important yet difficult problem due to the complicated nonlinear interactions. In this paper, we develop a general framework for decomposing a Kuramoto network into smaller directed acyclic subnetworks that will form the foundation of a divide-and-conquer approach for studying the frequency synchronization configurations of large Kuramoto networks.

math.CO↗

Three Formulations of the Kuramoto Model as a System of Polynomial Equations

We compare three formulations of stationary equations of the Kuramoto model as systems of polynomial equations. In the comparison, we present bounds on the numbers of real equilibria based on the work of Bernstein, Kushnirenko, and Khovanskii, and performance of methods for the optimisation over the set of equilibria based on the work of Lasserre, both of which could be of independent interest.

math.OC↗

A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes

The free sum is a basic geometric operation among convex polytopes. This note focuses on the relationship between the normalized volume of the free sum and that of the summands. In particular, we show that the normalized volume of the free sum of full dimensional polytopes is precisely the product of the normalized volumes of the summands.

math.CO↗

Unmixing the mixed volume computation

Computing mixed volume of convex polytopes is an important problem in computational algebraic geometry. This paper establishes sufficient conditions under which the mixed volume of several convex polytopes exactly equals the normalized volume of the convex hull of their union. Under these conditions the problem of computing mixed volume of several polytopes can be transformed into a volume computation problem for a single polytope in the same dimension. We demonstrate through problems from real world applications that substantial reduction in computational costs can be achieved via this transformation in situations where the convex hull of the union of the polytopes has less complex geometry than the original polytopes. We also discuss the important implications of this result in the polyhedral homotopy method for solving polynomial systems.

math.AG↗

Updated Core Libraries of the ALPS Project

The open source ALPS (Algorithms and Libraries for Physics Simulations) project provides a collection of physics libraries and applications, with a focus on simulations of lattice models and strongly correlated electron systems. The libraries provide a convenient set of well-documented and reusable components for developing condensed matter physics simulation codes, and the applications strive to make commonly used and proven computational algorithms available to a non-expert community. In this paper we present an update of the core ALPS libraries. We present in particular new Monte Carlo libraries and new Green's function libraries.

physics.comp-ph↗

The loss surface of deep linear networks viewed through the algebraic geometry lens

By using the viewpoint of modern computational algebraic geometry, we explore properties of the optimization landscapes of the deep linear neural network models. After clarifying on the various definitions of "flat" minima, we show that the geometrically flat minima, which are merely artifacts of residual continuous symmetries of the deep linear networks, can be straightforwardly removed by a generalized $L_2$ regularization. Then, we establish upper bounds on the number of isolated stationary points of these networks with the help of algebraic geometry. Using these upper bounds and utilizing a numerical algebraic geometry method, we find all stationary points of modest depth and matrix size. We show that in the presence of the non-zero regularization, deep linear networks indeed possess local minima which are not the global minima. Our computational results clarify certain aspects of the loss surfaces of deep linear networks and provide novel insights.

stat.ML↗

Excess electron screening of remote donors and mobility in modern GaAs/AlGaAs herostructures

In modern GaAs/Al$_x$Ga$_{1-x}$As heterostructures with record high mobilities, a two-dimensional electron gas (2DEG) in a quantum well is provided by two remote donor $δ$-layers placed on both sides of the well. Each $δ$-layer is located within a narrow GaAs layer, flanked by narrow AlAs layers which capture excess electrons from donors but leave each of them localized in a compact dipole atom with a donor. Still excess electrons can hop between host donors to minimize their Coulomb energy. As a result they screen the random potential of donors dramatically. We numerically model the pseudoground state of excess electrons at a fraction $f$ of filled donors and find both the mobility and the quantum mobility limited by scattering on remote donors as universal functions of $f$. We repeat our simulations for devices with additional disorder such as interface roughness of the doping layers, and find the quantum mobility is consistent with measured values. Thus, in order to increase the quantum mobility this additional disorder should be minimized.

cond-mat.mes-hall↗

Memory effect and phase transition in a hierarchical trap model for spin glass

We introduce an efficient dynamical tree method that enables us, for the first time, to explicitly demonstrate thermo-remanent magnetization memory effect in a hierarchical energy landscape. Our simulation nicely reproduces the nontrivial waiting-time and waiting-temperature dependences in this non-equilibrium phenomenon. We further investigate the condensation effect, in which a small set of micro-states dominates the thermodynamic behavior, in the multi-layer trap model. Importantly, a structural phase transition of the tree is shown to coincide with the onset of condensation phenomenon. Our results underscore the importance of hierarchical structure and demonstrate the intimate relation between glassy behavior and structure of barrier trees.

cond-mat.dis-nn↗

Charge ordering and non-local correlations in the doped extended Hubbard model

We study the extended Hubbard model away from half-filling on a two-dimensional square lattice using cluster dynamical mean field theory on clusters of size $8$. We show that the model exhibits metallic, compressible charge ordered, and insulating charge ordered phases. We determine the location of the charge ordering phase transition line at finite temperature and the properties of the phases as a function of doping, temperature, local interaction, and nearest neighbor interaction. An analysis of the energetics of the charge order transition shows that the charge ordering transition mainly results in a rearrangement of local and non-local potential energy. We show the doping evolution of the spectral function from the isotropic metal via a charge ordered metal to a charge ordered insulator with a big gap, and study finite size effects of the approximation.

cond-mat.str-el↗

Counting equilibria of the Kuramoto model using birationally invariant intersection index

Synchronization in networks of interconnected oscillators is a fascinating phenomenon that appear naturally in many independent fields of science and engineering. A substantial amount of work has been devoted to understanding all possible synchronization configurations on a given network. In this setting, a key problem is to determine the total number of such configurations. Through an algebraic formulation, for tree and cycle graphs, we provide an upper bound on this number using the birationally invariant intersection index of a system of rational functions on a toric variety.

math.AG↗

Aging, memory, and nonhierarchical energy landscape of spin jam

The notion of complex energy landscape underpins the intriguing dynamical behaviors in many complex systems ranging from polymers, to brain activity, to social networks and glass transitions. The spin glass state found in dilute magnetic alloys has been an exceptionally convenient laboratory frame for studying complex dynamics resulting from a hierarchical energy landscape with rugged funnels. Here, we show, by a bulk susceptibility and Monte Carlo simulation study, that densely populated frustrated magnets in a spin jam state exhibit much weaker memory effects than spin glasses, and the characteristic properties can be reproduced by a nonhierarchical landscape with a wide and nearly flat but rough bottom. Our results illustrate that the memory effects can be used to probe different slow dynamics of glassy materials, hence opening a window to explore their distinct energy landscapes.

cond-mat.dis-nn↗

Fixed Points of Belief Propagation -- An Analysis via Polynomial Homotopy Continuation

Belief propagation (BP) is an iterative method to perform approximate inference on arbitrary graphical models. Whether BP converges and if the solution is a unique fixed point depends on both the structure and the parametrization of the model. To understand this dependence it is interesting to find \emph{all} fixed points. In this work, we formulate a set of polynomial equations, the solutions of which correspond to BP fixed points. To solve such a nonlinear system we present the numerical polynomial-homotopy-continuation (NPHC) method. Experiments on binary Ising models and on error-correcting codes show how our method is capable of obtaining all BP fixed points. On Ising models with fixed parameters we show how the structure influences both the number of fixed points and the convergence properties. We further asses the accuracy of the marginals and weighted combinations thereof. Weighting marginals with their respective partition function increases the accuracy in all experiments. Contrary to the conjecture that uniqueness of BP fixed points implies convergence, we find graphs for which BP fails to converge, even though a unique fixed point exists. Moreover, we show that this fixed point gives a good approximation, and the NPHC method is able to obtain this fixed point.

stat.ML↗

Charge ordering and correlation effects in the extended Hubbard model

We study the half filled extended Hubbard model on a two-dimensional square lattice using cluster dynamical mean field theory on clusters of size 8-20. We show that the model exhibits metallic, Mott insulating, and charge ordered phases, and determine the location of the charge ordering phase transition line and the properties of the ordered and disordered phase as a function of temperature, local interaction, and nearest neighbor interaction. We find strong non-local correlations in the uniform phase and a pronounced screening effect in the vicinity of the phase transition, where non-local interactions push the system towards metallic behavior. In contrast, correlations in the ordered phase are mostly local to the unit cell. Extrapolation to the thermodynamic limit and control of all sources of errors allow us to assess the regime of applicability of simpler approximation schemes for systems with non-local interactions. We also demonstrate how strong non-local electron-electron interactions can increase electron mobility by turning a charge ordered insulator into a metal.

cond-mat.str-el↗

Enhancement of hopping conductivity by spontaneous fractal ordering of low-energy sites

Variable-range hopping conductivity has long been understood in terms of a canonical prescription for relating the single-particle density of states to the temperature-dependent conductivity. Here we demonstrate that this prescription breaks down in situations where a large and long-ranged random potential develops. In particular, we examine a canonical model of a completely compensated semiconductor, and we show that at low temperatures hopping proceeds along self-organized, low-dimensional subspaces having fractal dimension $d = 2$. We derive and study numerically the spatial structure of these subspaces, as well as the conductivity and density of states that result from them. One of our prominent findings is that fractal ordering of low energy sites greatly enhances the hopping conductivity, and allows Efros-Shklovskii type conductivity to persist up to unexpectedly high temperatures.

cond-mat.dis-nn↗