Topological phases of the two-leg Kitaev ladder

We study the phase diagram of the two-leg Kitaev model. Different topological phases can be characterized by either the number of Majorana modes for a deformed chain of the open ladder, or by a winding number related to the ‘h  -loop’ in the momentum space. By adding a three-spin interaction term to break the time-reversal symmetry, two originally different phases are glued together, so that the number of Majorana modes reduce to 0 or 1, namely, the topological invariant collapses to _Z2$Z_2$ from an integer Z. These observations are consistent with a recent general study [S. Tewari, J.D. Sau, arXiv:1111.6592v2].     

Space-time geometry of topological phases

The 2 + 1 dimensional lattice models of Levin and Wen (2005) [1] provide the most general known microscopic construction of topological phases of matter. Based heavily on the mathematical structure of category theory, many of the special properties of these models are not obvious. In the current paper, we present a geometrical space-time picture of the partition function of the Levin-Wen models which can be described as doubles (two copies with opposite chiralities) of underlying anyon theories. Our space-time picture describes the partition function as a knot invariant of a complicated link, where both the lattice variables of the microscopic Levin-Wen model and the terms of the Hamiltonian are represented as labeled strings of this link. This complicated link, previously studied in the mathematical literature, and known as Chain-Mail, can be related directly to known topological invariants of 3-manifolds such as the so-called Turaev-Viro invariant and the Witten-Reshitikhin-Turaev invariant. We further consider quasi-particle excitations of the Levin-Wen models and we see how they can be understood by adding additional strings to the Chain-Mail link representing quasi-particle world-lines. Our construction gives particularly important new insight into how a doubled theory arises from these microscopic models.     

Topological susceptibility in lattice QCD with unimproved Wilson fermions

We address a long standing problem regarding topology in lattice simulations of QCD with unimproved Wilson fermions. Earlier attempt with unimproved Wilson fermions at β=5.6 to verify the suppression of topological susceptibility with decreasing quark mass (mq) was unable to unambiguously confirm the suppression. We carry out systematic calculations for two degenerate flavours at two different lattice spacings (β=5.6 and 5.8). The effects of quark mass, lattice volume and the lattice spacing on the spanning of different topological sectors are presented. We unambiguously demonstrate the suppression of the topological susceptibility with decreasing quark mass, expected from chiral Ward identity and chiral perturbation theory.     

Controllable simulation of topological phases and edge states with quantum walk

We simulate various topological phenomena in condense matter, such as formation of different topological phases, boundary and edge states, through two types of quantum walk with step-dependent coins. Particularly, we show that one-dimensional quantum walk with step-dependent coin simulates all types of topological phases in BDI family, as well as all types of boundary and edge states. In addition, we show that step-dependent coins provide the number of steps as a controlling factor over the simulations. In fact, with tuning number of steps, we can determine the occurrences of boundary, edge states and topological phases, their types and where they should be located. These two features make quantum walks versatile and highly controllable simulators of topological phases, boundary, edge states, and topological phase transitions. We also report on emergences of cell-like structures for simulated topological phenomena. Each cell contains all types of boundary (edge) states and topological phases of BDI family.     

Topological phases in Bi/Sb planar and buckled honeycomb monolayers

We investigate topological phases in two-dimensional Bi/Sb honeycomb crystals considering planar and buckled structures, both freestanding and deposited on a substrate. We use the multi-orbital tight-binding model and compare results with density functional theory calculations. We distinguish topological phases by calculating topological invariants, analyzing edge states properties of systems in a ribbon geometry and studying their entanglement spectra. We show that coupling to the substrate induces transition to the $Z_2$ topological insulator phase. It is observed that topological crystalline insulator (TCI) phase, found in planar crystals, exhibits an additional pair of edge states in both energy spectrum and entanglement spectrum. Transport calculations for TCI phase suggest robust quantized conductance even in the presence of crystal symmetry-breaking disorder.     

Topological states in the Hofstadter model on a honeycomb lattice

We provide a detailed analysis of a topological structure of a fermion spectrum in the Hofstadter model with different hopping integrals along the $x,y,z$-links ($t_x=t,t_y=t_z=1$), defined on a honeycomb lattice. We have shown that the chiral gapless edge modes are described in the framework of the generalized Kitaev chain formalism, which makes it possible to calculate the Hall conductance of subbands for different filling and an arbitrary magnetic flux ϕ. At half-filling the gap in the center of the fermion spectrum opens for $t>t_c=2^{\varphi }$, a quantum phase transition in the 2D-topological insulator state is realized at $t_c$. The phase state is characterized by zero energy Majorana states localized at the boundaries. Taking into account the on-site Coulomb repulsion U (where $U<<1$), the criterion for the stability of a topological insulator state is calculated at $t<<1$, $t\sim U$. Thus, in the case of $U>4\mathrm{\Delta }$, the topological insulator state, which is determined by chiral gapless edge modes in the gap Δ, is destroyed.     

On the stability of topological phases on a lattice

We study the stability of anyonic models on lattices to perturbations. We establish a cluster expansion for the energy of the perturbed models and use it to study the stability of the models to local perturbations. We show that the spectral gap is stable when the model is defined on a sphere, so that there is no ground state degeneracy. We then consider the toric code Hamiltonian on a torus with a class of abelian perturbations and show that it is stable when the torus directions are taken to infinity simultaneously, and is unstable when the thin torus limit is taken.     

Low-temperature phases obtained by linear programming: An application to a lattice system of model chiral molecules

A convenient, Peierls-type approach to obtain low-temperature phases is to use the method of an m-potential. In this paper we show that, for more complex systems where it may be rather difficult to rewrite the Hamiltonian as an m-potential and whose configurations are subject to linear constraints, the verification of the Peierls condition can be reformulated as a linear programming problem. Before introducing this novel strategy for a general lattice system, we compare it with the m-potential method for a specific model molecular system consisting of an equimolar mixture of a chiral molecule and its non-superimposable mirror image that occupy all the sites of a honeycomb lattice. In one range of interactions, we prove that a racemic low-temperature phase occurs (containing equal numbers of each enantiomer). However, in a neighboring range of interactions, we show that a homochiral low-temperature phase (containing a single enantiomer) exists, and thus chiral segregation occurs in the system. Our linear programming technique yields these results in wider ranges of interactions than the m-potential method.     

Topological phases and phase transitions on the honeycomb lattice

We investigate possible phase transitions among the different topological insulators in a honeycomb lattice under the combined influence of spin-orbit couplings and staggered magnetic flux. We observe a series of topological phase transitions when tuning the flux amplitude, and find topologically nontrivial phases with high Chern number or spin-Chern number. Through tuning the exchange field, we also find a new quantum state which exhibits the electronic properties of both the quantum spin Hall state and quantum anomalous Hall state. The topological characterization based on the Chern number and the spin-Chern number are in good agreement with the edge-state picture of various topological phases.     

Topological dynamics of flipping Lorentz lattice gas models

We study the topological dynamics of the flipping mirror model of Ruijgrok and Cohen with one or an infinite number of particles. In particular we prove the topological transitivity and topological mixing up to a natural first integral for the one-particle model.     

Topological phases of silicene and germanene in an external magnetic field: Quantitative results

We investigate the topological phases of silicene and germanene that arise due to the strong spin–orbit interaction in an external perpendicular magnetic field. Below and above a critical field of 10 T, respectively, we demonstrate for silicene under 3% tensile strain quantum spin Hall and quantum anomalous Hall phases. Not far above the critical field, and therefore in the experimentally accessible regime, we obtain an energy gap in the meV range, which shows that the quantum anomalous Hall phase can be realized experimentally in silicene, in contrast to graphene (tiny energy gap) and germanene (enormous field required). (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Theory of twist liquids: Gauging an anyonic symmetry

Topological phases in (2+1)$(2+1)$-dimensions are frequently equipped with global symmetries, like conjugation, bilayer or electric–magnetic duality, that relabel anyons without affecting the topological structures. Twist defects are static point-like objects that permute the labels of orbiting anyons. Gauging these symmetries by quantizing defects into dynamical excitations leads to a wide class of more exotic topological phases referred as twist liquids  , which are generically non-Abelian. We formulate a general gauging framework, characterize the anyon structure of twist liquids and provide solvable lattice models that capture the gauging phase transitions. We explicitly demonstrate the gauging of the Z₂${\mathbb{Z}}_2$-symmetric toric code, SO(2N)₁$SO{\left(2N\right)}_1$ and SU(3)₁$SU{\left(3\right)}_1$ state as well as the S₃$S_3$-symmetric SO(8)₁$SO{\left(8\right)}_1$ state and a non-Abelian chiral state we call the “4-Potts” state.     

凝聚体中的拓扑量子态

探寻拓扑上非平庸的凝聚体物质状态,特别是其电子结构和输运性质,是当前凝聚体物理 学领域非常重要的前沿研究方向。本文讨论的大多数主题都与电子波函数的拓扑性质有关。全文 除简短的引言外,包括拓扑量子现象、各种拓扑相、拓扑性准粒子的异常输运性质、拓扑性集 体激发和耦合激发,以及继续发展的拓扑量子态研究五个章节。这些章节着重反映拓扑量子态研 究的各个侧面,汇总起来方可以凸显凝聚体中拓扑量子态的全貌。     

Topological permutation entropy

Permutation entropy quantifies the diversity of possible ordering of the successively observed values a random or deterministic system can take, just as Shannon entropy quantifies the diversity of the values themselves. When the observable or state variable has a natural order relation, making permutation entropy possible to compute, then the asymptotic rate of growth in permutation entropy with word length forms an alternative means of describing the intrinsic entropy rate of a source. Herein, extending a previous result on metric entropy rate, we show that the topological permutation entropy rate for expansive maps equals the conventional topological entropy rate familiar from symbolic dynamics. This result is not limited to one-dimensional maps.     

Calculating Topological Entropy

The attempt to find effective algorithms for calculating the topological entropy of piecewise monotone maps of the interval having more than three monotone pieces has proved to be a difficult problem. The algorithm introduced here is motivated by the fact that if f: [0, 1] → [0, 1] is a piecewise monotone map of the unit interval into itself, thenh(f)=lim_n→∞ (1/n) log Var(f ⁿ), where h(f) is the topological entropy off, and Var(f ⁿ) is the total variation off ⁿ. We show that it is not feasible to use this formula directly to calculate numerically the topological entropy of a piecewise monotone function, because of the slow convergence. However, a close examination of the reasons for this failure leads ultimately to the modified algorithm which is presented in this paper. We prove that this algorithm is equivalent to the standard power method for finding eigenvalues of matrices (with shift of origin) in those cases for which the function is Markov, and present encouraging experimental evidence for the usefulness of the algorithm in general by applying it to several one-parameter families of test functions.     

A Topological Glass

We propose and study a model with glassy behavior. The state space of the model is given by all triangulations of a sphere with n nodes, half of which are red and half are blue. Red nodes want to have 5 neighbors while blue ones want 7. Energies of nodes with other numbers of neighbors are supposed to be positive. The dynamics is that of flipping the diagonal of two adjacent triangles, with a temperature dependent probability. We show that this system has an approach to a steady state which is exponentially slow, and show that the stationary state is unordered. We also study the local energy landscape and show that it has the hierarchical structure known from spin glasses. Finally, we show that the evolution can be described as that of a rarefied gas with spontaneous generation of particles and annihilating collisions.     

Variational approximations for square lattice models in statistical mechanics

This paper concerns a square lattice, Ising-type model with interactions between the four spins at the corners of each face. These may include nearest and next-nearest-neighbor interactions, and interactions with a magnetic field. Provided the Hamiltonian is symmetric with respect to both row reversal and column reversal, a rapidly convergent sequence of variational approximations is obtained, giving the free energy and other thermodynamic properties. For the usual Ising model, the lowest such approximations are those of Bethe and of Kramers and Wannier. The method provides a new definition of corner transfer matrices.     

Topological defects in cosmology

Present status of theories of topological defects in particle theory models of the early Universe is discussed. Various consequences of topological defects in cosmology, such as constraints on particle theory models, structure formation etc. are discussed.     

超冷原子中拓扑超流的发展现状

拓扑超流态是一种奇异物质态,它的内部受能隙保护,而在其系统边缘却可以容纳无能隙的Majorana 费米子。由于该粒子满足非阿贝尔统计,并且受拓扑保护具有良好的稳定性,用它 们携带量子化的信息,可以用于拓扑量子计算的研究。近年来,理论工作预测了各类系统中可能 存在的拓扑超流态。我们首先介绍了在各类光晶格模型中的拓扑超流, 光晶格的超冷原子具有良 好的可控性与普适性,是实现拓扑超流的理想模型系统。接下来我们介绍了自旋轨道耦合调控下 的拓扑超流,自旋轨道耦合效应是诱导拓扑相的重要条件,并且人们已经在实验上合成了人工自 旋轨道耦合,这为实验上观测拓扑超流取得了突破性的进展。随着近年来实验技术的提高,曾经 难以在实验中观测的,被人们所忽略的拓扑Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) 超流相也 成为了人们研究的热点,因此我们接下来介绍了拓扑的FFLO 超流。此外,我们还介绍了拓扑超 流其他方面的进展,包括孤子引诱的拓扑超流、三组分的拓扑超流、大陈数的拓扑超流以及拓扑 超流临界温度的提高。在实验中,如何检测与实现拓扑超流,是其研究的目的及意义所在,因 此我们在文章的最后介绍了拓扑超流的识别与实现。     

Topological rewriting and the geometrization of programming

Spatial computing is an emerging field that recognizes the importance of explicitly handling spatial relationships at three levels: computer architectures, programming languages and applications. In this context, we present MGS, an experimental programming language where data structures are fields on abstract spaces. In MGS, fields are transformed using rules. We show that this approach is able to unify, at least for programming purposes, several computational models like Lindenmayer systems and cellular automata. The MGS notions of topological collection and transformation are formalized using concepts developed in algebraic topology. We propose to use transformations in order to implement a discrete version of some differential operators. These transformations satisfy a Stokes-like theorem. This result constitutes a geometric view of programming where data are handled like fields in physics. The relevance of this approach for the design of autonomic software systems is discussed in the conclusion.