Invariant Lattice Research Articles

A Note on Some Equivalences of Operators and Topology of Invariant Subspaces

In this paper we investigate the invariant and hyperinvariant subspace lattices of some operators. We give a lattice-theoretic description of the lattice of hyperinvariant subspaces of an operator in terms of its lattice of invariant subspaces. We also study the structure of these lattices for operators in certain equivalence classes of some equivalence relations.

Disorder-induced topological phase transitions on Lieb lattices

Motivated by the very recent experimental realization of electronic Lieb lattices and research interest on topological states of matter, we study the topological phase transitions driven by Anderson disorder on spin-orbit coupled Lieb lattices in the presence of spin-independent and dependent potentials. By combining the numerical transport and self-consistent Born approximation methods, we found that both time-reversal invariant and broken Lieb lattices can host disorder-induced gapful topological phases, including the quantum spin Hall insulator (QSHI) and quantum anomalous Hall insulator (QAHI) phases. For the time-reversal invariant case, this disorder can induce a topological phase transition directly from normal insulator (NI) to the QSHI. While for the time-reversal broken case, the disorder can induce either a QAHI-QSHI phase transition or a NI-QAHI-QSHI phase transition. Remarkably, the time-reversal broken QSHI phase can be induced by Anderson disorder on the spin-orbit coupled Lieb lattices without time-reversal symmetry.

From quenched disorder to continuous time random walk.

This work focuses on quantitative representation of transport in systems with quenched disorder. Explicit mapping of the quenched trap model to continuous time random walk is presented. Linear temporal transformation, t→t/Λ^{1/α}, for a transient process in the subdiffusive regime is sufficient for asymptotic mapping. An exact form of the constant Λ^{1/α} is established. A disorder averaged position probability density function for a quenched trap model is obtained, and analytic expressions for the diffusion coefficient and drift are provided.

Ising model in planar lacunary and fractal lattices: A path counting approach

The method of counting loops for calculating the partition function of the Ising model on the two dimensional square lattice is extended to lacunary planar lattices, especially scale invariant fractal lattices, the Sierpi\'nsky carpets with different values of the scale invariance ratio and of the number of deleted sites. The critical temperature of the Ising model on these lattices is exactly calculated for finite iteration steps, and for a range of the scale invariance ratio $n$ from 3 to 1000 and of the number of deleted sites from $(n-2)^2$ to $(n-10)^2$. The critical temperature at the limit of an infinite number of iteration of the segmentation process is asymptotically extrapolated. Comparison is made with results obtained previously by numerical methods. Thermodynamical functions are also calculated and the fractal spectra of the Ising partition functions on several examples of Sierpi\'nski carpets are illustrated.

The complexity of translationally invariant low-dimensional spin lattices in 3D

In this theoretical paper, we consider spin systems in three spatial dimensions and consider the computational complexity of estimating the ground state energy, known as the local Hamiltonian problem, for translationally invariant Hamiltonians. We prove that the local Hamiltonian problem for 3D lattices with face-centered cubic unit cells and 4-local translationally invariant interactions between spin-3/2 particles and open boundary conditions is QMAEXP-complete, where QMAEXP is the class of problems which can be verified in exponential time on a quantum computer. We go beyond a mere embedding of past hard 1D history state constructions, for which the local spin dimension is enormous: even state-of-the-art constructions have local dimension 42. We avoid such a large local dimension by combining some different techniques in a novel way. For the verifier circuit which we embed into the ground space of the local Hamiltonian, we utilize a recently developed computational model, called a quantum ring machine, which is especially well suited for translationally invariant history state constructions. This is encoded with a new and particularly simple universal gate set, which consists of a single 2-qubit gate applied only to nearest-neighbour qubits. The Hamiltonian construction involves a classical Wang tiling problem as a binary counter which translates one cube side length into a binary description for the encoded verifier input and a carefully engineered history state construction that implements the ring machine on the cubic lattice faces. These novel techniques allow us to significantly lower the local spin dimension, surpassing the best translationally invariant result to date by two orders of magnitude (in the number of degrees of freedom per coupling). This brings our models on par with the best non-translationally invariant construction.

Fuzzy sets within Finitely Supported Mathematics

The classical theory of fuzzy sets is extended to a recently developed framework named Finitely Supported Mathematics in order to characterize fuzzy sets over infinite universes in a finitary manner by involving the concept of “finite support”. We prove some algebraic properties of the new fuzzy sets within Finitely Supported Mathematics (including some that cannot be obtained in Zermelo–Fraenkel mathematics), and introduce operations and extension principles over these fuzzy sets. We introduce a specific (infinite) membership-degree association, and connect it to the notions of invariant complete lattices and invariant monoids in Finitely Supported Mathematics.

The Nielsen–Ninomiya theorem, -invariant non-Hermiticity and single 8-shaped Dirac cone

The Nielsen–Ninomiya theorem implies that any local, Hermitian and translationally invariant lattice action in even-dimensional spacetime possesses an equal number of left- and right-handed chiral fermions. We argue that if one sacrifices the property of Hermiticity while keeping the locality and translation invariance, and imposing invariance of the action under the space-time () reversal symmetry, then the excitation spectrum of the theory may contain a non-equal number of left- and right-handed massless fermions with real-valued dispersion. We illustrate our statement in a simple 1 + 1 dimensional lattice model which exhibits a skewed 8-figure pattern in its energy spectrum. A drawback of the model is that the symmetry of the Hamiltonian is spontaneously broken implying that the energy spectrum contains complex branches. We also demonstrate that the Dirac cone is robust against local disorder so that the massless excitations in this invariant model are not gapped by random space-dependent perturbations in the couplings.

Electronic Transport Properties of 1D‐Defects in Graphene and Other 2D‐Systems

AbstractThe continuous progress in device miniaturization demands a thorough understanding of the electron transport processes involved. The influence of defects ‐ discontinuities in the perfect and translational invariant crystal lattice ‐ plays a crucial role here. For graphene in particular, they limit the carrier mobility often demanded for applications by contributing additional sources of scattering to the sample. Due to its two‐dimensional nature graphene serves as an ideal system to study electron transport in the presence of defects, because one‐dimensional defects like steps, grain boundaries and interfaces are easy to characterize and have profound effects on the transport properties. While their contribution to the resistance of a sample can be extracted by carefully conducted transport experiments, scanning probe methods are excellent tools to study the influence of defects locally. In this letter, the authors review the results of scattering at local defects in graphene and other 2D systems by scanning tunneling potentiometry, 4‐point‐probe microscopy, Kelvin probe force microscopy and conventional transport measurements. Besides the comparison of the different defect resistances important for device fabrication, the underlying scattering mechanisms are discussed giving insight into the general physics of electron scattering at defects.

Decomposition of Garside groups and self-similar L-algebras

Picantin's iterated crossed product representation of Garside monoids is extended and reproved for a wide class of not necessarily noetherian partially ordered groups with a right invariant lattice structure. It is shown that the tree-like structure of such an iterated crossed product is equivalent to a partial cycle set, closely related to a class of set-theoretic solutions of the quantum Yang–Baxter equation. The decomposition of finite square-free solutions is related to the crossed product representation of the corresponding structure group.

Ab initio optimization principle for the ground states of translationally invariant strongly correlated quantum lattice models.

In this work, a simple and fundamental numeric scheme dubbed as ab initio optimization principle (AOP) is proposed for the ground states of translational invariant strongly correlated quantum lattice models. The idea is to transform a nondeterministic-polynomial-hard ground-state simulation with infinite degrees of freedom into a single optimization problem of a local function with finite number of physical and ancillary degrees of freedom. This work contributes mainly in the following aspects: (1) AOP provides a simple and efficient scheme to simulate the ground state by solving a local optimization problem. Its solution contains two kinds of boundary states, one of which play the role of the entanglement bath that mimics the interactions between a supercell and the infinite environment, and the other gives the ground state in a tensor network (TN) form. (2) In the sense of TN, a novel decomposition named as tensor ring decomposition (TRD) is proposed to implement AOP. Instead of following the contraction-truncation scheme used by many existing TN-based algorithms, TRD solves the contraction of a uniform TN in an opposite way by encoding the contraction in a set of self-consistent equations that automatically reconstruct the whole TN, making the simulation simple and unified; (3) AOP inherits and develops the ideas of different well-established methods, including the density matrix renormalization group (DMRG), infinite time-evolving block decimation (iTEBD), network contractor dynamics, density matrix embedding theory, etc., providing a unified perspective that is previously missing in this fields. (4) AOP as well as TRD give novel implications to existing TN-based algorithms: A modified iTEBD is suggested and the two-dimensional (2D) AOP is argued to be an intrinsic 2D extension of DMRG that is based on infinite projected entangled pair state. This paper is focused on one-dimensional quantum models to present AOP. The benchmark is given on a transverse Ising chain and 2D classical Ising model, showing the remarkable efficiency and accuracy of the AOP.

Socle localement analytique I

Let L be a finite extension of Q p and n a positive integer. To each Hodge filtration with distinct Hodge-Tate weights on an n-dimensional sufficiently generic representation of the Weil-Deligne group of L, we associate a semi-simple finite length locally Q p -analytic representation of GL n (L). We show several properties of this representation of GL n (L). For instance, if it has an invariant lattice, then the starting Hodge filtration is weakly admissible.

The operator algebra generated by the translation, dilation and multiplication semigroups

The weak operator topology closed operator algebra on L 2 ( R ) generated by the one-parameter semigroups for translation, dilation and multiplication by e i λ x , λ ≥ 0 , is shown to be a reflexive operator algebra, in the sense of Halmos, with invariant subspace lattice equal to a binest. This triple semigroup algebra, A p h , is antisymmetric in the sense that A ph ∩ A p h ⁎ = C I , it has a nonzero proper weakly closed ideal generated by the finite-rank operators, and its unitary automorphism group is R . Furthermore, the 8 choices of semigroup triples provide 2 unitary equivalence classes of operator algebras, with A ph and A p h ⁎ being chiral representatives.

Supercooled spin liquid state in the frustrated pyrochlore Dy2Ti2O7

A "supercooled" liquid develops when a fluid does not crystallize upon cooling below its ordering temperature. Instead, the microscopic relaxation times diverge so rapidly that, upon further cooling, equilibration eventually becomes impossible and glass formation occurs. Classic supercooled liquids exhibit specific identifiers including microscopic relaxation times diverging on a Vogel-Tammann-Fulcher (VTF) trajectory, a Havriliak-Negami (HN) form for the dielectric function ε(ω, T), and a general Kohlrausch-Williams-Watts (KWW) form for time-domain relaxation. Recently, the pyrochlore Dy2Ti2O7 has become of interest because its frustrated magnetic interactions may, in theory, lead to highly exotic magnetic fluids. However, its true magnetic state at low temperatures has proven very difficult to identify unambiguously. Here, we introduce high precision, boundary-free magnetization transport techniques based upon toroidal geometries and gain an improved understanding of the time- and frequency-dependent magnetization dynamics of Dy2Ti2O7. We demonstrate a virtually universal HN form for the magnetic susceptibility χ (ω, T), a general KWW form for the realtime magnetic relaxation, and a divergence of the microscopic magnetic relaxation rates with the VTF trajectory. Low-temperature Dy2Ti2O7 therefore exhibits the characteristics of a supercooled magnetic liquid. One implication is that this translationally invariant lattice of strongly correlated spins may be evolving toward an unprecedented magnetic glass state, perhaps due to many-body localization of spin.

Permutation invariant lattices

We say that a Euclidean lattice in Rn is permutation invariant if its automorphism group has non-trivial intersection with the symmetric group Sn, i.e., if the lattice is closed under the action of some non-identity elements of Sn. Given a fixed element τ∈Sn, we study properties of the set of all lattices closed under the action of τ: we call such lattices τ-invariant. These lattices naturally generalize cyclic lattices introduced by Micciancio (2002, 2007), which we previously studied in Fukshansky and Sun (2014). Continuing our investigation, we discuss some basic properties of permutation invariant lattices, in particular proving that the subset of well-rounded lattices in the set of all τ-invariant lattices in Rn has positive co-dimension (and hence comprises zero proportion) for all τ different from an n-cycle.

Invariant subspaces of composition operators

The invariant subspace lattices of composition operators acting on H 2 , the Hilbert-Hardy space over the unit disc, are characterized in select cases. The lattice of all spaces left invariant by both a composition operator and the unilateral shift Mz (the multiplication operator induced by the co- ordinate function), is shown to be nontrivial and is completely described in particular cases. Given an analytic selfmap j of the unit disc, we prove that j has an angular derivative at some point on the unit circle if and only if Cj, the composition operator induced by j, maps certain subspaces in the invariant subspace lattice of Mz into other such spaces. A similar characterization of the existence of angular derivatives of j, this time in terms of Aj, the Aleksandrov operator induced by j, is obtained.

Mutual information area laws for thermal free fermions

We provide a rigorous and asymptotically exact expression of the mutual information of translationally invariant free fermionic lattice systems in a Gibbs state. In order to arrive at this result, we introduce a novel framework for computing determinants of Töplitz operators with smooth symbols, and for treating Töplitz matrices with system size dependent entries. The asymptotically exact mutual information for a partition of the 1D lattice satisfies an area law, with a prefactor which we compute explicitly. As examples, we discuss the fermionic XX model in one dimension and free fermionic models on the torus in higher dimensions in detail. Special emphasis is put on the discussion of the temperature dependence of the mutual information, scaling like the logarithm of the inverse temperature, hence confirming an expression suggested by conformal field theory. We also comment on the applicability of the formalism to treat open systems driven by quantum noise. In the appendix, we derive useful bounds to the mutual information in terms of purities. Finally, we provide a detailed error analysis for finite system sizes. This analysis is valuable in its own right for the abstract theory of Töplitz determinants.

Crystallographic analysis of the FCC → BCC martensitic transformation in high-carbon steel

Based on the phenomenological crystallographic theory of martensitic transformations, the following crystallographic characteristics of the tetragonal martensite in high-carbon steel have been calculated: orientation relationships between the crystal lattices of the bct martensite and fcc austenite; the magnitude and direction of the macroscopic shear; the habit plane; the angle and the axis of rotation of the crystal lattice of the martensite. The calculation was performed for three variants of lattice deformation: Bain deformation; two-shear Kurdjumov-Sachs deformation; and the deformation we suggested upon the analysis of the fcc-bcc transformation. In the last variant, a minimum rotation of the crystal lattice of martensite is required; consequently, this variant is closest to the real mechanism of the martensitic transformation. An expression has been derived that describes the interrelation between the degree of tetragonality of the crystal lattice of martensite and the magnitude of the shear deformation of the lattice. It has been shown that the 12 crystal-lographically equivalent variants of shear upon the formation of the lattice of the tetragonal martensite form three groups in each of which the martensite has the same tetragonality axis. For each variant of the shear, we have two equivalent variants of deformation of the martensite with invariant lattice. This results in 24 variants of orientation relationships.

Search for localized Wannier functions of topological band structures via compressed sensing

We investigate the interplay of band structure topology and localization properties of Wannier functions. To this end, we extend a recently proposed compressed sensing based paradigm for the search for maximally localized Wannier functions [Ozolins et al., PNAS 110, 18368 (2013)]. We develop a practical toolbox that enables the search for maximally localized Wannier functions which exactly obey the underlying physical symmetries of a translationally invariant quantum lattice system under investigation. Most saliently, this allows us to systematically identify the most localized representative of a topological equivalence class of band structures, i.e., the most localized set of Wannier functions that is adiabatically connected to a generic initial representative. We also elaborate on the compressed sensing scheme and find a particularly simple and efficient implementation in which each step of the iteration is an $O(N \log N)$ algorithm in the number of lattice sites $N$. We present benchmark results on one-dimensional topological superconductors demonstrating the power of these tools. Furthermore, we employ our method to address the open question whether compact Wannier functions can exist for symmetry protected topological states like topological insulators in two dimensions. The existence of such functions would imply exact flat band models with strictly finite range hopping. Here, we find numerical evidence for the absence of such functions. We briefly discuss applications in dissipative state preparation and in devising variational sets of states for tensor network methods.

Entanglement dynamics in harmonic-oscillator chains

We study the long-time evolution of the bipartite entanglement in translationally invariant one-dimensional harmonic lattice systems. We show that for Gaussian states, in quadratic interactions with periodic boundary conditions, there exists a lower bound for the von Neumann entropy which increases linearly in time. This implies that the dynamics of harmonic lattice systems can in general not efficiently be simulated by algorithms based on matrix-product decompositions of the quantum state, and interactions are needed to suppress the entanglement growth with time.

Reflexive Index of a Family of Sets

As a further study on re∞exive families of subsets, we introduce the re∞exive index for a family of subsets of a given set and show that the index of a flnite family of subsets of a flnite or countably inflnite set is always flnite. The re∞exive indices of some special families are also considered. Given a set X, let Sub(X) denote the set of all subsets of X and End(X) denote the set of all endomappings f : X i! X. For any A µ Sub(X) and F µ End(X) deflne Alg(A) = ff 2 End(X) : f(A) µ A for all A 2 Ag; Lat(F) = fA 2 Sub(X) : f(A) µ A for all f 2 Fg: A family A µ Sub(X) is called re∞exive if A = Lat(Alg(A)), or equivalently, A = Lat(F) for some F µ End(X). As was shown in (9), A µ Sub(X) is re∞exive ifi it is closed under arbitrary unions and intersections and contains the empty set and X. The re∞exive families F µ End(X) were also introduced and characterized as those subsemigroups L of (End(X);-) such that L is a lower set and contains all existing suprema of subsets of L with respect to a naturally deflned partial order on End(X). The similar work in functional analysis is on the re∞exive invariant subspace lattices and re∞exive operator algebras (1-6). For any A µ Sub(X), let ^ A = Lat(Alg(A)). Then ^ A is the smallest family of subsets containing A which is closed under arbitrary unions and intersections containing empty set ; and X, and ^