Nontrivial Symmetry Research Articles

Constructions of L∞ Algebras and Their Field Theory Realizations

We construct L∞ algebras for general “initial data” given by a vector space equipped with an antisymmetric bracket not necessarily satisfying the Jacobi identity. We prove that any such bracket can be extended to a 2-term L∞ algebra on a graded vector space of twice the dimension, with the 3-bracket being related to the Jacobiator. While these L∞ algebras always exist, they generally do not realize a nontrivial symmetry in a field theory. In order to define L∞ algebras with genuine field theory realizations, we prove the significantly more general theorem that if the Jacobiator takes values in the image of any linear map that defines an ideal there is a 3-term L∞ algebra with a generally nontrivial 4-bracket. We discuss special cases such as the commutator algebra of octonions, its contraction to the “R-flux algebra,” and the Courant algebroid.

Topology-based crystal structure generator

Crystal structure prediction is a new dynamically developing field. Two main types of crystal structure prediction methods exist: (1) based on global optimization and (2) based on data mining. It seems promising to hybridize data mining and global optimization techniques. The former generally involve no empirical information and are truly predictive, the latter rely on the databases of existing crystal structures, and are relatively fast, but prone to error, because the databases are far from complete. Furthermore, the theorist’s dream is to be as little dependent on empirical data as possible and be always capable of predicting new structures, not contained in the databases. Here we present an approach to generate an infinite number of crystal structures from a finite set of idealized periodic nets. The resulting structures are highly ordered, possess nontrivial symmetries, and often low energy. Topologically generated structures can be used for initializing evolutionary crystal structure prediction calculations, or on their own — as an extended data mining approach. The efficiency of the proposed approach in both scenarios is confirmed by a series of tests, which we also present here. As an additional enhancement to evolutionary algorithm we introduce a technique for adjusting fractions of variation operators on the fly. Tests show significant performance improvements due to both of these developments.

Topological Phase Transition andZ2Index forS=1Quantum Spin Chains

We study $S=1$ quantum spin systems on the infinite chain with short ranged Hamiltonians which have certain rotational and discrete symmetry. We define a $\mathbb{Z}_2$ index for any gapped unique ground state, and prove that it is invariant under smooth deformation. By using the index, we provide the first rigorous proof of the existence of a "topological" phase transition, which cannot be characterized by any conventional order parameters, between the AKLT ground state and trivial ground states. This rigorously establishes that the AKLT model is in a nontrivial symmetry protected topological phase.

Symmetry Indicators and Anomalous Surface States of Topological Crystalline Insulators

The rich variety of crystalline symmetries in solids leads to a plethora of topological crystalline insulators (TCIs) featuring distinct physical properties, which are conventionally understood in terms of bulk invariants specialized to the symmetries at hand. While isolated examples of TCI have been identified and studied, the same variety demands a unified theoretical framework. In this work, we show how the surfaces of TCIs can be analyzed within a general surface theory with multiple flavors of Dirac fermions, whose mass terms transform in specific ways under crystalline symmetries. We identify global obstructions to achieving a fully gapped surface, which typically lead to gapless domain walls on suitably chosen surface geometries. We perform this analysis for all 32 point groups, and subsequently for all 230 space groups, for spin-orbit-coupled electrons. We recover all previously discussed TCIs in this symmetry class, including those with "hinge" surface states. Finally, we make connections to the bulk band topology as diagnosed through symmetry-based indicators. We show that spin-orbit-coupled band insulators with nontrivial symmetry indicators are always accompanied by surface states that must be gapless somewhere on suitably chosen surfaces. We provide an explicit mapping between symmetry indicators, which can be readily calculated, and the characteristic surface states of the resulting TCIs.

Absence of nontrivial symmetries to the heat equation in Goursat groups of dimension at least 4

Absence of nontrivial symmetries to the heat equation in Goursat groups of dimension at least 4

Exact solutions of the models of nonlinear diffusion in an inhomogeneous medium

The main purpose of our research is to find as many as possible the solutions of the equation of the general model of nonlinear diffusion in an inhomogeneous medium and to establish their physical meaning and to apply the obtained solutions to the description nonlinear diffusion processes in an inhomogeneous medium. To achieve of this purpose the basic submodels (possessing nontrivial symmetry properties) of the general model are obtained and researched. The formulas of the production of the new solutions for the equations of these submodels are obtained. For these submodels all invariant submodels are found. The essentially distinct invariant solutions (not connected with a help of the point transformations) describing these invariant submodels are found either explicitly, or their search is reduced to the solving of the nonlinear integral equations. The physical meanings of these solutions are established. Some of the 25 explicitly found solutions describe a diffusion process only for a finite period of time, others — for an infinite period of time. Some solutions describe a nonlinear diffusion process either with fixed or evolving ”black holes”, in the vicinity of which the concentration infinitely increases. The presence of the arbitrary constants in the integral equations, that determine other 27 solutions opens up the new opportunities for analytical and numerical study of the boundary value problems for the received submodels, and, thus, for the original model of the nonlinear diffusion process. For such invariant submodels, we are studied the diffusion processes, for which at the initial instant of the time at a fixed point either a concentration and rate of its change, or concentration and its gradient are given. The solving of the boundary value problems describing these processes reduces to the solving of nonlinear integral equations. The existence and uniqueness of the solutions of these boundary value problems under certain conditions are established. A mechanical relevance of the obtained solutions is as follows: 1) these solutions describe specific nonlinear diffusion processes in an inhomogeneous medium, 2) these solutions can be used as test solutions in the numerical calculations, which perform in the studies of the real diffusion processes, 3) these solutions make it possible to assess the degree of adequacy of a given mathematical models to the real physical processes, after carrying out experiments corresponding to these solutions, and estimating the resulting deviations.The obtained results can be used to study the diffusion of substances, the diffusion of conduction electrons and other particles, the diffusion of physical fields, the propagation of heat in an inhomogeneous medium.

Invariant submodels and exact solutions of the generalization of the Leith model of the wave turbulence

A generalization of Leith’s model of the phenomenological theory of the wave turbulence is investigated. With the methods of group analysis, the basic models possessing non-trivial symmetries are obtained. For each model, all the invariant submodels are found. For nonlinear differential equations describing these models, formulas for the production of new solutions containing arbitrary constants are obtained. By virtue of these formulas, each solution generates a family of the new solutions. In an explicit form, some invariant solutions (not connected by point transformations) describing invariant submodels are found. The physical meaning of these solutions is obtained. In particular, with the help of these solutions the turbulent processes for which there are “destructive waves” both with fixed wave numbers and with varying wave numbers are described. On the example of an invariant solution of rank 1, it was shown that the search of the invariant solutions of rank 1, which cannot be found explicitly, can be reduced to solve the integral equations. For this solution, turbulent processes are investigated for which at the initial instant of a time and for a fixed value of the wave number either the turbulence energy and rate of its change or the turbulence energy and its gradient are given. Under certain conditions, the existence and uniqueness of the solutions of the boundary value problems describing these processes are established.

Transformations among Pure Multipartite Entangled States via Local Operations are Almost Never Possible

Local operations assisted by classical communication (LOCC) constitute the free operations in entanglement theory. Hence, the determination of LOCC transformations is crucial for the understanding of entanglement. We characterize here almost all LOCC transformations among pure multipartite multilevel states. Combined with the analogous results for qubit states shown by Gour \emph{et al.} [J. Math. Phys. 58, 092204 (2017)], this gives a characterization of almost all local transformations among multipartite pure states. We show that nontrivial LOCC transformations among generic, fully entangled, pure states are almost never possible. Thus, almost all multipartite states are isolated. They can neither be deterministically obtained from local-unitary-inequivalent (LU-inequivalent) states via local operations, nor can they be deterministically transformed to pure, fully entangled LU-inequivalent states. In order to derive this result, we prove a more general statement, namely, that, generically, a state possesses no nontrivial local symmetry. We discuss further consequences of this result for the characterization of optimal, probabilistic single copy and probabilistic multi-copy LOCC transformations and the characterization of LU-equivalence classes of multipartite pure states.

Time-reversal symmetry, anomalies, and dualities in (2+1)$d$

We study continuum quantum field theories in 2+1 dimensions with time-reversal symmetry \cal T. The standard relation {\cal T}^2=(-1)^F is satisfied on all the “perturbative operators” i.e. polynomials in the fundamental fields and their derivatives. However, we find that it is often the case that acting on more complicated operators {\cal T}^2=(-1)^F {\cal M} with \cal M a non-trivial global symmetry. For example, acting on monopole operators, \cal M could be \pm1±1 depending on the magnetic charge. We study in detail U(1)U(1) gauge theories with fermions of various charges. Such a modification of the time-reversal algebra happens when the number of odd charge fermions is 2 ~{\rm mod }~4, e.g. in QED with two fermions. Our work also clarifies the dynamics of QED with fermions of higher charges. In particular, we argue that the long-distance behavior of QED with a single fermion of charge 22 is a free theory consisting of a Dirac fermion and a decoupled topological quantum field theory. The extension to an arbitrary even charge is straightforward. The generalization of these abelian theories to SO(N)SO(N) gauge theories with fermions in the vector or in two-index tensor representations leads to new results and new consistency conditions on previously suggested scenarios for the dynamics of these theories. Among these new results is a surprising non-abelian symmetry involving time-reversal.

Symmetry protected topological phases in two-orbital SU(4) fermionic atoms

We study one-dimensional systems of two-orbital SU(4) fermionic cold atoms. In particular, we focus on an SU(4) spin model [named SU(4) $e$-$g$ spin model] that is realized in a low-energy state in the Mott insulator phase at the filling $n_g=3, n_e=1$ ($n_g, n_e$: numbers of atoms in ground and excited states, respectively). Our numerical study with the infinite-size density matrix renormalization group shows that the ground state of SU(4) $e$-$g$ spin model is a nontrivial symmetry protected topological (SPT) phase protected by $Z_4 \times Z_4$ symmetry. Specifically, we find that the ground state belongs to an SPT phase with the topological index $2\in\mathbb{Z}_4$ and show sixfold degenerate edge states. This is topologically distinct from SPT phases with the index $1\in\mathbb{Z}_4$ that are realized in the SU(4) bilinear model and the SU(4) Affleck-Kennedy-Lieb-Tasaki (AKLT) model. We explore the phase diagram of SU(4) spin models including $e$-$g$ spin model, bilinear-biquadratic model, and AKLT model, and identify that antisymmetrization effect in neighboring spins (that we quantify with Casimir operators) is the driving force of the phase transitions. Furthermore, we demonstrate by using the matrix product state how the $\mathbb{Z}_4$ SPT state with six edge states appears in the SU(4) $e$-$g$ spin model.

Assessing the viability of A4 , S4 , and A5 flavor symmetries for description of neutrino mixing

We consider the $A_4$, $S_4$ and $A_5$ discrete lepton flavour symmetries in the case of 3-neutrino mixing, broken down to non-trivial residual symmetries in the charged lepton and neutrino sectors in such a way that at least one of them is a $Z_2$. Such symmetry breaking patterns lead to predictions for some of the three neutrino mixing angles and/or the leptonic Dirac CP violation phase $\delta$ of the neutrino mixing matrix. We assess the viability of these predictions by performing a statistical analysis which uses as an input the latest global data on the neutrino mixing parameters. We find 14 phenomenologically viable cases providing distinct predictions for some of the mixing angles and/or the Dirac phase $\delta$. Employing the current best fit values of the three neutrino mixing angles, we perform a statistical analysis of these cases taking into account the prospective uncertainties in the determination of the mixing angles, planned to be achieved in currently running (Daya Bay) and the next generation (JUNO, T2HK, DUNE) of neutrino oscillation experiments. We find that only six cases would be compatible with these prospective data. We show that this number is likely to be further reduced by a precision measurement of $\delta$.

Orbital angular momentum symmetry in a driven optical parametric oscillator.

We investigate the dynamics of a driven optical parametric oscillator under the injection of orbital angular momentum. The injected mode is adiabatically driven through arbitrary transformations on the Poincaré sphere of first-order paraxial beams. As a result, the down-converted beam conjugated to the seed is shown to follow a path imposed by a nontrivial symmetry on the Poincaré sphere. This symmetry allows controllable distinguishability between the spatial modes of the down-converted beams. In this Letter, we provide convincing experimental evidence of this effect.

Approximate conditions admitted by classes of the Lagrangian [formula omitted

We investigate a class of Lagrangians that admit a type of perturbed harmonic oscillator which occupies a special place in the literature surrounding perturbation theory. We establish explicit and generalized geometric conditions for the symmetry determining equations. The explicit scheme provided can be followed and specialized for any concrete perturbed differential equation possessing the Lagrangian. A systematic solution of the conditions generate nontrivial approximate symmetries and transformations. Detailed cases are discussed to illustrate the relevance of the conditions, namely (a) G1 as a quadratic polynomial, (b) the Klein–Gordon equation of a particle in the context of Generalized Uncertainty Principle and (c) an orbital equation from an embedded Reissner–Nordström black hole.

Equivalence and symmetries for variable coefficient linear heat type equations. I

A systematic and unified approach to transformations and symmetries of a class of variable coefficient heat type linear partial differential equations (PDEs) is presented. The complete symmetry group classification is re-performed in the current context. A useful criterion which is necessary and sufficient for being reducible to the standard heat equation by point transformations is established. A similar criterion is also valid for the equations to have a four- or six-dimensional symmetry group (nontrivial symmetry groups). In this situation, the basis elements are listed in terms of coefficients. A number of illustrative examples are given. In particular, some applications from the recent literature are re-examined in our new approach. Multidimensional parabolic PDEs of heat and Schrödinger type are also considered.

Engineering frequency-dependent superfluidity in Bose-Fermi mixtures

Unconventional superconductivity or superfluidity are among the most exciting and fascinating quantum states in condensed matter physics. Usually these states are characterized by non-trivial spatial symmetry of the pairing order parameter, such as in $^{3}He$ and high-$T_{c}$ cuprates. Besides spatial dependence the order parameter could have unconventional frequency dependence, which is also allowed by Fermi-Dirac statistics. For instance, odd-frequency pairing is an exciting paradigm when discussing exotic superfluidity or superconductivity and is yet to be realized in the experiments. In this paper we propose a symmetry-based method of controlling frequency dependence of the pairing order parameter via manipulating the inversion symmetry of the system. First, a toy model is introduced to illustrate that frequency dependence of the order parameter can be adjusted by controlling the inversion symmetry of the system. Second, taking advantage of the recent rapid developments of shaken optical lattices in ultracold gases, we propose a Bose-Fermi mixture to realize such frequency dependent superfluids. The key idea is introducing the frequency-dependent attraction between Fermions mediated by Bogoliubov phonons with asymmetric dispersion. Our proposal should pave an alternative way for exploring frequency-dependent superconductors or superfluids with cold atoms.

Integrable Variable Dissipation Systems on the Tangent Bundle of a Multi-Dimensional Sphere and Some Applications

This paper is a survey of integrable cases in dynamics of a multi-dimensional rigid body under the action of a nonconservative force field. We review both new results and results obtained earlier. Problems examined are described by dynamical systems with so-called variable dissipation with zero mean. The problem of the search for complete sets of transcendental first integrals of systems with dissipation is quite topical; a large number of works are devoted to it. We introduce a new class of dynamical systems that have a periodic coordinate. Due to the existence of nontrivial symmetry groups of such systems, we can prove that these systems possess variable dissipation with zero mean, which means that on the average for a period with respect to the periodic coordinate, the dissipation in the system is equal to zero, although in various domains of the phase space, either energy pumping or dissipation can occur. Based on the facts obtained, we analyze dynamical systems that appear in the dynamics of a multi-dimensional rigid body and obtain a series of new cases of complete integrability of the equations of motion in transcendental functions, which can be expressed through a finite combination of elementary functions. As applications, we study dynamical equations of motion arising in the study of the plane and spatial dynamics of a rigid body interacting with a medium and also a possible generalization of the obtained methods to the study of general systems arising in the qualitative theory of ordinary differential equations, in the theory of dynamical systems, and also in oscillation theory.

Lie point symmetries and ODEs passing the Painlevé test

The Lie point symmetries of ordinary differential equations (ODEs) that are candidates for having the Painlevé property are explored for ODEs of order n = 2,…, 5. Among the 6 ODEs identifying the Painlevé transcendents only PIII, PV and PVI have nontrivial symmetry algebras and that only for very special values of the parameters. In those cases the transcendents can be expressed in terms of simpler functions, i.e. elementary functions, solutions of linear equations, elliptic functions or Painlevé transcendents occurring at lower order. For higher order or higher degree ODEs that pass the Painlevé test only very partial classifications have been published. We consider many examples that exist in the literature and show how their symmetry groups help to identify those that may define genuinely new transcendents.

Composite symmetry-protected topological order and effective models

Strongly correlated quantum many-body systems at low dimension exhibit a wealth of phenomena, ranging from features of geometric frustration to signatures of symmetry-protected topological order. In suitable descriptions of such systems, it can be helpful to resort to effective models which focus on the essential degrees of freedom of the given model. In this work, we analyze how to determine the validity of an effective model by demanding it to be in the same phase as the original model. We focus our study on one-dimensional spin-1/2 systems and explain how non-trivial symmetry protected topologically ordered (SPT) phases of an effective spin 1 model can arise depending on the couplings in the original Hamiltonian. In this analysis, tensor network methods feature in two ways: On the one hand, we make use of recent techniques for the classification of SPT phases using matrix product states in order to identify the phases in the effective model with those in the underlying physical system, employing Kuenneth's theorem for cohomology. As an intuitive paradigmatic model we exemplify the developed methodology by investigating the bi-layered delta-chain. For strong ferromagnetic inter-layer couplings, we find the system to transit into exactly the same phase as an effective spin 1 model. However, for weak but finite coupling strength, we identify a symmetry broken phase differing from this effective spin-1 description. On the other hand, we underpin our argument with a numerical analysis making use of matrix product states.

Integrable deformations of the [formula omitted] coset CFTs

We study the effective action for the integrable λ-deformation of the Gk1×Gk2/Gk1+k2 coset CFTs. For unequal levels theses models do not fall into the general discussion of λ-deformations of CFTs corresponding to symmetric spaces and have many attractive features. We show that the perturbation is driven by parafermion bilinears and we revisit the derivation of their algebra. We uncover a non-trivial symmetry of these models parametric space, which has not encountered before in the literature. Using field theoretical methods and the effective action we compute the exact in the deformation parameter β-function and explicitly demonstrate the existence of a fixed point in the IR corresponding to the Gk1−k2×Gk2/Gk1 coset CFTs. The same result is verified using gravitational methods for G=SU(2). We examine various limiting cases previously considered in the literature and found agreement.

Quantum Systems Theory Viewed from Kossakowski-Lindblad Lie Semigroups — andVice Versa

The solutions to the celebrated Kossakowski-Lindblad equation extended by coherent controls yield Markovian quantum maps. More precisely, the set of all its solutions forms a semigroup of completely positive trace-preserving maps taking the specific form of a Lie semigroup. Non-trivial symmetries of these semigroups are shown to preclude accessibility in Markovian dissipative systems. This is the open-system analogue to closed systems, where triviality of (quadratic) symmetries of the Hamiltonian part suffices to decide that the system is fully controllable. The findings are placed into a unifying Lie frame of quantum systems and control theory alongside with illustrating examples.