Irreducible Sectors Research Articles

Deformed symmetry structures and quantum many-body scar subspaces

A quantum many-body scar system usually contains a special non-thermal subspace (approximately) decoupled from the rest of the Hilbert space. In this work, we propose a general structure called deformed symmetric spaces for the decoupled subspaces hosting quantum many-body scars, which are irreducible sectors of simple Lie groups transformed by matrix-product operators (or projected entangled pair operators), of which the entanglement entropies are proved to obey sub-volume-law scaling and thus violate the eigenstate thermalization hypothesis. A deformed symmetric space, in general, is required to have at least a U(1) sub-Lie-group symmetry to allow coherent periodic dynamics from certain low-entangled initial states. We enumerate several possible deforming transformations based on the sub-group symmetry requirement and recover many existing models whose scar states are not connected by symmetry. In particular, a two-dimensional scar model is proposed, which hosts a periodic dynamical trajectory on which all states are topologically ordered.

Many-body localization transition from flat-band fine tuning

Translationally invariant flatband Hamiltonians with interactions lead to a many-body localization transition. Our models are obtained from single-particle lattices hosting a mix of flat and dispersive bands, and equipped with fine-tuned two--body interactions. Fine-tuning of the interaction results in an extensive set of local conserved charges and a fragmentation of the Hilbert space into irreducible sectors. In each sector, the conserved charges originate from the flatband and act as an effective disorder inducing a transition between ergodic and localized phases upon variation of the interaction strength. Such fine-tuning is possible in arbitrary lattice dimensions and for any many-body statistics. We present computational evidence for this transition with spinless fermions.

Gaussian orthogonal ensemble for quasiperiodic tilings without unfolding: r -value statistics

We study the level-spacing statistics for noninteracting Hamiltonians defined on the two-dimensional quasiperiodic Ammann-Beenker (AB) tiling. When applying the numerical procedure of ``unfolding,'' these spectral properties in each irreducible sector are known to be well described by the universal Gaussian orthogonal random matrix ensemble. However, the validity and numerical stability of the unfolding procedure has occasionally been questioned due to the fractal self-similarity in the density of states for such quasiperiodic systems. Here, using the so-called $r$-value statistics for random matrices, $P(r)$, for which no unfolding is needed, we show that the Gaussian orthogonal ensemble again emerges as the most convincing level statistics for each irreducible sector. The results are extended to random-AB tilings where random flips of vertex connections lead to the irreducibility.

Mathcal{N} = 2 extended MacDowell-Mansouri supergravity

We construct a gauge theory based in the supergroup G = SU(2, 2|2) that generalizes MacDowell-Mansouri supergravity. This is done introducing an extended notion of Hodge operator in the form of an outer automorphism of su(2, 2|2)-valued 2-form tensors. The model closely resembles a Yang-Mills theory — including the action principle, equations of motion and gauge transformations — which avoids the use of the otherwise complicated component formalism. The theory enjoys H = SO(3, 1) × ℝ × U(1) × SU(2) off-shell symmetry whilst the broken symmetries G/H, translation-type symmetries and supersymmetry, can be recovered on surface of integrability conditions of the equations of motion, for which it suffices the Rarita-Schwinger equation and torsion-like constraints to hold. Using the matter ansatz —projecting the 1 ⊗ 1/2 reducible representation into the spin-1/2 irreducible sector — we obtain (chiral) fermion models with gauge and gravity interactions.

Lorentz Group Projector Technique for Decomposing Reducible Representations and Applications to High Spins

The momentum-independent Casimir operators of the homogeneous spin-Lorentz group are employed in the construction of covariant projector operators, which can decompose anyone of its reducible finite-dimensional representation spaces into irreducible components. One of the benefits from such operators is that any one of the finite-dimensional carrier spaces of the Lorentz group representations can be equipped with Lorentz vector indices because any such space can be embedded in a Lorentz tensor of a properly-designed rank and then be unambiguously found by a projector. In particular, all the carrier spaces of the single-spin-valued Lorentz group representations, which so far have been described as 2 ( 2 j + 1 ) column vectors, can now be described in terms of Lorentz tensors for bosons or Lorentz tensors with the Dirac spinor component, for fermions. This approach facilitates the construct of covariant interactions of high spins with external fields in so far as they can be obtained by simple contractions of the relevant S O ( 1 , 3 ) indices. Examples of Lorentz group projector operators for spins varying from 1 / 2 –2 and belonging to distinct product spaces are explicitly worked out. The decomposition of multiple-spin-valued product spaces into irreducible sectors suggests that not only the highest spin, but all the spins contained in an irreducible carrier space could correspond to physical degrees of freedom.

Bosonic and fermionic Weinberg-Joos (j,0) ⊕ (0,j) states of arbitrary spins as Lorentz tensors or tensor-spinors and second-order theory

We propose a general method for the description of arbitrary single spin-j states transforming according to (j, 0) ⊕ (0, j) carrier spaces of the Lorentz algebra in terms of Lorentz tensors for bosons, and tensor-spinors for fermions, and by means of second-order Lagrangians. The method allows to avoid the cumbersome matrix calculus and higher ∂2j order wave equations inherent to the Weinberg-Joos approach. We start with reducible Lorentz tensor (tensor-spinor) representation spaces hosting one sole (j, 0) ⊕ (0, j) irreducible sector and design there a representation reduction algorithm based on one of the Casimir invariants of the Lorentz algebra. This algorithm allows us to separate neatly the pure spin-j sector of interest from the rest, while preserving the separate Lorentz and Dirac indexes. However, the Lorentz invariants are momentum independent and do not provide wave equations. Genuine wave equations are obtained by conditioning the Lorentz tensors under consideration to satisfy the Klein-Gordon equation. In so doing, one always ends up with wave equations and associated Lagrangians that are of second order in the momenta. Specifically, a spin-3/2 particle transforming as (3/2, 0) ⊕ (0, 3/2) is comfortably described by a second-order Lagrangian in the basis of the totally anti-symmetric Lorentz tensor-spinor of second rank, Ψ [μν]. Moreover, the particle is shown to propagate causally within an electromagnetic background. In our study of (3/2, 0) ⊕ (0, 3/2) as part of Ψ [μν] we reproduce the electromagnetic multipole moments known from the Weinberg-Joos theory. We also find a Compton differential cross-section that satisfies unitarity in forward direction. The suggested tensor calculus presents itself very computer friendly with respect to the symbolic software FeynCalc.

Charge superselection sectors for QCD on the lattice

We study quantum chromodynamics (QCD) on a finite lattice Λ in the Hamiltonian approach. First, we present the field algebra AΛ as comprising a gluonic part, with basic building block being the crossed product C*-algebra C(G)⊗αG, and a fermionic (CAR-algebra) part generated by the quark fields. By classical arguments, AΛ has a unique (up to unitary equivalence) irreducible representation. Next, the algebra OΛi of internal observables is defined as the algebra of gauge invariant fields, satisfying the Gauss law. In order to take into account correlations of field degrees of freedom inside Λ with the “rest of the world,” we must extend OΛi by tensorizing with the algebra of external gauge invariant operators. This way we construct the full observable algebra OΛ. It is proved that its irreducible representations are labelled by Z3-valued boundary flux distributions. Then, it is shown that there exist unitary operators (charge carrying fields), which intertwine between irreducible sectors leading to a classification of irreducible representations in terms of the Z3-valued global boundary flux. By the global Gauss law, these three inequivalent charge superselection sectors can be labeled in terms of the global color charge (triality) carried by quark fields. Finally, OΛ is discussed in terms of generators and relations.

Topological Sectors and a Dichotomy in Conformal Field Theory

Let A be a local conformal net of factors on the circle with the split property. We provide a topological construction of soliton representations of the tensor product of n copies of A, that restrict to true representations of subnet inviant under cyclic permutations (cyclic orbifold). We prove a quantum index theorem for our sectors relating the Jones index to a topological degree. Then A is not completely rational iff the the cyclic orbifold has an irreducible representation with infinite index. This implies the following dichotomy: if all irreducible sectors of A have a conjugate sector then either A is completely rational or A has uncountably many different irreducible sectors. Thus A is rational iff A is completely rational. In particular, if the mu-index of A finite then A turns out to be strongly additive. By [KLM], if A is rational then the tensor category of representations of A is automatically modular, namely the braiding symmetry is non-degenerate. In interesting cases, we compute the fusion rules of the topological solitons and show that they determine all twisted sectors of the cyclic orbifold.

Universal quantum control in irreducible state-space sectors: Application to bosonic and spin-boson systems

We analyze the dynamical-algebraic approach to universal quantum control introduced in P. Zanardi, S. Lloyd, quant-ph/0305013. The quantum state-space $\cal H$ encoding information decomposes into irreducible sectors and subsystems associated to the group of available evolutions. If this group coincides with the unitary part of the group-algebra $\CC{\cal K}$ of some group $\cal K$ then universal control is achievable over the ${\cal K}$-irreducible components of $\cal H$. This general strategy is applied to different kind of bosonic systems. We first consider massive bosons in a double-well and show how to achieve universal control over all finite-dimensional Fock sectors. We then discuss a multi-mode massless case giving the conditions for generating the whole infinite-dimensional multi-mode Heisenberg-Weyl enveloping-algebra. Finally we show how to use an auxiliary bosonic mode coupled to finite-dimensional systems to generate high-order non-linearities needed for universal control.

A Modular Functor Which is Universal¶for Quantum Computation

We show that the topological modular functor from Witten-Chern-Simons theory is universal for quantum computation in the sense that a quantum circuit com- putation can be efficiently approximated by an intertwining action of a braid on the functor's state space. A computational model based on Chern-Simons theory at a fifth root of unity is defined and shown to be polynomially equivalent to the quantum circuit model. The chief technical advance: the density of the irreducible sectors of the Jones representation has topological implications which will be considered elsewhere.

Multi-Interval Subfactors and Modularity¶of Representations in Conformal Field Theory

We describe the structure of the inclusions of factors A(E) contained in A(E')' associated with multi-intervals E of R for a local irreducible net A of von Neumann algebras on the real line satisfying the split property and Haag duality. In particular, if the net is conformal and the subfactor has finite index, the inclusion associated with two separated intervals is isomorphic to the Longo-Rehren inclusion, which provides a quantum double construction of the tensor category of superselection sectors of A. As a consequence, the index of A(E) in A(E')' coincides with the global index associated with all irreducible sectors, the braiding symmetry associated with all sectors is non-degenerate, namely the representations of A form a modular tensor category, and every sector is a direct sum of sectors with finite dimension. The superselection structure is generated by local data. The same results hold true if conformal invariance is replaced by strong additivity and there exists a modular PCT symmetry.

Covariant Sectors with Infinite Dimension¶and Positivity of the Energy

Let ? be a local conformal net of von Neumann algebras on S 1 and ρ a Möbius covariant representation of ?, possibly with infinite dimension. If ρ has finite index, ρ has automatically positive energy. If ρ has infinite index, we show the spectrum of the energy always to contain the positive real line, but, as seen by an example, it may contain negative values. We then consider nets with Haag duality on ℝ, or equivalently sectors with non-solitonic extension to the dual net; we give a criterion for irreducible sectors to have positive energy, namely this is the case iff there exists an unbounded Möbius covariant left inverse. As a consequence the class of sectors with positive energy is stable under composition, conjugation and direct integral decomposition.

Generalized squeezing, Bogoliubov quasiparticles, and information in two- and three-mode systems.

It is shown that many types of two-mode squeezing are based on reducible representations of the SU(1,1) or SU(2) groups. In these cases it is possible to squeeze independently each irreducible sector and generalize the concept of squeezing. The related concept of Bogoliubov quasiparticles is also generalized. The information transmitted by two-mode states is calculated as a sum of the information in the two modes and the information in the correlation. Special emphasis is given in the effect of squeezing on the correlation between the two modes. Higher correlations in three-mode systems are also studied in the context of information theory.

The frequency distribution of the lattice vibrations of sodium

The frequencies of the latrice vibrations of metallic sodium have been calculated by Toya's method at 1661 points within the irreducible sector of the first Brillouin zone. From these the vibrational frequency distribution has been obtained and this shows good agreement with the distribution obtained by Dixon et al. by fitting force constants to the results of neutron scattering experiments. The distribution has been used to obtain values of the specific heat.