arXiv subjects

Juan Moreno

Publications and source records attributed to Juan Moreno.

5 recordsLinked to original sources

arXiv2024

Homotopical Foundations of Parametrized Quantum Spin Systems

In this paper, we present a homotopical framework for studying invertible gapped phases of matter from the point of view of infinite spin lattice systems, using the framework of algebraic quantum mechanics. We define the notion of quantum state types. These are certain lax-monoidal functors from the category of finite dimensional Hilbert spaces to the category of topological spaces. The universal example takes a finite dimensional Hilbert space to the pure state space of the quasi-local algebra of the quantum spin system with this Hilbert space at each site of a specified lattice. The lax-monoidal structure encodes the tensor product of states, which corresponds to stacking for quantum systems. We then explain how to formally extract parametrized phases of matter from quantum state types, and how they naturally give rise to $\mathscr{E}_\infty$-spaces for an operad we call the "multiplicative" linear isometry operad. We define the notion of invertible quantum state types and explain how the passage to phases for these is related to group completion. We also explain how invertible quantum state types give rise to loop-spectra. Our motivation is to provide a framework for constructing Kitaev's loop-spectrum of bosonic invertible gapped phases of matter. Finally, as a first step towards understanding the homotopy types of the loop-spectra associated to invertible quantum state types, we prove that the pure state space of any UHF algebra is simply connected.

math-ph↗

arXiv2022

Continuous Dependence on the Initial Data in the Kadison Transitivity Theorem and GNS Construction

We consider how the outputs of the Kadison transitivity theorem and Gelfand-Naimark-Segal construction may be obtained in families when the initial data are varied. More precisely, for the Kadison transitivity theorem, we prove that for any nonzero irreducible representation $(\mathcal{H}, π)$ of a $C^*$-algebra $\mathfrak{A}$ and $n \in \mathbb{N}$, there exists a continuous function $A:X \rightarrow \mathfrak{A}$ such that $π(A(\mathbf{x}, \mathbf{y}))x_i = y_i$ for all $i \in \{1, \ldots, n\}$, where $X$ is the set of pairs of $n$-tuples $(\mathbf{x}, \mathbf{y}) \in \mathcal{H}^n \times \mathcal{H}^n$ such that the components of $\mathbf{x}$ are linearly independent. Versions of this result where $A$ maps into the self-adjoint or unitary elements of $\mathfrak{A}$ are also presented. Regarding the Gelfand-Naimark-Segal construction, we prove that given a topological $C^*$-algebra fiber bundle $p:\mathfrak{A} \rightarrow Y$, one may construct a topological fiber bundle $\mathscr{P}(\mathfrak{A}) \rightarrow Y$ whose fiber over $y \in Y$ is the space of pure states of $\mathfrak{A}_y$ (with the norm topology), as well as bundles $\mathscr{H} \rightarrow \mathscr{P}(\mathfrak{A})$ and $\mathscr{N} \rightarrow \mathscr{P}(\mathfrak{A})$ whose fibers $\mathscr{H}_ω$ and $\mathscr{N}_ω$ over $ω\in \mathscr{P}(\mathfrak{A})$ are the GNS Hilbert space and closed left ideal, respectively, corresponding to $ω$. When $p:\mathfrak{A} \rightarrow Y$ is a smooth fiber bundle, we show that $\mathscr{P}(\mathfrak{A}) \rightarrow Y$ and $\mathscr{H}\rightarrow \mathscr{P}(\mathfrak{A})$ are also smooth fiber bundles; this involves proving that the group of $*$-automorphisms of a $C^*$-algebra is a Banach-Lie group. In service of these results, we review the geometry of the topology and pure state space. A simple non-interacting quantum spin system is provided as an example.

math.OA↗

arXiv2022

Flow of (higher) Berry curvature and bulk-boundary correspondence in parametrized quantum systems

This paper is concerned with the physics of parametrized gapped quantum many-body systems, which can be viewed as a generalization of conventional topological phases of matter. In such systems, rather than considering a single Hamiltonian, one considers a family of Hamiltonians that depend continuously on some parameters. After discussing the notion of phases of parametrized systems, we formulate a bulk-boundary correspondence for an important bulk quantity, the Kapustin-Spodyneiko higher Berry curvature, first in one spatial dimension and then in arbitrary dimension. This clarifies the physical interpretation of the higher Berry curvature, which in one spatial dimension is a flow of (ordinary) Berry curvature. In d dimensions, the higher Berry curvature is a flow of (d-1)-dimensional higher Berry curvature. Based on this, we discuss one-dimensional systems that pump Chern number to/from spatial boundaries, resulting in anomalous boundary modes featuring isolated Weyl points. In higher dimensions, there are pumps of the analogous quantized invariants obtained by integrating the higher Berry curvature. We also discuss the consequences for parametrized systems of Kitaev's proposal that invertible phases are classified by a generalized cohomology theory, and emphasize the role of the suspension isomorphism in generating new examples of parametrized systems from known invertible phases. Finally, we present a pair of general quantum pumping constructions, based on physical pictures introduced by Kitaev, which take as input a d-dimensional parametrized system, and produce new (d+1)-dimensional parametrized systems. These constructions are useful for generating examples, and we conjecture that one of the constructions realizes the suspension isomorphism in a generalized cohomology theory of invertible phases.

cond-mat.str-el↗

arXiv2019

Anisotropic shear stress $σ_{xy}$ effects in the basal plane of Sr$_2$RuO$_4$

In this short note, we repeat the calculations the jumps for the specific heat C$_{σ_{xy}}$, the elastic compliance S$_{xyxy}^{σ_{xy}}$ and the thermal expansion $α_{σ_{xy}}$ due to a shear stress $σ_{xy}$ in the basal plane of $Sr_2RuO_4$. Henceforth we clarify some issues regarding the elastic theoretical framework suitable to explain the sound speed experiments of Lupien et al. (2001,2002), and partially the strain experiments of Hicks et al. (2014), and Steppke et al. (2016) in strontium ruthenate. We continue to propose that the discontinuity in the elastic constant C$_{xyxy}$ of this tetragonal crystal gives unambiguous experimental evidence that the superconducting order parameter $Ψ$ has two components with a broken time-reversal symmetry state, and that the $γ$ band couples the anisotropic electron-phonon interaction to the $[xy]$ in-plane shear stress according to Walker and collaborators [4] and [3]. Some importants words about the roll of the spin equal to one for the transversal phonons are added in the conclusion following Levine [34].

cond-mat.supr-con↗

arXiv2019

A non-linear minimization calculation of the renormalized frequency $\tildeω$ in dirty d-wave superconductors

This work performs a comparative numerical study of the impurity average self-frequency $\tildeω$ in an unconventional superconducting alloy with non-magnetic impurities. Two methods are used: the Levenberg-Marquardt algorithm as a non-linear minimization problem, and a fixed-point iteration procedure. The unconventional superconducting renormalized by impurities \bw is a self-consistent complex non-linear equation with two varying parameters: the impurity concentration $Γ^+$ and the strength of the impurities c, for which its numerical solution is a computational challenge. Throughout this study $\tildeω$ is the renormalized frequency, \bg represents the inverse of the residual average lifetime $τ$ at zero frequency, and $N/N_0$ is the normalized superconducting density of states (DOS). This study uses an order parameter that corresponds to the high-temperature superconducting ceramics (HTS) with a well-established gap symmetry. The results reveal the computational efficiency of the non-linear minimization technique by improving the calculations of the $\tildeω$ computation when using a 2D parameter space ($Γ^+$, c), particularly in the unitary regime, where the imaginary part of $\tildeω$ is a complicated expression of those parameters; this allows to enhance the study of the universal behavior of this particular quantum mechanical state.

cond-mat.supr-con↗