arXiv subjects

Gerardo Ortiz

Publications and source records attributed to Gerardo Ortiz.

At least 19 recordsLinked to original sources

arXiv2026

Unveiling Topological Fusion in Quantum Hall Systems from Microscopic Principles

Establishing the fusion rules of anyonic quasiparticles in fractional quantum Hall fluids is essential for understanding their underlying topological order. Building on the conjecture that key topological properties are encoded in the "DNA" of candidate many-body wave functions - that is, the pattern of dominant orbital occupations restricted to a finite number of lowest Landau levels - we propose a combinatorial framework that derives these fusion rules directly from microscopic data. By extending Schrieffer's counting argument and introducing classes of topological excitations, our framework provides a unified route to the fusion rules for both Abelian and non-Abelian excitations. This approach elucidates the emergence of topological features from first principles in both fermionic and bosonic systems.

cond-mat.mes-hall↗

arXiv2026

Generalized Wigner theorem for non-invertible symmetries

We establish the conditions under which a conservation law associated with a non-invertible operator may be realized as a symmetry in quantum physics. As established by Wigner, all quantum symmetries must be represented by either unitary or antiunitary transformations. Relinquishing an implicit assumption of invertibility, we demonstrate that the fundamental invariance of quantum transition probabilities under the application of symmetries mandates that all non-invertible symmetries may only correspond to {\it projective} unitary or antiunitary transformations, i.e., {\it partial isometries}. This extends the notion of physical states beyond conventional rays in Hilbert space to equivalence classes in an {\it extended, gauged Hilbert space}, thereby broadening the traditional understanding of symmetry transformations in quantum theory. Our generalized theorem applies irrespective of the origin of the (non)invertible symmetry, holds in arbitrary spatial dimensions, and is independent of the Hamiltonian or action. We explore its physical consequences and, using simple model systems, illustrate how the distinction between invertible and non-invertible symmetries can sometimes be tied to the choice of boundary conditions.

quant-ph↗

arXiv2026

Integrable Model of a Superconductor with non-Fermi liquid and Mott Phases

We present and analyze an exactly solvable interacting fermionic pairing model, which features interactions that entangle states at momenta $\mathbf{k}$ and $-\mathbf{k}$. These interactions give rise to novel correlated ground states, leading to a rich phase diagram that includes superconducting, multiple metallic, and Mott-insulating phases. At finite interaction strengths, we observe the emergence of multiple many-body Fermi surfaces, which violate Luttinger's theorem and challenge the conventional Landau-Fermi liquid paradigm. A distinguishing feature of our model is that it remains quantum integrable, even with the addition of pairing interactions of various symmetries, setting it apart from the Hatsugai-Kohmoto model. Our results provide an analytically tractable framework for studying strong correlation effects that give rise to fractionalized excitations and unconventional superconductivity, offering valuable insights into a broad class of integrable many-body systems.

cond-mat.str-el↗

arXiv2025

A Minimal Network of Brain Dynamics: Hierarchy of Approximations to Quasi-critical Neural Network Dynamics

We present an interacting model of neural network dynamics that incorporates key biological features, including multiple forms of inhibitory interactions. We develop a hierarchy of analytical mean-field approximations to characterize nonequilibrium phase transitions between ordered, disordered, and chaotic regimes, complemented by a detailed stability analysis. We show that inhibition generically enhances the stability of network dynamics. The model is consistent with the quasi-criticality hypothesis, exhibiting regions of maximal dynamical susceptibility and mutual information, modulated by the strength of external stimuli. We further demonstrate that, at the mean-field level, the critical transition belongs to the mean-field directed percolation universality class, in agreement with prior experimental and theoretical studies. More broadly, our framework may offer insights into neurological disorders, with the unstable regime exhibiting chaotic dynamics that may be associated with epileptic seizures.

cond-mat.dis-nn↗

arXiv2025

Algebraic Fusion in a (2+1)-dimensional Lattice Model with Generalized Symmetries

The notion of quantum symmetry has recently been extended to include reduced-dimensional transformations and algebraic structures beyond groups. Such generalized symmetries lead to exotic phases of matter and excitations that defy Landau's original paradigm. Here, we develop an algebraic framework for systematically deriving the fusion rules of topological defects in higher-dimensional lattice systems with non-invertible generalized symmetries, and focus on a (2+1)-dimensional quantum Ising plaquette model as a concrete illustration. We show that bond-algebraic automorphisms, when combined with the so-called half-gauging procedure, reveal the structure of the non-invertible duality symmetry operators, which can be explicitly represented as a sequential quantum circuit. The resulting duality defects are constrained by the model's rigid higher symmetries (lower-dimensional subsystem symmetries), leading to restricted mobility. We establish the fusion algebra of these defects. Finally, in constructing the non-invertible duality transformation, we explicitly verify that it acts as a partial isometry on the physical Hilbert space, thereby satisfying a recent generalization of Wigner's theorem applicable to non-invertible symmetries.

quant-ph↗

arXiv2025

The Physics of Local Optimization in Complex Disordered Systems

Limited resources motivate decomposing large-scale problems into smaller,``local" subsystems and stitching together the so-found solutions. We explore the physics underlying this approach and discuss the concept of ``local hardness", i.e., the complexity of predicting local properties of the solution from local information, for the ground-state problem of both P- and NP-hard spin-glasses and related frustrated spin systems. Depending on the model considered, we observe varying scaling behaviors in how errors associated with local predictions decay as a function of the size of the solved subsystem. These errors are intimately connected to global critical threshold instabilities, characterized by gapless, avalanche-like excitations that follow scale-invariant size distributions. Away from criticality, local solvers quickly achieve high accuracy, aligning closely with the results of the computationally much more expensive global minimization. We leverage these findings to introduce a heuristic contraction-based algorithm for globally studying spin-glass ground states. The local solvers further display sharp imprints of the phase transition from the spin-glass to the ferromagnetic phase as the distribution of spin-glass couplings is shifted, as well as characteristic differences for the infinite-range model, implying the existence of specific classes of local hardness. Our findings shed light on how Nature may operate solely through local actions at her disposal.

cond-mat.dis-nn↗

arXiv2025

An Industry-Academia Partnership for Advancing Quantum Frontiers: Perspective from the U.S. Center for Quantum Technologies

The U.S. Center for Quantum Technologies (CQT) is a multi-university consortium established under the National Science Foundation's (NSF) Industry-University Cooperative Research Centers (IUCRC) program. Led jointly by Purdue University, Indiana University (both Bloomington and Indianapolis campuses), and the University of Notre Dame, CQT integrates academic research with industrial and governmental collaboration to accelerate quantum innovation. This perspective outlines the consortium's strategic mission, interdisciplinary research agenda, and its role in shaping the future of quantum-enabled technologies through collaborative development, translational impact, and workforce cultivation.

quant-ph↗

arXiv2025

Toward a Physics of Deep Learning and Brains

Deep neural networks and brains both learn and share superficial similarities: processing nodes are likened to neurons and adjustable weights are likened to modifiable synapses. But can a unified theoretical framework be found to underlie them both? Here we show that the equations used to describe neuronal avalanches in living brains can also be applied to cascades of activity in deep neural networks. These equations are derived from non-equilibrium statistical physics and show that deep neural networks learn best when poised between absorbing and active phases. Because these networks are strongly driven by inputs, however, they do not operate at a true critical point but within a quasi-critical regime -- one that still approximately satisfies crackling noise scaling relations. By training networks with different initializations, we show that maximal susceptibility is a more reliable predictor of learning than proximity to the critical point itself. This provides a blueprint for engineering improved network performance. Finally, using finite-size scaling we identify distinct universality classes, including Barkhausen noise and directed percolation. This theoretical framework demonstrates that universal features are shared by both biological and artificial neural networks.

cond-mat.dis-nn↗

arXiv2025

Emergence of a Boundary-Sensitive Phase in Hyperbolic Ising Models

Physical systems defined on hyperbolic lattices may exhibit phases of matter that only emerge due to negative curvature. We focus on the case of the Ising model under open boundary conditions and show that an ``intermediate'' phase emerges in addition to standard (high-temperature) paramagnetic and (low-temperature) ferromagnetic phases. When performing the Kramers-Wannier duality the fact that it alters boundary conditions becomes crucial, since a finite fraction of lattice sites lie on the boundary. We propose to characterize this ``intermediate'' phase by its sensitivity to boundary conditions, wherein bulk ordering is not spontaneous but rather induced by boundary effects, setting it apart from the Landau paradigm of spontaneous symmetry breaking. By developing a $\mathbb{Z}_2$ symmetry restricted extension of the Corner Transfer Matrix Renormalization Group method, we provide numerical evidence for the existence of all three distinct phases and their corresponding two-stage phase transitions, thereby establishing the complete phase diagram. We also establish how the (spontaneous) intermediate-to-ferromagnetic and the (induced) paramagnetic-to-intermediate transition points are related by the Kramers-Wannier duality relation. We discuss a holographic correspondence between boundary and bulk behaviors and derive exact expressions for boundary correlation functions on Cayley trees.

cond-mat.stat-mech↗

arXiv2024

Universal fragility of spin-glass ground-states under single bond changes

We consider the effect of perturbing a single bond on ground-states of nearest-neighbor Ising spin-glasses, with a Gaussian distribution of the coupling constants, across various two and three-dimensional lattices and regular random graphs. Our results reveal that the ground-states are strikingly susceptible to such changes. Altering the strength of only a single bond beyond a critical threshold value leads to a new ground-state that differs from the original one by a droplet of flipped spins whose boundary and volume diverge with the system size -- an effect that is reminiscent of the more familiar phenomenon of disorder chaos. These elementary fractal-boundary zero-energy droplets and their composites feature robust characteristics and provide the lowest-energy macroscopic spin-glass excitations. Remarkably, within numerical accuracy, the size of such droplets conforms to a nearly universal power-law distribution with exponents dependent on the spatial dimension of the system. Furthermore, the critical coupling strengths adhere to a stretched Gaussian distribution that is predominantly determined by the local coordination number.

cond-mat.dis-nn↗

arXiv2023

Fusion mechanism for quasiparticles and topological quantum order in the lowest Landau level

Starting from Halperin multilayer systems we develop a hierarchical scheme that generates, bosonic and fermionic, single-layer quantum Hall states (or vacua) of arbitrary filling factor. Our scheme allows for the insertion of quasiparticle excitations with either Abelian or non-Abelian statistics and quantum numbers that depend on the nature of the original vacuum. Most importantly, it reveals a fusion mechanism for quasielectrons and magnetoexcitons that generalizes ideas about particle fractionalization introduced in A. Bochniak, Z. Nussinov, A. Seidel, and G. Ortiz, Commun. Phys. 5, 171 (2022) for the case of Laughlin fluids. In addition, in the second quantization representation, we uncover the inherent topological quantum order characterizing these vacua. In particular, we illustrate the methodology by constructing generalized composite (generalized Read) operators for the non-Abelian Pfaffian and Hafnian quantum fluid states.

cond-mat.str-el↗

arXiv2023

Emergent magnetic order in the antiferromagnetic Kitaev model with a [111] field

The Kitaev spin liquid, stabilized as the ground state of the Kitaev honeycomb model, is a paradigmatic example of a topological $\mathbb{Z}_2$ quantum spin liquid. The fate of the Kitaev spin liquid in presence of an external magnetic field is a topic of current interest due to experiments, which apparently unveil a $\mathbb{Z}_2$ topological phase in the so-called Kitaev materials, and theoretical studies predicting the emergence of an intermediate quantum phase of debated nature before the appearance of a trivial partially polarized phase. In this work, we employ hierarchical mean-field theory, an algebraic and numerical method based on the use of clusters preserving relevant symmetries and short-range quantum correlations, to investigate the quantum phase diagram of the antiferromagnetic Kitaev's model in a [111] field. By using clusters of 24 sites, we predict that the Kitaev spin liquid transits through two intermediate phases characterized by stripe and chiral order, respectively, before entering the trivial partially polarized phase, differing from previous studies. We assess our results by performing exact diagonalization and computing the scaling of different observables, including the many-body Chern number and other topological quantities, thus establishing hierarchical mean-field theory as a method to study topological quantum spin liquids.

cond-mat.str-el↗

arXiv2023

Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners

We present a systematic recipe for generating and classifying duality transformations in one-dimensional quantum lattice systems. Our construction emphasizes the role of global symmetries, including those described by (non)-abelian groups but also more general categorical symmetries. These symmetries can be realized as matrix product operators which allow the extraction of a fusion category that characterizes the algebra of all symmetric operators commuting with the symmetry. Known as the bond algebra, its explicit realizations are classified by module categories over the fusion category. A duality is then defined by a pair of distinct module categories giving rise to dual realizations of the bond algebra, as well as dual Hamiltonians. Symmetries of dual models are in general distinct but satisfy a categorical Morita equivalence. A key novelty of our categorical approach is the explicit construction of matrix product operators that intertwine dual bond algebra realizations at the level of the Hilbert space, and in general map local order operators to non-local string-order operators. We illustrate this approach for known dualities such as Kramers-Wannier, Jordan-Wigner, Kennedy-Tasaki and the IRF-vertex correspondence, a new duality of the $t$-$J_z$ chain model, and dualities in models with the exotic Haagerup symmetry. Finally, we comment on generalizations to higher dimensions.

quant-ph↗

arXiv2023

Wiener-Hopf factorization approach to a bulk-boundary correspondence and stability conditions for topological zero-energy modes

Both the physics and applications of fermionic symmetry-protected topological phases rely heavily on a principle known as bulk-boundary correspondence, which predicts the emergence of protected boundary-localized energy excitations (boundary states) if the bulk is topologically non-trivial. Current theoretical approaches formulate a bulk-boundary correspondence as an equality between a bulk and a boundary topological invariant, where the latter is a property of boundary states. However, such an equality does not offer insight about the stability or the sensitivity of the boundary states to external perturbations. To solve this problem, we adopt a technique known as the Wiener-Hopf factorization of matrix functions. Using this technique, we first provide an elementary proof of the equality of the bulk and the boundary invariants for one-dimensional systems with arbitrary boundary conditions in all Altland-Zirnbauer symmetry classes. This equality also applies to quasi-one-dimensional systems (e.g., junctions) formed by bulks belonging to the same symmetry class. We then show that only topologically non-trivial Hamiltonians can host stable zero-energy edge modes, where stability refers to continuous deformation of zero-energy excitations with external perturbations that preserve the symmetries of the class. By leveraging the Wiener-Hopf factorization, we establish bounds on the sensitivity of such stable zero-energy modes to external perturbations. Our results show that the Wiener-Hopf factorization is a natural tool to investigate bulk-boundary correspondence in quasi-one-dimensional fermionic symmetry-protected topological phases. Our results on the stability and sensitivity of zero modes are especially valuable for applications, including Majorana-based topological quantum computing.

cond-mat.stat-mech↗

arXiv2023

Spin-Textured Neutron Beams with Orbital Angular Momentum

We present a rigorous theoretical framework underpinning the technique of spin-echo modulated small-angle neutron scattering (SEMSANS), and show how the technique can be extended in order to generate spin-textured neutron beams with orbital angular momentum (OAM) via birefringent neutron spin-polarization devices known as magnetic Wollaston prisms. Neutron OAM beams are mathematically characterized by a ``cork-screw'' phase singularity $e^{i \ell ϕ}$ about the propagation axis where $\ell$ is the OAM quantum number. To understand the precise relationship between the emergent OAM state and the variety of spin textures realized by various setups, we have developed a path-integral approach that in the interferometric limit makes a judicious use of magnetic Snell's law. We show that our proposed technique produces a complex two-dimensional pattern of spin-OAM entangled states which may be useful as a probe of quantum magnetic materials. We compare our path-integral approach to the well-known single-path Larmor precession model and present a pedagogical derivation of magnetic Snell's law of refraction for both massive and massless particles based on Maupertuis's action principle.

quant-ph↗

arXiv2023

Quantum Quenches of an SO(5) Pseudospin Reveal Higgs Bosons

Controlled dynamical probe measurement of complex order parameter fluctuations may reveal its massive collective excitations. Here, we design dynamical quench protocols to excite independently all ten midgap Higgs bosons in the isotropic Balian--Werthamer state of a spinfull $p$-wave superfluid or superconductor. The analysis is based on microscopic equations of motion of an SO(5) pseudospin, an extension of the usual Bloch equation to a five dimensional space. Key to these protocols is the realization of quenches that break the rotational symmetry of the kinetic energy and exploit the irreducible representation of the angular momentum $J=2$. For perturbative quenches, we find (non-decaying) periodic oscillations in time of these Higgs modes. Experiments to realize our proposal for either superfluids or superconductors are considered.

cond-mat.supr-con↗

arXiv2022

A Theorem on Extensive Spectral Degeneracy for Systems with Higher Symmetries in General Dimensions

We establish, in the spirit of the Lieb-Schultz-Mattis theorem, lower bounds on the spectral degeneracy of quantum systems with higher (Gauge Like) symmetries with rather generic physical boundary conditions in an arbitrary number of spatial dimensions. Contrary to applying twists or equivalent adiabatic operations, we exploit the effects of modified boundary conditions. When a general choice of boundary geometry is immaterial in approaching the thermodynamic limit, systems that exhibit non-commuting Gauge Like symmetries, such as the orbital compass model, must have an exponential (in the size of the boundary) degeneracy of each of their spectral levels. We briefly discuss why, in spite of the proven large degeneracy associated with infrared-ultraviolet mixing, some systems may still exhibit conventional physical behaviors, i.e., of those of systems with non-extensive degeneracies, due to entropic "order by disorder" type effects.

quant-ph↗

arXiv2022

Retrodictive Quantum Computing

Quantum models of computation are widely believed to be more powerful than classical ones. Efforts center on proving that, for a given problem, quantum algorithms are more resource efficient than any classical one. All this, however, assumes a standard predictive paradigm of reasoning where, given initial conditions, the future holds the answer. How about bringing information from the future to the present and exploit it to one's advantage? This is a radical new approach for reasoning, so-called Retrodictive Computation, that benefits from the specific form of the computed functions. We demonstrate how to use tools of symbolic computation to realize retrodictive quantum computing at scale and exploit it to efficiently, and classically, solve instances of the quantum Deutsch-Jozsa, Bernstein-Vazirani, Simon, Grover, and Shor's algorithms.

quant-ph↗