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Ivan Horváth

Publications and source records attributed to Ivan Horváth.

At least 19 recordsLinked to original sources
arXiv2026

Glue Condensate, Quark Condensate and Dirac Spectral Density

I derive the regularized formula for glue scalar density (gluon condensate) in terms of Dirac spectral density [arXiv:2509.03509], and elaborate on its uses and meaning. Particular attention is given to understanding of what this new formula reveals about the relation between glue and quark scalar densities, how it relates to IR phase, how it clarifies the distinction between anomalous and spontaneous ways of breaking symmetries, and what it says about the relation between UV and IR in QCD.

hep-lat↗
arXiv2025

Gluon Condensate via Dirac Spectral Density: IR Phase, Scale Anomaly and IR Decoupling

Quark and gluon scalar densities, $\langle \barψ ψ\rangle$ and $\langle F^2 \rangle$, reflect the degree of scale-invariance violations in SU(N) gauge theories with fundamental quarks. It is known that $\langle \barψ ψ\rangle$ can be usefully scale-decomposed via spectral density $ρ(λ)$ of Dirac modes. Here I give such formula for $\langle F^2 \rangle$, which reveals that gluon condensate is a strictly UV quantity. For the recently-found IR phase [1,2], where the infrared (IR) degrees of freedom separate out and become independent of the system's bulk, it implies that $\langle F^2 \rangle$ due to this IR part vanishes. Its glue thus doesn't contribute to scale anomaly of the entire system and is, in this sense, scale invariant consistently with the original claim. Associated formulas are used to define IR decoupling of glue, which may serve as an alternative indicator of IR phase transition. Using the simplest form of coherent lattice QCD, we express the effective action of full QCD entirely via Dirac spectral density.

hep-lat↗
arXiv2025

The Infrared Phase of QCD and Anderson Localization

When Anderson localization entered the QCD landscape, it was almost immediately thought about in connection with thermal phases, namely as a factor in the chiral transition. However, recent developments revealed an additional structure that made Anderson-like features central to the genesis of the entirely new thermal phase: the IR phase. I will explain these developments.

hep-lat↗
arXiv2024

Separation of Infrared and Bulk in Thermal QCD

A new thermal regime of QCD, featuring decoupled scale-invariant infrared glue, has been proposed to exist both in pure-glue (N$_f$=0) and ``real-world" (N$_f$=2+1 at physical quark masses) QCD. In this {\it IR phase}, elementary degrees of freedom flood the infrared, forming a distinct component independent from the bulk. This behavior necessitates non-analyticities in the theory. In pure-glue QCD, such non-analyticities have been shown to arise via Anderson-like mobility edges in Dirac spectra ($λ_{\rm IR} \!=\! 0$, $\pm λ_\text{A} \!\neq\! 0$), as manifested in the dimension function $d_{\rm IR} (λ)$. Here, we present the first evidence, based on lattice QCD calculation at $a$=0.105 fm, that this mechanism is also at work in real-world QCD, thus supporting the existence of the proposed IR regime in nature. An important aspect of our results is that, while at $T\!=\!234\,$MeV we find a dimensional jump between zero modes and lowest near-zero modes very close to unity ($d_{\rm IR} \!=\!3$ to $d_{\rm IR} \!\simeq\! 2$), similar to the IR phase of pure-glue QCD, at $T\!=\!187\,$MeV we observe a continuous $λ$-dependence. This suggests that thermal states just {\it above} the chiral crossover are non-analytically (in $T$) connected to thermal state at $T\!=\!234\,$MeV, supporting the key original proposition that the transition into the IR regime occurs at a temperature strictly above the chiral crossover.

hep-lat↗
arXiv2024

Dirac Spectral Density in N$_f$=2+1 QCD at T=230 MeV

We compute the renormalized Dirac spectral density in $N_f = 2+1$ QCD at physical quark masses, temperature $T = 230$ MeV and system size $L_s = 3.4$ fm. To that end, we perform a point-wise continuum limit of the staggered density in lattice QCD with staggered quarks. We find, for the first time, that a clear infrared structure (IR peak) emerges in the density of Dirac operator describing dynamical quarks. We also provide numerical evidence that a component of this peak, which becomes dominant in the thermodynamic limit, is due to a non-trivial accumulation of near-zero modes. Features of this structure are consistent with those previously attributed to the recently-proposed IR phase of thermal QCD. Our results (i) provide the only complete first-principles evidence that these IR features exist and are physical; (ii) improve the upper bound for IR-phase transition temperature $T_{\mathrm{IR}}$ so that the new window is $200 < T_{\mathrm{IR}} < 230\,$MeV; (iii) are consistent with non-restoration of anomalous U$_{\mathrm A}$(1) symmetry (chiral limit) below $T = 230$ MeV.

hep-lat↗
arXiv2024

Localized Modes in the IR Phase of QCD

Infrared (IR) dimension function $d_\text{IR}(λ)$ characterizes the space effectively utilized by QCD quarks at Dirac scale $λ$, and indirectly the space occupied by glue fields. It was proposed that its non-analytic behavior in thermal infrared phase reflects the separation of QCD system into an IR component and an independent bulk. Here we study the ``plateau modes" in IR component, whose dimensional properties were puzzling. Indeeed, in the recent metal-to-critical scenario of transition to IR phase, this low-dimensional plateau connects the Anderson-like mobility edge $λ_\text{IR}=0$ in Dirac spectrum with mobility edges $\pm λ_\text{A}$. For this structure to be truly Anderson-like, plateau modes have to be exponentially localized, implying that both the effective distances $L_\text{eff} \propto L^γ$ and the effective volumes $V_\text{eff} \propto L^{d_\text{IR}}$ in these modes grow slower than any positive power of IR cutoff $L$. Although $γ=0$ was confirmed in the plateau, it was found that $d_\text{IR}\approx 1$. Here we apply the recently proposed multidimension technique to the problem. We conclude that a plateau mode of pure-glue QCD at UV cutoff $a \!=\! 0.085\,$fm occupies a subvolume of IR dimension zero with probability at least 0.9999, substantiating this aspect of metal-to-critical scenario to a respective degree.

hep-lat↗
arXiv2023

Low-Dimensional Life of Critical Anderson Electron

We show that critical Anderson electron in 3 dimensions is present in its spatial effective support, which was recently determined to be a region of fractal dimension $\approx \! 8/3$, with probability 1 in infinite volume. Hence, its physics is fully confined to space of this lower dimension. Stated differently, effective description of space occupied by critical Anderson electron becomes a full description in infinite volume. We then show that it is a general feature of the effective counting dimension underlying these concepts, that its subnominal value implies an exact description by effective support.

cond-mat.dis-nn↗
arXiv2023

Counting-Based Effective Dimension and Discrete Regularizations

Fractal-like structures of varying complexity are common in nature, and measure-based dimensions (Minkowski, Hausdorff) supply their basic geometric characterization. However, at the level of fundamental dynamics, which is quantum, structure does not enter via geometry of fixed sets but is encoded in probability distributions on associated spaces. The question then arises whether a robust notion of fractal measure-based dimension exists for structures represented in this way. Starting from effective number theory, we construct all counting-based schemes to select effective supports on collections of objects with probabilities and associate the effective counting dimension (ECD) with each. We then show that ECD is scheme-independent and, thus, a well-defined measure-based dimension with meaning analogous to the Minkowski dimension of fixed sets. In physics language, ECD characterizes probabilistic descriptions arising in a theory or model via discrete ``regularization''. For example, our analysis makes recent surprising results on effective spatial dimensions in quantum chromodynamics and Anderson models well founded. We discuss how to assess the reliability of regularization removals in practice and perform such analysis in the context of 3d Anderson criticality.

hep-lat↗
arXiv2022

Topological Dimensions from Disorder and Quantum Mechanics?

We have recently shown that critical Anderson electron in $D=3$ dimensions effectively occupies a spatial region of infrared (IR) scaling dimension $d_\text{IR} \approx 8/3$. Here we inquire about the dimensional substructure involved. We partition space into regions of equal quantum occurrence probability, such that points comprising a region are of similar relevance, and calculate the IR scaling dimension $d$ of each. This allows us to infer the probability density $p(d)$ for dimension $d$ to be accessed by electron. We find that $p(d)$ has a strong peak at $d$ very close to 2. In fact, our data suggests that $p(d)$ is non-zero on the interval $[d_\text{min}, d_\text{max}] \approx [4/3,8/3]$ and may develop a discrete part ($δ$-function) at $d=2$ in infinite-volume limit. The latter invokes the possibility that combination of quantum mechanics and pure disorder can lead to emergence of topological dimensions. Although $d_\text{IR}$ is based on effective counting of which $p(d)$ has no a priori knowledge, $d_\text{IR} \ge d_\text{max}$ is an exact feature of the ensuing formalism. Possible connection of our results to recent findings of $d_\text{IR} \approx 2$ in Dirac near-zero modes of thermal quantum chromodynamics is emphasized.

cond-mat.dis-nn↗
arXiv2022

Super-Universality in Anderson Localization

We calculate the effective spatial dimension $d_\text{IR}$ of electron modes at critical points of 3D Anderson models in various universality classes (O,U,S,AIII). The results are equal within errors, and suggest the super-universal value $d_\text{IR} \!=\! 2.665(3) \!\approx\! 8/3$. The existence of such a unique marker may help identify natural processes driven by Anderson localization, and provide new insight into the spatial geometry of Anderson transitions. The recently introduced $d_\text{IR}$ is a measure-based dimension of Minkowski/Hausdorff type, designed to characterize probability-induced effective subsets.

cond-mat.dis-nn↗
arXiv2022

Anderson Metal-to-Critical Transition in QCD

A picture of thermal QCD phase change based on the analogy with metal-to-insulator transition of Anderson type was proposed in the past. In this picture, a low-$T$ thermal state is akin to a metal with deeply infrared (IR) Dirac modes abundant and extended, while a high-$T$ state is akin to an insulator with IR modes depleted and localized below a mobility edge $λ_{\text A} > 0$. Here we argue that, while $λ_{\text A}$ exists in QCD, a high-$T$ state is not an insulator in such an analogy. Rather, it is a critical state arising due to a new singular mobility edge at $λ_{\text IR}=0$. This new mobility edge appears upon the transition into the recently proposed IR phase. As a key part of such a metal-to-critical scenario, we present evidence using pure-glue QCD that deeply infrared Dirac modes in the IR phase extend to arbitrarily long distances. This is consistent with our previous suggestion that the IR phase supports scale invariance in the infrared. We discuss the role of Anderson-like aspects in this thermal regime and emphasize that the combination of gauge field topology and disorder plays a key role in shaping its IR physics. Our conclusions are conveyed by the structure of Dirac spectral non-analyticities.

hep-lat↗
arXiv2021

Unusual Features of QCD Low-Energy Modes in IR Phase

It was recently proposed that there is a phase in thermal QCD (IR phase) at temperatures well above the chiral crossover, featuring elements of scale invariance in the infrared (IR). Here we study the effective spatial dimensions, $d_{IR}$, of Dirac low-energy modes in this phase, in the context of pure-glue QCD. Our $d_{IR}$ is based on the scaling of mode support toward thermodynamic limit, and hence is an IR probe. Ordinary extended modes, such as those at high energy, have $d_{IR}=3$. We find $d_{IR}<3$ in the spectral range whose lower edge coincides with $λ_{IR}=0$, the singularity of spectral density defining the IR phase, and the upper edge with $λ_A$, the previously identified Anderson-like non-analyticity. Details near $λ_{IR}$ are unexpected in that only exact zero modes are $d_{IR}=3$, while a thin spectral layer near zero is $d_{IR}=2$, followed by an extended layer of $d_{IR}=1$ modes. With only integer values appearing, $d_{IR}$ may have topological origin. We find similar structure at $λ_A$, and associate its adjacent thin layer ($d_{IR} >\approx 2$) with Anderson-like criticality. Our analysis reveals the manner in which non-analyticities at $λ_{IR}$ and $λ_A$, originally identified in other quantities, appear in $d_{IR}(λ)$. This dimension structure may be important for understanding the near-perfect fluidity of the quark-gluon medium seen in accelerator experiments. The role of $λ_A$ in previously conjectured decoupling of IR component is explained.

hep-lat↗
arXiv2021

The Measure Aspect of Quantum Uncertainty, of Entanglement, and the Associated Entropies

Indeterminacy associated with probing of a quantum state is commonly expressed through spectral distances (metric) featured in the outcomes of repeated experiments. Here we express it as an effective amount (measure) of distinct outcomes instead. The resulting $μ$-uncertainties are described by the effective number theory [1] whose central result, the existence of a minimal amount, leads to a well-defined notion of intrinsic irremovable uncertainty. We derive $μ$-uncertainty formulas for arbitrary set of commuting operators, including the cases with continuous spectra. The associated entropy-like characteristics, the $μ$-entropies, convey how many degrees of freedom are effectively involved in a given measurement process. In order to construct quantum $μ$-entropies, we are led to quantum effective numbers designed to count independent, mutually orthogonal states effectively comprising a density matrix. This concept is basis-independent and leads to a measure-based characterization of entanglement.

quant-ph↗
arXiv2020

Effective Number Theory: Counting the Identities of a Quantum State

Quantum physics frequently involves a need to count the states, subspaces, measurement outcomes, and other elements of quantum dynamics. However, with quantum mechanics assigning probabilities to such objects, it is often desirable to work with the notion of a "total" that takes into account their varied relevance. For example, such an effective count of position states available to a lattice electron could characterize its localization properties. Similarly, the effective total of outcomes in the measurement step of a quantum computation relates to the efficiency of the quantum algorithm. Despite a broad need for effective counting, a well-founded prescription has not been formulated. Instead, the assignments that do not respect the measure-like nature of the concept, such as versions of the participation number or exponentiated entropies, are used in some areas. Here, we develop the additive theory of effective number functions (ENFs), namely functions assigning consistent totals to collections of objects endowed with probability weights. Our analysis reveals the existence of a minimal total, realized by the unique ENF, which leads to effective counting with absolute meaning. Touching upon the nature of the measure, our results may find applications not only in quantum physics, but also in other quantitative sciences.

quant-ph↗
arXiv2019

Possible New Phase of Thermal QCD

Using lattice simulations, we show that there is a phase of thermal QCD, where the spectral density $ρ(λ)$ of Dirac operator changes as $1/λ$ for the infrared eigenvalues $λ<T$. This behavior persists over the entire low energy band we can resolve accurately, over three orders of magnitude on our largest volumes. We propose that in this "IR phase", the well-known non-interacting scale invariance at very short distances (UV, $λ\rightarrow \infty$, asymptotic freedom), coexists with very different interacting type of scale invariance at long distances (IR, $λ<T$). Such dynamics may be responsible for the unusual fluidity properties of the medium observed at RHIC and LHC. We point out its connection to the physics of Banks-Zaks fixed point, leading to the possibility of massless glueballs in the fluid. Our results lead to the classification of thermal QCD phases in terms of IR scale invariance. The ensuing picture naturally subsumes the standard chiral crossover feature at $"\!T_c\!" \,\approx 155$ MeV. Its crucial new aspect is the existence of temperature $T_{IR}$ (200 MeV $< T_{IR} < $ 250 MeV) marking the onset of IR phase and possibly a true phase transition.

hep-lat↗
arXiv2019

A Different Angle on Quantum Uncertainty (Measure Angle)

The uncertainty associated with probing the quantum state is expressed as the effective abundance (measure) of possibilities for its collapse. New kinds of uncertainty limits entailed by quantum description of the physical system arise in this manner.

quant-ph↗
arXiv2017

Locality and Efficient Evaluation of Lattice Composite Fields: Overlap-Based Gauge Operators

We propose a novel general approach to locality of lattice composite fields, which in case of QCD involves locality in both quark and gauge degrees of freedom. The method is applied to gauge operators based on the overlap Dirac matrix elements, showing for the first time their local nature on realistic path-integral backgrounds. The framework entails a method for efficient evaluation of such non-ultralocal operators, whose computational cost is volume-indepenent at fixed accuracy, and only grows logarithmically as this accuracy approaches zero. This makes computation of useful operators, such as overlap-based topological density, practical. The key notion underlying these features is that of exponential insensitivity to distant fields, made rigorous by introducing the procedure of statistical regularization. The scales associated with insensitivity property are useful characteristics of non-local continuum operators.

hep-lat↗