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 dIR = 3 . We find dIR < 3 in the spectral range whose lower edge coincides with λIR = 0 , the singularity of spectral density defining the IR phase, and the upper edge with λA, the previously identified Anderson-like nonanalyticity. Details near λ IR are unexpected in that only exact zero modes are dIR = 3, while a thin spectral layer near zero is d IR = 2 , followed by an extended layer of dIR = 1 modes. With only integer values appearing, d IR may have a topological origin. We find similar structure at λ A , and associate its adjacent thin layer ( dIR ⪆ 2 ) with Anderson-like criticality. Our analysis reveals the manner in which nonanalyticities at λIR and λA, originally identified in other quantities, appear in dIR (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 λA in previously conjectured decoupling of IR component is explained.
Possible new phase of thermal QCD Alexandru, Andrei; Horváth, Ivan
Physical review. D,
11/2019, Volume:
100, Issue:
9
Journal Article
Peer reviewed
Open access
Using lattice simulations, we show that there is a phase of thermal QCD, where the spectral density ρ(λ) of the Dirac operator changes as 1/λ for infrared eigenvalues λ<T. This behavior persists over ...the entire low energy band we can resolve accurately, over 3 orders of magnitude on our largest volumes. We propose that in this "IR phase," the well-known noninteracting scale invariance at very short distances (UV, λ→∞, asymptotic freedom), coexists with a 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 the 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 "Tc" ≈ 155 MeV. Its crucial new aspect is the existence of temperature TIR (200 MeV < TIR < 250 MeV) marking the onset of the IR phase and possibly a true phase transition.
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 λA>0. Here we argue that, while λA 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 λIR=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.
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 ≈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.
•At the critical point, electron is fully contained in its effective support with dimension ≈8/3 in the thermodynamic limit.•Subnominal value of effective counting dimension implies exact description by effective support.•The results provide for novel description of critical electronic state and critical transport.
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.
We have recently shown that the critical Anderson electron in D=3 dimensions effectively occupies a spatial region of the infrared (IR) scaling dimension dIR≈8/3. Here, we inquire about the ...dimensional substructure involved. We partition space into regions of equal quantum occurrence probabilities, such that the 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 the electron. We find that p(d) has a strong peak at d very close to two. In fact, our data suggest that p(d) is non-zero on the interval dmin,dmax≈4/3,8/3 and may develop a discrete part (δ-function) at d=2 in the infinite-volume limit. The latter invokes the possibility that a combination of quantum mechanics and pure disorder can lead to the emergence of integer (topological) dimensions. Although dIR is based on effective counting, of which p(d) has no a priori knowledge, dIR≥dmax is an exact feature of the ensuing formalism. A possible connection of our results to the recent findings of dIR≈2 in Dirac near-zero modes of thermal quantum chromodynamics is emphasized.
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 geometric features of fixed sets but is encoded in probability distributions on associated spaces. The question then arises whether a robust notion of the 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
(ECD) with each. We then show that the ECD is scheme-independent and, thus, a well-defined measure-based dimension whose meaning is 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.
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