The Ihara Zeta Function of the Infinite Grid

Clair, Bryan

doi:10.37236/3561

Cited by 11 publications

(20 citation statements)

References 15 publications

(27 reference statements)

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“…The simplest kicked rotor is subject to force pulses P (q) = µ 2 q proportional to the angular displacement q: in that case, the map (32,33) is of form…”

Section: Cat Mapmentioning

confidence: 99%

“…The 2-component field at the temporal lattice site t, φ t = (q t , p t ) ∈ (0, 1] × (0, 1] is kicked rotor's the angular position and momentum. Hamilton's equations (32,33) induce forward-in-time evolution on a 2-torus (q t , p t ) phase space. Eliminating the momentum by Hamilton's discrete time velocity eq.…”

Section: Temporal Catmentioning

confidence: 99%

“…Just as the kicked rotor (32,33), the map can be interpreted as a kicked driven anaharmonic oscillator [94], with the nonlinear, cubic Biham-Wenzel [17] lattice site potential ( 48)…”

Section: A φ 3 Field Theorymentioning

confidence: 99%

“…giving rise to kicking pulse (33), so we refer to this field theory as φ 3 theory. For a sufficiently large stretching parameter a, lattice site field values of this φ 3 theory are in one-to-one correspondence to the unimodal Hénon map Smale horseshoe repeller, cleanly split into the 'left', positive stretching and 'right', negative stretching lattice site field values.…”

Section: A φ 3 Field Theorymentioning

confidence: 99%

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A chaotic lattice field theory in one dimension

Liang,

Cvitanović

2022

Preprint

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Motivated by Gutzwiller's semiclassical quantization, in which unstable periodic orbits of low-dimensional deterministic dynamics serve as a WKB 'skeleton' for chaotic quantum mechanics, we construct the corresponding deterministic skeleton for infinite-dimensional lattice-discretized scalar field theories. In the field-theoretical formulation, there is no evolution in time, and there is no 'Lyapunov horizon'; there is only an enumeration of lattice states that contribute to the theory's partition sum, each a global spatiotemporal solution of system's deterministic Euler-Lagrange equations.The reformulation aligns 'chaos theory' with the standard solid state, field theory, and statistical mechanics. In a spatiotemporal, crystallographer formulation, the timeperiodic orbits of dynamical systems theory are replaced by periodic d-dimensional Bravais cell tilings of spacetime, each weighted by the inverse of its instability, its Hill determinant. Hyperbolic shadowing of large cells by smaller ones ensures that the predictions of the theory are dominated by the smallest Bravais cells.The form of the partition function of a given field theory is determined by the group of its spatiotemporal symmetries, that is, by the space group of its lattice discretization, best studied on its reciprocal lattice. Already 1-dimensional lattice discretization is of sufficient interest to be the focus of this paper. In particular, from a spatiotemporal field theory perspective, 'time'-reversal is a purely crystallographic notion, a reflection point group, leading to a novel, symmetry quotienting perspective of time-reversible theories and associated topological zeta functions.

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“…The simplest kicked rotor is subject to force pulses P (q) = µ 2 q proportional to the angular displacement q: in that case, the map (32,33) is of form…”

Section: Cat Mapmentioning

confidence: 99%

Section: Temporal Catmentioning

confidence: 99%

“…Just as the kicked rotor (32,33), the map can be interpreted as a kicked driven anaharmonic oscillator [94], with the nonlinear, cubic Biham-Wenzel [17] lattice site potential ( 48)…”

Section: A φ 3 Field Theorymentioning

confidence: 99%

Section: A φ 3 Field Theorymentioning

confidence: 99%

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A chaotic lattice field theory in one dimension

Liang,

Cvitanović

2022

Preprint

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“…The SAM of the infinite 2D graphite lattice or the honeycomb lattice was obtained by Coulson and Rushbrooke [5] and, later, was precisely studied by Horiguchi in terms of the lattice Green function method [3,10]. The DOS is explicitly written by It is also noted that Ihara's zeta function of the infinite lattice is related to the elliptic integral [4].…”

Section: The Tba and Theory Of Graphmentioning

confidence: 99%

A novel conductivity mechanism of highly disordered carbon systems based on an investigation of graph zeta function

Matsutani

Sato

2017

Physics Letters A

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In the previous report [Phys. Rev. B 62 13812 (2000)], by proposing the mechanism under which electric conductivity is caused by the activational hopping conduction with the Wigner surmise of the level statistics, the temperature-dependent of electronic conductivity of a highly disordered carbon system was evaluated including apparent metal-insulator transition. Since the system consists of small pieces of graphite, it was assumed that the reason why the level statistics appears is due to the behavior of the quantum chaos in each granular graphite. In this article, we revise the assumption and show another origin of the Wigner surmise, which is more natural for the carbon system based on a recent investigation of graph zeta function in graph theory.

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