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Vertex representation of hyperbolic tensor networks.

Matej Mosko1, Maria Polackova1,2, Roman Krcmar1

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We introduce a vertex tensor network (TN) for hyperbolic lattices, finding entanglement entropy uniquely identifies geometry. This method accurately models phase transitions and lowest-energy quantum states.

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Area of Science:

  • Condensed Matter Physics
  • Quantum Information Theory
  • Computational Physics

Background:

  • Classical spin systems are often studied on regular lattices.
  • Tensor networks (TNs) provide a powerful framework for simulating quantum many-body systems.
  • Hyperbolic geometry presents unique challenges and opportunities for physical systems.

Purpose of the Study:

  • To develop a vertex representation of tensor networks (TNs) for classical spin systems on hyperbolic lattices.
  • To analyze the behavior of multistate spin systems within this hyperbolic TN framework.
  • To investigate the sensitivity of entanglement entropy and thermodynamic quantities to hyperbolic geometry.

Main Methods:

  • Formulating a vertex TN with regular p-sided polygons (p>4) and coordination number 4.
  • Analyzing the parameter space of multistate spin systems on the hyperbolic TN.
  • Calculating entanglement entropy and thermodynamic quantities.
  • Testing numerical accuracy across different orders of phase transitions.

Main Results:

  • Entanglement entropy effectively distinguishes between different hyperbolic geometries, unlike other thermodynamic quantities.
  • The hyperbolic structure of TNs leads to noncritical bulk properties.
  • Boundary conditions significantly influence the total free energy in the thermodynamic limit.
  • Vertex TNs demonstrate numerical accuracy in phase transitions, particularly at maximal entanglement entropy.

Conclusions:

  • The developed vertex TN is suitable for studying lowest-energy quantum states on hyperbolic lattices.
  • Entanglement entropy serves as a crucial indicator of hyperbolic geometry in these systems.
  • The TN framework offers insights into the interplay between geometry, thermodynamics, and quantum states.