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Tensor network simulations for nonorientable surfaces.

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This study introduces a novel tensor network approach for geometric constructions of the Klein bottle and real projective plane (RP^{2}). The method efficiently computes boundary terms and partition functions on these surfaces, advancing tensor renormalization group methodology.

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Area of Science:

  • Theoretical Physics
  • Condensed Matter Physics
  • Quantum Information

Background:

  • Tensor networks are powerful tools for studying complex quantum systems.
  • Boundary matrix product states have been used to analyze boundaries in quantum systems.
  • Understanding non-orientable surfaces like the Klein bottle and real projective plane (RP^{2}) is crucial in various physics domains.

Purpose of the Study:

  • To develop an efficient tensor network method for geometric constructions of the Klein bottle and RP^{2}.
  • To incorporate crosscap and rainbow boundaries into the tensor renormalization group (TRG) framework.
  • To enable efficient computation of physical quantities on these non-orientable surfaces.

Main Methods:

  • Utilizing tensor networks and TRG methodology.
  • Implementing an efficient representation of a spatial reflection operator.
  • Applying boundary matrix product state techniques within the TRG framework.

Main Results:

  • Efficient computation of crosscap and rainbow free energy terms on RP^{2}.
  • Calculation of the one-point function on RP^{2} for larger system sizes.
  • Capability to compute partition functions under isotropic conditions and construct other non-orientable surfaces.

Conclusions:

  • The proposed tensor network approach enhances the efficiency of studying non-orientable surfaces.
  • This method provides a versatile framework for exploring topological phases and quantum systems with non-trivial boundaries.
  • The advancement facilitates deeper investigations into the physics of surfaces with genus greater than zero.