The A-Cycle Problem for Transverse Ising Ring

Abstract

Traditionally, the transverse Ising model is mapped to the fermionic c-cycle problem, which neglects the boundary effect due to thermodynamic limit. If persisting on a perfect periodic boundary condition, we can get a so-called a-cycle problem that has not been treated seriously so far (Lieb et al., 1961 \textit{Ann. of Phys.} \textbf{16} 407). In this work, we show a little surprising but exact result in this respect. We find the odevity of the number of lattice sites, $N$, in the a-cycle problem plays an unexpected role even in the thermodynamic limit, $N\to \infty$, due to the boundary constraint. We pay a special attention to the system with $N(\in Odd)\to \infty$, which is in contrast to the one with $N(\in Even)\to \infty$, because the former suffers a ring frustration. As a new effect, we find the ring frustration induces a low-energy gapless spectrum above the ground state. By proving a theorem for a new type of Toeplitz determinant, we demonstrate that the ground state in the gapless region exhibits a peculiar longitudinal spin-spin correlation. The entangled nature of the ground state is also disclosed by the evaluation of its entanglement entropy. At low temperatures, new behavior of specific heat is predicted. We also propose an experimental protocol for observing the new phenomenon due to the ring frustration.

arXiv comment24 pages, 9 figures

References in corpus (12)

• Entanglement in quantum critical phenomena
• Geometric and Renormalized Entropy in Conformal Field Theory
• Surprises on the Way from 1D to 2D Quantum Magnets: the Novel Ladder Materials
• Controlling Spin Exchange Interactions of Ultracold Atoms in Optical Lattices
• Universality of Entropy Scaling in 1D Gap-less Models
• Quantum Criticality: competing ground states in low dimensions
• Quantum many particle systems in ring-shaped optical lattices
• On the definition of a unique effective temperature for non-equilibrium critical systems
• Quantum Simulation and Phase Diagram of the Transverse Field Ising Model with Three Atomic Spins
• Finite-size scaling of the entanglement entropy of the quantum Ising chain with homogeneous, periodically modulated and random couplings
• Haldane-like antiferromagnetic spin chain in the large anisotropy limit
• Antiferromagnetic molecular nanomagnets with odd-numbered coupled spins

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