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arXiv:1512.06964

Frustration-Induced Gaplessness in the Frustrated Transverse Ising Ring

Jianjun Dong, Li, Peng, Qihui Chen

Published Dec 22, 2015• 0 citations
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Abstract

New effects in the frustrated transverse Ising ring are predicted. The system is solved based on a mapping of Pauli spin operators to the Jordan-Wigner fermions. We group the low-lying energy levels into bands after imposing appropriate parity constraint, which projects out the redundant degrees of freedom brought about by the Jordan-Wigner fermions. In the region of strong antiferromagnetic coupling, we uncover an unusual gapless phase induced by the ring frustration. We demonstrate that its ground state exhibits a strong longitudinal spin-spin correlation and possesses a considerably large entropy of entanglement. The low-lying energy levels evolve adiabatically in the gapless phase, which facilitates us to work out new behaviors of density of states, low-temperature correlation functions and specific heat. We also propose an experimental protocol for observing this peculiar gapless phase.

DOI
10.48550/arxiv.1512.06964

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New effects in the frustrated transverse Ising ring are predicted. The system is solved based on a mapping of Pauli spin operators to the Jordan-Wigner fermions. We group the low-lying energy levels into bands after imposing appropriate parity constraint, which projects out the redundant degrees of freedom brought about by the Jordan-Wigner fermions. In the region of strong antiferromagnetic coupling, we uncover an unusual gapless phase induced by the ring frustration. We demonstrate that its ground state exhibits a strong longitudinal spin-spin correlation and possesses a considerably large entropy of entanglement. The low-lying energy levels evolve adiabatically in the gapless phase, which facilitates us to work out new behaviors of density of states, low-temperature correlation functions and specific heat. We also propose an experimental protocol for observing this peculiar gapless phase.
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 Ann. Phys . 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 → ∞ , due to the boundary constraint. We pay special attention to the system with N ( ∈ O d d ) → ∞ , which is in contrast to the one with N ( ∈ E v e n ) → ∞ , 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 temperature, new behavior of specific heat is predicted. We also propose an experimental protocol for observing the new phenomenon due to the ring frustration.
On a ring, a single Jordan-Wigner transformation between the Kitaev model and the spin model suffers redundant degrees of freedom. However, we can establish an exact quaternary Jordan-Wigner mapping involving two Kitaev rings and two spin rings with periodic or antiperiodic boundary conditions. This mapping allows us to demonstrate exactly how a topological extended-kink (TEK) phase develops in the interacting Kitaev ring with odd number of lattice sites. The emergence of this new phase is attributed to the effect of geometrical ring frustration. Unlike the usual topological phases protected by an energy gap in noninteracting systems, the TEK phase is gapless. And because the spectra of low-energy excitations are quadratic, the specific heat per site approaches half of the Boltzmann constant near absolute zero temperature. More interestingly, the ground state is unique, immune to spontaneous symmetry breaking. It exhibits a long-range correlation function with a nonlocal factor, but no local order parameter can be defined. As a concomitant effect, a special kind of localized kink zero mode (KZM) takes place if we introduce a type of bond defect. We also show that the KZM is robust against moderate disorders.
An unusual correlation function was conjectured by Campostrini et al. [Phys. Rev. E 91, 042123 (2015)PLEEE81539-375510.1103/PhysRevE.91.042123] for the ground state of a transverse Ising chain with geometrical frustration. Later, we provided a rigorous proof for it and demonstrated its nonlocal nature based on an evaluation of a Toeplitz determinant in the thermodynamic limit [J. Stat. Mech. (2016) 11310210.1088/1742-5468/2016/11/113102]. In this paper, we further prove that all the low excited energy states forming the gapless kink phase share the same asymptotic correlation function with the ground state. As a consequence, the thermal correlation function almost remains constant at low temperatures if one assumes a canonical ensemble.
Abstract Advance in quantum simulations using trapped ions or superconducting elements allows detailed analysis of the transverse field Ising model (TFIM), which can exhibit a quantum phase transition and has been a paradigm in exactly solvable quantum systems. The Jordan–Wigner transformation maps the one-dimensional TFIM to a fermion model, but additional complications arise in finite systems and introduce a fermion-number parity constraint when periodic boundary condition is imposed. By constructing the free energy and spin correlations with the fermion-number parity constraint and comparing the results to the TFIM with open boundary condition, we show that the boundary effects can become significant for the anti-ferromagnetic TFIM with odd number of sites at low temperature.
Traditionally, the transverse spin-[Formula: see text] XY model is mapped to a fermionic “c-cycle” problem, where the prior periodic boundary condition is applied to the fermionic chain and the additional boundary term has been neglected. However, the “a-cycle” problem (the original problem without any approximation) has not been treated seriously up to now. In this paper, we consider the XY model with ring frustration and diagonalize it without any approximation with the help of parity constraint. Then two peculiar gapless phases have been found.
A disordered spin state, the nearest-neighbor virtual bonding state of Wigner-Jordan fermions, of the quantum antiferromagnetic Heisenberg model on a triangular lattice is found to have a zero-temperature energy -1.106J per spin, lower than that calculated from spin-wave theories, and in excellent agreement with that from numerical calculations, (-1.094\ifmmode\pm\else\textpm\fi{}0.018)J per spin. The excitations of the state are Wigner-Jordan fermions with a gapless spectrum. The low-temperature specific heat is predicted to be proportional to ${\mathit{T}}^{\ensuremath{\nu}}$, with \ensuremath{\nu} between 2/3 and 1, significantly deviating from the spin-wave prediction, T2{\mathit{T}}^{2}T2. At higher temperature, a broad peak in the specific heat near T/2J\ensuremath{\approxeq}0.75 is also predicted.