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arXiv:1703.00595 · Modern Physics Letters B

The a-cycle problem in XY model with ring frustration

Jianjun Dong, Peng Li

Published Feb 23, 2017• 14 citations
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Abstract

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.

DOI
10.1142/s0217984917500610

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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.
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.
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 Landau theory’s implicit assumption that microscopic details cannot affect the global phases has been challenged only recently in systems such as antiferromagnetic quantum spin chains with periodic boundary conditions. The new paradigm of topological frustration has been accordingly put forward. In this context, the quantum phases exhibit a dependence on the boundary conditions. In this work, we show that topological frustration modifies the zero temperature phase diagram of the XY chain in a transverse magnetic field by inducing new boundary quantum phase transitions. Here, by boundary quantum phase transitions we mean non-analyticities that are non-extensive in the number of particles. Such transitions are both of first and second order. Moreover, we come across the first case of second order boundary quantum phase transition characterized by a quartic dispersion relation. Our analytical results are supported by both analytical calculations and numerical investigations, and lay the foundation for understanding the phase diagram of frustrated model.
Ordering of frustrated S=1/2 and 1 XY and Heisenberg spin chains with the competing nearest- and next-nearest-neighbor antiferromagnetic couplings is studied by exact diagonalization and density-matrix renormalization-group methods. It is found that the S=1 XY chain exhibits both gapless and gapped `chiral' phases characterized by the spontaneous breaking of parity, in which the long-range order parameter is a chirality, κi=SixSi+1y−SiySi+1xκ_i = S_i^xS_{i+1}^y-S_i^yS_{i+1}^xκi​=Six​Si+1y​−Siy​Si+1x​, whereas the spin correlation decays either algebraically or exponentially. Such chiral phases are not realized in the S=1/2 XY chain nor in the Heisenberg chains.
Tackling the critical transverse Ising ring with or without ring frustration, we establish the concept of nonlocality in a many-body system in the thermodynamic limit by calculating the nonlocal factors embedded in the factorizable correlation function. Through this exactly solvable prototype, we clearly show the intriguing difference between the periodic chains with odd and even numbers of lattice sites even in the thermodynamic limit. In the context of nonlocality, we also address the important application of finite-size scaling analysis by numerically working out the nonlocal factors of the isotropic XY model and the spin-1/2 Heisenberg model.
Recently it was highlighted that one-dimensional antiferromagnetic spin models with frustrated boundary conditions, i.e. periodic boundary conditions in a ring with an odd number of elements, may show very peculiar behavior. Indeed the presence of frustrated boundary conditions can destroy the local magnetic orders presented by the models when different boundary conditions are taken into account and induce novel phase transitions. Motivated by these results, we analyze the effects of the introduction of frustrated boundary conditions on several models supporting (symmetry protected) topological orders, and compare our results with the ones obtained with different boundary conditions. None of the topological order phases analyzed are altered by this change. This observation leads naturally to the conjecture that topological phases of one-dimensional systems are in general not affected by topological frustration.