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Register a new userGeometric phase of the one-dimensional Ising chain in a longitudinal field
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For the one-dimensional Ising chain with spin-$1/2$ and exchange couple $J$ in a steady transverse field(TF), an analytical theory has well been developed in terms of some topological order parameters such as Berry phase(BP). For a TF Ising chain, the nonzero BP which depends on the exchange couple and the field strength characterizes the corresponding symmetry breaking of parity and time reversal(PT). However, there seems to exist a topological phase transition for the one-dimensional Ising chain in a longitudinal field(LF) with the reduced field strength $epsilon$. If the LF is added at zero temperature, researchers believe that the LF also could influence the PT-symmetry and there exists the discontinuous BP. But the theoretic characterization has not been well founded. This paper tries to aim at this problem. With the Jordan-Wigner transformation, we give the four-fermion interaction form of the Hamiltonian in the one-dimensional Ising chain with a LF. Further by the method of Wicks theorem and the mean-field theory, the four-fermion interaction is well dealt with. We solve the ground state energy and the ground wave function in the momentum space. We discuss the BP and suggest that there exist nonzero BPs when $epsilon=0$ in the paramagnetic case where $J<0$ and when $-1<epsilon<1$, in the diamagnetic case where $J>0$.
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In this paper, we determine the geometric phase for the one-dimensional $XXZ$ Heisenberg chain with spin-$1/2$, the exchange couple $J$ and the spin anisotropy parameter $Delta$ in a longitudinal field(LF) with the reduced field strength $h$. Using the Jordan-Wigner transformation and the mean-field theory based on the Wicks theorem, a semi-analytical theory has been developed in terms of order parameters which satisfy the self-consistent equations. The values of the order parameters are numerically computed using the matrix-product-state(MPS) method. The validity of the mean-filed theory could be checked through the comparison between the self-consistent solutions and the numerical results. Finally, we draw the the topological phase diagrams in the case $J<0$ and the case $J>0$.
The aim of this work is to present a formulation to solve the one-dimensional Ising model using the elementary technique of mathematical induction. This formulation is physically clear and leads to the same partition function form as the transfer matrix method, which is a common subject in the introductory courses of statistical mechanics. In this way our formulation is a useful tool to complement the traditional more abstract transfer matrix method. The method can be straightforwardly generalized to other short-range chains, coupled chains and is also computationally friendly. These two approaches provide a more complete understanding of the system, and therefore our work can be of broad interest for undergraduate teaching in statistical mechanics.
The Ising model, with short-range interactions between constituents, is a basic mathematical model in statistical mechanics. It has been widely used to describe collective phenomena such as order-disorder phase transitions in various physical, biological, economical, and social systems. However, it was proven that spontaneous phase transitions do not exist in the one-dimensional Ising models. Besides low dimensionality, frustration is the other well-known suppressor of phase transitions. Here I show that surprisingly, a strongly frustrated one-dimensional two-leg ladder Ising model can exhibit a marginal finite-temperature phase transition. It features a large latent heat, a sharp peak in specific heat, and unconventional order parameters, which classify the transition as involving an entropy-favored intermediate-temperature ordered state and further unveil a crossover to an exotic normal state in which frustration effectively decouples the two strongly interacted legs in a counterintuitive non-mean-field way. These exact results expose a mathematical structure that has not appeared before in phase-transition problems, and shed new light on our understanding of phase transitions and the dynamical actions of frustration. Applications of this model and its mechanisms to various systems with extensions to consider higher dimensions, quantum characters, or external fields, etc. are anticipated and briefly discussed---with insights into the puzzling phenomena of strange strong frustration and intermediate-temperature orders such as the Bozin-Billinge orbital-degeneracy-lifting recently discovered in real materials.
We study a one-dimensional chain of corner-sharing triangles with antiferromagnetic Ising interactions along its bonds. Classically, this system is highly frustrated with an extensive entropy at T = 0 and exponentially decaying spin correlations. We show that the introduction of a quantum dynmamics via a transverse magnetic field removes the entropy and opens a gap, but leaves the ground state disordered at all values of the transverse field, thereby providing an analog of the disorder by disorder scenario first proposed by Anderson and Fazekas in their search for resonating valence bond states. Our conclusion relies on exact diagonalization calculations as well as on the analysis of a 14th order series expansion about the large transverse field limit. This test suggests that the series method could be used to search for other instances of quantum disordered states in frustrated transverse field magnets in higher dimensions.
We study the time evolution of the local magnetization in the critical Ising chain in a transverse field after a sudden change of the parameters at a defect. The relaxation of the defect magnetization is algebraic and the corresponding exponent, which is a continuous function of the defect parameters, is calculated exactly. In finite chains the relaxation is oscillating in time and its form is conjectured on the basis of precise numerical calculations.
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