Exact Solution of the 1D Ising Chain with Periodic Fixed Impurities: A Comprehensive Analysis

Malek Telfah, Abdalla A. Obeidat

Xuchu Huang

Telfah, Malek, Obeidat, Abdalla

We present an exact analytical solution for the one-dimensional Ising model in the presence of an external magnetic field applied periodically to every $k$-th site. The problem is handled using the symmetrized transfer matrix approach, we derive a compact closed-form expression for the system's eigenvalues for arbitrary period $k$. From the resulting free energy, we obtain exact expressions for the magnetization and zero-field susceptibility. Explicit results are presented for $k = 1$, $k = 2$, and $k = 3$ which is considered a novel result. We further analyze the spin-spin correlation functions, deriving the correlation length and the set of position-dependent correlation strength prefactors, $A_{ij}$. The framework highlights how impurity spacing suppresses thermodynamic responses, with susceptibility scaling as $χ\sim β/ k$ for large $k$, offering insights into diluted magnetic systems and serving as a benchmark for quasiperiodic modulations. The correlation strengths exhibit a strong anisotropy, revealing a complex, non-local structure of spin fluctuations. These results provide a complete and rigorous benchmark for understanding the effects of periodic modulation in 1D systems.

Malte Henkel, A. Patkós, Martin Schlottmann

Haopeng Tian, Tianhao He, Xintian Wu

The half-chain entanglement entropy (EE) of the one-dimensional $XX$ (isotropic $XY$) chain and quantum Ising chain (QIC) at the critical point are calculated numerically for large lattices. The periodic, free-free, fixed-fixed ($++$) and ($+\ensuremath{-}$), and fixed-free boundary conditions (BCs) are studied. The models are transformed into a fermion system with a Jordan-Wigner transformation. The correlation functions of these fermions are obtained numerically. With the correlation functions, the entanglement entropy is obtained. The numerical calculation is performed with quadruple precision. The numerical results agree with the exact relationship with an accuracy of ${10}^{\ensuremath{-}28}$ between the EEs of the $XX$ chain and QICs: (1) the EE of the $XX$ chain with periodic BCs is the sum of two QICs with periodic BCs; (2) the EE of the $XX$ chain with free-free BCs is the sum of two QICs with free-free BCs; (3) the EE of the $XX$ chain with fixed-free BCs is the sum of the EE of a QIC with fixed-free BCs and that with free-free BCs; (4) the EE of the $XX$ chain with fixed-fixed ($++$) BCs is the sum of the EE of a QIC with fixed-fixed ($++$) BCs and that with free-free BCs; and (5) the EE of the $XX$ chain with fixed-fixed ($+\ensuremath{-}$) BCs is the sum of the EE of a QIC with fixed-fixed ($++$) BCs and that of the first excited state of a QIC with free-free BCs. We fit the half-chain EE with finite-size scaling expansion of terms ${L}^{\ensuremath{-}n}$ and ${L}^{\ensuremath{-}n}lnL$, where $L$ is the length of the spin chain. We find that the fitted parameters for the leading term $lnL$ and the constant term are consistent with conformal field theory and that the exact solutions are highly accurate. The fitting accuracy is so high that the fitted central charge of the Ising model is $0.5\ifmmode\pm\else\textpm\fi{}1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}26}$ for the periodic boundary condition. From the higher-order corrections in the fitted expansion, we find that there exist ${L}^{\ensuremath{-}4}lnL$ and higher-order corrections for the fixed-fixed ($+\ensuremath{-}$) boundary conditions, whereas there exist ${L}^{\ensuremath{-}3}lnL$ and higher-order corrections for the fixed-free boundary conditions. These types of corrections mean corrections of the central charge.

Tuncer Kaya

An alternative exact explicit solution of 1D Ising chain is presented without using any boundary conditions (or free boundary condition) by the help of applying successively block-spin transformation. Exact relation are obtained between spin-spin correlation functions in the absence of external field. To evaluate average magnetization (or the order parameter), it is assumed that the average magnetization can be related to infinitely apart two spin correlation function as $^{2}=$. A discussion to justify this consideration is given in the introduction with a relevant manner. It is obtained that $=\tanh{K}$, which is exactly the same relation as the previously derived relation by considering the configurational space equivalence of $\{σ_{i}σ_{i+1}\}=\{s_{i}\}$ and the result of transfer matrix method in the absence of external field. By applying further block-spin transformation, it is obtained that $\{σ_{0}σ_{j}\}=(\tanh{K})^{j}$, here $j$ assumes the values of $j=2^n$, here $n$ is an integer numbers. We believe that this result is really important in that it is the only exact unique treatment of the 1D Ising chain beside with the transfer matrix method. It is also pointed out the irrelevances of some of the alternative derivation appearing in graduate level text books. The obtained unique correlation relation in this this study leads in the limit of $N\rightarrow\infty$ to $=(\tanh{K})^{N/2}$, indicating that the second order phase transition is only possible in the limit of $K\rightarrow\infty$. The obtained results in this work are exactly the same as those of obtained by the transfer matrix method which considers periodic boundary condition.