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arXiv:0709.0337 · Physical Review Letters

Rigorous Solution of the Spin-1 Quantum Ising Model with Single-Ion Anisotropy

Zhihua Yang, Li‐Ping Yang, Jianhui Dai, Tao Xiang

Published Feb 13, 2008• 37 citations
arXivHTMLPDF

Abstract

We solve the spin-1 quantum Ising model with single-ion anisotropy by mapping it onto a series of segmented spin-1/2 transverse Ising chains, separated by the S(z)=0 states called holes. A recursion formula is derived for the partition function to simplify the summation of hole configurations. This allows the thermodynamic quantities of this model to be rigorously determined in the thermodynamic limit. The low temperature behavior is governed by the interplay between the hole excitations and the fermionic excitations within each spin-1/2 Ising segment. The quantum critical fluctuations around the Ising critical point of the transverse Ising model are strongly suppressed by the hole excitations.

DOI
10.1103/physrevlett.100.067203

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We solve the spin-1 quantum Ising model with single-ion anisotropy by mapping it onto a series of segmented spin-1/2 transverse Ising chains, separated by the S(z)=0 states called holes. A recursion formula is derived for the partition function to simplify the summation of hole configurations. This allows the thermodynamic quantities of this model to be rigorously determined in the thermodynamic limit. The low temperature behavior is governed by the interplay between the hole excitations and the fermionic excitations within each spin-1/2 Ising segment. The quantum critical fluctuations around the Ising critical point of the transverse Ising model are strongly suppressed by the hole excitations.
We propose a hole decomposition scheme to exactly solve a class of spin-1 quantum Ising models with transverse or longitudinal single-ion anisotropy. In this scheme, the spin-1 model is mapped onto a family of the S=1/2S=1/2S=1/2 transverse Ising models, characterized by the total number of holes. A recursion formula is derived for the partition function based on the reduced S=1/2S=1/2S=1/2 Ising model. This simplifies greatly the summation over all the hole configurations. It allows the thermodynamic quantities to be rigorously determined in the thermodynamic limit. The ground-state phase diagram is determined for both the uniform and dimerized spin chains. The corresponding thermodynamic properties are calculated and discussed.
The low-density linked-cluster expansion of the free energy of the ferromagnetic Ising model with single-ion anisotropy, i.e., the $\ensuremath{\Delta} \ensuremath{\Sigma}{i}^{}S_{\mathrm{zi}}^{}{}_{}{}^{2}$ term, is given by $\ensuremath{-}\ensuremath{\beta}f(\ensuremath{\beta}J,\ensuremath{\beta}\ensuremath{\Delta},\ensuremath{\beta}h)=\frac{1}{2}qJ+h\ensuremath{-}\ensuremath{\Delta}+\ensuremath{\Sigma}{l=1}^{\ensuremath{\infty}}({u}^{12}\ensuremath{\mu})^{l}L_{l}(u,\ensuremath{\eta})$, with $u=\mathrm{exp}(\ensuremath{-}\frac{J}{{k}_{B}T})$, $\ensuremath{\mu}=\mathrm{exp}(\ensuremath{-}\frac{2mh}{{k}_{B}T})$, and $\ensuremath{\eta}=\mathrm{exp}(\frac{\ensuremath{\Delta}}{{k}_{B}T})$. This is a low-temperature expansion with each spin Si{S}_{i}Si​ having magnitude one. The sixth-order series available in the literature is analyzed by evaluating the zeros of ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{L}}_{l}={u}^{12}^{l}L_{l}$ as a function of $\ensuremath{\Delta}$, and from that the critical temperature is deduced. The knowledge of the critical temperature as a function of the anisotropy leads to the second-order part of the phase boundary of $\ensuremath{\beta}J$ with $\ensuremath{\beta}\ensuremath{\Delta}$ which estimates a tricritical value of $\ensuremath{\Delta}$. The asymptotic behavior of ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{L}}_{l}$ is studied and from that the critical exponent $\ensuremath{\delta}$ of the $M\ensuremath{-}h$ isotherm as $h\ensuremath{\rightarrow}0$ is found. It is observed that the anisotropy determines the transition temperature, but the exponent $\ensuremath{\delta}$ is independent of the same as expected from the universality hypothesis.
We investigate a ferromagnetic spin-1 Ising model with a transverse single-ion crystal-field term. By means of a mapping to the usual spin-1/2 transverse Ising model, we determine the location of the T=0T=0T=0 quantum critical point exactly in one dimension, and to high precision in higher dimensions. The nature of the excited states is also explored. We show how this approach can be generalized to higher spin values.
The effect of uniaxial single-ion anisotropy on magnetic and critical properties of the mixed spin-1/2 and spin-S (S > 1/2) Ising model on a three-coordinated Bethe lattice is rigorously examined with the help of star-triangle transformation and exact recursion relations.In particular, our attention is focused on the ferrimagnetic version of the model, which exhibits diverse temperature dependences of the total and both sublattice magnetizations.It is shown that the critical behavior of the mixed-spin Ising model on the Bethe lattice basically depends on whether the quantum spin number S is integer or half-odd-integer.
Partition functions are ubiquitous in physics: They are important in determining the thermodynamic properties of many-body systems and in understanding their phase transitions. As shown by Lee and Yang, analytically continuing the partition function to the complex plane allows us to obtain its zeros and thus the entire function. Moreover, the scaling and nature of these zeros can elucidate phase transitions. Here, we show how to find partition function zeros on noisy intermediate-scale trapped-ion quantum computers in a scalable manner, using the XXZ spin chain model as a prototype, and observe their transition from XY-like behavior to Ising-like behavior as a function of the anisotropy. While quantum computers cannot yet scale to the thermodynamic limit, our work provides a pathway to do so as hardware improves, allowing the future calculation of critical phenomena for systems beyond classical computing limits.