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arXiv:2403.08389

Order parameters in quasi-1D spin systems

Garry Goldstein

Published Mar 13, 2024• 0 citations
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Abstract

In this work we extend the notion of what is meant by a meanfield. Meanfields are approximately maps - through some self consistency relation - of a complex, usually manybody, problem to a simpler more readily solvable problem. This mapping can then be solved to represent properties of the complex many body problem using some self consistency relations. Prototypical examples of simpler meanfield problems (meanfield systems) are the single site and free particle problems. Here we propose a new class of simple meanfield systems where the simple problem to be solved is a 1D spin chain. These meanfields are particularly useful for studying quasi-1D models, where there is a 3D system composed of weakly coupled 1D spin chains with the coupling in the transverse direction weaker than in the 1D direction. We illustrate this idea by considering meanfields for the Ising (of any coupling sign) and ferromagnetic Heisenberg models with one direction coupled much more strongly then the other directions (quasi-1D systems) which map at meanfield level onto the 1D Ising and 1D ferromagnetic Heisenberg models. We also consider more exotic models to illustrate other methods of solving 1D systems, namely the $N$-state Potts model. Magnetic phase transition temperatures and are obtained for all three models, we see that they significantly differ from the usual meanfield estimates. Indeed if the 1D direction has coupling $Γ$ and the transverse directions have coupling $J$ with $λ\sim\fracΓ{J}\gg1$ then regular meanfield would predict the transition temperature to be $k_{B}T_{c}\simΓ$ for all three models while 1D meanfield predicts temperatures of $k_{B}T_{c}\sim\fracΓ{\log\left(λ\right)}$ for the Ising and Potts models and $k_{B}T_{c}\sim\fracΓ{\sqrtλ}$ for the ferromagnetic Heisenberg model. Cluster 1D meanfield extensions are also proposed.

DOI
10.48550/arxiv.2403.08389

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In this work we extend the notion of what is meant by a meanfield. Meanfields are approximately maps - through some self consistency relation - of a complex, usually manybody, problem to a simpler more readily solvable problem. This mapping can then be solved to represent properties of the complex many body problem using some self consistency relations. Prototypical examples of simpler meanfield problems (meanfield systems) are the single site and free particle problems. Here we propose a new class of simple meanfield systems where the simple problem to be solved is a 1D spin chain. These meanfields are particularly useful for studying quasi-1D models, where there is a 3D system composed of weakly coupled 1D spin chains with the coupling in the transverse direction weaker than in the 1D direction. We illustrate this idea by considering meanfields for the Ising (of any coupling sign) and ferromagnetic Heisenberg models with one direction coupled much more strongly then the other directions (quasi-1D systems) which map at meanfield level onto the 1D Ising and 1D ferromagnetic Heisenberg models. We also consider more exotic models to illustrate other methods of solving 1D systems, namely the NNN-state Potts model. Magnetic phase transition temperatures and are obtained for all three models, we see that they significantly differ from the usual meanfield estimates. Indeed if the 1D direction has coupling ΓΓΓ and the transverse directions have coupling JJJ with λ∼ΓJ≫1λ\sim\fracΓ{J}\gg1λ∼JΓ​≫1 then regular meanfield would predict the transition temperature to be kBTc∼Γk_{B}T_{c}\simΓkB​Tc​∼Γ for all three models while 1D meanfield predicts temperatures of kBTc∼Γlog⁡(λ)k_{B}T_{c}\sim\fracΓ{\log\left(λ\right)}kB​Tc​∼log(λ)Γ​ for the Ising and Potts models and kBTc∼Γλk_{B}T_{c}\sim\fracΓ{\sqrtλ}kB​Tc​∼λ​Γ​ for the ferromagnetic Heisenberg model. Cluster 1D meanfield extensions are also proposed.
The ferromagnetic transition is considered in detail for the Heisenberg Hamiltonian with all spins coupled equally. For this special model the statistical problem is solved exactly. The transition develops very slowly as the number NNN of particles in the system is increased; the spin order at the nominal Curie temperature is proportional to ${N}^{\ensuremath{-}\frac{1}{4}}$. The order in the most probable state differs from the mean value of the order, except for $N\ensuremath{\rightarrow}\ensuremath{\infty}$. The connections are studied with the molecular-field, spin-wave, and self-consistent approximations. The failure of the self-consistent approximation is relatively severe as compared with the nearest-neighbor problem. The antiferromagnetic ground state is very close in energy to the N\'eel state.
The ground-state properties of the quasi-one-dimensional spin-1/21/21/2 antiferromagnetic Heisenberg model is investigated by using a variational method. Spins on chains along the xxx direction are antiferromagnetically coupled with exchange J>0J>0J>0, while spins between chains in the yyy direction are coupled either ferromagnetically (${J}^{\ensuremath{'}}<0$) or antiferromagnetically (${J}^{\ensuremath{'}}>0$). The staggered and the colinear antiferromagnetic magnetizations are computed and their dependence on the anisotropy parameter $\ensuremath{\lambda}=|{J}^{\ensuremath{'}}|/J$ is analyzed. It is found that an infinitesimal interchain coupling parameter is sufficient to stabilize a long-range order with either a staggered magnetization ms{m}_{s}ms​ (${J}^{\ensuremath{'}}>0$) or a colinear antiferromagnetic magnetization mcaf{m}_{caf}mcaf​ (${J}^{\ensuremath{'}}<0$), both behaving as $\ensuremath{\simeq}{\ensuremath{\lambda}}^{1/2}$ for $\ensuremath{\lambda}\ensuremath{\rightarrow}0$.
The prohibition of finite-temperature phase transition in one-dimensional (1D) Ising models and 1D/2D quantum Heisenberg models with short-range interactions fundamentally constrains the application potentials of low-dimensional magnetic materials. Recently, ultranarrow phase crossover (UNPC), which can approach a transition at a desirable finite temperature T0T_0T0​ arbitrarily closely, was discovered in 1D decorated Ising chains and ladders. Here we present a theoretical study of similarly decorated, yet much more challenging, quantum Heisenberg ferrimagnets in a magnetic field, which features ferromagnetic backbone exchange JJJ, antiferromagnetic site-decoration coupling JAFJ_{AF}JAF​, and different magnetic moments for the backbone and decorating spins μaSa<μbSbμ_aS_a<μ_bS_bμa​Sa​<μb​Sb​. We exactly solved the model in the large JJJ limit -- as a central-macrospin model -- and found two finite-temperature second-order transitions; just above Tc2T_{c2}Tc2​ a ``half-ice, half-fire'' regime appears. Finite-JJJ weak-field results follow from an effective-field mapping, suggesting the emergence of UNPC at finite T0T_0T0​ in 2D square lattices thanks to its exponentially strong initial magnetic susceptibility χ0∝e4πSa2J/T0χ_0\propto e^{4πS_a^2 J/T_0}χ0​∝e4πSa2​J/T0​, though less likely in 1D chains where χ0∝J/T0χ_0\propto J/T_0χ0​∝J/T0​. These results may shed light on new technological applications of low-dimensional quantum spin systems and attract experimental and computational tests.
Pairing and topological phases in cold atoms with long range interactions Chapter 1 Introduction The systematic study and classification of phase transitions has become a popular research area in physics over the past decades. The most familiar phenomena associated to the concept of phase transition are the ones giving rise to macroscopic changes in the properties of strongly correlated many-body systems due to thermal fluctuations. The most prominent examples of this kind of behaviour are the ferromagnetic-paramagnetic transition in lattice spin models and the liquid-gas transition. On the other hand, when dealing with quantum-mechanical systems changes of matter from one state to another can arise as the result of the variation of parameters other than temperature. In particular, when a given many-body quantum problem is studied at zero temperature, thereby suppressing thermal fluctuations, it is the interplay between kinetic energy and interactions that can cause the system to jump into different phases as the model parameters are varied. Phase transitions occurring at zero temperature are widely known as quantum phase transitions. In the framework of the physics of phase transitions, the role of dimensionality is a noticeable one. The major challenge in this perspective is the treatment of low dimensional systems, where standard mean-field treatments and ordinary perturbation theory are known to fail due to the enhancement of fluctuations. The goal of describing collective behaviours in reduced dimensionality is usually pursued in one-dimensional (1D) systems, which turn out to be simple enough to be studied in an effective way through both numerical and analytical techniques, while exhibiting a rich and deeply fascinating phenomenology. As a matter of fact, since particles confined to one dimension cannot avoid interactions with each other, the typical low energy excitations of the system are represented by collective density waves, which substitute nearly free quasiparticle excitations characterizing the behaviour of three-dimensional Fermi liquids in the description of the effects of the interplay between kinetic fluctuations and interactions. On the other hand, the tools employed in the activity of tackling 1D manybody problems are efficient and well developed. The analytical techniques aremainly based on the bosonization technique, which allows for the reformulation of strongly correlated 1D systems onto free bosonic theories, which can then be analyzed by means of standard path integral techniques. Meanwhile, the density matrix renormalization group (DMRG) algorithm represents the state-of-theart numerical machinery in the study of 1D physics, since it allows to efficiently extract the low energy properties of the model Hamiltonian and consequently characterize the corresponding phase diagram by means of the behaviour of properly chosen order parameters. As a final remark pointing out the relevance of the research effort in the direction of a better understanding of 1D many-body quantum systems, it is worth noticing that 1D systems do not represent only a playground for theorists, but have been experimentally realized by means of setups with cold atoms trapped in optical lattices, where the experimentalist is able to tune the model parameters by varying the characteristic features of the trapping laser beams and meanwhile has a comfortable access to the most relevant observables.
Mean-field theory is a widely used approximation for describing phase transitions, particularly effective above the upper critical dimension. Its origins can be traced back to the van der Waals theory of the liquid-gas transition and Weiss's molecular field theory of the paramagnetic-ferromagnetic transition. However, it was Lev D. Landau who provided a unifying and general framework applicable to a broad class of physical systems. The mean-field approach typically involves neglecting thermal fluctuations, which is a reasonable assumption in many classical contexts. However, its application to quantum phase transitions at zero temperature is less common. The aim of this short pedagogical paper is to explore and clarify the use of the mean-field approach in the less familiar domain of quantum phase transitions. We specifically consider the Ising model and the Blume-Capel model.
A multichain mean-field theory is developed and applied to a two-dimensional system of weakly coupled S=1/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2S=1/2 Heisenberg chains. The environment of a chain C0{C}_{0}C0​ is modeled by a number of neighboring chains ${C}_{\ensuremath{\delta}}$, $\ensuremath{\delta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ifmmode\pm\else\textpm\fi{}1,\dots{},\ifmmode\pm\else\textpm\fi{}n$, with the edge chains ${C}_{\ifmmode\pm\else\textpm\fi{}n}$ coupled to a staggered field. Using a quantum Monte Carlo method, the effective (2n+1)(2n+1)(2n+1)-chain Hamiltonian is solved self-consistently for nnn up to 444. The results are compared with simulation results for the original Hamiltonian on large rectangular lattices. Both methods show that the staggered magnetization MMM for small interchain couplings $\ensuremath{\alpha}$ behaves as $M\ensuremath{\sim}\sqrt{\ensuremath{\alpha}}$ enhanced by a multiplicative logarithmic correction.