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arXiv:1609.04990 · Condensed Matter Physics

The large-m limit, and spin liquid correlations in kagome-like spin models

Yavors'kii

Published Mar 1, 2017• 0 citations
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Abstract

It is noted that the pair correlation matrix χ of the nearest neighbor Ising model on periodic three-dimensional (d = 3) kagome-like lattices of corner-sharing triangles can be calculated partially exactly.Specifically, a macroscopic number 1/3 N + 1 out of N eigenvalues of χ are degenerate at all temperatures T , and correspond to an eigenspace L -of χ, independent of T .Degeneracy of the eigenvalues, and L -are an exact result for a complex d = 3 statistical physical model.It is further noted that the eigenvalue degeneracy describing the same Lis exact at all T in an infinite spin dimensionality m limit of the isotropic m-vector approximation to the Ising models.A peculiar match of the opposite m = 1 and m → ∞ limits can be interpreted that the m → ∞ considerations are exact for m = 1.It is not clear whether the match is coincidental.It is then speculated that the exact eigenvalues degeneracy in L -in the opposite limits of m can imply their quasi-degeneracy for intermediate 1 m < ∞.For an anti-ferromagnetic nearest neighbor coupling, that renders kagome-like models highly geometrically frustrated, these are spin states largely from L -that for m 2 contribute to χ at low T .The m → ∞ formulae can be thus quantitatively correct in description of χ and clarifying the role of perturbations in kagome-like systems deep in the collective paramagnetic regime.An exception may be an interval of T , where the order-by-disorder mechanisms select sub-manifolds of L -.

DOI
10.5488/cmp.20.13701

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Journal of Applied Physics•Mar 1, 1970•21 citations
It is noted that the pair correlation matrix χ of the nearest neighbor Ising model on periodic three-dimensional (d = 3) kagome-like lattices of corner-sharing triangles can be calculated partially exactly.Specifically, a macroscopic number 1/3 N + 1 out of N eigenvalues of χ are degenerate at all temperatures T , and correspond to an eigenspace L -of χ, independent of T .Degeneracy of the eigenvalues, and L -are an exact result for a complex d = 3 statistical physical model.It is further noted that the eigenvalue degeneracy describing the same Lis exact at all T in an infinite spin dimensionality m limit of the isotropic m-vector approximation to the Ising models.A peculiar match of the opposite m = 1 and m → ∞ limits can be interpreted that the m → ∞ considerations are exact for m = 1.It is not clear whether the match is coincidental.It is then speculated that the exact eigenvalues degeneracy in L -in the opposite limits of m can imply their quasi-degeneracy for intermediate 1 m < ∞.For an anti-ferromagnetic nearest neighbor coupling, that renders kagome-like models highly geometrically frustrated, these are spin states largely from L -that for m 2 contribute to χ at low T .The m → ∞ formulae can be thus quantitatively correct in description of χ and clarifying the role of perturbations in kagome-like systems deep in the collective paramagnetic regime.An exception may be an interval of T , where the order-by-disorder mechanisms select sub-manifolds of L -.
We use the coupled-cluster method to study the zero-temperature properties of an extended two-dimensional Heisenberg antiferromagnet formed from spin-12\frac{1}{2}21​ moments on an infinite spatially anisotropic kagome lattice of corner-sharing isosceles triangles, with nearest-neighbor bonds only. The bonds have exchange constants J1>0{J}_{1}>0J1​>0 along two of the three lattice directions and ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$ along the third. In the classical limit, the ground-state (GS) phase for $\ensuremath{\kappa}<\frac{1}{2}$ has collinear ferrimagnetic (N\'eel${}^{\ensuremath{'}}$) order where the J2{J}_{2}J2​-coupled chain spins are ferromagnetically ordered in one direction with the remaining spins aligned in the opposite direction, while for $\ensuremath{\kappa}>\frac{1}{2}$ there exists an infinite GS family of canted ferrimagnetic spin states, which are energetically degenerate. For the spin-12\frac{1}{2}21​ case, we find that quantum analogs of both these classical states continue to exist as stable GS phases in some regions of the anisotropy parameter $\ensuremath{\kappa}$, namely, for $0<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{1}}$ for the N\'eel${}^{\ensuremath{'}}$ state and for (at least part of) the region $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}$ for the canted phase. However, they are now separated by a paramagnetic phase without either sort of magnetic order in the region ${\ensuremath{\kappa}}_{{c}_{1}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{2}}$, which includes the isotropic kagome point $\ensuremath{\kappa}=1$ where the stable GS phase is now believed to be a topological (Z2{\mathbb{Z}}_{2}Z2​) spin liquid. Our best numerical estimates are ${\ensuremath{\kappa}}_{{c}_{1}}=0.515\ifmmode\pm\else\textpm\fi{}0.015$ and ${\ensuremath{\kappa}}_{{c}_{2}}=1.82\ifmmode\pm\else\textpm\fi{}0.03$.
Ever since the introduction of resonating valence bond states by P.\\ W.\\ Anderson in 1970s, the search for exotic many--body quantum behavior in peculiar states of matter known as quantum spin liquids has been a central challenge in the field of condensed matter physics. An important searching ground for such \\emph{quantum--ness} is the low energy physics of frustrated magnets. In a quantum system, frustration---the inability of a physical system to achieve a global state that consistently minimizes its energy locally---enables the unavoidable quantum fluctuations to hybridize an extensive number of classical states into highly entangled quantum states at low temperatures. It can be argued that this is the most practical approach to achieve genuine many--body quantum behavior at large scales. The fundamental source of frustration in many physical systems, is the existence of local non--commuting terms. In the case of antiferromagnet spin theories the simplest examples are triangles of nearest--neighboring sites. Consequently, lattices with triangles as building blocks have long been studied in hopes of discovering spin liquids physics. In particular the kagome lattice, which is composed of corner--sharing triangles, has been the center of at least four decades of spin liquid research. This is due to the underconstrained nature of system made out of corner--sharing triangles. After the first three chapters, introduction in chapter~\\ref{chap:intro}, preliminary physics in chapter~\\ref{chap:prelim}, and a brief introduction to spin--liquids in chapter~\\ref{chap:QSLs}, in chapter~\\ref{chap:kagome} I tackle a simpler version of the long-standing problem of spin--1/21/21/2 Heisenberg antiferromagnets by considering a quasi--1D lattice consisting entirely of corner--sharing triangles, \\emph{kagome strip}. I illustrate that the standard Heisenberg antiferromagnet Hamiltonian over kagome strip is an extended gapless quantum phase, that is well characterized by two fermionic/bosonic gapless modes and power--law decaying spin and bond--energy correlations. I also demonstrate that the correlation functions oscillate at tunably incommensurate wave vectors. It turns out that this phase can be identified by a particular marginal instability of a two-band spinon Fermi surface coupled to an emergent U(1) gauge field. This interpretation is supported by analytic Abelian bosonization and with extensive numerical large--scale density matrix renormalization group study as well as variational Monte Carlo calculations on Gutzwiller ans\\"atz wave functions. This intriguing result is the first numerical demonstration of emergent fermionic spinons in a simple SU(2) invariant nearest-neighbor Heisenberg model beyond the strictly 1D (Bethe chain) limit. The unexpected success of the fermionic spinons in describing the low energy physics of kagome antiferromagnet, reintroduces the questions about the validity of spinon physics as an effective theory of quantum spin liquids. However, numerical methods are still far behind the theoretical advances. And still even today, an accurate description of the quantum spin liquid states using tensor network methods is notoriously challenging. It is known that for large quasi-1D systems, the density matrix renormalization group and related methods usually require significant computational resources and sometimes fail to converge to a satisfactory state. On the other hand, variational wavefunctions acquired from the Gutzwiller projection of gaussian fermionic theories have long served as both a theoretical starting point for the construction of such spin liquid states and as an inspiration for numerical variational Monte Carlo (VMC) to calculate observables of interest. Noting this observation I examine a different method by exploring the possibility of constructing a matrix product state (MPS) representation for a Gutzwiller--projected state from two given MPS representations of gaussian fermionic theories in the \\ref{chap:gutz} chapter. I investigate the complexity of different approaches to achieve Gutzwiller projection for MPSs and introduce the novel algorithm which we call the Guzwiller zipper method. The performance of the algorithm is tested against two copies of a single half-filled band of spin-1/2 fermionic spinons. In a successful attempt to describe the nature of spin liquid states on quasi-1D strips of triangular and kagome-like lattices, I apply this method to two MPS of multi-band fermionic spinon theories and compare with the complexity of the traditional VMC approach. In particular, we methodically disprove the conjecture of a spinon fermi surface spin liquid for the triangular lattice. Finally, I conclude this thesis by laying out a bird's eye view of the current and future of spin liquid physics in kagome and triangular lattices and pointing out numerous possible applications of the novel Gutzwiller zipper method. In addition, I would also discuss its possible extensions to more complicated tensor networks as well as versions for higher spatial dimensions.
Exactly solvable models play a special role in condensed matter physics, serving as secure theoretical starting points for investigation of new phenomena. Changlani et al. [Phys. Rev. Lett. 120, 117202 (2018)] have discovered a limit of the XXZ model for S=12S=\frac{1}{2}S=21​ spins on the kagome lattice, which is not only exactly solvable, but features a huge degeneracy of exact ground states corresponding to solutions of a three-coloring problem. This special point of the model was proposed as a parent for multiple phases in the wider phase diagram, including quantum spin liquids. Here, we show that the construction of Changlani et al. can be extended to more general forms of anisotropic exchange interaction, finding a line of parameter space in an XYZ model which maintains both the macroscopic degeneracy and the three-coloring structure of solutions. We show that the ground states along this line are partially ordered, in the sense that infinite-range correlations of some spin components coexist with a macroscopic number of undetermined degrees of freedom. We therefore propose the exactly solvable limit of the XYZ model on corner-sharing triangle-based lattices as a tractable starting point for discovery of quantum spin systems which mix ordered and spin-liquid-like properties.
We investigate a nearest-neighbor Ising model on the kagome lattice coupled to elastic lattice distortions. By integrating out the lattice degrees of freedom, we derive an approximate spin-only description in terms of a kagome J1J_1J1​-J2J_2J2​-J3J_{3}J3​ Ising model with interactions extending to third nearest-neighbors. Using tensor network methods complemented by classical Monte Carlo simulations, we revisit its zero-temperature phase diagram and characterize the ground states studied more than thirty years ago. We discover three phases with extensive zero-temperature entropy and algebraically decaying spin correlations. Importantly, these states are not confined to fine-tuned points or frontiers between phases, but occupy extended areas of parameter space. Their quasi long-range order is reflected in Bragg-like peaks in the structure factors; we relate this, and other properties, to diverse phases found in Ising and Heisenberg models. Finally, we trace the trajectories generated by the magnetoelastic coupling through the effective-model phase diagram, explaining the emergence of self-frustration for the unfrustrated magnetoelastic model with ferromagnetic interactions.
A two-dimensional Ising model on a particular topological structure, a ``triangles-in-triangles'' Kagom\'e structure, which is called a triangular-Kagom\'e lattice, is investigated. In this model, we consider two interaction parameters Jaa{J}_{aa}Jaa​ and Jab{J}_{ab}Jab​, corresponding to the interactions among spins on the inner sites and between spins on the inner and vertex sites of the triangular in the Kagom\'e lattice, respectively. By summing over all spins at inner sites in the partition function, we arrive at the partition function of the Kagom\'e lattice. The effective interaction of the corresponding Kagom\'e lattice is always ferromagnetic, even for antiferromagnetic Jaa{J}_{aa}Jaa​. The critical properties of the system depend only on ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}$. When ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}>\ensuremath{-}1$, the system has long range order at low temperature. However, when ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}<\ensuremath{-}1$, the partition function of the triangular-Kagom\'e lattice can be related to that of the Kagom\'e lattice with effective interaction at a temperature higher than its critical temperature. Therefore, the system will be in paramagnetic phase at all temperatures for ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}<\ensuremath{-}1$. The phase diagram for ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}$ is given exactly.
Some years ago Ashkin and Lamb noted that the phase transition in the two-dimensional Ising model with nearest-neighbor interaction was characterized mathematically by an asymptotic degeneracy of the largest eigenvalue of a linear operator. More recently Kac and Thompson showed this eigenvalue degeneracy also characterized the phase transition in the Kac model (with weak, long-range forces), suggesting that this ``mathematical mechanism'' is not restricted to systems with short-range forces. However both the Kac model and the Ising model consider the ``spins'' to be one-dimensional unit vectors assuming only the discrete values +1 and −1. We are therefore led to consider the nature of the phase transition in one of the few exactly soluble models in which the spins can assume an entire continuum of orientations-the closed linear chain of classical spins of arbitrary spin dimensionality interacting isotropically through the Hamiltonian Hν=− ∑ i=1N Ji,i+1Si(ν)·Si+1(ν),where the exchange constants Ji,i+1 are arbitrary numbers. We find for all spin dimensionalities ν that the two-spin correlation function may be expressed as ρN(ν) (γ) = (λ1(ν)/λ0(ν))r, where λ0(ν) and λ1(ν) are the largest and next-largest eigenvalues of a certain linear operator. Thus the onset of longrange order, limr→∞ρN(ν) (γ)≠0, is characterized by the degeneracy of λ0(ν) and λ1(ν). This onset of long-range order occurs at a critical temperature Tc(ν)=0, for all values of spin dimensionality ν.