arXiv subjects

R. F. Bishop

Publications and source records attributed to R. F. Bishop.

At least 19 recordsLinked to original sources

arXiv2022

Generalized Phase-Space Techniques to Explore Quantum Phase Transitions in Critical Quantum Spin Systems

We apply the generalized Wigner function formalism to detect and characterize a range of quantum phase transitions in several cyclic, finite-length, spin-$\frac{1}{2}$ one-dimensional spin-chain models, viz., the Ising and anisotropic $XY$ models in a transverse field, and the $XXZ$ anisotropic Heisenberg model. We make use of the finite system size to provide an exhaustive exploration of each system's single-site, bipartite and multi-partite correlation functions. In turn, we are able to demonstrate the utility of phase-space techniques in witnessing and characterizing first-, second- and infinite-order quantum phase transitions, while also enabling an in-depth analysis of the correlations present within critical systems. We also highlight the method's ability to capture other features of spin systems such as ground-state factorization and critical system scaling. Finally, we demonstrate the generalized Wigner function's utility for state verification by determining the state of each system and their constituent sub-systems at points of interest across the quantum phase transitions, enabling interesting features of critical systems to be intuitively analyzed.

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arXiv2022

Frustrated spin-$\frac{1}{2}$ Heisenberg magnet on an $AA$-stacked honeycomb bilayer: High-order study of the collinear magnetic phases of the $J_{1}$--$J_{2}$--$J_{1}^{\perp}$ model

The zero-temperature phase diagram of the frustrated spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{1}^{\perp}$ Heisenberg magnet on an $AA$-stacked honeycomb bilayer lattice is studied using the coupled cluster method implemented to very high orders. On each monolayer the spins interact via nearest-neighbor (NN) and frustrating next-nearest-neighbor isotropic antiferromagnetic Heisenberg interactions with respective strength parameters $J_{1}>0$ and $J_{2}\equivκJ_{1}>0$. The two layers are coupled such that NN interlayer pairs of spins also interact via a similar isotropic Heisenberg interaction of strength $J_{1}^{\perp}\equiv δJ_{1}$, which may be of either sign. In particular, we locate with high accuracy the complete phase boundaries in the $κ$-$δ$ half-plane with $κ>0$ of the two quasiclassical collinear antiferromagnetic phases with Néel or Néel-II magnetic order in each monolayer, and the interlayer NN pairs of spins either aligned (for $δ<0$) or anti-aligned (for $δ> 0$) to one another. Compared to the two-sublattice Néel order, in which all NN intralayer pairs of spins are antiparallel to one another, the four-sublattice Néel-II order is characterized by NN intralayer pairs of spins on the honeycomb lattice being antiparallel to one another along zigzag (or sawtooth) chains in a specified direction from among the three equivalent honeycomb-lattice directions, and parallel to one another for the corresponding interchain pairs.

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arXiv2020

Non-Hermitian coupled cluster method for non-stationary systems and its interaction-picture reinterpretation

The interaction picture in a non-Hermitian realization is discussed in detail and considered for its practical use in many-body quantum physics. The resulting non-Hermitian interaction-picture (NHIP) description of dynamics, in which both the wave functions and operators belonging to physical observables cease to remain constant in time, is a non-Hermitian generalization of the traditional Dirac picture of standard quantum mechanics, which itself is widely used in quantum field theory calculations. Particular attention is paid here to the variational (or, better, bivariational) and dynamical (i.e., non-stationary) aspects that are characteristic of the coupled cluster method (CCM) techniques that nowadays form one of the most versatile and most accurate of all available formulations of quantum many-body theory. In so doing we expose and exploit multiple parallels between the NHIP and the CCM in its time-dependent versions.

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arXiv2019

Frustrated spin-$\frac{1}{2}$ Heisenberg magnet on a square-lattice bilayer: High-order study of the quantum critical behavior of the $J_{1}$--$J_{2}$--$J_{1}^{\perp}$ model

The zero-temperature phase diagram of the spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{1}^{\perp}$ model on an $AA$-stacked square-lattice bilayer is studied using the coupled cluster method implemented to very high orders. Both nearest-neighbor (NN) and frustrating next-nearest-neighbor Heisenberg exchange interactions, of strengths $J_{1}>0$ and $J_{2} \equiv κJ_{1}>0$, respectively, are included in each layer. The two layers are coupled via a NN interlayer Heisenberg exchange interaction with a strength $J_{1}^{\perp} \equiv δJ_{1}$. The magnetic order parameter $M$ (viz., the sublattice magnetization) is calculated directly in the thermodynamic (infinite-lattice) limit for the two cases when both layers have antiferromagnetic ordering of either the Néel or the striped kind, and with the layers coupled so that NN spins between them are either parallel (when $δ< 0$) or antiparallel (when $δ> 0$) to one another. Calculations are performed at $n$th order in a well-defined sequence of approximations, which exactly preserve both the Goldstone linked cluster theorem and the Hellmann-Feynman theorem, with $n \leq 10$. The sole approximation made is to extrapolate such sequences of $n$th-order results for $M$ to the exact limit, $n \to \infty$. By thus locating the points where $M$ vanishes, we calculate the full phase boundaries of the two collinear AFM phases in the $κ$--$δ$ half-plane with $κ> 0$. In particular, we provide the accurate estimate, ($κ\approx 0.547,δ\approx -0.45$), for the position of the quantum triple point (QTP) in the region $δ< 0$. We also show that there is no counterpart of such a QTP in the region $δ> 0$, where the two quasiclassical phase boundaries show instead an ``avoided crossing'' behavior, such that the entire region that contains the nonclassical paramagnetic phases is singly connected.

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arXiv2019

Collinear antiferromagnetic phases of a frustrated spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{1}^{\perp}$ Heisenberg model on an $AA$-stacked bilayer honeycomb lattice

The zero-temperature quantum phase diagram of the spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{1}^{\perp}$ model on an $AA$-stacked bilayer honeycomb lattice is investigated using the coupled cluster method (CCM). The model comprises two monolayers in each of which the spins, residing on honeycomb-lattice sites, interact via both nearest-neighbor (NN) and frustrating next-nearest-neighbor isotropic antiferromagnetic (AFM) Heisenberg exchange iteractions, with respective strengths $J_{1} > 0$ and $J_{2} \equiv κJ_{1}>0$. The two layers are coupled via a comparable Heisenberg exchange interaction between NN interlayer pairs, with a strength $J_{1}^{\perp} \equiv δJ_{1}$. The complete phase boundaries of two quasiclassical collinear AFM phases, namely the Néel and Néel-II phases, are calculated in the $κδ$ half-plane with $κ> 0$. Whereas on each monolayer in the Néel state all NN pairs of spins are antiparallel, in the Néel-II state NN pairs of spins on zigzag chains along one of the three equivalent honeycomb-lattice directions are antiparallel, while NN interchain spins are parallel. We calculate directly in the thermodynamic (infinite-lattice) limit both the magnetic order parameter $M$ and the excitation energy $Δ$ from the $s^{z}_{T}=0$ ground state to the lowest-lying $|s^{z}_{T}|=1$ excited state (where $s^{z}_{T}$ is the total $z$ component of spin for the system as a whole, and where the collinear ordering lies along the $z$ direction) for both quasiclassical states used (separately) as the CCM model state, on top of which the multispin quantum correlations are then calculated to high orders ($n \leq 10$) in a systematic series of approximations involving $n$-spin clusters. The sole approximation made is then to extrapolate the sequences of $n$th-order results for $M$ and $Δ$ to the exact limit, $n \to \infty$.

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arXiv2018

Non-Coplanar Model States in Quantum Magnetism Applications of the High-Order Coupled Cluster Method

Coplanar model states for applications of the coupled cluster method (CCM) to problems in quantum magnetism are those in which all spins lie in a plane, whereas three-dimensional (3D) model states are, by contrast, non-coplanar ones in which all the spins do not lie in any single plane. Here we extend the CCM to non-coplanar / 3D model states and we present results for three cases: (a) the spin-half one-dimensional Ising ferromagnet in an applied transverse magnetic field (as an exactly solvable test model to use as a yardstick for the viability and accuracy of our new methodology); (b) the spin-half triangular-lattice Heisenberg antiferromagnet in the presence of an external magnetic field; and (c) the spin-$S$ triangular-lattice {\it XXZ} antiferromagnet in the presence of an external magnetic field, for the cases $\frac{1}{2} \leq S \leq5 $. For 3D model states the sets of algebraic CCM equations for the ket- and bra-state correlation coefficients become complex-valued, but ground-state expectation values of all physical observables are manifestly real numbers, as required. Excellent correspondence is seen with the results of other methods, where they exist, for these systems. CCM results demonstrate explicitly that coplanar ordering is favoured over non-coplanar ordering for the triangular-lattice spin-half Heisenberg antiferromagnet at all values of the applied external magnetic field, whereas for the anisotropic {\it XXZ} model non-coplanar ordering can be favoured in some regions of the parameter space. Specifically, we present a precise determination of the boundary (i.e., the critical value of the {\it XXZ} anisotropy parameter $Δ$) between a 3D ground state and a coplanar ground state for the {\it XXZ} model for values for the external magnetic field near to saturation, for values of the spin quantum number $S \leq 5$.

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arXiv2018

The Interplay Between Lattice Topology, Frustration, and Spin Quantum Number in Quantum Antiferromagnets on Archimedean Lattices

The interplay between lattice topology, frustration, and spin quantum number, $s$, is explored for the Heisenberg antiferromagnet (HAFM) on the eleven two-dimensional Archimedean lattices (square, honeycomb, CaVO, SHD, SrCuBO, triangle, bounce, trellis, maple-leaf, star, and kagome). We show the CCM provides consistently accurate results when compared to the results of other approximate methods. The $\sqrt{3}\times\sqrt{3}$ model state provides lower ground-state energies than those of the $q=0$ model state for the kagome and star lattices for most values of $s$. The $q=0$ model state provides lower ground-state energies only for $s=1/2$ for the kagome lattice and $s=1/2$ and $s=1$ for the star lattice. The kagome and star lattices demonstrate the least amount of magnetic ordering and the unfrustrated lattices (square, honeycomb, SHD, and CaVO) demonstrate the most magnetic ordering for all values of $s$. The SrCuBO and triangular lattices also demonstrate high levels of magnetic ordering, while the remaining lattices (bounce, maple-leaf, and trellis) tend to lie between these extremes, again for all values of $s$. These results also clearly reflect the strong increase in magnetic order with increasing spin quantum number $s$ for all lattices. The ground-state energy, $E_g/(NJs^2)$, scales with $s^{-1}$ to first order, as expected from spin-wave theory, although the order parameter, $M/s$, scales with $s^{-1}$ for most of the lattices only. Self-consistent spin-wave theory calculations indicated previously that $M/s$ scales with $s^{-2/3}$ for the kagome lattice HAFM, whereas previous CCM results (replicated here also) suggested that $M/s$ scales with $s^{-1/2}$. By using similar arguments, we find here also that $M/s$ scales with $s^{-1/3}$ on the star lattice and with $s^{-2/3}$ on the SrCuBO lattice.

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arXiv2018

A general approach to quantum mechanics as a statistical theory

Since the very early days of quantum theory there have been numerous attempts to interpret quantum mechanics as a statistical theory. This is equivalent to describing quantum states and ensembles together with their dynamics entirely in terms of phase-space distributions. Finite dimensional systems have historically been an issue. In recent works [Phys. Rev. Lett. 117, 180401 and Phys. Rev. A 96, 022117] we presented a framework for representing any quantum state as a complete continuous Wigner function. Here we extend this work to its partner function -- the Weyl function. In doing so we complete the phase-space formulation of quantum mechanics -- extending work by Wigner, Weyl, Moyal, and others to any quantum system. This work is structured in three parts. Firstly we provide a brief modernized discussion of the general framework of phase-space quantum mechanics. We extend previous work and show how this leads to a framework that can describe any system in phase space -- putting it for the first time on a truly equal footing to Schrödinger's and Heisenberg's formulation of quantum mechanics. Importantly, we do this in a way that respects the unifying principles of "parity" and "displacement" in a natural broadening of previously developed phase space concepts and methods. Secondly we consider how this framework is realized for different quantum systems; in particular we consider the proper construction of Weyl functions for some example finite dimensional systems. Finally we relate the Wigner and Weyl distributions to statistical properties of any quantum system or set of systems.

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arXiv2017

A high-order study of the quantum critical behavior of a frustrated spin-$\frac{1}{2}$ antiferromagnet on a stacked honeycomb bilayer

We study a frustrated spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{3}$--$J_{1}^{\perp}$ Heisenberg antiferromagnet on an $AA$-stacked bilayer honeycomb lattice. In each layer we consider nearest-neighbor (NN), next-nearest-neighbor, and next-next-nearest-neighbor antiferromagnetic (AFM) exchange couplings $J_{1}$, $J_{2}$, and $J_{3}$, respectively. The two layers are coupled with an AFM NN exchange coupling $J_{1}^{\perp}\equivδJ_{1}$. The model is studied for arbitrary values of $δ$ along the line $J_{3}=J_{2}\equivαJ_{1}$ that includes the most highly frustrated point at $α=\frac{1}{2}$, where the classical ground state is macroscopically degenerate. The coupled cluster method is used at high orders of approximation to calculate the magnetic order parameter and the triplet spin gap. We are thereby able to give an accurate description of the quantum phase diagram of the model in the $αδ$ plane in the window $0 \leq α\leq 1$, $0 \leq δ\leq 1$. This includes two AFM phases with Néel and striped order, and an intermediate gapped paramagnetic phase that exhibits various forms of valence-bond crystalline order. We obtain accurate estimations of the two phase boundaries, $δ= δ_{c_{i}}(α)$, or equivalently, $α= α_{c_{i}}(δ)$, with $i=1$ (Néel) and 2 (striped). The two boundaries exhibit an "avoided crossing" behavior with both curves being reentrant.

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arXiv2017

Gapped paramagnetic state in a frustrated spin-$\frac{1}{2}$ Heisenberg antiferromagnet on the cross-striped square lattice

We implement the coupled cluster method to very high orders of approximation to study the spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$ Heisenberg model on a cross-striped square lattice. Every nearest-neighbour pair of sites on the square lattice has an isotropic antiferromagnetic exchange bond of strength $J_{1}>0$, while the basic square plaquettes in alternate columns have either both or neither next-nearest-neighbour (diagonal) pairs of sites connected by an equivalent frustrating bond of strength $J_{2} \equiv αJ_{1} > 0$. By studying the magnetic order parameter (i.e., the average local on-site magnetization) in the range $0 \leq α\leq 1$ of the frustration parameter we find that the quasiclassical antiferromagnetic Néel and (so-called) double Néel states form the stable ground-state phases in the respective regions $α< α_{1a}^{c} = 0.46(1)$ and $α> α_{1b}^{c} = 0.615(5)$. The double Néel state has Néel ($\cdots\uparrow\downarrow\uparrow\downarrow\cdots$) ordering along the (column) direction parallel to the stripes of squares with both or no $J_{2}$ bonds, and spins alternating in a pairwise ($\cdots\uparrow\uparrow\downarrow\downarrow\uparrow\uparrow\downarrow\downarrow\cdots$) fashion along the perpendicular (row) direction, so that the parallel pairs occur on squares with both $J_{2}$ bonds present. Further explicit calculations of both the triplet spin gap and the zero-field uniform transverse magnetic susceptibility provide compelling evidence that the ground-state phase over all or most of the intermediate regime $α_{1a}^{c} < α< α_{1b}^{c}$ is a gapped state with no discernible long-range magnetic order.

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arXiv2017

Transverse Magnetic Susceptibility of a Frustrated Spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{1}^{\perp}$ Heisenberg Antiferromagnet on a Bilayer Honeycomb Lattice

We use the coupled cluster method (CCM) to study a frustrated spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{1}^{\perp}$ Heisenberg antiferromagnet on a bilayer honeycomb lattice with $AA$ stacking. Both nearest-neighbor (NN) and frustrating next-nearest-neighbor antiferromagnetic (AFM) exchange interactions are present in each layer, with respective exchange coupling constants $J_{1}>0$ and $J_{2} \equiv κJ_{1} > 0$. The two layers are coupled with NN AFM exchanges with coupling strength $J_{1}^{\perp}\equiv δJ_{1}>0$. We calculate to high orders of approximation within the CCM the zero-field transverse magnetic susceptibility $χ$ in the Néel phase. We thus obtain an accurate estimate of the full boundary of the Néel phase in the $κδ$ plane for the zero-temperature quantum phase diagram. We demonstrate explicitly that the phase boundary derived from $χ$ is fully consistent with that obtained from the vanishing of the Néel magnetic order parameter. We thus conclude that at all points along the Néel phase boundary quasiclassical magnetic order gives way to a nonclassical paramagnetic phase with a nonzero energy gap. The Néel phase boundary exhibits a marked reentrant behavior, which we discuss in detail.

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arXiv2017

A frustrated honeycomb-bilayer Heisenberg antiferromagnet: The spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{1}^{\perp}$ model

We use the coupled cluster method to study the zero-temperature quantum phase diagram of the spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{1}^{\perp}$ model on the honeycomb bilayer lattice. In each layer we include both nearest-neighbor and frustrating next-nearest-neighbor antiferromagnetic exchange couplings, of strength $J_{1}>0$ and $J_{2} \equiv κJ_{1} > 0$, respectively. The two layers are coupled by an interlayer nearest-neighbor exchange, with coupling constant $J_{1}^{\perp} \equiv δJ_{1}>0$. We calculate directly in the infinite-lattice limit both the ground-state energy per spin and the Néel magnetic order parameter, as well as the triplet spin gap. By implementing the method to very high orders of approximation we obtain an accurate estimate for the full boundary of the Néel phase in the $κδ$ plane. For each value $δ< δ_{c}^{>}(0) \approx 1.70(5)$ we find an upper critical value $κ_{c}(δ)$, such that Néel order is present for $κ< κ_{c}(δ)$. Conversely, for each value $κ< κ_{c}(0) \approx 0.19(1)$ we find an upper critical value $δ_{c}^{>}(κ)$, such that Néel order persists for $0 < δ< δ_{c}^{>}(κ)$. Most interestingly, for values of $κ$ in the range $κ_{c}(0) < κ< κ^{>} \approx 0.215(2)$ we find a reentrant behavior such that Néel order exists only in the range $δ_{c}^{<}(κ) < δ< δ_{c}^{>}(κ)$, with $δ_{c}^{<}(κ)>0$. These latter upper and lower critical values coalesce when $κ= κ^{>}$, such that $δ_{c}^{<}(κ^{>}) = δ_{c}^{>}(κ^{>}) \approx 0.25(5)$.

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arXiv2016

Ground-state phases of the spin-1 $J_{1}$--$J_{2}$ Heisenberg antiferromagnet on the honeycomb lattice

We study the zero-temperature quantum phase diagram of a spin-1 Heisenberg antiferromagnet on the honeycomb lattice with both nearest-neighbor exchange coupling $J_{1}>0$ and frustrating next-nearest-neighbor coupling $J_{2} \equiv κJ_{1} > 0$, using the coupled cluster method implemented to high orders of approximation, and based on model states with different forms of classical magnetic order. For each we calculate directly in the bulk thermodynamic limit both ground-state low-energy parameters (including the energy per spin, magnetic order parameter, spin stiffness coefficient, and zero-field uniform transverse magnetic susceptibility) and their generalized susceptibilities to various forms of valence-bond crystalline (VBC) order, as well as the energy gap to the lowest-lying spin-triplet excitation. In the range $0 < κ< 1$ we find evidence for four distinct phases. Two of these are quasiclassical phases with antiferromagnetic long-range order, one with 2-sublattice Néel order for $κ< κ_{c_{1}} = 0.250(5)$, and another with 4-sublattice Néel-II order for $κ> κ_{c_{2}} = 0.340(5)$. Two different paramagnetic phases are found to exist in the intermediate region. Over the range $κ_{c_{1}} < κ< κ^{i}_{c} = 0.305(5)$ we find a gapless phase with no discernible magnetic order, which is a strong candidate for being a quantum spin liquid, while over the range $κ^{i}_{c} < κ< κ_{c_{2}}$ we find a gapped phase, which is most likely a lattice nematic with staggered dimer VBC order that breaks the lattice rotational symmetry.

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arXiv2016

Large-$s$ expansions for the low-energy parameters of the honeycomb-lattice Heisenberg antiferromagnet with spin quantum number $s$

The coupled cluster method (CCM) is employed to very high orders of approximation to study the ground-state (GS) properties of the spin-$s$ Heisenberg antiferromagnet (with isotropic interactions, all of equal strength, between nearest-neighbour pairs only) on the honeycomb lattice. We calculate with high accuracy the complete set of GS parameters that fully describes the low-energy behaviour of the system, in terms of an effective magnon field theory, viz., the energy per spin, the magnetic order parameter (i.e., the sublattie magnetization), the spin stiffness and the zero-field (uniform, transverse) magnetic susceptibility, for all values of the spin quantum number $s$ in the range $\frac{1}{2} \leq s \leq \frac{9}{2}$. The CCM data points are used to calculate the leading quantum corrections to the classical ($s \rightarrow \infty$) values of these low-energy parameters, considered as large-$s$ asymptotic expansions.

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arXiv2015

Spin-gap study of the spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$ model on the triangular lattice

We use the coupled cluster method implemented at high orders of approximation to study the spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$ model on the triangular lattice with Heisenberg interactions between nearest-neighbour and next-nearest-neighbour pairs of spins, with coupling strengths $J_{1}>0$ and $J_{2} \equiv κJ_{1} >0$, respectively. In the window $0 \leq κ\leq 1$ we find that the 3-sublattice 120$^{\circ}$ Néel-ordered and 2-sublattice 180$^{\circ}$ stripe-ordered antiferromagnetic states form the stable ground-state phases in the regions $κ< κ^{c}_{1} = 0.060(10)$ and $κ> κ^{c}_{2} = 0.165(5)$, respectively. The spin-triplet gap is found to vanish over essentially the entire region $κ^{c}_{1} < κ< κ^{c}_{2}$ of the intermediate phase.

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arXiv2015

Frustrated Heisenberg antiferromagnet on the honeycomb lattice: Spin gap and low-energy parameters

We use the coupled cluster method implemented to high orders of approximation to investigate the frustrated spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{3}$ antiferromagnet on the honeycomb lattice with isotropic Heisenberg interactions of strength $J_{1} > 0$ between nearest-neighbor pairs, $J_{2}>0$ between next-nearest-neighbor pairs, and $J_{3}>0$ between next-next-neareast-neighbor pairs of spins. In particular, we study both the ground-state (GS) and lowest-lying triplet excited-state properties in the case $J_{3}=J_{2} \equiv κJ_{1}$, in the window $0 \leq κ\leq 1$ of the frustration parameter, which includes the (tricritical) point of maximum classical frustration at $κ_{\rm cl} = \frac{1}{2}$. We present GS results for the spin stiffness, $ρ_{s}$, and the zero-field uniform magnetic susceptibility, $χ$, which complement our earlier results for the GS energy per spin, $E/N$, and staggered magnetization, $M$, to yield a complete set of accurate low-energy parameters for the model. Our results all point towards a phase diagram containing two quasiclassical antiferromagnetic phases, one with Néel order for $κ< κ_{c_{1}}$, and the other with collinear striped order for $κ> κ_{c_{2}}$. The results for both $χ$ and the spin gap $Δ$ provide compelling evidence for a quantum paramagnetic phase that is gapped over a considerable portion of the intermediate region $κ_{c_{1}} < κ< κ_{c_{2}}$, especially close to the two quantum critical points at $κ_{c_{1}}$ and $κ_{c_{2}}$. Each of our fully independent sets of results for the low-energy parameters is consistent with the values $κ_{c_{1}} = 0.45 \pm 0.02$ and $κ_{c_{2}} = 0.60 \pm 0.02$, and with the transition at $κ_{c_{1}}$ being of continuous (and probably of the deconfined) type and that at $κ_{c_{2}}$ being of first-order type.

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arXiv2015

Ground-state phase structure of the spin-$\frac{1}{2}$ anisotropic planar pyrochlore

We study the zero-temperature ground-state (GS) properties of the spin-$\frac{1}{2}$ anisotropic planar pyrochlore, using the coupled cluster method (CCM) implemented to high orders of approximation. The system comprises a $J_{1}$--$J_{2}$ model on the checkerboard lattice, with isotropic Heisenberg interactions of strength $J_{1}$ between all nearest-neighbour pairs of spins on the square lattice, and of strength $J_{2}$ between half of the next-nearest-neighbour pairs (in the checkerboard pattern). We calculate results for the GS energy and average local GS on-site magnetization, using various antiferromagnetic classical ground states as CCM model states. We also give results for the susceptibility of one of these states against the formation of crossed-dimer valence-bond crystalline (CDVBC) ordering. The complete GS phase diagram is presented for arbitrary values of the frustration parameter $κ\equiv J_{2}/J_{1}$, and when each of the exchange couplings can take either sign.

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arXiv2014

Highly frustrated spin-lattice models of magnetism and their quantum phase transitions: A microscopic treatment via the coupled cluster method

We outline how the coupled cluster method of microscopic quantum many-body theory can be utilized in practice to give highly accurate results for the ground-state properties of a wide variety of highly frustrated and strongly correlated spin-lattice models of interest in quantum magnetism, including their quantum phase transitions. The method itself is described, and it is shown how it may be implemented in practice to high orders in a systematically improvable hierarchy of (so-called LSUB$m$) approximations, by the use of computer-algebraic techniques. The method works from the outset in the thermodynamic limit of an infinite lattice at all levels of approximation, and it is shown both how the "raw" LSUB$m$ results are themselves generally excellent in the sense that they converge rapidly, and how they may accurately be extrapolated to the exact limit, $m \rightarrow \infty$, of the truncation index $m$, which denotes the {\it only} approximation made. All of this is illustrated via a specific application to a two-dimensional, frustrated, spin-half $J^{XXZ}_{1}$--$J^{XXZ}_{2}$ model on a honeycomb lattice with nearest-neighbor and next-nearest-neighbor interactions with exchange couplings $J_{1}>0$ and $J_{2} \equiv κJ_{1} > 0$, respectively, where both interactions are of the same anisotropic $XXZ$ type. We show how the method can be used to determine the entire zero-temperature ground-state phase diagram of the model in the range $0 \leq κ\leq 1$ of the frustration parameter and $0 \leq Δ\leq 1$ of the spin-space anisotropy parameter. In particular, we identify a candidate quantum spin-liquid region in the phase space.

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