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

DAMIAN J. J. FARNELL, R. F. Bishop, Johannes Richter

Published Jul 1, 2019• 1 citation
OpenAlex

Abstract

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. A crucial first step in applying the CCM to any such lattice quantum spin system is to perform a passive rotation of the local spin axes so that all spins in the model state appear mathematically to point in the same (say, downwards z-)direction. Whereas this process leads to terms with only real coefficients in the rotated Hamiltonian for coplanar model states, an additional complication arises for 3D model states where the corresponding coefficients can become complex-valued. We show here for the first time how high-order implementations of the CCM can be performed for such Hamiltonians. We explain in detail why the extension of the computational implementation of the CCM when going from coplanar to 3D model states is a non-trivial task that has not hitherto been undertaken. To illustrate these new developments, 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 XXZ antiferromagnet in the presence of an external magnetic field, for the cases <= S <= 5. 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, and as we explicitly demonstrate in all three applications. Indeed, excellent correspondence is seen with the results of other methods, where they exist, for these systems. In particular, our 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 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 XXZ anisotropy parameter Delta) between a 3D ground state and a coplanar ground state for the XXZ model for values for the external magnetic field near to saturation, for values of the spin quantum number S <= 5. Although the CCM calculations are computationally intensive for this frustrated model, especially for high spin quantum numbers, our accurate new results certainly improve our understanding of it.

Related papers

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

DAMIAN J. J. FARNELL, R. F. Bishop, Johannes Richter

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. A crucial first step in applying the CCM to any such lattice quantum spin system is to perform a passive rotation of the local spin axes so that all spins in the model state appear mathematically to point in the same (say, downwards z-)direction. Whereas this process leads to terms with only real coefficients in the rotated Hamiltonian for coplanar model states, an additional complication arises for 3D model states where the corresponding coefficients can become complex-valued. We show here for the first time how high-order implementations of the CCM can be performed for such Hamiltonians. We explain in detail why the extension of the computational implementation of the CCM when going from coplanar to 3D model states is a non-trivial task that has not hitherto been undertaken. To illustrate these new developments, 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 XXZ antiferromagnet in the presence of an external magnetic field, for the cases $$\frac{1}{2} \le S \le 5 $$ . 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, and as we explicitly demonstrate in all three applications. Indeed, excellent correspondence is seen with the results of other methods, where they exist, for these systems. In particular, our 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 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 XXZ anisotropy parameter $$\Delta $$ ) between a 3D ground state and a coplanar ground state for the XXZ model for values for the external magnetic field near to saturation, for values of the spin quantum number $$S \le 5$$ . Although the CCM calculations are computationally intensive for this frustrated model, especially for high spin quantum numbers, our accurate new results certainly improve our understanding of it.

arXiv:1811.04228•Journal of Statistical Physics•May 2, 2019•6 citations

High-Order Coupled Cluster Method Calculations using Three-Dimensional Model States: An Illustration for the Triangular-Lattice Antiferromagnet in an External Field

DAMIAN J. J. FARNELL, Andrew I. Croudace

The coupled cluster method (CCM) has previously been applied to study the ground- and excited-state properties of many different types of frustrated and unfrustrated quantum spin systems. A common feature in the application of the CCM is to rotate the local spin axes of the (often classical) model state so that (notationally only) the spins all appear to point in the downwards $z$-direction. Hitherto, we remark that only coplanar model states have been used because they do not lead to imaginary terms in the new Hamiltonian. By contrast, non-coplanar "three-dimensional" (3D) model states can lead to imaginary terms in the new Hamiltonian after rotation. In principle, however, macroscopic quantities predicted by the CCM (such as the ground-state energy and order parameter) should still be real (even though the Hamiltonian may be complex) because the transformations of local spin axes are unitary. Here we explain how we may use such 3D model states for the CCM and how we may solve for the (now possibly complex) CCM correlation coefficients. We present results for the spin-half triangular-lattice antiferromagnet in an external magnetic field. We use both coplanar model states and a single 3D non-coplanar model state. We calculate the ground-state energy and the lattice magnetisation as a function of the magnetic field strength. We find that the energies of the coplanar states lie lower than that of the non-coplanar state for all values of the external field, as seen in other approximate studies for this model. We find that the non-coplanar state does not detect the well-known spin plateau occurring in this model, although (as seen before) it is clearly observed for the coplanar model states using the CCM. These results are an excellent initial validation of the new approach for the application of the CCM using 3D model states.

arXiv:1312.0785•Dec 3, 2013•1 citation

High-order coupled-cluster method for general spin-lattice problems: An illustration via the anisotropic Heisenberg model

DAMIAN J. J. FARNELL, Klaus A. Gernoth, R. F. Bishop

A high-order coupled cluster method (CCM) for the zero-temperature ground states of quantum lattice spin systems of general spin quantum number $(s>~1/2)$ is described and illustrated via the spin-1 anisotropic Heisenberg antiferromagnet on the square lattice. Results for the ground-state energy and sublattice magnetization are seen to be in excellent agreement with the best of other approximate methods. Furthermore, we are able to follow the solution to the CCM equations at a given level of approximation with respect to the anisotropy parameter $\ensuremath{\Delta}$ from the trivial Ising limit $(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Delta}}\ensuremath{\infty})$ down to a critical value ${\ensuremath{\Delta}}_{c},$ at which point the physical solution terminates. The results for the ground-state energy, sublattice magnetization, and phase transition point thus support our choice of this model as a good nontrivial test case of our CCM formalism.

Physical review. B, Condensed matter•Oct 15, 2001•30 citations

High-Order Coupled Cluster Method (CCM) Formalism 2 -- "Generalised" Expectation Values: Spin-Spin Correlation Functions for Frustrated and Unfrustrated 2D Antiferromagnets

DAMIAN J. J. FARNELL

Recent developments of high-order CCM have been to extend existing formalism and codes to $s \ge \frac 12$ for both the ground and excited states. In this article, we describe how "generalised" expectation values for a wide range of one- and two-body spin operators may also be determined using existing the CCM code for the ground state. We present new results for the spin-spin correlation functions of the spin-half square- and triangular-lattice antiferromagnets by using the LSUB$m$ approximation. We show that the absolute values of the spin-spin correlation functions $| < \bf{s}(0).\bf{s}(r) > |$ converge with increasing approximation level for both lattices. We believe that the LSUB$m$ approximation provides reasonable results for the correlation functions for lattice separations roughly of order $r \approx m$ for the square lattice. We compare qualitatively our results for the square lattice to those results of quantum Monte Carlo (QMC) and we see that good correspondence is observed. Indeed as seen by QMC, the spin-spin correlation function initially decays strongly with $|r|$ before becoming constant for larger values of $|r|$. CCM results are also compared to results of exact diagonalisations for both lattices. ED results demonstrate a strong finite-size effects at lattice separations $r=L/2$ (where $N=L \times L$) for both lattices. The behaviour of the correlation function for the triangular lattice is qualitatively similar to that of the square lattice, namely, that it decays strongly at first before becoming constant. This is in keeping with the behaviour of both models, which are believed strongly to be Néel-ordered from approximate studies.

arXiv:0911.5150•Nov 27, 2009•0 citations

Coupled cluster theory of strongly correlated spin- and electron-lattice systems: an illustration via a model exhibiting competition between magnetic order and dimerization

R. F. Bishop, Sven E Kr ger

The coupled cluster method (CCM) of microscopic quantum many-body theory has become an ab initio method of first choice in quantum chemistry and many fields of nuclear, subnuclear and condensed matter physics, when results of high accuracy are required. In recent years it has begun to be applied with equal success to strongly correlated systems of electrons or quantum spins defined on a regular spatial lattice. One regularly finds that the CCM is able to describe accurately the various zero-temperature phases and the quantum phase transitions between them, even when frustration is present and other methods such as quantum Monte Carlo often fail. We illustrate the use and powerfulness of the method here by applying it to a square-lattice spin-half Heisenberg model where frustration is introduced by competing nearest neighbour bonds. The model exhibits the physically interesting phenomenon of competition between magnetic order and dimerization. Results obtained for the model with the CCM are compared with those found from spin-wave theory and from extrapolating the results of exact diagonalizations of small lattices. We show that the CCM is essentially unique among available methods in being able both to describe accurately all phases of this complex model and to provide accurate predictions of the various phase boundaries and the order of the corresponding transitions.

Journal of Physics A Mathematical and General•Aug 20, 2003•0 citations

A microscopic theory of magnetic order in strongly correlated quantum spin lattices

R. F. Bishop, DAMIAN J. J. FARNELL

Quantum spins arranged on regular geometrical lattices, and interacting through a wide diversity of specified model Hamiltonians, provide a rich variety of magnetically ordered phases with many subtle properties. The zero-temperature quantum phase transitions between states of different order can be explored by the variation of internal control parameters within the interaction Hamiltonian. We illustrate how the systematic inclusion of multi-spin correlations for such lattice systems may be efficiently and effectively implemented via the coupled cluster method (CCM), which has been widely demonstrated on many previous occasions to be one of the most versatile methods of microscopic quantum many-body theory. Its power is illustrated by application to the spin-½ anisotropic Heisenberg (or XXZ) model on an infinite square lattice. The results obtained for such properties as the ground-state energy and sublattice magnetization are found to be as accurate as those obtained by any available alternative method. We show also how the CCM can now provide accurate ab initio estimates for the critical values of the parameters which mark the positions of phase boundaries between states of different quantum order, as well as for the critical indices associated with the singular behaviour near these transition points of various physical quantities.

•Jan 1, 1998•0 citations
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. A crucial first step in applying the CCM to any such lattice quantum spin system is to perform a passive rotation of the local spin axes so that all spins in the model state appear mathematically to point in the same (say, downwards z-)direction. Whereas this process leads to terms with only real coefficients in the rotated Hamiltonian for coplanar model states, an additional complication arises for 3D model states where the corresponding coefficients can become complex-valued. We show here for the first time how high-order implementations of the CCM can be performed for such Hamiltonians. We explain in detail why the extension of the computational implementation of the CCM when going from coplanar to 3D model states is a non-trivial task that has not hitherto been undertaken. To illustrate these new developments, 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 XXZ antiferromagnet in the presence of an external magnetic field, for the cases <= S <= 5. 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, and as we explicitly demonstrate in all three applications. Indeed, excellent correspondence is seen with the results of other methods, where they exist, for these systems. In particular, our 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 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 XXZ anisotropy parameter Delta) between a 3D ground state and a coplanar ground state for the XXZ model for values for the external magnetic field near to saturation, for values of the spin quantum number S <= 5. Although the CCM calculations are computationally intensive for this frustrated model, especially for high spin quantum numbers, our accurate new results certainly improve our understanding of it.
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. A crucial first step in applying the CCM to any such lattice quantum spin system is to perform a passive rotation of the local spin axes so that all spins in the model state appear mathematically to point in the same (say, downwards z-)direction. Whereas this process leads to terms with only real coefficients in the rotated Hamiltonian for coplanar model states, an additional complication arises for 3D model states where the corresponding coefficients can become complex-valued. We show here for the first time how high-order implementations of the CCM can be performed for such Hamiltonians. We explain in detail why the extension of the computational implementation of the CCM when going from coplanar to 3D model states is a non-trivial task that has not hitherto been undertaken. To illustrate these new developments, 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 XXZ antiferromagnet in the presence of an external magnetic field, for the cases 12≤S≤5\frac{1}{2} \le S \le 5 21​≤S≤5 . 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, and as we explicitly demonstrate in all three applications. Indeed, excellent correspondence is seen with the results of other methods, where they exist, for these systems. In particular, our 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 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 XXZ anisotropy parameter Δ\Delta Δ ) between a 3D ground state and a coplanar ground state for the XXZ model for values for the external magnetic field near to saturation, for values of the spin quantum number S≤5S \le 5S≤5 . Although the CCM calculations are computationally intensive for this frustrated model, especially for high spin quantum numbers, our accurate new results certainly improve our understanding of it.
The coupled cluster method (CCM) has previously been applied to study the ground- and excited-state properties of many different types of frustrated and unfrustrated quantum spin systems. A common feature in the application of the CCM is to rotate the local spin axes of the (often classical) model state so that (notationally only) the spins all appear to point in the downwards zzz-direction. Hitherto, we remark that only coplanar model states have been used because they do not lead to imaginary terms in the new Hamiltonian. By contrast, non-coplanar "three-dimensional" (3D) model states can lead to imaginary terms in the new Hamiltonian after rotation. In principle, however, macroscopic quantities predicted by the CCM (such as the ground-state energy and order parameter) should still be real (even though the Hamiltonian may be complex) because the transformations of local spin axes are unitary. Here we explain how we may use such 3D model states for the CCM and how we may solve for the (now possibly complex) CCM correlation coefficients. We present results for the spin-half triangular-lattice antiferromagnet in an external magnetic field. We use both coplanar model states and a single 3D non-coplanar model state. We calculate the ground-state energy and the lattice magnetisation as a function of the magnetic field strength. We find that the energies of the coplanar states lie lower than that of the non-coplanar state for all values of the external field, as seen in other approximate studies for this model. We find that the non-coplanar state does not detect the well-known spin plateau occurring in this model, although (as seen before) it is clearly observed for the coplanar model states using the CCM. These results are an excellent initial validation of the new approach for the application of the CCM using 3D model states.
A high-order coupled cluster method (CCM) for the zero-temperature ground states of quantum lattice spin systems of general spin quantum number (s> 1/2)(s>~1/2)(s> 1/2) is described and illustrated via the spin-1 anisotropic Heisenberg antiferromagnet on the square lattice. Results for the ground-state energy and sublattice magnetization are seen to be in excellent agreement with the best of other approximate methods. Furthermore, we are able to follow the solution to the CCM equations at a given level of approximation with respect to the anisotropy parameter $\ensuremath{\Delta}$ from the trivial Ising limit $(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Delta}}\ensuremath{\infty})$ down to a critical value ${\ensuremath{\Delta}}_{c},$ at which point the physical solution terminates. The results for the ground-state energy, sublattice magnetization, and phase transition point thus support our choice of this model as a good nontrivial test case of our CCM formalism.
Recent developments of high-order CCM have been to extend existing formalism and codes to s≥12s \ge \frac 12s≥21​ for both the ground and excited states. In this article, we describe how "generalised" expectation values for a wide range of one- and two-body spin operators may also be determined using existing the CCM code for the ground state. We present new results for the spin-spin correlation functions of the spin-half square- and triangular-lattice antiferromagnets by using the LSUBmmm approximation. We show that the absolute values of the spin-spin correlation functions ∣<s(0).s(r)>∣| < \bf{s}(0).\bf{s}(r) > |∣<s(0).s(r)>∣ converge with increasing approximation level for both lattices. We believe that the LSUBmmm approximation provides reasonable results for the correlation functions for lattice separations roughly of order r≈mr \approx mr≈m for the square lattice. We compare qualitatively our results for the square lattice to those results of quantum Monte Carlo (QMC) and we see that good correspondence is observed. Indeed as seen by QMC, the spin-spin correlation function initially decays strongly with ∣r∣|r|∣r∣ before becoming constant for larger values of ∣r∣|r|∣r∣. CCM results are also compared to results of exact diagonalisations for both lattices. ED results demonstrate a strong finite-size effects at lattice separations r=L/2r=L/2r=L/2 (where N=L×LN=L \times LN=L×L) for both lattices. The behaviour of the correlation function for the triangular lattice is qualitatively similar to that of the square lattice, namely, that it decays strongly at first before becoming constant. This is in keeping with the behaviour of both models, which are believed strongly to be Néel-ordered from approximate studies.
The coupled cluster method (CCM) of microscopic quantum many-body theory has become an ab initio method of first choice in quantum chemistry and many fields of nuclear, subnuclear and condensed matter physics, when results of high accuracy are required. In recent years it has begun to be applied with equal success to strongly correlated systems of electrons or quantum spins defined on a regular spatial lattice. One regularly finds that the CCM is able to describe accurately the various zero-temperature phases and the quantum phase transitions between them, even when frustration is present and other methods such as quantum Monte Carlo often fail. We illustrate the use and powerfulness of the method here by applying it to a square-lattice spin-half Heisenberg model where frustration is introduced by competing nearest neighbour bonds. The model exhibits the physically interesting phenomenon of competition between magnetic order and dimerization. Results obtained for the model with the CCM are compared with those found from spin-wave theory and from extrapolating the results of exact diagonalizations of small lattices. We show that the CCM is essentially unique among available methods in being able both to describe accurately all phases of this complex model and to provide accurate predictions of the various phase boundaries and the order of the corresponding transitions.
Quantum spins arranged on regular geometrical lattices, and interacting through a wide diversity of specified model Hamiltonians, provide a rich variety of magnetically ordered phases with many subtle properties. The zero-temperature quantum phase transitions between states of different order can be explored by the variation of internal control parameters within the interaction Hamiltonian. We illustrate how the systematic inclusion of multi-spin correlations for such lattice systems may be efficiently and effectively implemented via the coupled cluster method (CCM), which has been widely demonstrated on many previous occasions to be one of the most versatile methods of microscopic quantum many-body theory. Its power is illustrated by application to the spin-½ anisotropic Heisenberg (or XXZ) model on an infinite square lattice. The results obtained for such properties as the ground-state energy and sublattice magnetization are found to be as accurate as those obtained by any available alternative method. We show also how the CCM can now provide accurate ab initio estimates for the critical values of the parameters which mark the positions of phase boundaries between states of different quantum order, as well as for the critical indices associated with the singular behaviour near these transition points of various physical quantities.