Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Solid state

Infinite-order cumulant expansion for spins

Michele Parrinello, Tadashi Arai

Published Jul 1, 1974• 41 citations
Publisher

Abstract

The cumulant expansion for spins developed previously is rearranged and extended up to an infinite order in perturbation. A pair of spin-deviation lines is considered as a propagator and, as an additional propagator appears and interacts with the original one, that part of the diagram is defined as a self-energy correction. All cumulant corrections are decomposed and added to the corresponding self-energy corrections, so that self-energy corrections and their cumulant corrections are automatically calculated together exactly. The contribution of a self-energy correction to a diagram depends on the geometry of the incoming and outgoing spin-deviation lines of the self-energy diagram, yielding a new type of perturbation expansion. Numerical results suggest that the nature of the spin-spin correlation in the spin-1/2 case is distinct from that in other cases and that spins behave like fermions rather than bosons in this limit.

DOI
10.1103/physrevb.10.265

Related papers

Cumulant Expansion and Wick's Theorem for Spins. Application to the Antiferromagnetic Ground State

Tadashi Arai, Bernard Albert Goodman

A form of Wick's theorem is derived which is applicable to spin operators of arbitrary magnitude in a manner analogous to the decomposition of fermion and boson operator products. Use of the theorem together with Kubo's cumulant rearrangement of perturbation theory leads to a compact prescription for the calculation of the ground-state energy of spin systems. This method is parallel to Goldstone's linked-cluster expansion for fermions, but the expansion for spins, as well as for bosons, contains cumulant correction terms. The Green functions are shown to be expanded similarly in terms of cumulants. The method is illustrated by the calculation of the antiferromagnetic ground state, and circumvents the involved development in the previous work of Davis and Boon. It is found that our procedure gives better convergence than Davis's expansion and that Boon's final formulas as well as his numerical results contain some errors.

Physical Review•Mar 10, 1967•20 citations

Many-body perturbation expansions without diagrams. I. Normal states

Behnam Farid

On the basis of an exact perturbational expression for the interacting one-particle Green function $G$ corresponding to bosons / fermions in terms of the bare interaction potential $v$ and permanents / determinants of the non-interacting one-particle Green function $G_0$, we deduce four recursive perturbation expansions for the self-energy $Σ$. With $W$ denoting the dynamic screened interaction potential, these perturbation expansions are identical to those of $Σ$ in terms of (i) self-energy diagrams and $(v, G_0)$, (ii) $G$-skeleton self-energy diagrams and $(v, G)$, (iii) $W$-skeleton self-energy diagrams and $(W, G_0)$, and (iv) $G$- and $W$-skeleton self-energy diagrams and $(W,G)$. For the calculation of $W$, we rely on a similar exact perturbational expression for the interacting two-particle Green function $G_2$ as for $G$. From this expression, we deduce four recursive perturbation expansions for the polarization function $P$, necessary for the calculation of $W$, that are similar to those for $Σ$ specified above. The correlation functions considered in this paper may be corresponding to ground states, and thermal ensemble of states. For thermal ensemble of states, we consider both the imaginary-time formalism of Matsubara, and the real-time formalism of thermo-field dynamics (TFD). The latter is advantageous for the direct calculation of dynamic correlation functions. In an appendix, we apply the formalisms presented in this paper to the Hubbard Hamiltonian for spin-$\tfrac{1}{2}$ fermions on a lattice in arbitrary $d$ spatial dimensions. In two further appendices, we present methods and short programs for determining the $ν$th-order diagrams corresponding to the perturbation expansions of $G$ in terms of $(v,G_0)$, and $Σ$ in terms of $(v,G_0)$ and $(v,G)$ on the basis of the cycle decompositions of the elements of the symmetric group $S_{2ν}$.

arXiv:1912.00474•Dec 1, 2019•2 citations

The cumulant expansion in renormalization group transformations: the convergence question and the dependence upon cell size

Per Chr. Hemmer, Manuel G. Velárde

The Niemeijer-van Leeuwen (1973) cumulant expansion for calculation of renormalized interactions is carried to third order for the Ising model on the triangular lattice. Kadanoff cells containing three spins are used. Five types of pair interactions and three types of four-spin interaction enter the calculations. The renormalization equations have one non-trivial fixed point. The accuracy of the result for the corresponding relevant eigenvalues is, however, less than in second order. The idea that the cumulant expansion is an asymptotic expansion has thus been given numerical support.

Journal of Physics A Mathematical and General•Oct 1, 1976•6 citations

Loop expansion around the Bethe approximation through the M -layer construction

Ada Altieri, Maria Chiara Angelini, Carlo Lucibello +3 more

Abstract For every physical model defined on a generic graph or factor graph, the Bethe M -layer construction allows building a different model for which the Bethe approximation is exact in the large M limit, and coincides with the original model for M = 1 . The 1 / M perturbative series is then expressed by a diagrammatic loop expansion in terms of so-called fat diagrams. Our motivation is to study some important second-order phase transitions that do exist on the Bethe lattice, but are either qualitatively different or absent in the corresponding fully connected case. In this case, the standard approach based on a perturbative expansion around the naive mean field theory (essentially a fully connected model) fails. On physical grounds, we expect that when the construction is applied to a lattice in finite dimension there is a small region of the external parameters, close to the Bethe critical point, where strong deviations from mean-field behavior will be observed. In this region, the 1 / M expansion for the corrections diverges, and can be the starting point for determining the correct non-mean-field critical exponents using renormalization group arguments. In the end, we will show that the critical series for the generic observable can be expressed as a sum of Feynman diagrams with the same numerical prefactors of field theories. However, the contribution of a given diagram is not evaluated by associating Gaussian propagators to its lines, as in field theories: one has to consider the graph as a portion of the original lattice, replacing the internal lines with appropriate one-dimensional chains, and attaching to the internal points the appropriate number of infinite-size Bethe trees to restore the correct local connectivity of the original model. The actual contribution of each (fat) diagram is the so-called line-connected observable, which also includes contributions from sub-diagrams with appropriate prefactors. In order to compute the corrections near to the critical point, Feynman diagrams (with their symmetry factors) can be read directly from the appropriate field-theoretical literature; the computation of momentum integrals is also quite similar; the extra work consists of computing the line-connected observable of the associated fat diagram in the limit of all lines becoming infinitely long.

arXiv:1707.08499•Journal of Statistical Mechanics Theory and Experiment•Nov 1, 2017•27 citations

Diagrammatic Expansion for the Ising Model with Arbitrary Spin and Range of Interaction

Gerald Horwitz, Herbert B. Callen

A new and simple diagrammatic expansion is developed for the free energy of the Ising model with arbitrary spin and range of interaction. The contribution of each diagram is a product of (1) the reciprocal of the order of the symmetry group of the diagram, (2) a product of semi-invariants, with a factor ${M}_{n}$ for each spin in the diagram $n$ being the number of bonds joined to that spin) and (3) a sum of products $\ensuremath{\Sigma}{i,j,k,\ensuremath{\cdots}}^{}(2\ensuremath{\beta}{J}_{\mathrm{ij}})(2\ensuremath{\beta}{J}_{\mathrm{kl}})\ensuremath{\cdots}$, where $2\ensuremath{\beta}{J}_{\mathrm{ij}}$ corresponds to a bond between spins $i$ and $j$, and where the summation is carried out with no restrictions on the summation indices (i.e., no "excluded volume" corrections). The expansion variables are the spin deviation operators ${\ensuremath{\sigma}}_{i}\ensuremath{\equiv}\overline{S}\ensuremath{-}{S}_{\mathrm{iz}}$. The quantity $\overline{S}$ is chosen to eliminate a large class of diagrams; this choice also minimizes the corresponding free-energy contribution and implies $\overline{S}=〈{S}_{\mathrm{iz}}〉$. By further renormalization of the vertices all reducible (i.e., simply articulated) diagrams are eliminated. To zero order the molecular field solution is obtained. The next approximation consists of the summation of renormalized simple loop diagrams. The justification of this choice of diagrams rests, at low temperature, on the decrease of the value of the higher order semi-invariants, on the relationship of these loop diagrams to the random phase approximation, and on the agreement with the rigorous low-temperature series result. At the Curie temperature the same choice of diagrams is justified by a modification of Brout's $\frac{1}{z}$ criterion, so that the expansion can be viewed as an expansion in $\frac{1}{z}$, where $z$ is the effective number of spins interacting with a given spin. Finally the choice of loop diagrams is justified at high temperature by exact agreement to second order in $\frac{{T}_{c}}{T}$ (and by a very close agreement to fourth order) with the rigorous high-temperature series expansion. Thus the theory agrees with rigorous results in the low-temperature and paramagnetic regions and has a well-defined criterion of validity in the intermediate temperature region. For nearest-neighbor interactions and spin \textonehalf{} the specific heat is continuous through the Curie temperature, with an infinite slope on the low-temperature side of the transition.

Physical Review•Dec 15, 1961•174 citations

Determining the validity of cumulant expansions for central spin models

Piper Fowler-Wright, Kristín B. Arnardóttir, Peter Kirton +2 more

For a model with many-to-one connectivity it is widely expected that mean-field theory captures the exact many-particle $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, and that higher-order cumulant expansions of the Heisenberg equations converge to this same limit whilst providing improved approximations at finite $N$. Here we show that this is in fact not always the case. Instead, whether mean-field theory correctly describes the large-$N$ limit depends on how the model parameters scale with $N$, and the convergence of cumulant expansions may be nonuniform across even and odd orders. Further, even when a higher-order cumulant expansion does recover the correct limit, the error is not monotonic with $N$ and may exceed that of mean-field theory.

arXiv:2303.04410•Physical Review Research•Sep 1, 2023•21 citations
The cumulant expansion for spins developed previously is rearranged and extended up to an infinite order in perturbation. A pair of spin-deviation lines is considered as a propagator and, as an additional propagator appears and interacts with the original one, that part of the diagram is defined as a self-energy correction. All cumulant corrections are decomposed and added to the corresponding self-energy corrections, so that self-energy corrections and their cumulant corrections are automatically calculated together exactly. The contribution of a self-energy correction to a diagram depends on the geometry of the incoming and outgoing spin-deviation lines of the self-energy diagram, yielding a new type of perturbation expansion. Numerical results suggest that the nature of the spin-spin correlation in the spin-1/2 case is distinct from that in other cases and that spins behave like fermions rather than bosons in this limit.
A form of Wick's theorem is derived which is applicable to spin operators of arbitrary magnitude in a manner analogous to the decomposition of fermion and boson operator products. Use of the theorem together with Kubo's cumulant rearrangement of perturbation theory leads to a compact prescription for the calculation of the ground-state energy of spin systems. This method is parallel to Goldstone's linked-cluster expansion for fermions, but the expansion for spins, as well as for bosons, contains cumulant correction terms. The Green functions are shown to be expanded similarly in terms of cumulants. The method is illustrated by the calculation of the antiferromagnetic ground state, and circumvents the involved development in the previous work of Davis and Boon. It is found that our procedure gives better convergence than Davis's expansion and that Boon's final formulas as well as his numerical results contain some errors.
On the basis of an exact perturbational expression for the interacting one-particle Green function GGG corresponding to bosons / fermions in terms of the bare interaction potential vvv and permanents / determinants of the non-interacting one-particle Green function G0G_0G0​, we deduce four recursive perturbation expansions for the self-energy ΣΣΣ. With WWW denoting the dynamic screened interaction potential, these perturbation expansions are identical to those of ΣΣΣ in terms of (i) self-energy diagrams and (v,G0)(v, G_0)(v,G0​), (ii) GGG-skeleton self-energy diagrams and (v,G)(v, G)(v,G), (iii) WWW-skeleton self-energy diagrams and (W,G0)(W, G_0)(W,G0​), and (iv) GGG- and WWW-skeleton self-energy diagrams and (W,G)(W,G)(W,G). For the calculation of WWW, we rely on a similar exact perturbational expression for the interacting two-particle Green function G2G_2G2​ as for GGG. From this expression, we deduce four recursive perturbation expansions for the polarization function PPP, necessary for the calculation of WWW, that are similar to those for ΣΣΣ specified above. The correlation functions considered in this paper may be corresponding to ground states, and thermal ensemble of states. For thermal ensemble of states, we consider both the imaginary-time formalism of Matsubara, and the real-time formalism of thermo-field dynamics (TFD). The latter is advantageous for the direct calculation of dynamic correlation functions. In an appendix, we apply the formalisms presented in this paper to the Hubbard Hamiltonian for spin-12\tfrac{1}{2}21​ fermions on a lattice in arbitrary ddd spatial dimensions. In two further appendices, we present methods and short programs for determining the νννth-order diagrams corresponding to the perturbation expansions of GGG in terms of (v,G0)(v,G_0)(v,G0​), and ΣΣΣ in terms of (v,G0)(v,G_0)(v,G0​) and (v,G)(v,G)(v,G) on the basis of the cycle decompositions of the elements of the symmetric group S2νS_{2ν}S2ν​.
The Niemeijer-van Leeuwen (1973) cumulant expansion for calculation of renormalized interactions is carried to third order for the Ising model on the triangular lattice. Kadanoff cells containing three spins are used. Five types of pair interactions and three types of four-spin interaction enter the calculations. The renormalization equations have one non-trivial fixed point. The accuracy of the result for the corresponding relevant eigenvalues is, however, less than in second order. The idea that the cumulant expansion is an asymptotic expansion has thus been given numerical support.
Abstract For every physical model defined on a generic graph or factor graph, the Bethe M -layer construction allows building a different model for which the Bethe approximation is exact in the large M limit, and coincides with the original model for M = 1 . The 1 / M perturbative series is then expressed by a diagrammatic loop expansion in terms of so-called fat diagrams. Our motivation is to study some important second-order phase transitions that do exist on the Bethe lattice, but are either qualitatively different or absent in the corresponding fully connected case. In this case, the standard approach based on a perturbative expansion around the naive mean field theory (essentially a fully connected model) fails. On physical grounds, we expect that when the construction is applied to a lattice in finite dimension there is a small region of the external parameters, close to the Bethe critical point, where strong deviations from mean-field behavior will be observed. In this region, the 1 / M expansion for the corrections diverges, and can be the starting point for determining the correct non-mean-field critical exponents using renormalization group arguments. In the end, we will show that the critical series for the generic observable can be expressed as a sum of Feynman diagrams with the same numerical prefactors of field theories. However, the contribution of a given diagram is not evaluated by associating Gaussian propagators to its lines, as in field theories: one has to consider the graph as a portion of the original lattice, replacing the internal lines with appropriate one-dimensional chains, and attaching to the internal points the appropriate number of infinite-size Bethe trees to restore the correct local connectivity of the original model. The actual contribution of each (fat) diagram is the so-called line-connected observable, which also includes contributions from sub-diagrams with appropriate prefactors. In order to compute the corrections near to the critical point, Feynman diagrams (with their symmetry factors) can be read directly from the appropriate field-theoretical literature; the computation of momentum integrals is also quite similar; the extra work consists of computing the line-connected observable of the associated fat diagram in the limit of all lines becoming infinitely long.
A new and simple diagrammatic expansion is developed for the free energy of the Ising model with arbitrary spin and range of interaction. The contribution of each diagram is a product of (1) the reciprocal of the order of the symmetry group of the diagram, (2) a product of semi-invariants, with a factor Mn{M}_{n}Mn​ for each spin in the diagram nnn being the number of bonds joined to that spin) and (3) a sum of products $\ensuremath{\Sigma}{i,j,k,\ensuremath{\cdots}}^{}(2\ensuremath{\beta}{J}_{\mathrm{ij}})(2\ensuremath{\beta}{J}_{\mathrm{kl}})\ensuremath{\cdots}$, where $2\ensuremath{\beta}{J}_{\mathrm{ij}}$ corresponds to a bond between spins iii and jjj, and where the summation is carried out with no restrictions on the summation indices (i.e., no "excluded volume" corrections). The expansion variables are the spin deviation operators ${\ensuremath{\sigma}}_{i}\ensuremath{\equiv}\overline{S}\ensuremath{-}{S}_{\mathrm{iz}}$. The quantity S‾\overline{S}S is chosen to eliminate a large class of diagrams; this choice also minimizes the corresponding free-energy contribution and implies S‾=〈Siz〉\overline{S}=〈{S}_{\mathrm{iz}}〉S=〈Siz​〉. By further renormalization of the vertices all reducible (i.e., simply articulated) diagrams are eliminated. To zero order the molecular field solution is obtained. The next approximation consists of the summation of renormalized simple loop diagrams. The justification of this choice of diagrams rests, at low temperature, on the decrease of the value of the higher order semi-invariants, on the relationship of these loop diagrams to the random phase approximation, and on the agreement with the rigorous low-temperature series result. At the Curie temperature the same choice of diagrams is justified by a modification of Brout's 1z\frac{1}{z}z1​ criterion, so that the expansion can be viewed as an expansion in 1z\frac{1}{z}z1​, where zzz is the effective number of spins interacting with a given spin. Finally the choice of loop diagrams is justified at high temperature by exact agreement to second order in TcT\frac{{T}_{c}}{T}TTc​​ (and by a very close agreement to fourth order) with the rigorous high-temperature series expansion. Thus the theory agrees with rigorous results in the low-temperature and paramagnetic regions and has a well-defined criterion of validity in the intermediate temperature region. For nearest-neighbor interactions and spin \textonehalf{} the specific heat is continuous through the Curie temperature, with an infinite slope on the low-temperature side of the transition.
For a model with many-to-one connectivity it is widely expected that mean-field theory captures the exact many-particle $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, and that higher-order cumulant expansions of the Heisenberg equations converge to this same limit whilst providing improved approximations at finite NNN. Here we show that this is in fact not always the case. Instead, whether mean-field theory correctly describes the large-NNN limit depends on how the model parameters scale with NNN, and the convergence of cumulant expansions may be nonuniform across even and odd orders. Further, even when a higher-order cumulant expansion does recover the correct limit, the error is not monotonic with NNN and may exceed that of mean-field theory.