Physical Review A

Commutator and Anticommutator Green's Functions, Zero-Frequency Poles, and Long-Time Correlations

Philip E. Bloomfield, N. Nafari

Published Feb 1, 1972• 17 citations

Abstract

Previous studies of the commutator-type Green's functions are extended to the anticommutator type. The existence of an $\ensuremath{\omega}=0$ pole in the latter but not former case is demonstrated. Its residue is simply related to the long-time behavior of the corresponding correlation function.

DOI: 10.1103/physreva.5.806

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Physical Review A

Commutator and Anticommutator Green's Functions, Zero-Frequency Poles, and Long-Time Correlations

Philip E. Bloomfield, N. Nafari

Published Feb 1, 1972• 17 citations

Publisher

Abstract

Previous studies of the commutator-type Green's functions are extended to the anticommutator type. The existence of an

\ensuremath{\omega}=0

pole in the latter but not former case is demonstrated. Its residue is simply related to the long-time behavior of the corresponding correlation function.

DOI: 10.1103/physreva.5.806

Related papers

Note on the zero-frequency pole in Green's functions

J F Fernandez, H. A. Gersch

A simple method to treat consistently the zero-frequency pole in the spectral representation of the correlation functions in equilibrium statistical mechanics is presented.

Proceedings of the Physical SocietyJun 1, 196718 citations

Correlation functions and Green functions: zero-frequency anomalies

P C Kwok, T D Schultz

Journal of Physics C Solid State PhysicsJul 1, 196943 citations

Commutator and Anticommutator Green Functions in the Ising Model with Transverse Field

V. L. Aksionov, Henryk Konwent, Juergen Schreiber

physica status solidi (b)Jul 1, 19781 citation

Absence of long-range order in an antiferromagnetic chain with long-range interactions: Green’s function approach

Ren-Gui Zhu, An Min Wang

The antiferromagnetic spin- $1∕2$ chain with power-law decaying interactions $\ensuremath{\mid}{J}_{n}\ensuremath{\mid}\ensuremath{\propto}1∕{n}^{p}$ is studied by a Green's function method. The dispersion relation ${\ensuremath{\omega}}_{k}$ and an expression for the critical temperature ${T}_{N}$ as a function of $p$ are presented. By analyzing the small- $k$ behavior of ${\ensuremath{\omega}}_{k}$, it is shown that there will be no long-range order at finite temperature when $p\ensuremath{\geqslant}2$, which is in good agreement with Bruno's rigorous proof. Our work refurbishes a previous work also by Green's function method, and validates the Tyablikov decoupling approximation to long-range interactions with $1<p<2$ .The antiferromagnetic spin-$1∕2$ chain with power-law decaying interactions $\ensuremath{\mid}{J}_{n}\ensuremath{\mid}\ensuremath{\propto}1∕{n}^{p}$ is studied by a Green's function method. The dispersion relation ${\ensuremath{\omega}}_{k}$ and an expression for the critical temperature ${T}_{N}$ as a function of $p$ are presented. By analyzing the small-$k$ behavior of ${\ensuremath{\omega}}_{k}$, it is shown that there will be no long-range order at finite temperature when $p\ensuremath{\geqslant}2$, which is in good agreement with Bruno's rigorous proof. Our work refurbishes a previous work also by Green's function method, and validates the Tyablikov decoupling approximation to long-range interactions with $1<p<2$.

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Recent attempts to develop double-time Green's-function theories for the transverse Ising model (TIM) have been subject to certain difficulties; the appearance of zero-frequency poles in the commutator Green's function (CGF) led to the avoidance of CGF and an incomplete treatment based only upon anticommutator Green's functions (AGF), and ambiguities in the determination of some thermal averages. Upon decoupling the equations-of-motion hierarchy, we develop a consistent Green's-function theory of the TIM. The AGF and CGF are shown to provide consistent results, the problem of zero-frequency poles in CGF being treated appropriately. We resolve the ambiguity problem and obtain reasonable values for the Weiss critical field and the ground-state magnetization.Recent attempts to develop double-time Green's-function theories for the transverse Ising model (TIM) have been subject to certain difficulties; the appearance of zero-frequency poles in the commutator Green's function (CGF) led to the avoidance of CGF and an incomplete treatment based only upon anticommutator Green's functions (AGF), and ambiguities in the determination of some thermal averages. Upon decoupling the equations-of-motion hierarchy, we develop a consistent Green's-function theory of the TIM. The AGF and CGF are shown to provide consistent results, the problem of zero-frequency poles in CGF being treated appropriately. We resolve the ambiguity problem and obtain reasonable values for the Weiss critical field and the ground-state magnetization.

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Andrea Blason, Michele Fabrizio

Green's function zeros, which can emerge only if correlation is strong, have been for long overlooked and believed to be devoid of any physical meaning, unlike Green's function poles. Here, we prove that Green's function zeros instead contribute on the same footing as poles to determine the topological character of an insulator. The key to the proof, worked out explicitly in two dimensions but easily extendable in three dimensions, is to express the topological invariant in terms of a quasiparticlethermal Green's function matrix ${G}_{*}(i\ensuremath{\epsilon},\mathbf{k})=1/[i\ensuremath{\epsilon}\ensuremath{-}{H}_{*}(\ensuremath{\epsilon},\mathbf{k})]$, with Hermitian ${H}_{*}(\ensuremath{\epsilon},\mathbf{k})$, by filtering out the positive-definite quasiparticle residue. In that way, the topological invariant is easily found to reduce to the Thouless, Kohmoto, Nightingale, and den Nijs formula for quasiparticles described by the noninteracting Hamiltonian ${H}_{*}(0,\mathbf{k})$ . Since the poles of the quasiparticle Green's function ${G}_{*}(\ensuremath{\epsilon},\mathbf{k})$ on the real frequency axis correspond to poles and zeros of the physical-particle Green's function $G(\ensuremath{\epsilon},\mathbf{k})$, both of them equally determine the topological character of an insulator.Green's function zeros, which can emerge only if correlation is strong, have been for long overlooked and believed to be devoid of any physical meaning, unlike Green's function poles. Here, we prove that Green's function zeros instead contribute on the same footing as poles to determine the topological character of an insulator. The key to the proof, worked out explicitly in two dimensions but easily extendable in three dimensions, is to express the topological invariant in terms of a quasiparticlethermal Green's function matrix ${G}_{*}(i\ensuremath{\epsilon},\mathbf{k})=1/[i\ensuremath{\epsilon}\ensuremath{-}{H}_{*}(\ensuremath{\epsilon},\mathbf{k})]$, with Hermitian ${H}_{*}(\ensuremath{\epsilon},\mathbf{k})$, by filtering out the positive-definite quasiparticle residue. In that way, the topological invariant is easily found to reduce to the Thouless, Kohmoto, Nightingale, and den Nijs formula for quasiparticles described by the noninteracting Hamiltonian ${H}_{*}(0,\mathbf{k})$. Since the poles of the quasiparticle Green's function ${G}_{*}(\ensuremath{\epsilon},\mathbf{k})$ on the real frequency axis correspond to poles and zeros of the physical-particle Green's function $G(\ensuremath{\epsilon},\mathbf{k})$, both of them equally determine the topological character of an insulator.

arXiv:2304.08180Physical review. B./Physical review. BSep 11, 202340 citations

Previous studies of the commutator-type Green's functions are extended to the anticommutator type. The existence of an

\ensuremath{\omega}=0

pole in the latter but not former case is demonstrated. Its residue is simply related to the long-time behavior of the corresponding correlation function.

The antiferromagnetic spin-

1∕2

chain with power-law decaying interactions $\ensuremath{\mid}{J}_{n}\ensuremath{\mid}\ensuremath{\propto}1∕{n}^{p}$ is studied by a Green's function method. The dispersion relation ${\ensuremath{\omega}}_{k}$ and an expression for the critical temperature

{T}_{N}

as a function of

p

are presented. By analyzing the small-

k

behavior of ${\ensuremath{\omega}}_{k}$, it is shown that there will be no long-range order at finite temperature when $p\ensuremath{\geqslant}2$, which is in good agreement with Bruno's rigorous proof. Our work refurbishes a previous work also by Green's function method, and validates the Tyablikov decoupling approximation to long-range interactions with

1<p<2

{H}_{*}(0,\mathbf{k})

. Since the poles of the quasiparticle Green's function ${G}_{*}(\ensuremath{\epsilon},\mathbf{k})$ on the real frequency axis correspond to poles and zeros of the physical-particle Green's function $G(\ensuremath{\epsilon},\mathbf{k})$, both of them equally determine the topological character of an insulator.