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

arXiv:cond-mat/0407255 · Physical Review B

Higher order corrections to effective low-energy theories for strongly correlated electron systems

A. L. Chernyshev, D. Galanakis, Philip Phillips, A. V. Rozhkov, A.–M. S. Tremblay

Published Dec 7, 2004• 65 citations
arXivHTMLPDF

Abstract

Three well-known perturbative approaches to deriving low-energy effective theories, the degenerate Brillouin-Wigner perturbation theory (projection method), the canonical transformation, and the resolvent methods, are compared. We use the Hubbard model as an example to show how, to fourth order in hopping $t$, all methods lead to the same effective theory, namely the $t\text{\ensuremath{-}}J$ model with ring exchange and various correlated hoppings. We emphasize subtle technical difficulties that make such a derivation less trivial to carry out for orders higher than second. We also show that in higher orders, different approaches can lead to seemingly different forms for the low-energy Hamiltonian. All of these forms are equivalent since they are connected by an additional unitary transformation whose generator is given explicitly. The importance of transforming the operators is emphasized and the equivalence of their transformed structure within the different approaches is also demonstrated.

DOI
10.1103/physrevb.70.235111

Related papers

Mapping the Hubbard model to the t-J model using ground state unitary transformations

Yifan Tian

The effective low-energy models of the Hubbard model are usually derived from perturbation theory. Here we derive the effective model of the Hubbard model in spin space and t-J space using a unitary transformation from numerical optimization. We represent the Hamiltonian as Matrix product state(MPO) and represent the unitary transformation using gates according to tensor network methods. We obtain this unitary transformation by optimizing the unitary transformation between the ground state of the Hubbard model and the projection of the Hubbard model ground state into spin space and t-J space. The unitary transformation we get from numerical optimization yields effective models that are in line with perturbation theories. This numerical optimization method starting from ground state provides another approach to analyze effective low-energy models of strongly correlated electron systems.

arXiv:1908.03979•Aug 12, 2019•0 citations

Canonical perturbation expansion of the Hubbard model

K. A. Chao, Józef Spałek, Andrzej M. Oleś

We have incorporated the projection operators to the canonical transformation to derive an analytical infinite perturbation-expansion series. This canonical perturbation expansion (CPE) is valid if the unperturbed Hamiltonian ${H}_{0}$ and the perturbation ${H}_{1}$ can be expressed as ${H}_{0}=\ensuremath{\Sigma}{j}^{}{P}_{j}H{P}_{j}$ and ${H}_{1}=\ensuremath{\Sigma}{j\ensuremath{\ne}k}^{}{P}_{j}H{P}_{k}$, where ${P}_{j}$ is the projection operator corresponding to a group of closely spaced effective one-electron orbital energies ${E}_{j\ensuremath{\mu}}$ with $\ensuremath{\mu}=1, 2, \dots{}, {d}_{j}$, and if $|{E}_{j\ensuremath{\mu}}\ensuremath{-}{E}_{j\ensuremath{\nu}}|\ensuremath{\ll}|{E}_{j\ensuremath{\omega}}\ensuremath{-}{E}_{k\ensuremath{\delta}}|$ with $j\ensuremath{\ne}k$. We have shown that the CPE is equivalent to the time-dependent perturbation theory. An extremely simple effective Hamiltonian $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ is obtained when the CPE is applied to the $s$-band Hubbard model at the atomic limit. An explicit form of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ to the eighth order is given, and the magnetic interaction in $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ is of the form of Heisenberg exchange ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{j}$, including far neighbors. We then use this form to compute the antiferromagnetic groundstate energy to the seventh order. Our result is compared with other works.

Physical review. B, Condensed matter•Oct 1, 1978•199 citations

Effective Hamiltonian for cuprate superconductors

C. D. Batista, Armando A. Aligia

We compare numerical results of O and Cu occupations and optical and magnetic excitations for the three-band Hubbard model in a ${\mathrm{Cu}}_{4}$${\mathrm{O}}_{8}$ cluster, with those obtained with an effective Hamiltonian ${\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}$ in which the Cu-O hopping ${\mathit{t}}_{\mathit{p}\mathit{d}}$ is eliminated by means of a canonical transformation. This transformation up to order ${\mathit{t}}_{\mathit{p}\mathit{d}}^{4}$ is not quantitatively valid when the relevant energy differences are of the order of \ensuremath{\surd}8 ${\mathit{t}}_{\mathit{p}\mathit{d}}$ or less. However, an ${\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}$ with the form of the second-order one, but which retains all higher-order terms contained in a ${\mathrm{CuO}}_{4}$ (${\mathrm{Cu}}_{2}$${\mathrm{O}}_{7}$) cluster for zero or one added hole (one added electron) to the undoped system, leads to a very good agreement with the results using the full Hilbert space. This ${\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}$ makes possible the numerical study of the three-band Hubbard model for realistic parameters, using larger clusters.

Physical review. B, Condensed matter•Apr 1, 1993•43 citations

Low-energy theory of the t−t′−t″−U Hubbard model at half-filling: Interaction strengths in cuprate superconductors and an effective spin-only description of La2CuO4

Jean-Yves P. Delannoy, Michel J. P. Gingras, P. C. W. Holdsworth +1 more

Spin-only descriptions of the half-filled one-band Hubbard model are relevant for a wide range of Mott insulators. In addition to the usual Heisenberg exchange, many other types of interactions, including ring exchange, appear in the effective Hamiltonian in the intermediate coupling regime. In order to improve on the quantitative description of magnetic excitations in the insulating antiferromagnetic phase of copper-oxide (cuprate) materials, and to be consistent with band-structure calculations and photoemission experiments on these systems, we include second- and third-neighbor hopping parameters, ${t}^{\ensuremath{'}}$ and ${t}^{\ensuremath{''}}$, into the Hubbard Hamiltonian. A unitary transformation method is used to find systematically the effective Hamiltonian and any operator in the spin-only representation. The results include all closed four-hop electronic pathways in the canonical transformation. The method generates many ring exchange terms that play an important role in the comparison with experiments on ${\text{La}}_{2}{\text{CuO}}_{4}$. Performing a spin-wave analysis, we calculate the magnon dispersion as a function of $U$, $t$, ${t}^{\ensuremath{'}}$, and ${t}^{\ensuremath{''}}$. The four parameters are estimated by fitting the magnon dispersion to the experimental results of Coldea et al. [Phys. Rev. Lett. 86, 5377 (2001)] for ${\text{La}}_{2}{\text{CuO}}_{4}$. The ring exchange terms are found essential, in particular to determine the relative sign of ${t}^{\ensuremath{'}}$ and ${t}^{\ensuremath{''}}$, with the values found in good agreement with independent theoretical and experimental estimates for other members of the cuprate family. The zero-temperature sublattice magnetization is calculated using these parameters and also found to be in good agreement with the experimental value estimated by Lee et al. [Phys. Rev. B 60, 3643 (1999)]. We find a value of the interaction strength $U\ensuremath{\simeq}8t$ consistent with Mott insulating behavior.

arXiv:0808.3167•Physical Review B•Jun 24, 2009•57 citations

Operator Projection Method Applied to the Single-Particle Green's Function in the Hubbard Model

Shigeki Onoda, Masatoshi Imada

A new non-perturbative framework for many-body correlated systems is formulated by extending the operator projection method (OPM). This method offers a systematic expansion which enables us to project into the low-energy structure after extracting the higher-energy hierarchy. This method also opens a way to systematically take into account the effects of collective excitations. The Mott-Hubbard metal-insulator transition in the Hubbard model is studied by means of this projection beyond the second order by taking into account magnetic and charge fluctuations in the presence of the high-energy Mott-Hubbard structure. At half filling, the Mott-Hubbard gap is correctly eproduced between the separated two bands. Near half filling, a strongly renormalized low-energy single-particle excitations coexisting with the Mott-Hubbard bands are shown to appear. Signifcance of momentum-dependent self-energy in the results is stressed.

arXiv:cond-mat/0012376•Journal of the Physical Society of Japan•Mar 15, 2001•18 citations

Derivation of Effective Models using Self-Similar Continuous Unitary Transformations in Real Space

Alexander Reischl

In this thesis the method of self-similar continuous unitary transformations (CUTs) in real space is developed. It is applied to various models from the field of strongly correlated solid state physics and optical lattices. The CUT method maps a given Hamiltonian to an effective Hamiltonian which is in some respect simpler. In a self-similar CUT the truncation scheme relies on the structure of the operators. The real space approach put forward in this thesis truncates the operators according to their extension in real space. We mainly use a MKU generator to derive an effective model which is block-diagonal in the number of quasiparticles. We give a detailed description of the method. The transformation of the Hamiltonian and of the observables is explained. The implementation on a computer and the optimization of the programs are extensively discussed. The first application of the method deals with the fermionic Hubbard model. We derive an effective Hamiltonian, a generalized t-J-model, which conserves the number of doubly occupied sites. The effective model includes beyond the well-known Heisenberg term also further two-spin and four-spin terms. In addition, the charge motion and interaction within the effective model is addressed. In the context of the Hubbard model we discuss the question if an effective model that conserves the number of double occupancies can be derived in principle. The antiferromagnetic spin ladder and spin chain are treated from the viewpoint of a dimerized system. For the spin ladder we calculate the dispersion, the two-particle energies including bound states and the multi-particle continua. The discussion of multi-particle continua shows that the order of the energy levels is important for the convergence of the transformation. For the spin chain the dispersion is calculated. In addition, the transformation of an observable yields the corresponding spectral weights. The spectral weights are compared to results from a perturbative CUT. For the bosonic Hubbard model we calculate the dispersion and determine the phase diagram. The spectral weight of a certain observable is calculated in order to explain the outcome of a recent experiment that measures the spectroscopic response of atoms confined in optical traps. The distribution of spectral weight is analyzed also at finite temperatures. We find evidence for a substantial temperature of the size of the microscopic parameters. This finding is supported by purely thermodynamic considerations.

•Jan 1, 2006•1 citation
Three well-known perturbative approaches to deriving low-energy effective theories, the degenerate Brillouin-Wigner perturbation theory (projection method), the canonical transformation, and the resolvent methods, are compared. We use the Hubbard model as an example to show how, to fourth order in hopping ttt, all methods lead to the same effective theory, namely the t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model with ring exchange and various correlated hoppings. We emphasize subtle technical difficulties that make such a derivation less trivial to carry out for orders higher than second. We also show that in higher orders, different approaches can lead to seemingly different forms for the low-energy Hamiltonian. All of these forms are equivalent since they are connected by an additional unitary transformation whose generator is given explicitly. The importance of transforming the operators is emphasized and the equivalence of their transformed structure within the different approaches is also demonstrated.
The effective low-energy models of the Hubbard model are usually derived from perturbation theory. Here we derive the effective model of the Hubbard model in spin space and t-J space using a unitary transformation from numerical optimization. We represent the Hamiltonian as Matrix product state(MPO) and represent the unitary transformation using gates according to tensor network methods. We obtain this unitary transformation by optimizing the unitary transformation between the ground state of the Hubbard model and the projection of the Hubbard model ground state into spin space and t-J space. The unitary transformation we get from numerical optimization yields effective models that are in line with perturbation theories. This numerical optimization method starting from ground state provides another approach to analyze effective low-energy models of strongly correlated electron systems.
We have incorporated the projection operators to the canonical transformation to derive an analytical infinite perturbation-expansion series. This canonical perturbation expansion (CPE) is valid if the unperturbed Hamiltonian H0{H}_{0}H0​ and the perturbation H1{H}_{1}H1​ can be expressed as ${H}_{0}=\ensuremath{\Sigma}{j}^{}{P}_{j}H{P}_{j}$ and ${H}_{1}=\ensuremath{\Sigma}{j\ensuremath{\ne}k}^{}{P}_{j}H{P}_{k}$, where Pj{P}_{j}Pj​ is the projection operator corresponding to a group of closely spaced effective one-electron orbital energies ${E}_{j\ensuremath{\mu}}$ with $\ensuremath{\mu}=1, 2, \dots{}, {d}_{j}$, and if $|{E}_{j\ensuremath{\mu}}\ensuremath{-}{E}_{j\ensuremath{\nu}}|\ensuremath{\ll}|{E}_{j\ensuremath{\omega}}\ensuremath{-}{E}_{k\ensuremath{\delta}}|$ with $j\ensuremath{\ne}k$. We have shown that the CPE is equivalent to the time-dependent perturbation theory. An extremely simple effective Hamiltonian $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ is obtained when the CPE is applied to the sss-band Hubbard model at the atomic limit. An explicit form of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ to the eighth order is given, and the magnetic interaction in $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ is of the form of Heisenberg exchange ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{j}$, including far neighbors. We then use this form to compute the antiferromagnetic groundstate energy to the seventh order. Our result is compared with other works.
We compare numerical results of O and Cu occupations and optical and magnetic excitations for the three-band Hubbard model in a Cu4{\mathrm{Cu}}_{4}Cu4​O8{\mathrm{O}}_{8}O8​ cluster, with those obtained with an effective Hamiltonian Heff{\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}Heff​ in which the Cu-O hopping tpd{\mathit{t}}_{\mathit{p}\mathit{d}}tpd​ is eliminated by means of a canonical transformation. This transformation up to order tpd4{\mathit{t}}_{\mathit{p}\mathit{d}}^{4}tpd4​ is not quantitatively valid when the relevant energy differences are of the order of \ensuremath{\surd}8 tpd{\mathit{t}}_{\mathit{p}\mathit{d}}tpd​ or less. However, an Heff{\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}Heff​ with the form of the second-order one, but which retains all higher-order terms contained in a CuO4{\mathrm{CuO}}_{4}CuO4​ (Cu2{\mathrm{Cu}}_{2}Cu2​O7{\mathrm{O}}_{7}O7​) cluster for zero or one added hole (one added electron) to the undoped system, leads to a very good agreement with the results using the full Hilbert space. This Heff{\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}Heff​ makes possible the numerical study of the three-band Hubbard model for realistic parameters, using larger clusters.
Spin-only descriptions of the half-filled one-band Hubbard model are relevant for a wide range of Mott insulators. In addition to the usual Heisenberg exchange, many other types of interactions, including ring exchange, appear in the effective Hamiltonian in the intermediate coupling regime. In order to improve on the quantitative description of magnetic excitations in the insulating antiferromagnetic phase of copper-oxide (cuprate) materials, and to be consistent with band-structure calculations and photoemission experiments on these systems, we include second- and third-neighbor hopping parameters, ${t}^{\ensuremath{'}}$ and ${t}^{\ensuremath{''}}$, into the Hubbard Hamiltonian. A unitary transformation method is used to find systematically the effective Hamiltonian and any operator in the spin-only representation. The results include all closed four-hop electronic pathways in the canonical transformation. The method generates many ring exchange terms that play an important role in the comparison with experiments on La2CuO4{\text{La}}_{2}{\text{CuO}}_{4}La2​CuO4​. Performing a spin-wave analysis, we calculate the magnon dispersion as a function of UUU, ttt, ${t}^{\ensuremath{'}}$, and ${t}^{\ensuremath{''}}$. The four parameters are estimated by fitting the magnon dispersion to the experimental results of Coldea et al. [Phys. Rev. Lett. 86, 5377 (2001)] for La2CuO4{\text{La}}_{2}{\text{CuO}}_{4}La2​CuO4​. The ring exchange terms are found essential, in particular to determine the relative sign of ${t}^{\ensuremath{'}}$ and ${t}^{\ensuremath{''}}$, with the values found in good agreement with independent theoretical and experimental estimates for other members of the cuprate family. The zero-temperature sublattice magnetization is calculated using these parameters and also found to be in good agreement with the experimental value estimated by Lee et al. [Phys. Rev. B 60, 3643 (1999)]. We find a value of the interaction strength $U\ensuremath{\simeq}8t$ consistent with Mott insulating behavior.
A new non-perturbative framework for many-body correlated systems is formulated by extending the operator projection method (OPM). This method offers a systematic expansion which enables us to project into the low-energy structure after extracting the higher-energy hierarchy. This method also opens a way to systematically take into account the effects of collective excitations. The Mott-Hubbard metal-insulator transition in the Hubbard model is studied by means of this projection beyond the second order by taking into account magnetic and charge fluctuations in the presence of the high-energy Mott-Hubbard structure. At half filling, the Mott-Hubbard gap is correctly eproduced between the separated two bands. Near half filling, a strongly renormalized low-energy single-particle excitations coexisting with the Mott-Hubbard bands are shown to appear. Signifcance of momentum-dependent self-energy in the results is stressed.
In this thesis the method of self-similar continuous unitary transformations (CUTs) in real space is developed. It is applied to various models from the field of strongly correlated solid state physics and optical lattices. The CUT method maps a given Hamiltonian to an effective Hamiltonian which is in some respect simpler. In a self-similar CUT the truncation scheme relies on the structure of the operators. The real space approach put forward in this thesis truncates the operators according to their extension in real space. We mainly use a MKU generator to derive an effective model which is block-diagonal in the number of quasiparticles. We give a detailed description of the method. The transformation of the Hamiltonian and of the observables is explained. The implementation on a computer and the optimization of the programs are extensively discussed. The first application of the method deals with the fermionic Hubbard model. We derive an effective Hamiltonian, a generalized t-J-model, which conserves the number of doubly occupied sites. The effective model includes beyond the well-known Heisenberg term also further two-spin and four-spin terms. In addition, the charge motion and interaction within the effective model is addressed. In the context of the Hubbard model we discuss the question if an effective model that conserves the number of double occupancies can be derived in principle. The antiferromagnetic spin ladder and spin chain are treated from the viewpoint of a dimerized system. For the spin ladder we calculate the dispersion, the two-particle energies including bound states and the multi-particle continua. The discussion of multi-particle continua shows that the order of the energy levels is important for the convergence of the transformation. For the spin chain the dispersion is calculated. In addition, the transformation of an observable yields the corresponding spectral weights. The spectral weights are compared to results from a perturbative CUT. For the bosonic Hubbard model we calculate the dispersion and determine the phase diagram. The spectral weight of a certain observable is calculated in order to explain the outcome of a recent experiment that measures the spectroscopic response of atoms confined in optical traps. The distribution of spectral weight is analyzed also at finite temperatures. We find evidence for a substantial temperature of the size of the microscopic parameters. This finding is supported by purely thermodynamic considerations.