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:1209.6018

Two-particle Irreducible Effective Action Approach to Correlated Electron Systems

Wei‐jie Fu

Published Sep 26, 2012• 0 citations
arXivHTMLPDF

Abstract

The two-particle irreducible (2PI) effective action theories are employed to study the strongly fluctuating electron systems, under the formalism of the two-dimensional Hubbard model. We obtain the corresponding quantum 2PI effective action after the original classic action of the Hubbard model is bosonized. In our actual calculations, the 2PI effective action is expanded to three loops, in which the leading order (LO) and next-to-leading order (NLO) quantum fluctuations are included. Numerical calculations indicate that the NLO fluctuations should not be neglected when the Coulomb on-site repulsion energy is larger than two times the nearest-neighbor hopping energy.

DOI
10.48550/arxiv.1209.6018

Related papers

Extended dynamical mean field theory combined with the two-particle irreducible functional renormalization-group approach as a tool to study strongly correlated systems

A. A. Katanin

We propose an approach describing correlated electronic systems within the two-particle irreducible functional renormalization-group (2PI-fRG) approach, which uses initial self-energy and the two-particle irreducible vertices, obtained from (extended) dynamical mean-field theory [E(DMFT)]. Using the 2PI-fRG approach allows us to treat both local and nonlocal interactions. In the case of purely local interaction the corresponding equations have similar (although not identical) structure to the earlier developed DMFT+fRG $({\mathrm{DMF}}^{2}\mathrm{RG})$ approach. In a simplest truncation, neglecting scale dependence of the two-particle irreducible vertices, we reproduce the results for the two-particle vertices/susceptibilities in the ladder approximation of the dual-boson (DB) or the dynamic vertex approximation approach; in more sophisticated truncations the method allows us to consider nonlocal corrections to the self-energy as well as the interplay of charge and spin correlations. The proposed scheme is tested on the two-dimensional standard and extended $U\text{\ensuremath{-}}V$ half-filled Hubbard models. For the standard Hubbard model we obtain nonlocal self-energy, which is in agreement with numerical studies; for the extended Hubbard model we find the boundary of charge instability, which agrees well with the results of the DB approach. We also ascertain that the effect of spin correlations on electron interaction in the charge channel, not considered previously in the DB approach, only slightly reduces critical next-nearest-neighbor interaction of charge instability of the extended Hubbard model at the considered finite low temperature, yielding better agreement with dynamic cluster approximation. The considered method is rather general and can be applied to study various phenomena in strongly correlated electronic systems.

arXiv:1604.01702•Physical review. B./Physical review. B•Mar 8, 2019•25 citations

Effective action for superfluid Fermi systems in the strong-coupling limit

N. Dupuis

We derive the low-energy effective action for three-dimensional superfluid Fermi systems in the strong-coupling limit, where superfluidity originates from Bose-Einstein condensation of composite bosons. Taking into account density and pairing fluctuations on the same footing, we show that the effective action involves only the fermion density ${\ensuremath{\rho}}_{\mathbf{r}}$ and its conjugate variable, the phase ${\ensuremath{\theta}}_{\mathbf{r}}$ of the pairing order parameter ${\ensuremath{\Delta}}_{\mathbf{r}}$. We recover the standard action of a Bose superfluid of density ${\ensuremath{\rho}}_{\mathbf{r}}∕2$, where the bosons have a mass ${m}_{B}=2m$ and interact via a repulsive contact potential with amplitude ${g}_{B}=4\ensuremath{\pi}{a}_{B}∕{m}_{B},{a}_{B}=2a$ ($a$ the $s$-wave scattering length associated to the fermion-fermion interaction in vacuum). For lattice models, the derivation of the effective action is based on the mapping of the attractive Hubbard model onto the Heisenberg model in a uniform magnetic field, and a coherent state path integral representation of the partition function. The effective description of the Fermi superfluid in the strong-coupling limit is a Bose-Hubbard model with an intersite hopping amplitude ${t}_{B}=J∕2$ and an on-site repulsive interaction ${U}_{B}=2Jz$, where $J=4{t}^{2}∕U$ ($t$ and $\ensuremath{-}U$ are the intersite hopping amplitude and the on-site attraction in the (fermionic) Hubbard model, $z$ the number of nearest-neighbor sites).

arXiv:cond-mat/0412563•Physical Review A•Jul 11, 2005•5 citations

2PI effective action for the SYK model and tensor field theories

Dario Benedetti, Razvan Gurău

Abstract We discuss the two-particle irreducible (2PI) effective action for the SYK model and for tensor field theories. For the SYK model the 2PI effective action reproduces the bilocal reformulation of the model without using replicas. In general tensor field theories the 2PI formalism is the only way to obtain a bilocal reformulation of the theory, and as such is a precious instrument for the identification of soft modes and for possible holographic interpretations. We compute the 2PI action for several models, and push it up to fourth order in the 1/N expansion for the model proposed by Witten in [1], uncovering a one-loop structure in terms of an auxiliary bilocal action.

arXiv:1802.05500•Journal of High Energy Physics•May 1, 2018•55 citations

A STRONG-COUPLING EXPANSION FOR THE HUBBARD MODEL

N. Dupuis, Stéphane Pairault

We reconsider the strong-coupling expansion for the Hubbard model recently introduced by Sarker and Pairault et al. By introducing slave particles that act as projection operators onto the empty, singly occupied and doubly occupied atomic states, the perturbation theory around the atomic limit distinguishes between processes that do conserve or do not conserve the total number of doubly occupied sites. This allows for a systematic t/U expansion that does not break down at low temperature (t being the intersite hopping amplitude and U the local Coulomb repulsion). The fermionic field becomes a two-component field, which reflects the presence of the two Hubbard bands. The single-particle propagator is naturally expressed as a function of a 2×2 matrix self-energy. Furthermore, by introducing a time- and space-fluctuating spin-quantization axis in the functional integral, we can expand around a "non-degenerate" ground-state where each singly occupied site has a well defined spin direction (which may fluctuate in time). This formalism is used to derive the effective action of charge carriers in the lower Hubbard band to first order in t/U. We recover the action of the t–J model in the spin-hole coherent-state path integral. We also compare our results with those previously obtained by studying fluctuations around the large-U Hartree–Fock saddle point.

arXiv:cond-mat/0008447•International Journal of Modern Physics B•Sep 30, 2000•6 citations

Effective pairing interaction in the two-dimensional Hubbard model within a spin rotationally invariant approach

Vardan Apinyan, T. K. Kopeć

We implement the rotationally invariant formulation of the two-dimensional Hubbard model, with nearest-neighbors hopping $t$, which allows for the analytical study of the system in the low-energy limit. Both U(1) and SU(2) gauge transformations are used to factorize the charge and spin contributions to the original electron operator in terms of the corresponding gauge fields. The Hubbard Coulomb energy $U$ term is then expressed in terms of quantum phase variables conjugate to the local charge and variable spin-quantization axis, providing a useful representation of strongly correlated systems. It is shown that these gauge fields play a similar role as phonons in the BCS theory: they act as the ``glue'' for fermion pairing. By tracing out gauge degrees of freedom, the form of paired states is established and the strength of the pairing potential is determined. It is found that the attractive pairing potential in the effective low-energy fermionic action is nonzero in a rather narrow range of $U/t$.

arXiv:0812.1461•Physical Review B•Nov 13, 2008•2 citations

Nonequilibrium dynamics of condensates in a lattice with the two-particle-irreducible effective action in the 1∕N expansion

Kristan Temme, Thomas Gasenzer

The dynamical evolution of a Bose-Einstein condensate trapped in a one-dimensional lattice potential is investigated theoretically in the framework of the Bose-Hubbard model. The emphasis is set on the far-from-equilibrium evolution in a case where the gas is strongly interacting. This is realized by an appropriate choice of the parameters in the Hamiltonian, and by starting with an initial state, where one lattice well contains a Bose-Einstein condensate while all other wells are empty. Oscillations of the condensate as well as noncondensate fractions of the gas between the different sites of the lattice are found to be damped as a consequence of the collisional interactions between the atoms. Functional integral techniques involving self-consistently determined mean fields as well as two-point correlation functions are used to derive the two-particle-irreducible (2PI) effective action. The action is expanded in inverse powers of the number of field components $\mathcal{N}$, and the dynamic equations are derived from it to next-to-leading order in this expansion. This approach reaches considerably beyond the Hartree-Fock-Bogoliubov mean-field theory, and its results are compared to the exact quantum dynamics obtained by Rey et al. [Phys. Rev. A 69, 033610 (2004)] for small atom numbers.

arXiv:cond-mat/0607116•Physical Review A•Nov 2, 2006•16 citations
The two-particle irreducible (2PI) effective action theories are employed to study the strongly fluctuating electron systems, under the formalism of the two-dimensional Hubbard model. We obtain the corresponding quantum 2PI effective action after the original classic action of the Hubbard model is bosonized. In our actual calculations, the 2PI effective action is expanded to three loops, in which the leading order (LO) and next-to-leading order (NLO) quantum fluctuations are included. Numerical calculations indicate that the NLO fluctuations should not be neglected when the Coulomb on-site repulsion energy is larger than two times the nearest-neighbor hopping energy.
We propose an approach describing correlated electronic systems within the two-particle irreducible functional renormalization-group (2PI-fRG) approach, which uses initial self-energy and the two-particle irreducible vertices, obtained from (extended) dynamical mean-field theory [E(DMFT)]. Using the 2PI-fRG approach allows us to treat both local and nonlocal interactions. In the case of purely local interaction the corresponding equations have similar (although not identical) structure to the earlier developed DMFT+fRG (DMF2RG)({\mathrm{DMF}}^{2}\mathrm{RG})(DMF2RG) approach. In a simplest truncation, neglecting scale dependence of the two-particle irreducible vertices, we reproduce the results for the two-particle vertices/susceptibilities in the ladder approximation of the dual-boson (DB) or the dynamic vertex approximation approach; in more sophisticated truncations the method allows us to consider nonlocal corrections to the self-energy as well as the interplay of charge and spin correlations. The proposed scheme is tested on the two-dimensional standard and extended $U\text{\ensuremath{-}}V$ half-filled Hubbard models. For the standard Hubbard model we obtain nonlocal self-energy, which is in agreement with numerical studies; for the extended Hubbard model we find the boundary of charge instability, which agrees well with the results of the DB approach. We also ascertain that the effect of spin correlations on electron interaction in the charge channel, not considered previously in the DB approach, only slightly reduces critical next-nearest-neighbor interaction of charge instability of the extended Hubbard model at the considered finite low temperature, yielding better agreement with dynamic cluster approximation. The considered method is rather general and can be applied to study various phenomena in strongly correlated electronic systems.
We derive the low-energy effective action for three-dimensional superfluid Fermi systems in the strong-coupling limit, where superfluidity originates from Bose-Einstein condensation of composite bosons. Taking into account density and pairing fluctuations on the same footing, we show that the effective action involves only the fermion density ${\ensuremath{\rho}}_{\mathbf{r}}$ and its conjugate variable, the phase ${\ensuremath{\theta}}_{\mathbf{r}}$ of the pairing order parameter ${\ensuremath{\Delta}}_{\mathbf{r}}$. We recover the standard action of a Bose superfluid of density ${\ensuremath{\rho}}_{\mathbf{r}}∕2$, where the bosons have a mass mB=2m{m}_{B}=2mmB​=2m and interact via a repulsive contact potential with amplitude ${g}_{B}=4\ensuremath{\pi}{a}_{B}∕{m}_{B},{a}_{B}=2a$ (aaa the sss-wave scattering length associated to the fermion-fermion interaction in vacuum). For lattice models, the derivation of the effective action is based on the mapping of the attractive Hubbard model onto the Heisenberg model in a uniform magnetic field, and a coherent state path integral representation of the partition function. The effective description of the Fermi superfluid in the strong-coupling limit is a Bose-Hubbard model with an intersite hopping amplitude tB=J∕2{t}_{B}=J∕2tB​=J∕2 and an on-site repulsive interaction UB=2Jz{U}_{B}=2JzUB​=2Jz, where J=4t2∕UJ=4{t}^{2}∕UJ=4t2∕U (ttt and $\ensuremath{-}U$ are the intersite hopping amplitude and the on-site attraction in the (fermionic) Hubbard model, zzz the number of nearest-neighbor sites).
Abstract We discuss the two-particle irreducible (2PI) effective action for the SYK model and for tensor field theories. For the SYK model the 2PI effective action reproduces the bilocal reformulation of the model without using replicas. In general tensor field theories the 2PI formalism is the only way to obtain a bilocal reformulation of the theory, and as such is a precious instrument for the identification of soft modes and for possible holographic interpretations. We compute the 2PI action for several models, and push it up to fourth order in the 1/N expansion for the model proposed by Witten in [1], uncovering a one-loop structure in terms of an auxiliary bilocal action.
We reconsider the strong-coupling expansion for the Hubbard model recently introduced by Sarker and Pairault et al. By introducing slave particles that act as projection operators onto the empty, singly occupied and doubly occupied atomic states, the perturbation theory around the atomic limit distinguishes between processes that do conserve or do not conserve the total number of doubly occupied sites. This allows for a systematic t/U expansion that does not break down at low temperature (t being the intersite hopping amplitude and U the local Coulomb repulsion). The fermionic field becomes a two-component field, which reflects the presence of the two Hubbard bands. The single-particle propagator is naturally expressed as a function of a 2×2 matrix self-energy. Furthermore, by introducing a time- and space-fluctuating spin-quantization axis in the functional integral, we can expand around a "non-degenerate" ground-state where each singly occupied site has a well defined spin direction (which may fluctuate in time). This formalism is used to derive the effective action of charge carriers in the lower Hubbard band to first order in t/U. We recover the action of the t–J model in the spin-hole coherent-state path integral. We also compare our results with those previously obtained by studying fluctuations around the large-U Hartree–Fock saddle point.
We implement the rotationally invariant formulation of the two-dimensional Hubbard model, with nearest-neighbors hopping ttt, which allows for the analytical study of the system in the low-energy limit. Both U(1) and SU(2) gauge transformations are used to factorize the charge and spin contributions to the original electron operator in terms of the corresponding gauge fields. The Hubbard Coulomb energy UUU term is then expressed in terms of quantum phase variables conjugate to the local charge and variable spin-quantization axis, providing a useful representation of strongly correlated systems. It is shown that these gauge fields play a similar role as phonons in the BCS theory: they act as the ``glue'' for fermion pairing. By tracing out gauge degrees of freedom, the form of paired states is established and the strength of the pairing potential is determined. It is found that the attractive pairing potential in the effective low-energy fermionic action is nonzero in a rather narrow range of U/tU/tU/t.
The dynamical evolution of a Bose-Einstein condensate trapped in a one-dimensional lattice potential is investigated theoretically in the framework of the Bose-Hubbard model. The emphasis is set on the far-from-equilibrium evolution in a case where the gas is strongly interacting. This is realized by an appropriate choice of the parameters in the Hamiltonian, and by starting with an initial state, where one lattice well contains a Bose-Einstein condensate while all other wells are empty. Oscillations of the condensate as well as noncondensate fractions of the gas between the different sites of the lattice are found to be damped as a consequence of the collisional interactions between the atoms. Functional integral techniques involving self-consistently determined mean fields as well as two-point correlation functions are used to derive the two-particle-irreducible (2PI) effective action. The action is expanded in inverse powers of the number of field components N\mathcal{N}N, and the dynamic equations are derived from it to next-to-leading order in this expansion. This approach reaches considerably beyond the Hartree-Fock-Bogoliubov mean-field theory, and its results are compared to the exact quantum dynamics obtained by Rey et al. [Phys. Rev. A 69, 033610 (2004)] for small atom numbers.