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

Ward-Takahashi relations for SO(4) symmetry in the Hubbard model

Atsushi Masumizu, Kiyoshi Sogo

Published Sep 12, 2005• 7 citations
Publisher

Abstract

Ward-Takahashi relations for spin symmetry $\mathrm{SU}{(2)}_{\mathrm{S}}$ and pseudospin symmetry $\mathrm{SU}{(2)}_{\mathrm{C}}$ are derived for the simple Hubbard model in arbitrary dimensions. For this purpose we generally discuss local SO(4) symmetry in terms of Clifford algebra, and employ a gauge transformed form of the original Hubbard Hamiltonian, which makes the Ward-Takahashi relations much simpler. Derived Ward-Takahashi relations are rewritten as the relations between self-energy and vertex part to discuss spontaneously broken symmetry and Fermi liquid property of the system for zero temperature magnetism.

DOI
10.1103/physrevb.72.115107

Related papers

SO(4) Symmetry of the Hubbard Model and its Experimental Consequences

Shoucheng Zhang

The one band Hubbard model on a bi-partite lattice possess a global SO(4) symmetry group which is a direct product of the spin SU(2) and the pseudospin SU(2) symmetry group. Recently, the group theorectical consequence of this symmetry property has been investigated systematically. It is shown that this symmetry property can be used to reduce the complexity of diagonalizing the Hubbard Hamiltonian, and classify all the eigenstates according to the representation of the SO(4) group. It also leads to an exact one-to-one corespondence of the eigenstates at half-filling to that away from half-filling. Most importantly however, it is shown that this symmetry property leads to a direct experimental prediction for the high Tc superconductivity. If the high Tc superconductivity is correctly described by the Hubbard model, then the SO(4) symmetry dictates the existence of a collective mode that only appears below the superconducting transition temperature. Such a mode can be detected in an electron energy loss experiment as a sharp resonance feature. The location of the peak measure the parameters in the Hubbard model, the intensity is proportional to the superconducting order parameter and the width measures the departure of the Hubbard model from reality.

International Journal of Modern Physics B•Jan 1, 1991•28 citations

Symmetry properties of magnetization in the Hubbard model at finite temperatures

Gang Su, Masuo Suzuki

By making use of some symmetry properties of the relevant Hamiltonian, two fundamental relations between the ferromagnetic magnetization and a spin correlation function are derived for the $d(=1,2,3)$-dimensional Hubbard model at finite temperatures. These can be viewed as a kind of Ward-Takahashi identity. The properties of the magnetization as a function of the applied field are discussed. The results thus obtained hold true for both repulsive and attractive on-site Coulomb interactions, and for arbitrary electron fillings.

arXiv:cond-mat/9803106•Physical review. B, Condensed matter•Jun 1, 1998•4 citations

Finite-temperature charge and spin transport in the one-dimensional Hubbard model accounting for its global [ SU ( 2 ) × SU ( 2 ) × U ( 1 ) ]/ Z 2 2 symmetry

J. M. P. Carmelo, J. E. C. Carmelo

Most studies on the one-dimensional (1D) Hubbard model with transfer integral $t$ and on-site repulsion $U$ have assumed that for $u=U/4t>0$ and $h=\ensuremath{\mu}=0$, where $h$ denotes the magnetic field and $\ensuremath{\mu}$ the chemical potential, its global symmetry is $\mathrm{SO}(4)=[\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)]/{Z}_{2}$. However, both in 1D and for the $U>0$ Hubbard model on any bipartite lattice it is actually larger and given by $[\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)]/{Z}_{2}^{2}=[\mathrm{SO}(4)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)]/{Z}_{2}$. The studies of this paper account for the irreducible representations of that symmetry to identify both the spin and charge carriers that populate each of the energy eigenstates that span that model full Hilbert space. The global $\ensuremath{\tau}$-translational U(1) symmetry beyond $\mathrm{SO}(4)$ is found to describe the relative translational degrees of freedom of such spin and charge carriers, respectively. We show that important finite-temperature transport quantities as the $h>0$ spin and $\ensuremath{\mu}>0$ charge stiffnesses and the $h=0$ spin and $\ensuremath{\mu}=0$ charge diffusion constants are controlled by microscopic processes associated with the spin and charge elementary currents carried by these spin and charge carriers, respectively. We describe the model finite-temperature transport properties in terms of the microscopic processes associated with the spin and charge elementary currents carried by the corresponding carriers. We expect that the general $\ensuremath{\tau}$- and physical $\ensuremath{\alpha}$-spin representation used in our study is that suitable to address the open problems on finite-temperature transport in the 1D Hubbard model also discussed in this paper.

arXiv:2503.21573•Physical review. B./Physical review. B•Mar 26, 2025•7 citations

Spin analog method for collective modes of the Hubbard model

E. J. Melé

We make use of a canonical transformation relating the positive U and negative U sectors of the one band Hubbard model to study the global SU(2) pseudospin symmetry in the model, and to investigate the structure of the collective excitation spectrum from a superconducting ground state in the model. New collective modes in this model associated with the SU(2) pseudospin symmetry, recently discussed by Zhang, appear in the pseudo-spinwave spectrum of the transformed problem. Consequences of this mapping for a self dual generalization of the Hubbard model are also discussed.

Solid State Communications•Aug 1, 1991•0 citations

Symmetry breaking in the Hubbard model at zero temperature

Gang Su

It is shown within the pseudospin framework that the U(1) phase symmetry and the pseudospin SU(2) symmetry in the Hubbard model at zero temperature are spontaneously broken away from half filling, which implies that the Hubbard model probably possesses some massless Goldstone bosons in its ground state.

Physical review. B, Condensed matter•Feb 1, 1993•0 citations

SU(2) gauge theory of the Hubbard model and application to the honeycomb lattice

Michael Hermele

Motivated by recent experiments on the triangular lattice Mott-Hubbard system $\ensuremath{\kappa}\text{\ensuremath{-}}{(\mathrm{BEDT}\text{\ensuremath{-}}\mathrm{TTF})}_{2}{\mathrm{Cu}}_{2}{(\mathrm{C}\mathrm{N})}_{3}$, we develop a general formalism to investigate quantum spin liquid insulators adjacent to the Mott transition in Hubbard models. This formalism, dubbed the SU(2) slave-rotor formulation, is an extension of the SU(2) gauge theory of the Heisenberg model to the case of the Hubbard model. Furthermore, we propose the honeycomb lattice Hubbard model (at half filling) as a candidate for a spin liquid ground state near the Mott transition; this is an appealing possibility, as this model can be studied via quantum Monte Carlo simulation without a sign problem. The pseudospin symmetry of Hubbard models on bipartite lattices turns out to play a crucial role in our analysis, and we develop our formalism primarily for the case of a bipartite lattice. We also sketch its development for a general Hubbard model. We develop a mean-field theory to describe spin liquids and some competing states, and apply it to the honeycomb lattice. On the insulating side of the Mott transition, we find an SU(2) algebraic spin liquid (ASL), described by gapless $S=1∕2$ Dirac fermions (spinons) coupled to a fluctuating SU(2) gauge field. This result contrasts with that obtained via a U(1) slave-rotor approach, which instead found a U(1) ASL. That formulation does not respect the pseudospin symmetry, and is therefore not correct on the honeycomb lattice. We construct a low-energy effective theory describing the ASL phase, the conducting semimetal phase and the Mott transition between them. This physics can be detected in numerical simulations via the simultaneous presence of substantial antiferromagnetic and valence-bond solid correlations. In the SU(2) ASL, these observables have slowly decaying fluctuations in space and time, described by power laws with the same critical exponent. We recover the results of the U(1) slave-rotor formulation in the presence of a strong breaking of pseudospin symmetry. Our analysis suggests that both a third-neighbor electron hopping, and/or pseudospin-breaking terms such as a nearest-neighbor density interaction, may help to stabilize a spin liquid phase.

arXiv:cond-mat/0701134•Physical Review B•Jul 31, 2007•85 citations
Ward-Takahashi relations for spin symmetry SU(2)S\mathrm{SU}{(2)}_{\mathrm{S}}SU(2)S​ and pseudospin symmetry SU(2)C\mathrm{SU}{(2)}_{\mathrm{C}}SU(2)C​ are derived for the simple Hubbard model in arbitrary dimensions. For this purpose we generally discuss local SO(4) symmetry in terms of Clifford algebra, and employ a gauge transformed form of the original Hubbard Hamiltonian, which makes the Ward-Takahashi relations much simpler. Derived Ward-Takahashi relations are rewritten as the relations between self-energy and vertex part to discuss spontaneously broken symmetry and Fermi liquid property of the system for zero temperature magnetism.
The one band Hubbard model on a bi-partite lattice possess a global SO(4) symmetry group which is a direct product of the spin SU(2) and the pseudospin SU(2) symmetry group. Recently, the group theorectical consequence of this symmetry property has been investigated systematically. It is shown that this symmetry property can be used to reduce the complexity of diagonalizing the Hubbard Hamiltonian, and classify all the eigenstates according to the representation of the SO(4) group. It also leads to an exact one-to-one corespondence of the eigenstates at half-filling to that away from half-filling. Most importantly however, it is shown that this symmetry property leads to a direct experimental prediction for the high Tc superconductivity. If the high Tc superconductivity is correctly described by the Hubbard model, then the SO(4) symmetry dictates the existence of a collective mode that only appears below the superconducting transition temperature. Such a mode can be detected in an electron energy loss experiment as a sharp resonance feature. The location of the peak measure the parameters in the Hubbard model, the intensity is proportional to the superconducting order parameter and the width measures the departure of the Hubbard model from reality.
By making use of some symmetry properties of the relevant Hamiltonian, two fundamental relations between the ferromagnetic magnetization and a spin correlation function are derived for the d(=1,2,3)d(=1,2,3)d(=1,2,3)-dimensional Hubbard model at finite temperatures. These can be viewed as a kind of Ward-Takahashi identity. The properties of the magnetization as a function of the applied field are discussed. The results thus obtained hold true for both repulsive and attractive on-site Coulomb interactions, and for arbitrary electron fillings.
Most studies on the one-dimensional (1D) Hubbard model with transfer integral ttt and on-site repulsion UUU have assumed that for u=U/4t>0u=U/4t>0u=U/4t>0 and $h=\ensuremath{\mu}=0$, where hhh denotes the magnetic field and $\ensuremath{\mu}$ the chemical potential, its global symmetry is $\mathrm{SO}(4)=[\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)]/{Z}_{2}$. However, both in 1D and for the U>0U>0U>0 Hubbard model on any bipartite lattice it is actually larger and given by $[\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)]/{Z}_{2}^{2}=[\mathrm{SO}(4)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)]/{Z}_{2}$. The studies of this paper account for the irreducible representations of that symmetry to identify both the spin and charge carriers that populate each of the energy eigenstates that span that model full Hilbert space. The global $\ensuremath{\tau}$-translational U(1) symmetry beyond SO(4)\mathrm{SO}(4)SO(4) is found to describe the relative translational degrees of freedom of such spin and charge carriers, respectively. We show that important finite-temperature transport quantities as the h>0h>0h>0 spin and $\ensuremath{\mu}>0$ charge stiffnesses and the h=0h=0h=0 spin and $\ensuremath{\mu}=0$ charge diffusion constants are controlled by microscopic processes associated with the spin and charge elementary currents carried by these spin and charge carriers, respectively. We describe the model finite-temperature transport properties in terms of the microscopic processes associated with the spin and charge elementary currents carried by the corresponding carriers. We expect that the general $\ensuremath{\tau}$- and physical $\ensuremath{\alpha}$-spin representation used in our study is that suitable to address the open problems on finite-temperature transport in the 1D Hubbard model also discussed in this paper.
We make use of a canonical transformation relating the positive U and negative U sectors of the one band Hubbard model to study the global SU(2) pseudospin symmetry in the model, and to investigate the structure of the collective excitation spectrum from a superconducting ground state in the model. New collective modes in this model associated with the SU(2) pseudospin symmetry, recently discussed by Zhang, appear in the pseudo-spinwave spectrum of the transformed problem. Consequences of this mapping for a self dual generalization of the Hubbard model are also discussed.
Motivated by recent experiments on the triangular lattice Mott-Hubbard system $\ensuremath{\kappa}\text{\ensuremath{-}}{(\mathrm{BEDT}\text{\ensuremath{-}}\mathrm{TTF})}_{2}{\mathrm{Cu}}_{2}{(\mathrm{C}\mathrm{N})}_{3}$, we develop a general formalism to investigate quantum spin liquid insulators adjacent to the Mott transition in Hubbard models. This formalism, dubbed the SU(2) slave-rotor formulation, is an extension of the SU(2) gauge theory of the Heisenberg model to the case of the Hubbard model. Furthermore, we propose the honeycomb lattice Hubbard model (at half filling) as a candidate for a spin liquid ground state near the Mott transition; this is an appealing possibility, as this model can be studied via quantum Monte Carlo simulation without a sign problem. The pseudospin symmetry of Hubbard models on bipartite lattices turns out to play a crucial role in our analysis, and we develop our formalism primarily for the case of a bipartite lattice. We also sketch its development for a general Hubbard model. We develop a mean-field theory to describe spin liquids and some competing states, and apply it to the honeycomb lattice. On the insulating side of the Mott transition, we find an SU(2) algebraic spin liquid (ASL), described by gapless S=1∕2S=1∕2S=1∕2 Dirac fermions (spinons) coupled to a fluctuating SU(2) gauge field. This result contrasts with that obtained via a U(1) slave-rotor approach, which instead found a U(1) ASL. That formulation does not respect the pseudospin symmetry, and is therefore not correct on the honeycomb lattice. We construct a low-energy effective theory describing the ASL phase, the conducting semimetal phase and the Mott transition between them. This physics can be detected in numerical simulations via the simultaneous presence of substantial antiferromagnetic and valence-bond solid correlations. In the SU(2) ASL, these observables have slowly decaying fluctuations in space and time, described by power laws with the same critical exponent. We recover the results of the U(1) slave-rotor formulation in the presence of a strong breaking of pseudospin symmetry. Our analysis suggests that both a third-neighbor electron hopping, and/or pseudospin-breaking terms such as a nearest-neighbor density interaction, may help to stabilize a spin liquid phase.