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:0902.1627 · Journal of Physics Conference Series

Low-temperature thermodynamics of one class of flat-band models

Oleg Derzhko, Johannes Richter, A. Honecker

Published Jan 1, 2009• 2 citations
arXivHTMLPDF

Abstract

We consider the antiferromagnetic Heisenberg model and the repulsive Hubbard model for a class of frustrated lattices with a completely dispersionless (flat) lowest one-particle (either one-magnon or one-electron) band. We construct exact many-particle ground states for a wide range of particle densities, calculate their degeneracy, and, as a result, obtain closed-form expressions for the low-temperature thermodynamic quantities around a particular value of the magnetic field h sat or the chemical potential μ 0 . We confirm our analytic findings by numerical data for finite lattices.

DOI
10.1088/1742-6596/145/1/012059

Related papers

Low-temperature properties of the Hubbard model on highly frustrated one-dimensional lattices

Oleg Derzhko, Johannes Richter, A. Honecker +2 more

We consider the repulsive Hubbard model on three highly frustrated one-dimensional lattices---sawtooth chain and two kagom\'e chains---with completely dispersionless (flat) lowest single-electron bands. We construct the complete manifold of exact many-electron ground states at low electron fillings and calculate the degeneracy of these states. As a result, we obtain closed-form expressions for low-temperature thermodynamic quantities around a particular value of the chemical potential ${\ensuremath{\mu}}_{0}$. We discuss specific features of thermodynamic quantities of these ground-state ensembles such as residual entropy, an extra low-temperature peak in the specific heat, and the existence of ferromagnetism and paramagnetism. We confirm our analytical results by comparison with exact-diagonalization data for finite systems.

arXiv:0911.3596•Physical Review B•Jan 26, 2010•78 citations

Low-temperature thermodynamics for a flat-band ferromagnet: Rigorous versus numerical results

Oleg Derzhko, A. Honecker, Johannes Richter

The repulsive Hubbard model on a sawtooth chain exhibits the lowest single-electron band which is completely dispersionless (flat) for a specific choice of the hopping parameters. We construct exact many-electron ground states for electron fillings up to 1/4. We map the low-energy degrees of freedom of the electron model to a model of classical hard dimers on a chain and, as a result, obtain the ground-state degeneracy as well as closed-form expressions for the low-temperature thermodynamic quantities around a particular value of the chemical potential. We compare our analytical findings with complementary numerical data. Although we consider a specific model, we believe that some of our results such as a low-temperature peak in the specific heat are generic for flat-band ferromagnets.

arXiv:cond-mat/0703295•Physical Review B•Dec 6, 2007•38 citations

Exact low-temperature properties of a class of highly frustrated Hubbard models

Oleg Derzhko, A. Honecker, Johannes Richter

We study the repulsive Hubbard model both analytically and numerically on a family of highly frustrated lattices which have one-electron states localized on isolated trapping cells. We construct and count exact many-electron ground states for a wide range of electron densities and obtain closed-form expressions for the low-temperature thermodynamic quantities which are universal for all lattices of the family. Furthermore, we find that saturated ferromagnetism is obtained only for sufficiently high electron densities and large Hubbard repulsion $U$ while there is no finite average moment in the ground states at lower densities.

arXiv:0807.2991•Physical Review B•Feb 2, 2009•26 citations

Strongly correlated flat-band systems: The route from Heisenberg spins to Hubbard electrons

Oleg Derzhko, Johannes Richter, Mykola Maksymenko

On a large class of lattices (such as the sawtooth chain, the kagome and the pyrochlore lattices), the quantum Heisenberg and the repulsive Hubbard models may host a completely dispersionless (flat) energy band in the single-particle spectrum. The flat-band states can be viewed as completely localized within a finite volume (trap) of the lattice and allow for construction of many-particle states, roughly speaking, by occupying the traps with particles. If the flat-band happens to be the lowest-energy one, the manifold of such many-body states will often determine the ground-state and low-temperature physics of the models at hand even in the presence of strong interactions. The localized nature of these many-body states makes possible the mapping of this subset of eigenstates onto a corresponding classical hard-core system. As a result, the ground-state and low-temperature properties of the strongly correlated flat-band systems can be analyzed in detail using concepts and tools of classical statistical mechanics (e.g., classical lattice-gas approach or percolation approach), in contrast to more challenging quantum many-body techniques usually necessary to examine strongly correlated quantum systems. In this review, we recapitulate the basic features of the flat-band spin systems and briefly summarize earlier studies in the field. The main emphasis is made on recent developments which include results for both spin and electron flat-band models. In particular, for flat-band spin systems, we highlight field-driven phase transitions for frustrated quantum Heisenberg antiferromagnets at low temperatures, chiral flat-band states, as well as the effect of a slight dispersion of a previously strictly flat-band due to nonideal lattice geometry. For electronic systems, we discuss the universal low-temperature behavior of several flat-band Hubbard models, the emergence of ground-state ferromagnetism in the square-lattice Tasaki–Hubbard model and the related Pauli-correlated percolation problem, as well as the dispersion-driven ground-state ferromagnetism in flat-band Hubbard systems. Closely related studies and possible experimental realizations of the flat-band physics are also described briefly.

arXiv:1502.02729•International Journal of Modern Physics B•May 7, 2015•251 citations

THE SAWTOOTH CHAIN: FROM HEISENBERG SPINS TO HUBBARD ELECTRONS

Jonas Richter, Oleg Derzhko, A. Honecker

We report on recent studies of the spin-half Heisenberg and the Hubbard model on the sawtooth chain. For both models we construct a class of exact eigenstates which are localized due to the frustrating geometry of the lattice for a certain relation of the exchange (hopping) integrals. Although these eigenstates differ in details for the two models because of the different statistics, they share some characteristic features. The localized eigenstates are highly degenerate and become ground states in high magnetic fields (Heisenberg model) or at certain electron fillings (Hubbard model), respectively. They may dominate the low-temperature thermodynamics and lead to an extra low-temperature maximum in the specific heat. The ground-state degeneracy can be calculated exactly by a mapping of the manifold of localized ground states onto a classical hard-dimer problem, and explicit expressions for thermodynamic quantities can be derived which are valid at low temperatures near the saturation field for the Heisenberg model or around a certain value of the chemical potential for the Hubbard model, respectively.

arXiv:0806.0922•International Journal of Modern Physics B•Oct 20, 2008•29 citations

Enhancedlow-temperatureentropyandflat-bandferromagnetisminthe t − J modelonthesawtoothlattice

A. Honecker, Johannes Richter, Theoretische Physik

Using the example of the sawtooth chain, we argue that the t−J model shares important features with the Hubbard model on highly frustrated lattices. The lowest single-fermion band is completely flat (for a specific choice of the hopping parameters ti,j in the case of the sawtooth chain), giving rise to single-particle excitations which can be localized in real space. These localized excitations do not interact for sufficient spatial separations such that exact many-electron states can also be constructed. Furthermore, all these excitations acquire zero energy for a suitable choice of the chemical potential µ. This leads to: (i) a jump in the particle density at zero temperature, (ii) a finite zero-temperature entropy, (iii) a ferromagnetic ground state with a charge gap when the flat band is fully occupied and (iv) unusually large temperature variations when µ is varied adiabatically at finite temperature.

•Jan 1, 2007•0 citations
We consider the antiferromagnetic Heisenberg model and the repulsive Hubbard model for a class of frustrated lattices with a completely dispersionless (flat) lowest one-particle (either one-magnon or one-electron) band. We construct exact many-particle ground states for a wide range of particle densities, calculate their degeneracy, and, as a result, obtain closed-form expressions for the low-temperature thermodynamic quantities around a particular value of the magnetic field h sat or the chemical potential μ 0 . We confirm our analytic findings by numerical data for finite lattices.
We consider the repulsive Hubbard model on three highly frustrated one-dimensional lattices---sawtooth chain and two kagom\'e chains---with completely dispersionless (flat) lowest single-electron bands. We construct the complete manifold of exact many-electron ground states at low electron fillings and calculate the degeneracy of these states. As a result, we obtain closed-form expressions for low-temperature thermodynamic quantities around a particular value of the chemical potential ${\ensuremath{\mu}}_{0}$. We discuss specific features of thermodynamic quantities of these ground-state ensembles such as residual entropy, an extra low-temperature peak in the specific heat, and the existence of ferromagnetism and paramagnetism. We confirm our analytical results by comparison with exact-diagonalization data for finite systems.
The repulsive Hubbard model on a sawtooth chain exhibits the lowest single-electron band which is completely dispersionless (flat) for a specific choice of the hopping parameters. We construct exact many-electron ground states for electron fillings up to 1/4. We map the low-energy degrees of freedom of the electron model to a model of classical hard dimers on a chain and, as a result, obtain the ground-state degeneracy as well as closed-form expressions for the low-temperature thermodynamic quantities around a particular value of the chemical potential. We compare our analytical findings with complementary numerical data. Although we consider a specific model, we believe that some of our results such as a low-temperature peak in the specific heat are generic for flat-band ferromagnets.
We study the repulsive Hubbard model both analytically and numerically on a family of highly frustrated lattices which have one-electron states localized on isolated trapping cells. We construct and count exact many-electron ground states for a wide range of electron densities and obtain closed-form expressions for the low-temperature thermodynamic quantities which are universal for all lattices of the family. Furthermore, we find that saturated ferromagnetism is obtained only for sufficiently high electron densities and large Hubbard repulsion UUU while there is no finite average moment in the ground states at lower densities.
On a large class of lattices (such as the sawtooth chain, the kagome and the pyrochlore lattices), the quantum Heisenberg and the repulsive Hubbard models may host a completely dispersionless (flat) energy band in the single-particle spectrum. The flat-band states can be viewed as completely localized within a finite volume (trap) of the lattice and allow for construction of many-particle states, roughly speaking, by occupying the traps with particles. If the flat-band happens to be the lowest-energy one, the manifold of such many-body states will often determine the ground-state and low-temperature physics of the models at hand even in the presence of strong interactions. The localized nature of these many-body states makes possible the mapping of this subset of eigenstates onto a corresponding classical hard-core system. As a result, the ground-state and low-temperature properties of the strongly correlated flat-band systems can be analyzed in detail using concepts and tools of classical statistical mechanics (e.g., classical lattice-gas approach or percolation approach), in contrast to more challenging quantum many-body techniques usually necessary to examine strongly correlated quantum systems. In this review, we recapitulate the basic features of the flat-band spin systems and briefly summarize earlier studies in the field. The main emphasis is made on recent developments which include results for both spin and electron flat-band models. In particular, for flat-band spin systems, we highlight field-driven phase transitions for frustrated quantum Heisenberg antiferromagnets at low temperatures, chiral flat-band states, as well as the effect of a slight dispersion of a previously strictly flat-band due to nonideal lattice geometry. For electronic systems, we discuss the universal low-temperature behavior of several flat-band Hubbard models, the emergence of ground-state ferromagnetism in the square-lattice Tasaki–Hubbard model and the related Pauli-correlated percolation problem, as well as the dispersion-driven ground-state ferromagnetism in flat-band Hubbard systems. Closely related studies and possible experimental realizations of the flat-band physics are also described briefly.
We report on recent studies of the spin-half Heisenberg and the Hubbard model on the sawtooth chain. For both models we construct a class of exact eigenstates which are localized due to the frustrating geometry of the lattice for a certain relation of the exchange (hopping) integrals. Although these eigenstates differ in details for the two models because of the different statistics, they share some characteristic features. The localized eigenstates are highly degenerate and become ground states in high magnetic fields (Heisenberg model) or at certain electron fillings (Hubbard model), respectively. They may dominate the low-temperature thermodynamics and lead to an extra low-temperature maximum in the specific heat. The ground-state degeneracy can be calculated exactly by a mapping of the manifold of localized ground states onto a classical hard-dimer problem, and explicit expressions for thermodynamic quantities can be derived which are valid at low temperatures near the saturation field for the Heisenberg model or around a certain value of the chemical potential for the Hubbard model, respectively.
Using the example of the sawtooth chain, we argue that the t−J model shares important features with the Hubbard model on highly frustrated lattices. The lowest single-fermion band is completely flat (for a specific choice of the hopping parameters ti,j in the case of the sawtooth chain), giving rise to single-particle excitations which can be localized in real space. These localized excitations do not interact for sufficient spatial separations such that exact many-electron states can also be constructed. Furthermore, all these excitations acquire zero energy for a suitable choice of the chemical potential µ. This leads to: (i) a jump in the particle density at zero temperature, (ii) a finite zero-temperature entropy, (iii) a ferromagnetic ground state with a charge gap when the flat band is fully occupied and (iv) unusually large temperature variations when µ is varied adiabatically at finite temperature.