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Strong coupling expansion of the t-V model (further results)

Marcin Szyniszewski, Evgeni Burovski

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

We employ a strong coupling expansion - similar to the one used in the lattice field theory studies [1] - to solve the one-dimensional extended t-V model of fermions on a lattice [2]. This model is solved for a range of filling factors, including both commensurate - where a charge density wave is present - and incommensurate densities. The first set consists not only of a trivial case of half filling. The method allows us to trace the transition from a Luttinger liquid phase to a Mott insulating phase and calculate the critical parameter K. This simple yet powerful method is not based on Bethe ansatz and it works for both integrable and non-integrable systems. Furthermore, we investigate how tailoring the interaction can introduce other ordered phases of the system [3].References:[1] C.J. Hamer, Phys. Lett. B 82, 75-78 (1979); D.P. Crewther, C.J. Hamer, Nucl. Phys. B 170, 353-368 (1980).[2] G. Gómez-Santos, Phys. Rev. Lett. 70, 3780 (1993); R.G. Dias, Phys. Rev. B 62, 7791 (2000).[3] P. Schmitteckert, R. Werner, Phys. Rev. B 69, 195115 (2004); T. Mishra et al., Phys. Rev. B 84, 115135 (2011).

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Generalised t-V model in one dimension

Marcin Szyniszewski, Evgeni Burovski

We use a strong coupling expansion [1] to solve the one-dimensional extended t‑V model of fermions [2,3]. The model is solved for a range of densities, including both commensurate – where a charge density wave is present – and incommensurate densities. The first set consists not only of a trivial case of half filling. The method allows us to trace the transition from a Luttinger liquid phase to a Mott insulating phase. This simple yet powerful method is not based on Bethe ansatz and it works for both integrable and non-integrable systems.References:[1] C.J. Hamer, Phys. Lett. B, 82, 75-78 (1979).[2] G. Gómez-Santos, Phys. Rev. Lett., 70, 3780 (1993).[3] R.G. Dias, Phys. Rev. B, 62, 7791 (2000).

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The one-dimensional extended t-V model of fermions on a lattice is a model with long-range interactions that exhibits a transition between a Luttinger liquid conducting phase and a Mott insulating phase. It is known that by tailoring the potential energy of the insulating system, one can force a phase transition into another insulating phase. We show how to construct all possible charge density wave phases of the system with low critical densities in the atomic limit. Higher critical densities are investigated by a brute-force analysis of the basis of finite systems.

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The generalized t-V model in one dimension

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We develop a systematic strong coupling approach for studying an extended t-V model with interactions of a finite range. Our technique is not based on the Bethe ansatz and is applicable to both integrable and non-integrable models. We illustrate our technique by presenting analytic results for the ground state energy (up to order 7 in t/V ), the current density and density-density correlations for integrable and non-integrable models with commensurate filling factors. We further present preliminary numerical results for incommensurate non- integrable models.

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Quantum Monte Carlo study of the one-dimensional Holstein model of spinless fermions

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The Holstein model of spinless fermions interacting with dispersionless phonons in one dimension is studied by a Green's function Monte Carlo technique. The ground-state energy, first fermionic excited state, density wave correlations, and mean lattice displacements are calculated for lattices of up to 16 sites, for one fermion per two sites, i.e., a half-filled band. Results are obtained for values of the fermion hopping parameter of t=0.1\ensuremath{\omega}, \ensuremath{\omega}, and 10\ensuremath{\omega}, where \ensuremath{\omega} is the phonon frequency. At a finite fermion-phonon coupling g there is a transition from a metallic phase to an insulating phase in which there is charge-density-wave order. Finite-size scaling is found to hold in the metallic phase and is used to extract the coupling dependence of the Luttinger liquid parameters, ${\mathit{u}}_{\mathrm{\ensuremath{\rho}}}$ and ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$, the velocity of charge excitations, and the correlation exponent, respectively. For free fermions (g=0) and for strong coupling (${\mathit{g}}^{2}$\ensuremath{\gg}t\ensuremath{\omega}) our results agree well with known analytic results. \textcopyright{} 1996 The American Physical Society.

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The metal-insulator transition in one dimension

Heinz J. Schulz

The low--energy excited states of a system of interacting one--dimensional fermions in a conducting state are collective charge and spin density oscillations. The unusual physical properties of such a system (called ``Luttinger liquid'') are characterized by the velocities $u_ρ$ and $u_σ$ of the charge and spin excitations, as well as by a parameter $K_ρ$ that determines the power law behavior of correlation functions. Umklapp scattering occuring at half--filling or other commensurate bandfilling can lead to a transition into an insulating state, characterized in particular by a gap in the charge excitations (the Mott--Hubbard gap). The properties in the vicinity of the transition are shown to depend on both the way the transition is approached (constant bandfilling and varying interaction, or constant interaction and varying bandfilling) and on the ``order'' of the commensurability. In particular, even and odd fractional fillings show quite different behavior. This behavior is illustrated in detail using lattice models like the Hubbard model and its extensions.

arXiv:cond-mat/9412036•Dec 7, 1994•7 citations

Exact numerical and analytical results for correlated lattice electrons in one dimension

Satoshi Ejima

Since the advent of high-$T_c$ cuprate superconductors in 1986, strongly correlated electron systems have attracted much attention. Since the cuprates are essentially two-dimensional, low-dimensional systems have moved into the focus of condensed-matter theory. From a theoretical point of view, one-dimensional systems are of particular interest because there are exact numerical and analytical methods which permit detailed studies and deep insights into the many-body problem. In the first part of this thesis, using the numerically exact methods Exact Diagonalization and the Density-Matrix Renormalization Group (DMRG), we investigate the properties of the Tomonaga–Luttinger liquid which is the generic metallic state of matter in one dimension. In particular, we concentrate on the investigation of the so-called Tomonaga–Luttinger liquid parameter which determines the critical exponent $\alpha$ for the density of states near the Fermi energy. Experimental results for some quasi one-dimensional materials report $\alpha \gtrsim 1$, which would imply $K_\rho \lesssim 0.17$, a value which cannot be reconciled with the bare Hubbard model where $K_\rho^H \geq 0.5$, i.e., $\alpha^H \lesssim 1/8$. We develop new accurate numerical methods to obtain $K_\rho$ and investigate how to obtain such small values for $K_\rho$ for slightly doped charge-density-wave insulators. In the second part of this thesis, using the Thermodynamic Bethe Ansatz (TBA) as exact analytical method, we investigate the one-dimensional Hubbard model in the spindisordered regime, which is characterized by the temperature being much larger than the magnetic energy scale but small compared to the Mott–Hubbard gap. Our study is motivated by the controversy about the Mott–Hubbard insulator in infinite dimensions whose ground state is also spin-disordered. In this system the determination of the precise value of the critical interaction strength Uc where the Mott–Hubbard gap closes is still unsolved. Therefore, we provide an example of a Hubbard-type model with a disordered spin background which can be solved exactly. The thermodynamics of our model can be understood in terms of gapped charged excitations with an effective dispersion which are decoupled form the spin degrees of freedom; the latter contribute only entropically. An interpretation of this regime in terms of a putative interacting-electron system at zero temperature leads to a metal-insulator transition at a finite interaction strength above which the gap opens linearly. Our exact results indicate that the strong-coupling expansion of the ground-state energy cannot be used to locate $U_c$. However, the strong-coupling expansion of the gap permits a reliable extrapolation of the critical interaction strength.

•May 3, 2006•0 citations
We employ a strong coupling expansion - similar to the one used in the lattice field theory studies [1] - to solve the one-dimensional extended t-V model of fermions on a lattice [2]. This model is solved for a range of filling factors, including both commensurate - where a charge density wave is present - and incommensurate densities. The first set consists not only of a trivial case of half filling. The method allows us to trace the transition from a Luttinger liquid phase to a Mott insulating phase and calculate the critical parameter K. This simple yet powerful method is not based on Bethe ansatz and it works for both integrable and non-integrable systems. Furthermore, we investigate how tailoring the interaction can introduce other ordered phases of the system [3].References:[1] C.J. Hamer, Phys. Lett. B 82, 75-78 (1979); D.P. Crewther, C.J. Hamer, Nucl. Phys. B 170, 353-368 (1980).[2] G. Gómez-Santos, Phys. Rev. Lett. 70, 3780 (1993); R.G. Dias, Phys. Rev. B 62, 7791 (2000).[3] P. Schmitteckert, R. Werner, Phys. Rev. B 69, 195115 (2004); T. Mishra et al., Phys. Rev. B 84, 115135 (2011).
We use a strong coupling expansion [1] to solve the one-dimensional extended t‑V model of fermions [2,3]. The model is solved for a range of densities, including both commensurate – where a charge density wave is present – and incommensurate densities. The first set consists not only of a trivial case of half filling. The method allows us to trace the transition from a Luttinger liquid phase to a Mott insulating phase. This simple yet powerful method is not based on Bethe ansatz and it works for both integrable and non-integrable systems.References:[1] C.J. Hamer, Phys. Lett. B, 82, 75-78 (1979).[2] G. Gómez-Santos, Phys. Rev. Lett., 70, 3780 (1993).[3] R.G. Dias, Phys. Rev. B, 62, 7791 (2000).
The one-dimensional extended t-V model of fermions on a lattice is a model with long-range interactions that exhibits a transition between a Luttinger liquid conducting phase and a Mott insulating phase. It is known that by tailoring the potential energy of the insulating system, one can force a phase transition into another insulating phase. We show how to construct all possible charge density wave phases of the system with low critical densities in the atomic limit. Higher critical densities are investigated by a brute-force analysis of the basis of finite systems.
We develop a systematic strong coupling approach for studying an extended t-V model with interactions of a finite range. Our technique is not based on the Bethe ansatz and is applicable to both integrable and non-integrable models. We illustrate our technique by presenting analytic results for the ground state energy (up to order 7 in t/V ), the current density and density-density correlations for integrable and non-integrable models with commensurate filling factors. We further present preliminary numerical results for incommensurate non- integrable models.
The Holstein model of spinless fermions interacting with dispersionless phonons in one dimension is studied by a Green's function Monte Carlo technique. The ground-state energy, first fermionic excited state, density wave correlations, and mean lattice displacements are calculated for lattices of up to 16 sites, for one fermion per two sites, i.e., a half-filled band. Results are obtained for values of the fermion hopping parameter of t=0.1\ensuremath{\omega}, \ensuremath{\omega}, and 10\ensuremath{\omega}, where \ensuremath{\omega} is the phonon frequency. At a finite fermion-phonon coupling g there is a transition from a metallic phase to an insulating phase in which there is charge-density-wave order. Finite-size scaling is found to hold in the metallic phase and is used to extract the coupling dependence of the Luttinger liquid parameters, ${\mathit{u}}_{\mathrm{\ensuremath{\rho}}}$ and ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$, the velocity of charge excitations, and the correlation exponent, respectively. For free fermions (g=0) and for strong coupling (g2{\mathit{g}}^{2}g2\ensuremath{\gg}t\ensuremath{\omega}) our results agree well with known analytic results. \textcopyright{} 1996 The American Physical Society.
The low--energy excited states of a system of interacting one--dimensional fermions in a conducting state are collective charge and spin density oscillations. The unusual physical properties of such a system (called ``Luttinger liquid'') are characterized by the velocities uρu_ρuρ​ and uσu_σuσ​ of the charge and spin excitations, as well as by a parameter KρK_ρKρ​ that determines the power law behavior of correlation functions. Umklapp scattering occuring at half--filling or other commensurate bandfilling can lead to a transition into an insulating state, characterized in particular by a gap in the charge excitations (the Mott--Hubbard gap). The properties in the vicinity of the transition are shown to depend on both the way the transition is approached (constant bandfilling and varying interaction, or constant interaction and varying bandfilling) and on the ``order'' of the commensurability. In particular, even and odd fractional fillings show quite different behavior. This behavior is illustrated in detail using lattice models like the Hubbard model and its extensions.
Since the advent of high-TcT_cTc​ cuprate superconductors in 1986, strongly correlated electron systems have attracted much attention. Since the cuprates are essentially two-dimensional, low-dimensional systems have moved into the focus of condensed-matter theory. From a theoretical point of view, one-dimensional systems are of particular interest because there are exact numerical and analytical methods which permit detailed studies and deep insights into the many-body problem. In the first part of this thesis, using the numerically exact methods Exact Diagonalization and the Density-Matrix Renormalization Group (DMRG), we investigate the properties of the Tomonaga–Luttinger liquid which is the generic metallic state of matter in one dimension. In particular, we concentrate on the investigation of the so-called Tomonaga–Luttinger liquid parameter which determines the critical exponent α\alphaα for the density of states near the Fermi energy. Experimental results for some quasi one-dimensional materials report α≳1\alpha \gtrsim 1α≳1, which would imply Kρ≲0.17K_\rho \lesssim 0.17Kρ​≲0.17, a value which cannot be reconciled with the bare Hubbard model where KρH≥0.5K_\rho^H \geq 0.5KρH​≥0.5, i.e., αH≲1/8\alpha^H \lesssim 1/8αH≲1/8. We develop new accurate numerical methods to obtain KρK_\rhoKρ​ and investigate how to obtain such small values for KρK_\rhoKρ​ for slightly doped charge-density-wave insulators. In the second part of this thesis, using the Thermodynamic Bethe Ansatz (TBA) as exact analytical method, we investigate the one-dimensional Hubbard model in the spindisordered regime, which is characterized by the temperature being much larger than the magnetic energy scale but small compared to the Mott–Hubbard gap. Our study is motivated by the controversy about the Mott–Hubbard insulator in infinite dimensions whose ground state is also spin-disordered. In this system the determination of the precise value of the critical interaction strength Uc where the Mott–Hubbard gap closes is still unsolved. Therefore, we provide an example of a Hubbard-type model with a disordered spin background which can be solved exactly. The thermodynamics of our model can be understood in terms of gapped charged excitations with an effective dispersion which are decoupled form the spin degrees of freedom; the latter contribute only entropically. An interpretation of this regime in terms of a putative interacting-electron system at zero temperature leads to a metal-insulator transition at a finite interaction strength above which the gap opens linearly. Our exact results indicate that the strong-coupling expansion of the ground-state energy cannot be used to locate UcU_cUc​. However, the strong-coupling expansion of the gap permits a reliable extrapolation of the critical interaction strength.