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The Mott metal-insulator transition in the 1D Hubbard model in an external magnetic field

Holger Frahm, T. Vekua

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

We study the low energy behavior of the one dimensional Hubbard model across the Mott metal-insulator phase transition in an external magnetic field. In particular we calculate elements of the dressed charge matrix at the critical point of the Mott transition for arbitrary Hubbard repulsion and magnetization numerically and, in certain limiting cases, analytically. These results are combined with a non-perturbative effective field theory approach to reveal how the breaking of time reversal symmetry influences the Mott transition.

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The Mott metal-insulator transition in the two-band Hubbard model in infinite dimensions is studied by using the linearized dynamical mean-field theory. The discontinuity in the chemical potential for the change from hole to electron doping is calculated analytically as a function of the on-site Coulomb interaction $U$ at the $d$-orbital and the charge-transfer energy $Δ$ between the $d$- and $p$-orbitals. Critical behaviour of the quasiparticle weight is also obtained analytically as a function of $U$ and $Δ$. The analytic results are in good agreement with the numerical results of the exact diagonalization method.

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Scaling theory of the Mott-Hubbard metal-insulator transition in one dimension

Charles Allen Stafford, Andrew J. Millis

We use the Bethe ansatz equations to calculate the charge stiffness ${\mathit{D}}_{\mathit{c}}$=(L/2)${\mathit{d}}^{2}$${\mathit{E}}_{0}$/d${\mathrm{\ensuremath{\Phi}}}_{\mathit{c}}^{2}$${\mathrm{\ensuremath{\Vert}}}_{\mathrm{\ensuremath{\Phi}}\mathit{c}}$=0 of the one-dimensional repulsive-interaction Hubbard model for electron densities close to the Mott insulating value of one electron per site (n=1), where ${\mathit{E}}_{0}$ is the ground-state energy, L is the circumference of the system (assumed to have periodic boundary conditions), and (\ensuremath{\Elzxh}c/e)${\mathrm{\ensuremath{\Phi}}}_{\mathit{c}}$ is the magnetic flux enclosed. We obtain an exact result for the asymptotic form of ${\mathit{D}}_{\mathit{c}}$(L) as L\ensuremath{\rightarrow}\ensuremath{\infty} at n=1, which defines and yields an analytic expression for the correlation length \ensuremath{\xi} in the Mott insulating phase of the model as a function of the on-site repulsion U. In the vicinity of the zero-temperature critical point U=0, n=1, we show that the charge stiffness has the hyperscaling form ${\mathit{D}}_{\mathit{c}}$(n,L,U)=${\mathit{Y}}_{+}$(\ensuremath{\xi}\ensuremath{\delta},\ensuremath{\xi}/L), where \ensuremath{\delta}=\ensuremath{\Vert}1-n\ensuremath{\Vert} and ${\mathit{Y}}_{+}$ is a universal scaling function which we calculate. The physical significance of \ensuremath{\xi} in the metallic phase of the model is that it defines the characteristic size of the charge-carrying solitons, or holons. We construct an explicit mapping for arbitrary U and \ensuremath{\xi}\ensuremath{\delta}\ensuremath{\ll}1 of the holons onto weakly interacting spinless fermions, and use this mapping to obtain an asymptotically exact expression for the low-temperature thermopower near the metal-insulator transition, which is a generalization to arbitrary U of a result previously obtained using a weak-coupling approximation, and implies holelike transport for 01-n\ensuremath{\ll}${\ensuremath{\xi}}^{\mathrm{\ensuremath{-}}1}$.

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Metal-insulator transition in an Hubbard-like model with degeneracy in one dimension

P. Schlottmann

We consider the SU(N) generalization of the one-dimensional Hubbard model with arbitrary degeneracy N (spin and orbital degrees of freedom). This model is integrable and has several unusual properties at low temperatures. The Bethe-Ansatz equations at T=0 are analyzed in the thermodynamic limit in the absence of external fields. In the continuum limit, the effective interaction between the charge degrees of freedom corresponds to a potential of the form [sinh(ax)${]}^{\mathrm{\ensuremath{-}}2}$, where x is the distance between the particles involved and a is an inverse length scale. In the limit N\ensuremath{\rightarrow}\ensuremath{\infty} and in the continuum limit, the charges reduce to a Bose gas interacting via a \ensuremath{\delta}-function potential. We further address here the properties of the charge degrees of freedom for a band filling close to one electron per site. The charge excitations obey Fermi statistics. We find a Mott metal-insulator transition at a critical value ${\mathit{U}}_{\mathit{c}}$ of the Coulomb repulsion. ${\mathit{U}}_{\mathit{c}}$ depends on N (${\mathit{U}}_{\mathit{c}}$=0 for N=2). A qualitative change in the charge-rapidity distribution is found at ${\mathit{U}}_{\mathit{c}}$. The Fermi velocity is finite for U${\mathit{U}}_{\mathit{c}}$, diverges as U\ensuremath{\rightarrow}${\mathit{U}}_{\mathit{c}}$, and vanishes for U>${\mathit{U}}_{\mathit{c}}$.

Physical review. B, Condensed matter•Mar 15, 1992•21 citations
We study the low energy behavior of the one dimensional Hubbard model across the Mott metal-insulator phase transition in an external magnetic field. In particular we calculate elements of the dressed charge matrix at the critical point of the Mott transition for arbitrary Hubbard repulsion and magnetization numerically and, in certain limiting cases, analytically. These results are combined with a non-perturbative effective field theory approach to reveal how the breaking of time reversal symmetry influences the Mott transition.
We study the low energy behavior of the one dimensional Hubbard model across the Mott metal–insulator phase transition in an external magnetic field. In particular we calculate elements of the dressed charge matrix at the critical point of the Mott transition for arbitrary Hubbard repulsion and magnetization numerically and, in certain limiting cases, analytically. These results are combined with a non-perturbative effective field theory approach to reveal how the breaking of time reversal symmetry influences the Mott transition.
Using results on the scaling of energies with the size of the system and the principles of conformal quantum field theory, we calculate the asymptotics of correlation functions for the one-dimensional Hubbard model in the repulsive regime in the presence of an external magnetic field. The critical exponents are given in terms of a dressed charge matrix that is defined in terms of a set of integral equations obtained from the Bethe-Ansatz solution for the Hubbard model. An interpretation of this matrix in terms of thermodynamical coefficients is given, and several limiting cases are considered.
The paramagnetic phase of the one-band Hubbard model is studied at zero-temperature, within the framework of dynamical mean-field theory, and for general particle-hole asymmetry where a doping-induced Mott transition occurs. Our primary focus is the Mott insulator (MI) phase, and our main aim to establish what can be shown exactly about it. To handle the locally doubly-degenerate MI requires two distinct self-energies, which reflect the broken symmetry nature of the phase and together determine the standard single self-energy. Exact results are obtained for the local charge, local magnetic moment and associated spin susceptibilities, the interaction-renormalised levels, and the low-energy behaviour of the self-energy in the MI phase. The metallic phase is also considered briefly, and shown to acquire an emergent particle-hole symmetry as the Mott transition is approached. Throughout the metal, Luttinger's theorem is reflected in the vanishing of the Luttinger integral; for the generic MI by contrast this is shown to be non-vanishing, but again to have a universal magnitude. Numerical results are also obtained using NRG, for the metal/MI phase boundary, the scaling behaviour of the charge as the Mott transition is aproached from the metal, and associated universal scaling of single-particle dynamics as the low-energy Kondo scale vanishes.
The Mott metal-insulator transition in the two-band Hubbard model in infinite dimensions is studied by using the linearized dynamical mean-field theory. The discontinuity in the chemical potential for the change from hole to electron doping is calculated analytically as a function of the on-site Coulomb interaction UUU at the ddd-orbital and the charge-transfer energy ΔΔΔ between the ddd- and ppp-orbitals. Critical behaviour of the quasiparticle weight is also obtained analytically as a function of UUU and ΔΔΔ. The analytic results are in good agreement with the numerical results of the exact diagonalization method.
We use the Bethe ansatz equations to calculate the charge stiffness Dc{\mathit{D}}_{\mathit{c}}Dc​=(L/2)d2{\mathit{d}}^{2}d2E0{\mathit{E}}_{0}E0​/d${\mathrm{\ensuremath{\Phi}}}_{\mathit{c}}^{2}$${\mathrm{\ensuremath{\Vert}}}_{\mathrm{\ensuremath{\Phi}}\mathit{c}}$=0 of the one-dimensional repulsive-interaction Hubbard model for electron densities close to the Mott insulating value of one electron per site (n=1), where E0{\mathit{E}}_{0}E0​ is the ground-state energy, L is the circumference of the system (assumed to have periodic boundary conditions), and (\ensuremath{\Elzxh}c/e)${\mathrm{\ensuremath{\Phi}}}_{\mathit{c}}$ is the magnetic flux enclosed. We obtain an exact result for the asymptotic form of Dc{\mathit{D}}_{\mathit{c}}Dc​(L) as L\ensuremath{\rightarrow}\ensuremath{\infty} at n=1, which defines and yields an analytic expression for the correlation length \ensuremath{\xi} in the Mott insulating phase of the model as a function of the on-site repulsion U. In the vicinity of the zero-temperature critical point U=0, n=1, we show that the charge stiffness has the hyperscaling form Dc{\mathit{D}}_{\mathit{c}}Dc​(n,L,U)=Y+{\mathit{Y}}_{+}Y+​(\ensuremath{\xi}\ensuremath{\delta},\ensuremath{\xi}/L), where \ensuremath{\delta}=\ensuremath{\Vert}1-n\ensuremath{\Vert} and Y+{\mathit{Y}}_{+}Y+​ is a universal scaling function which we calculate. The physical significance of \ensuremath{\xi} in the metallic phase of the model is that it defines the characteristic size of the charge-carrying solitons, or holons. We construct an explicit mapping for arbitrary U and \ensuremath{\xi}\ensuremath{\delta}\ensuremath{\ll}1 of the holons onto weakly interacting spinless fermions, and use this mapping to obtain an asymptotically exact expression for the low-temperature thermopower near the metal-insulator transition, which is a generalization to arbitrary U of a result previously obtained using a weak-coupling approximation, and implies holelike transport for 01-n\ensuremath{\ll}${\ensuremath{\xi}}^{\mathrm{\ensuremath{-}}1}$.
We consider the SU(N) generalization of the one-dimensional Hubbard model with arbitrary degeneracy N (spin and orbital degrees of freedom). This model is integrable and has several unusual properties at low temperatures. The Bethe-Ansatz equations at T=0 are analyzed in the thermodynamic limit in the absence of external fields. In the continuum limit, the effective interaction between the charge degrees of freedom corresponds to a potential of the form [sinh(ax)${]}^{\mathrm{\ensuremath{-}}2}$, where x is the distance between the particles involved and a is an inverse length scale. In the limit N\ensuremath{\rightarrow}\ensuremath{\infty} and in the continuum limit, the charges reduce to a Bose gas interacting via a \ensuremath{\delta}-function potential. We further address here the properties of the charge degrees of freedom for a band filling close to one electron per site. The charge excitations obey Fermi statistics. We find a Mott metal-insulator transition at a critical value Uc{\mathit{U}}_{\mathit{c}}Uc​ of the Coulomb repulsion. Uc{\mathit{U}}_{\mathit{c}}Uc​ depends on N (Uc{\mathit{U}}_{\mathit{c}}Uc​=0 for N=2). A qualitative change in the charge-rapidity distribution is found at Uc{\mathit{U}}_{\mathit{c}}Uc​. The Fermi velocity is finite for UUc{\mathit{U}}_{\mathit{c}}Uc​, diverges as U\ensuremath{\rightarrow}Uc{\mathit{U}}_{\mathit{c}}Uc​, and vanishes for U>Uc{\mathit{U}}_{\mathit{c}}Uc​.