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arXiv:0801.2473 · Physical Review B

Fidelity approach to the Hubbard model

Lorenzo Campos Venuti, Marco Cozzini, Pierfrancesco Buonsante, Francesco Massel, Noah Bray-Ali, Paolo Zanardi A

Published Sep 9, 2008• 53 citations
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Abstract

We use the fidelity approach to quantum critical points to study the zero-temperature phase diagram of the one-dimensional Hubbard model. Using a variety of analytical and numerical techniques, we analyze the fidelity metric in various regions of the phase diagram with particular care to the critical points. Specifically we show that close to the Mott transition, taking place at on-site repulsion $U=0$ and electron density $n=1$, the fidelity metric satisfies an hyperscaling form which we calculate. This implies that in general, as one approaches the critical point $U=0$, $n=1$, the fidelity metric tends to a limit which depends on the path of approach. At half-filling, the fidelity metric is expected to diverge as ${U}^{\ensuremath{-}4}$ when $U$ is sent to zero.

DOI
10.1103/physrevb.78.115410

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We use the fidelity approach to quantum critical points to study the zero-temperature phase diagram of the one-dimensional Hubbard model. Using a variety of analytical and numerical techniques, we analyze the fidelity metric in various regions of the phase diagram with particular care to the critical points. Specifically we show that close to the Mott transition, taking place at on-site repulsion U=0U=0U=0 and electron density n=1n=1n=1, the fidelity metric satisfies an hyperscaling form which we calculate. This implies that in general, as one approaches the critical point U=0U=0U=0, n=1n=1n=1, the fidelity metric tends to a limit which depends on the path of approach. At half-filling, the fidelity metric is expected to diverge as U\ensuremath−4{U}^{\ensuremath{-}4}U\ensuremath−4 when UUU is sent to zero.
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 investigated the fidelity susceptibility in the one-dimension (1D) Hubbard model and the extended Hubbard model at half-filling via the density matrix renormalization group. From the numerical results, we argue that in the 1D Hubbard model, the fidelity susceptibility shows a divergence at two points which is infinitesimally close to the critical point while it is always extensive exactly at the critical point. For the extended Hubbard model, we found that for a properly chosen driving parameter, the fidelity susceptibility is able to reveal the quantum phase transitions between the PS (phase separation)-superconducting, superconducting-CDW (charge-density wave), CDW-SDW(spin-density wave), SDW-PS, CDW-BOW (bond-order wave), and the BOW-SDW phases.
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Even though the Hubbard model is one of the most fundamental models of highly correlated electrons, analytical and numerical data describing its thermodynamics at nonzero magnetization are relatively scarce. We present a detailed investigation of the thermodynamic properties for the one-dimensional repulsive Hubbard model in the presence of an arbitrary magnetic field for all values of the filling fraction and temperatures as low as $T\ensuremath{\sim}0.005t.$ Our analysis is based on the system of integral equations derived in the quantum transfer matrix framework. We determine the critical exponents of the quantum phase transitions and also provide analytical derivations for some of the universal functions characterizing the thermodynamics in the vicinities of the quantum critical points. Extensive numerical data for the specific heat, susceptibility, compressibility, and entropy are reported. The experimentally relevant double occupancy presents an interesting doubly nonmonotonic temperature dependence at intermediate values of the interaction strength and also at large repulsion and magnetic fields close to the critical value. The susceptibility in zero magnetic field has a logarithmic singularity at low temperatures for all filling factors similar to the behavior of the same quantity in the spin-1/2 isotropic Heisenberg model. We determine the density profiles for a harmonically trapped system and show that while the total density profile seems to depend mainly on the value of chemical potential at the center of the trap the distribution of phases in the inhomogeneous system changes dramatically as we increase the magnetic field.
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