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arXiv:cond-mat/0506576 · Physical Review Letters

Entanglement Scaling in the One-Dimensional Hubbard Model at Criticality

Daniel Larsson, Henrik Johannesson

Published Nov 3, 2005• 128 citations
arXivHTMLPDF

Abstract

We derive exact expressions for the local entanglement entropy $\mathcal{E}$ in the ground state of the one-dimensional Hubbard model at a quantum phase transition driven by a change in magnetic field $h$ or chemical potential $\ensuremath{\mu}$. The leading divergences of $\ensuremath{\partial}\mathcal{E}/\ensuremath{\partial}h$ and $\ensuremath{\partial}\mathcal{E}/\ensuremath{\partial}\ensuremath{\mu}$ are shown to be directly related to those of the zero-temperature spin and charge susceptibilities. Logarithmic corrections to scaling signal a change in the number of local states accessible to the system as it undergoes the transition.

DOI
10.1103/physrevlett.95.196406

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The phase diagram, the Bethe-ansatz ground-state properties, including the chemical potential μ, the spin (magnetic) and charge susceptibilities, are calculated within the one-dimensional Hubbard model in entire range of interaction strength (-∞< U/t<+∞), magnetic field (h≥ 0) and all electron concentrations (0≤n≤1). The continuous and smooth variation of μ with n and h in the vicinity of n=1 points on the gapless character of charge excitations at U<0 and provides rigorous upper and lower bounds for μ. The spin (magnetic) susceptability χ at half-filling changes discontinuously as U→0 and is strongly enhanced by electron repulsion, comparing with that of the non-interactig case. The compressibility κ ch increases with n at U<0 and shows non-monotonous behavior with a dramatic increase at U>0. Variations of κ ch -1 in both repulsive and attractive cases qualitatively well reproduces corresponding behavior of charge stiffness.
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.
The exact phase diagrams of the one-dimensional Hubbard model, both attractive and repulsive, in the presence of an arbitrary magnetic field h for various electron concentrations n and on-site interaction strengths U <0 or U >0 are calculated and investigated. The exact ground-state properties, namely, the ground-state energy, the average spin (magnetization), the concentration of the doubly occupied sites, the kinetic energy, the chemical potential, the spin (magnetic) susceptibility and the charge compressibility, are calculated and examined over a wide range of interaction strengths U for various h and n . It is found that the spin susceptibility at half-filling is non-analytic and changes discontinuously as U →0. The exact theory shows the absence of a charge energy gap in the U ⩽0 region for all n and provides, for the chemical potential, the rigorous upper and lower bounds for half-filled and empty bands respectively. The analytical results derived for the weak-coupling limit and asymptotic expansions for strong coupling are in full agreement with the numerical calculations.
[[abstract]]Cited: 0 References: 20 Citation Map Abstract: The phase diagram, the Bethe-ansatz ground-state properties, including the chemical potential mu, the spin (magnetic) and charge susceptibilities, are calculated within the one-dimensional Hubbard model in entire range of interaction strength (-infinity < U/t < + infinity), magnetic field (h greater than or equal to 0) and all electron concentrations (0 less than or equal to n less than or equal to 1). The continuous and smooth variation of mu with n and h in the vicinity of n = 1 points on the gapless character of charge excitations at U < <0 and provides rigorous upper and lower bounds for <mu>. The spin (magnetic) susceptibility chi at half-filling changes discontinuously as U --> 0 and is strongly enhanced by electron repulsion, comparing with that of the non-interacting case. The compressibility rc,h increases with n at U < 0 and shows non-monotonous behavior with a dramatic increase at U > 0. Variation of k(ch)(-1) in both repulsive and attractive cases qulitatively well reproduces corresponding behavior of charge stiffness.
We study the effects of limited entanglement in the one-dimensional Hubbard model by representing the ground states in the form of matrix product states. Finite-entanglement scaling behavior over a wide range is observed at half-filling. The critical exponents characterizing the length scale in terms of the size of the matrices used are obtained, confirming the theoretical prediction that the values of the exponents are solely determined by the central charge. The entanglement spectrum shows that a global double degeneracy occurs in the ground states with a charge gap. We also find that the Mott transition, tuned by changing the chemical potential, always occurs through a first-order transition, and the metallic phase has a few conducting states, including the states with a mean-field nature close to the critical point, as expected in variational matrix product states with a finite amount of entanglement.
In this paper, we study block-block entanglement in the ground state of a one-dimensional extended Hubbard model. Our results show that the phase diagram derived from the block-block entanglement manifests a richer structure than that of the local (single-site) entanglement because it comprises nonlocal correlation. Besides phases characterized by the charge-density wave, the spin-density wave, and phase separation, which can be sketched out by the local entanglement, singlet superconductivity phase could be identified on the contour map of the block-block entanglement if the block size is larger than 1. Scaling analysis shows that log2(l){\mathrm{log}}_{2}(l)log2​(l) behavior of the block-block entanglement may exist in both the noncritical and critical regions, while some local extremum are induced by the finite-size effect. We also study the block-block entanglement defined in the momentum space and discuss its relation to the phase transition from singlet superconducting state to the charge-density-wave state.