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arXiv:1402.6801 · Physical Review B

Universal logarithmic temperature dependence of magnetic susceptibility of one-dimensional electrons at critical values of magnetic field

T. Vekua

Published Mar 25, 2014• 3 citations
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Abstract

We study the leading low-temperature dependence of magnetic susceptibility of one-dimensional electrons with a fixed total number of particles at the magnetic fields equal to zero-temperature critical values where the magnetic field induces commensurate-incommensurate quantum phase transitions. For free and repulsively interacting electrons there is only one such critical field corresponding to the transition to the fully polarized state. For attractively interacting electrons besides saturation field there is another critical field equal to the spin gap where zero-temperature magnetization sets in. For all cases, except for the lattice models at half filling, the magnetic susceptibility at critical values of magnetic field has a universal logarithmic temperature dependence, $\ensuremath{\chi}(T)=\ensuremath{\chi}(0)(1+2/ln\phantom{\rule{0.16em}{0ex}}T+\ensuremath{\cdots})$ for $T\ensuremath{\rightarrow}0$.

DOI
10.1103/physrevb.89.121112

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We study the leading low-temperature dependence of magnetic susceptibility of one-dimensional electrons with a fixed total number of particles at the magnetic fields equal to zero-temperature critical values where the magnetic field induces commensurate-incommensurate quantum phase transitions. For free and repulsively interacting electrons there is only one such critical field corresponding to the transition to the fully polarized state. For attractively interacting electrons besides saturation field there is another critical field equal to the spin gap where zero-temperature magnetization sets in. For all cases, except for the lattice models at half filling, the magnetic susceptibility at critical values of magnetic field has a universal logarithmic temperature dependence, \ensuremathχ(T)=\ensuremathχ(0)(1+2/lnT+\ensuremath⋯)\ensuremath{\chi}(T)=\ensuremath{\chi}(0)(1+2/ln\phantom{\rule{0.16em}{0ex}}T+\ensuremath{\cdots})\ensuremathχ(T)=\ensuremathχ(0)(1+2/lnT+\ensuremath⋯) for T\ensuremath→0T\ensuremath{\rightarrow}0T\ensuremath→0.
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.
[[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.
The lowest energies of the one-dimensional half-filled Hubbard model as a function of the total spin are computed from the exact solution of one-dimensional interacting fermions. The lowest energies are used to calculate the magnetization curve at zero temperature. In the case of repulsive interaction our work is an extension of Griffiths' theory for the Heisenberg model to the itinerant electron model of antiferromagnet. In the case of attractive interaction it is shown that the magnetic susceptibility in zero field vanishes.
The lowest energies of the one-dimensional half-filled Hubbard model as a function of the total spin are computed from the exact solution of one-dimensional interacting fermions. The lowest energies are used to calculate the magnetization curve at zero temperature. In the case of repulsive interaction our work is an extension of Griffith's theory for the Heisenberg model to the itinerant electron model of antiferromagnet. In the case of attractive interaction it is shown that the magnetic susceptibility in zero field vanishes.
We study magnetic-field-induced critical singularities in the superconducting phase of the hole-doped Hubbard model of repulsively interacting electrons, defined on a two-leg ladder. We argue that, provided the low-energy spin excitations in doped ladders carry electric charge, the low-temperature thermodynamic quantities, such as the specific-heat coefficient and magnetic susceptibility, will show logarithmic singularities in the quantum critical regime. This behavior is in drastic contrast to the magnetic-field-induced criticality in undoped Mott insulator ladders, which is governed by the zero-scale-factor universality with its hallmark square-root singularities.
The exact phase diagram and the ground-state properties of the one-dimensional Hubbard model with arbitrary on-site interaction of electrons are calculated over a wide range of magnetic field and electron concentrations by means of the Bethe-ansatz formalism. The ground-state properties, including the total energy, the average spin (magnetization) and spin (magnetic) susceptibility are investigated for both signs of the interaction strength U/t. The critical behavior near the onset of magnetization and magnetic saturation are also analyzed. At the onset of magnetization and near the magnetic saturation the spin susceptibility χ diverges at all U/t for half-filling case n=1, whereas for n≠1 it is always finite. The reverse susceptibility χ-1(U) exhibits anomalous hump, which increases with h or n, and shows discontinuity as U/t→±0 at infinitesimal h→0. The analytical results for the ground-state properties in strong and weak interaction limits are in full agreement with our numerical calculations.