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arXiv:1611.10265 · Physical review. B./Physical review. B

Magnetic-field-induced criticality in superconducting two-leg ladders

T. Vekua

Published Mar 24, 2017• 0 citations
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Abstract

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.

DOI
10.1103/physrevb.95.115143

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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.
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We have investigated quantum critical behavior in the doped two-leg extended Hubbard ladder, by using a weak-coupling bosonization method. In the ground state, the dominant fluctuation changes from the conventional ddd-wave-like superconducting (SCd)(\mathrm{SC}d)(SCd) state into density-wave states, with increasing nearest-neighbor repulsions and/or decreasing doping rate. The competition between the SCd\mathrm{SC}dSCd state and the charge-density-wave state coexisting with the ppp-density-wave state becomes noticeable on the critical point, at which the gap for magnetic excitations vanishes. Based on the Majorana-fermion description of the effective theory, we calculate the temperature dependence of the magnetic response such as the spin susceptibility and the NMR relaxation rate, which exhibit unusual properties due to two kinds of spin excitation modes. On the quantum critical point, the spin susceptibility shows paramagnetic behavior with logarithmic corrections and the NMR relaxation rate also exhibits anomalous power-law behavior. We discuss the commensurability effect due to the umklapp scattering and relevance to the two-leg ladder compounds ${\mathrm{Sr}}_{14\ensuremath{-}x}{\mathrm{Ca}}_{x}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$.
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 influence of nonmagnetic doping on the thermodynamic properties of two-leg S = 1/2 spin ladders is studied in this paper. It is shown that, for a weak interchain coupling, the problem can be mapped onto a model of random mass Dirac (Majorana) fermions. We investigate in detail the structure of the fermionic states localized at an individual mass kink (zero modes) in the framework of a generalized Dirac model. The low-temperature thermodynamic properties are dominated by these zero modes. We use the single-fermion density of states, known to exhibit the Dyson singularity in the zero-energy limit, to construct the thermodynamics of the spin ladder. In particular, we find that the magnetic susceptibility χχχ diverges at T→0T\to 0T→0 as 1/Tln⁡2(1/T)1/T\ln^2(1/T)1/Tln2(1/T), and the specific heat behaves as C∝1/ln⁡3(1/T)C\propto 1/\ln^3(1/T)C∝1/ln3(1/T). The predictions on magnetic susceptibility are consistent with the most recent results of quantum Monte Carlo simulations on doped ladders with randomly distributed impurities. We also calculate the average staggered magnetic susceptibility induced in the system by such defects.
The degenerate Hubbard chain with N internal degrees of freedom is integrable if site occupations of more than two electrons are excluded. The ground state of the model has several unusual properties; e.g., the magnetic susceptibility has logarithmic singularities as the field H\ensuremath{\rightarrow}0 and also the specific-heat coefficient is singular. An extensive discussion of ground-state properties as a function of U and the band filling is presented. The elemental charge and spin excitations are derived for arbitrary U and band filling. If, on average, there is exactly one electron per site as a function of U, a qualitative change in the low-energy charge excitations is found at a critical value Uc{\mathit{U}}_{\mathit{c}}Uc​. This change is probably associated with a Mott metal-insulator transition (Uc{\mathit{U}}_{\mathit{c}}Uc​=0 for N=2). The Fermi velocity is finite for UUc{\mathit{U}}_{\mathit{c}}Uc​ but vanishes for U>Uc{\mathit{U}}_{\mathit{c}}Uc​ (for one electron per site). The spin-wave velocity is inversely proportional to the magnetic susceptibility for all U and band fillings. There are N-1 elemental spin-wave branches. For arbitrary band filling there are then N-1 soft modes at different finite momenta, all related to the Fermi surface. For exactly one electron per site, the spin-excitation spectrum can be mapped onto the one of the SU(N) Heisenberg chain. The implications of the hole states on superconductivity are also discussed.
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 $T\ensuremath{\rightarrow}0$.