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Leading and next-to-leading-order finite-size corrections to the ground and first excited states are calculated for the spin-1/2 anisotropic Heisenberg model in the critical region. The analytic results are compared to numerical data obtained for chains up to a length of N=1024. It is found that, near the isotropic point, the asymptotic region where the results obtained for N to infinity are applicable sets in at very large N values, and for obtaining good accuracy in fitting the numerical data one has to take into account several correction terms, even at large (N>100) chain lengths.

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We consider a generalization of the quantum Rabi model where the twolevel system and the single-mode cavity oscillator are coupled by an additional Starklike term. By adapting a method recently introduced by Braak [Phys. Rev. Lett. 107, 100401 (2011)], we solve the model exactly. The low-lying spectrum in the experimentally relevant ultrastrong and deep strong regimes of the Rabi coupling is found to exhibit two striking features absent from the original quantum Rabi model: avoided level crossings for states of the same parity and an anomalously rapid onset of two-fold near-degenerate levels as the Rabi coupling increases.

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