We study the spectrum of fermions hopping on a chain with a weak incommensuration close to dimerization; both
q, the deviation of the wave number from $\ensuremath{\pi},$ and $\ensuremath{\delta},$ the strength of the incommensuration, are small. For free fermions, we use a continuum Dirac theory to show that there are an infinite number of bands which meet at zero energy as q approaches zero. In the limit that the ratio $q/\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\delta}}0,$ the number of states lying inside the
q=0 gap is nonzero and equal to $2\ensuremath{\delta}/{\ensuremath{\pi}}^{2}.$ Thus the limit $\stackrel{\ensuremath{\rightarrow}}{q}0$ differs from
q=0; this can be seen clearly in the behavior of the specific heat at low temperature. For interacting fermions or the
XXZ spin-
1/2 chain close to dimerization, we use bosonization and a renormalization group analysis to argue that similar results hold; as $\stackrel{\ensuremath{\rightarrow}}{q}0,$ there is a nontrivial density of states near zero energy. However, the limit $\stackrel{\ensuremath{\rightarrow}}{q}0$ and
q=0 give the same results near commensurate wave numbers which are different from $\ensuremath{\pi};$ for both free and interacting fermions, we find that a nonzero value of q is necessary to close the gap. Our results for free fermions are applied to the Azbel-Hofstadter problem of electrons hopping on a two-dimensional lattice in the presence of a magnetic field. Finally, we discuss the complete energy spectrum of free fermions with incommensurate hopping by going up to higher orders in $\ensuremath{\delta}.$
We study the exactly solvable one-dimensional model: the dimerized
XY chain with uniform and staggered transverse fields, equivalent upon fermionization to the noninteracting dimerized Kitaev-Majorana chain with modulation. The model has three known gapped phases with local and nonlocal (string) orders, along with the gapless incommensurate (IC) phase in the
U(1) limit. The criticality is controlled by the properties of zeros of model's partition function, analytically continued onto the complex wave numbers. In the ground state they become complex zeros of the spectrum of the Hamiltonian. The analysis of those roots yields the phase diagram which contains continuous quantum phase transitions and weaker singularities known as disorder lines (DLs) or modulation transitions. The latter are shown to occur in two types: DLs of the first kind with continuous appearance of the IC oscillations, and DLs of the second kind corresponding to a jump of the wave number of oscillations. The salient property of zeros of the spectrum is that the ground state is shown to be separable (factorized), and the model is disentangled on a subset of the DLs. From analysis of those zeros we also find the Majorana edge states and their wave functions.
We investigate the zero magnetization and ground state phase diagram of a spin-1/2 chain with competing ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor exchange couplings in the strongly interacting regime. Using density matrix renormalization group (DMRG) simulations, we discover two successive commensurate-incommensurate transitions of the nonconformal Pokrovsky-Talapov universality class, occurring even at zero magnetic field. We argue that the first transition marks the proliferation of solitonic discommensurations (kink-antikink pairs) in a critical phase with central charge c=2, emerging from a period-4 gapped phase. At the second transition, an incommensurate quadrupolar (or nematic) Luttinger liquid forms out of a gapped phase separation, via the condensation of two-magnon bound pairs. In the strongly interacting limit, the location of the two Pokrovsky-Talapov transitions can be analytically predicted with degenerate perturbation theory. Our results demonstrate that frustration alone is sufficient to drive continuous commensurate-incommensurate transitions of Mott type and stabilize complex objects with incommensurate quasi-long-range order without doping.
Quantum phase transition in dimerized antiferromagnetic Heisenberg spin chain has been studied. A staircase structure in the variation of concurrence within strongly coupled pairs with that of external magnetic field has been observed indicating multiple critical points. Emergence of entanglement due to external magnetic field or magnetic entanglement is observed for weakly coupled spin pairs in the same dimer chain. Though closed dimerized isotropic XXX Heisenberg chains with different dimer strengths were mainly explored, analogous studies on open chains as well as closed anisotropic (XX interaction) chains with tilted external magnetic field have also been studied.
The behavior in an external magnetic field is studied exactly for a wide class of multichain quantum spin models. It is shown that the magnetic field together with the interchain couplings cause commensurate–incommensurate phase transitions between the gapless phases in the ground state. The conformal limit of these models is studied and it is shown that the low-lying excitations for the incommensurate phases are not independent, because they are governed by the same magnetic field (chemical potential for excitations). A scenario for the transition from one to two space dimensions for the exactly integrable multichain quantum spin models is proposed, and it is shown that the incommensurate phases in an external magnetic field disappear in the limit of an infinite number of coupled spin chains. The similarities in the external field behavior for the quantum multichain spin models and a wide class of quantum field theories are discussed. The scaling exponents for the appearance of the gap in the spectrum of low-lying excitations of the quantum multichain models due to the relevant perturbations of the integrable theories are calculated.