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arXiv:1908.06962 · Physical review. B./Physical review. B

String and conventional order parameters in the solvable modulated quantum chain

Gennady Y. Chitov, Toplal Pandey, P. N. Timonin

Published Sep 23, 2019• 13 citations
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Abstract

The phase diagram and the order parameters of the exactly solvable quantum one-dimensional model are analyzed. The model in its spin representation is the dimerized $XY$ spin chain in the presence of uniform and staggered transverse fields. In the fermionic representation this model is the dimerized noninteracting Kitaev chain with a modulated chemical potential. The model has a rich phase diagram which contains phases with local and nonlocal (string) orders. We have calculated within the same systematic framework the local order parameters (spontaneous magnetization) and the nonlocal string order parameters, along with the topological winding numbers for all domains of the phase diagram. The topologically nontrivial phase is shown to have a peculiar oscillating string order with the wave number $q=\ensuremath{\pi}/2$, awaiting for its experimental confirmation.

DOI
10.1103/physrevb.100.104428

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The phase diagram and the order parameters of the exactly solvable quantum one-dimensional model are analyzed. The model in its spin representation is the dimerized XYXYXY spin chain in the presence of uniform and staggered transverse fields. In the fermionic representation this model is the dimerized noninteracting Kitaev chain with a modulated chemical potential. The model has a rich phase diagram which contains phases with local and nonlocal (string) orders. We have calculated within the same systematic framework the local order parameters (spontaneous magnetization) and the nonlocal string order parameters, along with the topological winding numbers for all domains of the phase diagram. The topologically nontrivial phase is shown to have a peculiar oscillating string order with the wave number q=\ensuremathπ/2q=\ensuremath{\pi}/2q=\ensuremathπ/2, awaiting for its experimental confirmation.
The XYZ antiferromagnetic spin-1/21/21/2 chain with alternation of the exchange and anisotropy couplings in the presence of uniform and staggered axial magnetic fields is studied. The analysis is done using the effective quadratic fermionic Hamiltonian resulting from the Hartee-Fock approximation. Combining the exact and the mean-field methods, the local and string order parameters on the ground-state phase diagram of the model are identified and calculated. We found a topological phase with oscillating string order with a period of four lattice spacings, not reported before for this model. A detailed analysis of patterns of the string order is given. The special XXZ limit of the model with additional U(1) symmetry brings about, in agreement with the Lieb-Schultz-Mattis theorem and its extensions, plateaux of magnetization and some additional conserving quantities. We have shown that in the XYZ chain, where the plateaux are smeared, the robust oscillating string order parameter is continuously connected to its XXZ limit. Also, the nontrivial winding number and zero-energy localized Majorana edge states, as additional attributes of topological order, are robust in that phase, even off the line of U(1) symmetry.
We study the exactly solvable one-dimensional model: the dimerized XYXYXY 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)U(1)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.
Dimerized antiferromagnetic spin-1/2 chains and ladders demonstrate quantum critical phase transition, the existence or absence of which is dependent on the dimerization and the dimerization pattern of the chain and the ladder, respectively. The gapped phases can not be distinguished by the conventional Landau long-range order parameters. However, they possess non-local topological string order parameters which can be used to classify different phases. We utilize the self-consistent free fermionic approximation and some standard results for exactly solved models to analytically calculate the string order parameters of dimerized spin chains. As a complement parameter the gapped phases possess the topological number, called the winding number and they are characterized by different integer values of the winding number. In order to calculate the string order parameters and winding numbers in dimerized spin chains and two-leg ladders we use analytical methods such as the Jordan-Wigner transformation, mean-field approximation, duality transformations, and some standard results available for the exactly 1D solve models. It is shown that the winding number provides the complementary framework to the string order parameter to characterize the topological gapped phases.
We have calculated order parameters for the phases of the Kitaev chain with interaction and dimerization at a special symmetric point applying the Jordan-Wigner and other duality transformations. We use string order parameters (SOPs) defined via the correlation functions of the Majorana string operators. The SOPs are mapped onto the local order parameters of some dual Hamiltonians and easily calculated. We have shown that the phase diagram of the interacting dimerized chain comprises the phases with the conventional local order as well as the phases with nonlocal SOPs. From the results for the critical indices, we infer the two-dimensional Ising universality class of criticality at the particular symmetry point where the model is exactly solvable.
We study the spin-12\frac{1}{2}21​ Kitaev-Heisenberg (KJ) model in a two-leg ladder. Without a Heisenberg interaction, the Kitaev phase in the ladder model has Majorana fermions with local Z2{Z}_{2}Z2​ gauge fields, and is usually described as a disordered phase without any order parameter. Here we identify a long-range nonlocal string order parameter (SOP) in the Kitaev phase which survives with a finite Heisenberg interaction. The SOP is obtained by relating the Kitaev ladder, through a nonlocal unitary transformation, to a one-dimensional XYXYXY chain with an Ising coupling to a dangling spin at every site. This differentiates the Kitaev phases from other nearby phases including a rung singlet. Two phases with nonzero SOP corresponding to ferromagnetic and antiferromagnetic Kitaev interactions are identified. The full phase diagram of the KJ ladder is determined using exact diagonalization and density matrix renormalization group methods, which shows a striking similarity to the KJ model on a two-dimensional honeycomb lattice.
In this letter, we propose an application of string order parameter (SOP), commonly used in quantum spin systems, to identify symmetry-protected topological phase (SPT) in fermionic systems in the example of the dimerized fermionic chain. As a generalized form of dimerized model, we consider a one-dimensional spin-1/2 XX model with alternating spin couplings. We employ Jordan-Wigner fermionization to map this model to the spinless Su-Schrieffer-Heeger fermionic model (SSH) with generalized hopping signs. We demonstrate a phase transition between a trivial insulating phase and the Haldane phase by the exact analytical evaluation of reconstructed SOPs which are represented as determinants of Toeplitz matrices with the given generating functions. To get more insight into the topological quantum phase transition (tQPT) and microscopic correlations, we study the pairwise concurrence as a local entanglement measure of the model. We show that the first derivative of the concurrence has a non-analytic behaviour in the vicinity of the tQPT, like in the second order trivial QPTs.