The Jordan-Wigner transformation is frequently utilised to rewrite quantum spin chains in terms of fermionic operators. When the resulting Hamiltonian is bilinear in these fermions, i.e. the fermions are free, the exact spectrum follows from the eigenvalues of a matrix whose size grows only linearly with the volume of the system. However, several Hamiltonians that do not admit a Jordan-Wigner transformation to fermion bilinears still have the same type of free-fermion spectra. The spectra of such “free fermions in disguise” models can be found exactly by an intricate but explicit construction of the raising and lowering operators. We generalise the methods further to find a family of such spin chains. We compute the exact spectrum, and generalise an elegant graph-theory construction. We also explain how this family admits an N=2 lattice supersymmetry.
It is shown that a solvable Hamiltonian can be obtained from a series of operators satisfying specific commutation relations. A transformation that diagonalize the Hamiltonian is obtained simultaneously. The two-dimensional Ising model with periodic interactions, the one-dimensional XY model with period 2, the transverse Ising chain, the one-dimensional Kitaev model and the cluster model, and other composite quantum spin chains are diagonalized following this procedure. The Jordan–Wigner transformation, the transformation from the Pauli spin operators to the Majorana fermion used by Shankar and Murthy, and the transformation introduced by Nambu, are special cases of this treatment.
An infinite number of spin chains are solved and it is derived that the ground-state phase transitions belong to the universality classes with central charge c = m / 2 , where m is an integer. The models are diagonalized by automatically obtained transformations, many of which are different from the Jordan–Wigner transformation. The free energies, correlation functions, string order parameters, exponents, central charges, and the phase diagram are obtained. Most of the examples consist of the stabilizers of the cluster state. A unified structure of the one-dimensional XY and cluster-type spin chains is revealed, and other series of solvable models can be obtained through this formula.
The relationship between the eigenspectrum of Ising and XY quantum chains is well known. Although the Ising model has a Z(2) symmetry and the XY model a U(1) symmetry, both models are described in terms of free-fermionic quasiparticles. The fermionic quasienergies are obtained by means of a Jordan-Wigner transformation. On the other hand, there exists in the literature a huge family of Z(N) quantum chains whose eigenspectra, for N>2, are given in terms of free parafermions, and they are not derived from the standard Jordan-Wigner transformation. The first members of this family are the Z(N) free-parafermionic Baxter quantum chains. In this paper, we introduce a family of XY models that, beyond two-body, also have N-multispin interactions. Similar to the standard XY model, they have a U(1) symmetry and are also solved by the Jordan-Wigner transformation. We show that with appropriate choices of the N-multispin couplings, the eigenspectra of these XY models are given in terms of combinations of Z(N) free-parafermionic quasienergies. In particular, all the eigenenergies of the Z(N) free-parafermionic models are also present in the related free-fermionic XY models. The correspondence is established via the identification of the characteristic polynomial, which fixes the eigenspectrum. In the Z(N) free-parafermionic models, the quasienergies obey an exclusion circle principle that is not present in the related N-multispin XY models.
The relationship between the eigenspectrum of Ising and
XY quantum chains is well known. Although the Ising model has a
Z(2) symmetry and the
XY model a
U(1) symmetry, both models are described in terms of free-fermionic quasiparticles. The fermionic quasienergies are obtained by means of a Jordan-Wigner transformation. On the other hand, there exists in the literature a huge family of
Z(N) quantum chains whose eigenspectra, for
N>2, are given in terms of free parafermions, and they are not derived from the standard Jordan-Wigner transformation. The first members of this family are the
Z(N) free-parafermionic Baxter quantum chains. In this paper, we introduce a family of
XY models that, beyond two-body, also have
N-multispin interactions. Similar to the standard
XY model, they have a
U(1) symmetry and are also solved by the Jordan-Wigner transformation. We show that with appropriate choices of the
N-multispin couplings, the eigenspectra of these
XY models are given in terms of combinations of
Z(N) free-parafermionic quasienergies. In particular, all the eigenenergies of the
Z(N) free-parafermionic models are also present in the related free-fermionic
XY models. The correspondence is established via the identification of the characteristic polynomial, which fixes the eigenspectrum. In the
Z(N) free-parafermionic models, the quasienergies obey an exclusion circle principle that is not present in the related
N-multispin
XY models.
Abstract We establish a direct connection between inhomogeneous XX spin chains (or free fermion systems with nearest-neighbors hopping) and certain QES models on the line giving rise to a family of weakly orthogonal polynomials. We classify all such models and their associated XX chains, which include two families related to the Lamé (finite gap) quantum potential on the line. For one of these chains, we numerically compute the Rényi bipartite entanglement entropy at half filling and derive an asymptotic approximation thereof by studying the model’s continuum limit, which turns out to describe a massless Dirac fermion on a suitably curved background. We show that the leading behavior of the entropy is that of a c = 1 critical system, although there is a subleading log(log N ) correction (where N is the number of sites) unusual in this type of models.