For a class of Hamiltonians of XXZ spin chains in a uniform external magnetic field that are small quantum perturbations of an Ising Hamiltonian, it is shown that the spectral gap above the ground-state energy remains strictly positive when the perturbation is turned on, uniformly in the length of the chain. This result is proven for rather generic perturbations of both the ferromagnetic and the antiferromagnetic Ising Hamiltonian. In the antiferromagnetic case, the external magnetic field is required to be small. For a chain of an even number of sites, the two-fold degenerate ground-state energy of the unperturbed antiferromagnetic Hamiltonian may split into two energy levels separated by a very small gap. These results are proven by using a new, quite subtle refinement of a method developed in earlier work and used to iteratively block-diagonalize Hamiltonians of systems confined to ever larger subsets of a lattice by using strictly local unitary conjugations. The new method developed in this paper provides complete control of boundary effects on the low-energy spectrum of perturbed Ising chains uniformly in their length.For a class of Hamiltonians of XXZ spin chains in a uniform external magnetic field that are small quantum perturbations of an Ising Hamiltonian, it is shown that the spectral gap above the ground-state energy remains strictly positive when the perturbation is turned on, uniformly in the length of the chain. This result is proven for rather generic perturbations of both the ferromagnetic and the antiferromagnetic Ising Hamiltonian. In the antiferromagnetic case, the external magnetic field is required to be small. For a chain of an even number of sites, the two-fold degenerate ground-state energy of the unperturbed antiferromagnetic Hamiltonian may split into two energy levels separated by a very small gap. These results are proven by using a new, quite subtle refinement of a method developed in earlier work and used to iteratively block-diagonalize Hamiltonians of systems confined to ever larger subsets of a lattice by using strictly local unitary conjugations. The new method developed in this paper provides complete control of boundary effects on the low-energy spectrum of perturbed Ising chains uniformly in their length.

arXiv:2306.02772Advances in MathematicsMay 18, 20261 citation

XXZ spin- 12 representation of a finite- U Bose-Hubbard chain at half-integer filling

Domenico Giuliano, Davide Rossini, Pasquale Sodano +1 more

Using a similarity Hamiltonian renormalization procedure, we determine an effective spin- $\frac{1}{2}$ representation of the Bose-Hubbard model at half-integer filling and at a finite onsite interaction energy $U$ . By means of bosonization, we are able to recast the effective Hamiltonian as that of a spin- $\frac{1}{2}$ XXZ magnetic chain with pertinently renormalized coupling and anisotropy parameters. We use this mapping to provide analytical estimates of the correlation functions of the Bose-Hubbard model. We then compare such results with those based on density matrix renormalization group numerical simulations of the Bose-Hubbard model for various values of $U$ and for a number $L$ of lattice sites as low as $L\ensuremath{\sim}30$. We find an excellent agreement up to $10%$ between the output of analytical and numerical computations, even for relatively small values of $U$ . Our analysis implies that, also at finite $U$ , the one-dimensional Bose-Hubbard model with suitably chosen parameters may be seen as a quantum simulator of the XXZ chain.Using a similarity Hamiltonian renormalization procedure, we determine an effective spin-$\frac{1}{2}$ representation of the Bose-Hubbard model at half-integer filling and at a finite onsite interaction energy $U$. By means of bosonization, we are able to recast the effective Hamiltonian as that of a spin-$\frac{1}{2}$ XXZ magnetic chain with pertinently renormalized coupling and anisotropy parameters. We use this mapping to provide analytical estimates of the correlation functions of the Bose-Hubbard model. We then compare such results with those based on density matrix renormalization group numerical simulations of the Bose-Hubbard model for various values of $U$ and for a number $L$ of lattice sites as low as $L\ensuremath{\sim}30$. We find an excellent agreement up to $10%$ between the output of analytical and numerical computations, even for relatively small values of $U$. Our analysis implies that, also at finite $U$, the one-dimensional Bose-Hubbard model with suitably chosen parameters may be seen as a quantum simulator of the XXZ chain.

arXiv:1210.2685Physical Review BJan 3, 201321 citations

Statistical and dynamical properties in the energy spectra of XXZ spin chains

和恵工藤

Jan 1, 20050 citations

Thermodynamics of the quantum spin-S XXZ chain

Onofre Rojas, S. M. de Souza, E. V. Corrêa Silva +1 more

arXiv:cond-mat/0504527The European Physical Journal BAug 1, 20051 citation

Spinon excitations in the spin-1 XXZ chain and hidden supersymmetry

Chihiro Matsui

We study spinon excitations of the integrable spin-1 (Fateev–Zamolodchikov; FZ) chain and their relation to the hidden supersymmetry. Using the notion of the supercharges earlier introduced to the spin chains, which change the system length by one, we found that they nontrivially act on one of two kinds of the degrees of freedom for the FZ chain. Their actions were obtained to be the same as those of the supercharges defined on the supersymmetric sine-Gordon model, the low-energy effective field theory of the FZ chain. Moreover, we construct the eigenstates which are invariant under the supersymmetric Hamiltonian given as the anti-commutator of the supercharges.We study spinon excitations of the integrable spin-1 (Fateev–Zamolodchikov; FZ) chain and their relation to the hidden supersymmetry. Using the notion of the supercharges earlier introduced to the spin chains, which change the system length by one, we found that they nontrivially act on one of two kinds of the degrees of freedom for the FZ chain. Their actions were obtained to be the same as those of the supercharges defined on the supersymmetric sine-Gordon model, the low-energy effective field theory of the FZ chain. Moreover, we construct the eigenstates which are invariant under the supersymmetric Hamiltonian given as the anti-commutator of the supercharges.

arXiv:1607.04317Nuclear Physics BSep 13, 20165 citations

Using a similarity Hamiltonian renormalization procedure, we determine an effective spin-

\frac{1}{2}

representation of the Bose-Hubbard model at half-integer filling and at a finite onsite interaction energy

U

. By means of bosonization, we are able to recast the effective Hamiltonian as that of a spin-

\frac{1}{2}

XXZ magnetic chain with pertinently renormalized coupling and anisotropy parameters. We use this mapping to provide analytical estimates of the correlation functions of the Bose-Hubbard model. We then compare such results with those based on density matrix renormalization group numerical simulations of the Bose-Hubbard model for various values of

U

and for a number

L

of lattice sites as low as $L\ensuremath{\sim}30$. We find an excellent agreement up to

10%

between the output of analytical and numerical computations, even for relatively small values of

U

. Our analysis implies that, also at finite

U

, the one-dimensional Bose-Hubbard model with suitably chosen parameters may be seen as a quantum simulator of the XXZ chain.