In this paper we study the low-lying spectrum of the AKLT model perturbed by small, finite-range potentials and with open boundary conditions imposed at the edges of the chain. Our analysis is based on the local, iterative Lie Schwinger block-diagonalization method which allows us to control small interaction terms localized near the boundary of the chain that are responsible for the possible splitting of the ground-state energy of the AKLT Hamiltonian into energy levels separated by small gaps, for which we provide an explicit formula. This improves earlier results concerning the persistence of the so-called bulk gap in these models, besides illustrating the power of our general methods in a non-trivial application.In this paper we study the low-lying spectrum of the AKLT model perturbed by small, finite-range potentials and with open boundary conditions imposed at the edges of the chain. Our analysis is based on the local, iterative Lie Schwinger block-diagonalization method which allows us to control small interaction terms localized near the boundary of the chain that are responsible for the possible splitting of the ground-state energy of the AKLT Hamiltonian into energy levels separated by small gaps, for which we provide an explicit formula. This improves earlier results concerning the persistence of the so-called bulk gap in these models, besides illustrating the power of our general methods in a non-trivial application.

Journal of Functional AnalysisNov 4, 20250 citations

Low energy spectrum of the XXZ model coupled to a magnetic field

Simone Del Vecchio, Jürg Martin Fröhlich, Alessandro Pizzo +1 more

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

Local Topological Quantum Order and Spectral Gap Stability for the AKLT Models on the Hexagonal and Lieb Lattices

Thomas Jackson, Bruno Nachtergaele, Amanda Young

We prove that the ground state of the AKLT models on the hexagonal lattice and the Lieb lattice satisfy the local topological quantum order (LTQO) condition. This will be a consequence of proving that the finite volume ground states are indistinguishable from a unique infinite volume ground state. Concretely, we identify a sequence of increasing and absorbing finite volumes for which any finite volume ground state expectation is well approximated by the infinite volume state with error decaying at a uniform exponential rate in the distance between the support of the observable and boundary of the finite volume. As a corollary to the LTQO property, we obtain that the spectral gap above the ground state in these models is stable under general small perturbations of sufficient decay. We prove these results by a detailed analysis of the polymer representation of the ground states state derived by Kennedy, Lieb and Tasaki (1988) with the necessary modifications required for proving the strong form of ground state indistinguishability needed for LTQO.We prove that the ground state of the AKLT models on the hexagonal lattice and the Lieb lattice satisfy the local topological quantum order (LTQO) condition. This will be a consequence of proving that the finite volume ground states are indistinguishable from a unique infinite volume ground state. Concretely, we identify a sequence of increasing and absorbing finite volumes for which any finite volume ground state expectation is well approximated by the infinite volume state with error decaying at a uniform exponential rate in the distance between the support of the observable and boundary of the finite volume. As a corollary to the LTQO property, we obtain that the spectral gap above the ground state in these models is stable under general small perturbations of sufficient decay. We prove these results by a detailed analysis of the polymer representation of the ground states state derived by Kennedy, Lieb and Tasaki (1988) with the necessary modifications required for proving the strong form of ground state indistinguishability needed for LTQO.

math-phCross-listedarXiv:2605.12184Communications in Mathematical PhysicsMay 13, 20260 citations

Lie-Schwinger block-diagonalization and gapped quantum chains

Froehlich, Juerg, Alessandro Pizzo

We study quantum chains whose Hamiltonians are perturbations by bounded interactions of short range of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. We prove that, for small values of a coupling constant, the spectral gap of the perturbed Hamiltonian above its ground-state energy is bounded from below by a positive constant uniformly in the length of the chain. In our proof we use a novel method based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain.We study quantum chains whose Hamiltonians are perturbations by bounded interactions of short range of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. We prove that, for small values of a coupling constant, the spectral gap of the perturbed Hamiltonian above its ground-state energy is bounded from below by a positive constant uniformly in the length of the chain. In our proof we use a novel method based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain.

arXiv:1812.02457Dec 6, 20181 citation

Local determination of low-energy states in gapped fermion systems with weak long-range interactions

Stefan Teufel

We consider bulk-gapped fermion systems whose infinite-volume bulk is obtained by adding sufficiently weak even many-body interactions to a free Hamiltonian with spectral gap $g>0$ , allowing polynomial spatial decay of both parts. We prove quantitative local determination of low-energy states: whenever another interaction agrees with the bulk interaction inside a finite or infinite region $U$ , with bulk terms crossing its boundary unchanged or removed, all its states with excitation energies in $[0,3g/4]$ have local expectations close to those of the bulk ground state deep inside $U$ . The error decays with the distance to the complement of $U$ , independently of the interaction terms supported outside $U$ and of the chosen low-energy state. Quantitative local determination has strong consequences. (i) The bulk interaction has a unique algebraic ground state. (ii) A strong "local perturbations perturb locally" (LPPL) principle holds: arbitrary distant defects change local ground-state expectations only by a spatially decaying error. (iii) Low-energy excitations in finite samples remain localized near the boundary for all times, up to spatially decaying tails, uniformly over arbitrary even boundary terms. (iv) Correlations from deep inside $U$ to outside $U$ are small, with quantitative bounds in terms of spatial separation and distance to $U^c$ . To prove local determination of low-energy states, we combine quasi-adiabatic transport of the free ground state to the interacting bulk ground state with a new condition, local excitation detectability (LED). A companion paper establishes gap stability and constructs the transport for weak interactions, while we prove LED at sufficiently weak coupling. The abstract theorem is nonperturbative and applies whenever the transport and LED hypotheses hold.We consider bulk-gapped fermion systems whose infinite-volume bulk is obtained by adding sufficiently weak even many-body interactions to a free Hamiltonian with spectral gap $g>0$, allowing polynomial spatial decay of both parts. We prove quantitative local determination of low-energy states: whenever another interaction agrees with the bulk interaction inside a finite or infinite region $U$, with bulk terms crossing its boundary unchanged or removed, all its states with excitation energies in $[0,3g/4]$ have local expectations close to those of the bulk ground state deep inside $U$. The error decays with the distance to the complement of $U$, independently of the interaction terms supported outside $U$ and of the chosen low-energy state. Quantitative local determination has strong consequences. (i) The bulk interaction has a unique algebraic ground state. (ii) A strong "local perturbations perturb locally" (LPPL) principle holds: arbitrary distant defects change local ground-state expectations only by a spatially decaying error. (iii) Low-energy excitations in finite samples remain localized near the boundary for all times, up to spatially decaying tails, uniformly over arbitrary even boundary terms. (iv) Correlations from deep inside $U$ to outside $U$ are small, with quantitative bounds in terms of spatial separation and distance to $U^c$. To prove local determination of low-energy states, we combine quasi-adiabatic transport of the free ground state to the interacting bulk ground state with a new condition, local excitation detectability (LED). A companion paper establishes gap stability and constructs the transport for weak interactions, while we prove LED at sufficiently weak coupling. The abstract theorem is nonperturbative and applies whenever the transport and LED hypotheses hold.

math-phCross-listedarXiv:2610.40337Oct 30, 20260 citations

Local iterative block-diagonalization of gapped Hamiltonians: A new tool in singular perturbation theory

Simone Del Vecchio, Jürg Martin Fröhlich, Alessandro Pizzo +1 more

In this paper, the local iterative Lie–Schwinger block-diagonalization method, introduced and developed in our previous work for quantum chains, is extended to higher-dimensional quantum lattice systems with Hamiltonians that can be written as the sum of an unperturbed gapped operator, consisting of a sum of on-site terms, and a perturbation, consisting of bounded interaction potentials of short range multiplied by a real coupling constant t. Our goal is to prove that the spectral gap above the ground-state energy of such Hamiltonians persists for sufficiently small values of |t|, independently of the size of the lattice. New ideas and concepts are necessary to extend our method to systems in dimension d > 1: As in our earlier work, a sequence of local block-diagonalization steps based on judiciously chosen unitary conjugations of the original Hamiltonian is introduced. The supports of effective interaction potentials generated in the course of these block-diagonalization steps can be identified with what we call minimal rectangles contained in the lattice, a concept that serves to tackle combinatorial problems that arise in the course of iterating the block-diagonalization steps. For a given minimal rectangle, control of the effective interaction potentials generated in each block-diagonalization step with support in the given rectangle is achieved by exploiting a variety of rather subtle mechanisms, which include, for example, the use of weighted sums of paths consisting of overlapping rectangles and of large denominators, expressed in terms of sums of orthogonal projections, which serve to control analogous sums of projections in the numerators resulting from the unitary conjugations of the interaction potential terms involved in the local block-diagonalization step.In this paper, the local iterative Lie–Schwinger block-diagonalization method, introduced and developed in our previous work for quantum chains, is extended to higher-dimensional quantum lattice systems with Hamiltonians that can be written as the sum of an unperturbed gapped operator, consisting of a sum of on-site terms, and a perturbation, consisting of bounded interaction potentials of short range multiplied by a real coupling constant t. Our goal is to prove that the spectral gap above the ground-state energy of such Hamiltonians persists for sufficiently small values of |t|, independently of the size of the lattice. New ideas and concepts are necessary to extend our method to systems in dimension d > 1: As in our earlier work, a sequence of local block-diagonalization steps based on judiciously chosen unitary conjugations of the original Hamiltonian is introduced. The supports of effective interaction potentials generated in the course of these block-diagonalization steps can be identified with what we call minimal rectangles contained in the lattice, a concept that serves to tackle combinatorial problems that arise in the course of iterating the block-diagonalization steps. For a given minimal rectangle, control of the effective interaction potentials generated in each block-diagonalization step with support in the given rectangle is achieved by exploiting a variety of rather subtle mechanisms, which include, for example, the use of weighted sums of paths consisting of overlapping rectangles and of large denominators, expressed in terms of sums of orthogonal projections, which serve to control analogous sums of projections in the numerators resulting from the unitary conjugations of the interaction potential terms involved in the local block-diagonalization step.

arXiv:2007.07667Journal of Mathematical PhysicsJul 1, 20226 citations

We consider bulk-gapped fermion systems whose infinite-volume bulk is obtained by adding sufficiently weak even many-body interactions to a free Hamiltonian with spectral gap

g>0

, allowing polynomial spatial decay of both parts. We prove quantitative local determination of low-energy states: whenever another interaction agrees with the bulk interaction inside a finite or infinite region

U

, with bulk terms crossing its boundary unchanged or removed, all its states with excitation energies in

[0,3g/4]

have local expectations close to those of the bulk ground state deep inside

U

. The error decays with the distance to the complement of

U

, independently of the interaction terms supported outside

U

and of the chosen low-energy state. Quantitative local determination has strong consequences. (i) The bulk interaction has a unique algebraic ground state. (ii) A strong "local perturbations perturb locally" (LPPL) principle holds: arbitrary distant defects change local ground-state expectations only by a spatially decaying error. (iii) Low-energy excitations in finite samples remain localized near the boundary for all times, up to spatially decaying tails, uniformly over arbitrary even boundary terms. (iv) Correlations from deep inside

U

to outside

U

are small, with quantitative bounds in terms of spatial separation and distance to

U^c

. To prove local determination of low-energy states, we combine quasi-adiabatic transport of the free ground state to the interacting bulk ground state with a new condition, local excitation detectability (LED). A companion paper establishes gap stability and constructs the transport for weak interactions, while we prove LED at sufficiently weak coupling. The abstract theorem is nonperturbative and applies whenever the transport and LED hypotheses hold.