Boundary effects and the stability of the low energy spectrum of the AKLT model

To introduce some notation used throughout our paper we recapitulate the definition of the AKLT model and recall its main features.

Our main result is the following theorem proven in Section 3 (see Theorem 3.4).
Theorem. There exists some constant $\bar{t}>0$ independent of the number $N$ of sites in $\Lambda$ such that, for any real coupling constant $t$ with $|t|<\bar{t}$ and for all $1<N<\infty$,

1. (i)

   the spectrum of $K_{\Lambda}(t)$ is contained in two disjoint, $t$-dependent regions $\sigma^{+}$ and $\sigma^{-}$ separated by a gap $\Delta_{\Lambda}(t)\geq\frac{\varepsilon}{4}$, with $\varepsilon$ independent of $N$, as specified in Theorem 1.2; i.e., $E^{\prime}-E^{\prime\prime}>\Delta_{\Lambda}(t)$, for all $E^{\prime}\in\sigma^{+}$ and all $E^{\prime\prime}\in\sigma^{-}$;

2. (ii)

   for any $d\in\mathbb{N}\cap[1\,,\,\frac{N}{2})$, the eigenspace corresponding to the eigenvalues contained in $\sigma^{-}$ is four-dimensional; the gaps between the eigenvalues in $\sigma^{-}$ coincide with the gaps between the eigenvalues of the symmetric matrix

   $$P^{(-)}_{\Lambda}\,\Big{(}\,t\sum_{i=1}^{d}\,V_{i,i+1}+\,t\,\sum_{i=N-d}^{N-1}V_{i,i+1}\Big{)}\,P^{(-)}_{\Lambda}\,,$$

   (1.14)

   up to corrections bounded by

   $$|t|\cdot 3^{-(d-1)}\,+\,o(|t|)\,.$$

Acknowledgements. A.P. acknowledges support through the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006, and also support through GNFM - INDAM.

It will be convenient to think of a macroscopic (finite) lattice with left endpoint $X=1$, right endpoint $X=N$, and lattice spacing $\sqrt{t^{-1}}$. The $M^{th}$ site of this lattice is the point

of the microscopic lattice $\Lambda$. The set $\mathcal{I}_{K,J}$ is the interval (i.e., a subset of $\Lambda$ consisting of successive sites) whose endpoints coincide⁵⁵5In general, if the range, $\kappa$, of the interaction terms is larger than $1$, the intervals $\mathcal{I}_{K,J}$ are defined in such a way that they overlap, i.e., the endpoints coincide with the sites $M=J$ and $M=J+K$ only up to corrections depending on $\kappa$ and, consequently, $K\cdot\sqrt{t^{-1}}$ is its length up to corrections of the order the step length, $1$, of the microscopic lattice where the model is defined. with the sites $M=J$ and $M=J+K$ of the macroscopic lattice. Notice that it can be helpful to think of the sets $\mathcal{I}_{K,J}$ (and of some enlargements of these sets defined later on) as intervals contained in the real line; for examples, see Figures 1, 2, and 3, which are intended to display the overlap between such sets. As an interval of the real line, $\mathcal{I}_{K,J}$ has length $K$ in units of $\sqrt{t^{-1}}$.

In the following it will be useful to introduce an ordering relation amongst the intervals labeled by the pairs $(K,J)$ with the property that shorter intervals precede longer ones. This relation is specified as follows.

The interval $\mathcal{I}_{1,J}$ is the support of the operator

Thanks to (1.12) and to our definition of the size, $|\mathcal{I}_{1,J}|$, of the interval $\mathcal{I}_{1,J}$, namely $|\mathcal{I}_{1,J}|=\sqrt{t^{-1}}$, the following operator norm estimate holds,

After dividing them by $\sqrt{t^{-1}}$, we denote such a collection of potentials by $V_{\mathcal{I}_{1,J}}$; i.e.,

the support of $V_{\mathcal{I}_{1,J}}$ being $\mathcal{I}_{1,J}$. The observation in (2.6) implies that $\|\,V_{\mathcal{I}_{1,J}}\,\|\leq 1$. In order to implement the block-diagonalization procedure, it is convenient to re-write the Hamiltonian $K_{\Lambda}(t)$ using these definitions; i.e.,

The block-diagonalization is based on spectral projections, $P_{\mathcal{I}_{K,Q}}^{(\pm)}$, associated with intervals $\mathcal{I}_{K,Q}$, which we define next.

In order to explain how the method in [FP] has to be modified because of specific features of the AKLT model (as compared to the models with ultralocal Hamiltonians treated in [FP]), we first observe that the procedure presented in that reference is based on an iterative block-diagonalization of the perturbing potentials in the Hamiltonian of those models involving Lie-Schwinger conjugations, assuming that the coupling constant, $t(=|t|)$, of the perturbation is sufficiently small. In this paper, too, the block-diagonalization is implemented with the help of local Lie-Schwinger conjugations; and “local” means that each conjugation involves only operators supported in an interval $\mathcal{I}_{1,J}$ (of length 1 and with left endpoint in $J$ in the macroscopic lattice). More precisely, in the notations of Section 1.1, the local Hamiltonian supported in the interval $\mathcal{I}_{1,J}$ is conjugated by a suitable local unitary operator. Specifically,

(where $H^{0}_{\mathcal{I}_{1,J}}$ is defined in (1.5)) is conjugated by a suitably defined unitary operator $e^{Z_{\mathcal{I}_{1,J}}}$,

with the purpose to render the new potentials $V^{\prime}_{\mathcal{I}_{1,J}}\equiv V^{\prime}_{\mathcal{I}_{1,J}}(t)$ block-diagonal with respect to the projections $P^{(-)}_{\mathcal{I}_{1,J}}\,,\,P^{(+)}_{\mathcal{I}_{1,J}}$; see Definition 2.5.

In the algorithm designed in [FP], the steps of the block-diagonalization are indexed by pairs $(K,Q)$ labelling the intervals $\mathcal{I}_{K,Q}$ (for which we have introduced an ordering relation in Definition 2.4); that is, in step $(K,Q)$, the potential, $V_{\mathcal{I}_{K,Q}}^{\,(K,Q)_{-1}}$ – which is the potential obtained in the previous step (i.e., in step $(K,Q)_{-1}$) and is supported in $\mathcal{I}_{K,Q}$ – gets block-diagonalized. In the process new terms, given by

(where $ad$ stands for adjoint action; check (2.40) for its definition) are created that contribute to new interaction potentials, $V^{(K,Q)}_{\mathcal{I}_{K^{\prime},Q^{\prime}}\cup\mathcal{I}_{K,Q}}$, supported in larger intervals $\mathcal{I}_{K^{\prime},Q^{\prime}}\cup\mathcal{I}_{K,Q}$. All the terms created by the block-diagonalization procedure with support in the interval $\mathcal{I}_{K^{\prime\prime},Q^{\prime\prime}}:=\mathcal{I}_{K^{\prime},Q^{\prime}}\cup\mathcal{I}_{K,Q}$ are lumped together. To control the size of $V_{\mathcal{I}_{K^{\prime\prime},Q^{\prime\prime}}}$ one has to count certain growth processes (of intervals) yielding a given interval $\mathcal{I}_{K^{\prime\prime},Q^{\prime\prime}}:=\mathcal{I}_{K^{\prime},Q^{\prime}}\cup\mathcal{I}_{K,Q}$. The number of such growth processes can easily be estimated to be at most exponential, i.e., to be bounded above by $C^{K^{\prime\prime}}$, for some universal constant $C>1$. Since

it is then quite easy to inductively prove a bound of the type

for some constant $\rho$, with $0<\rho<\frac{1}{2}$, provided that $|t|$ is small enough, uniformly in the number $N$ of sites of the chain. In the following, we will always assume (w.l.o.g.) that $t\geq 0$, and our results will hold under the assumption that $t<\bar{t}$, for some constant $\bar{t}$ independent of $N(=|\Lambda|)$.

Before describing the structure of the Hamiltonian obtained in each step of the block-diagonali-zation procedure (see the definitions contained in Section 2.4), we discuss some new ingredients incorporated into the algorithm (see Section 2.5) yielding the new potentials in each step of the block-diagonalization. The need for new ingredients becomes apparent already in the steps performed to block-diagonalize the bare potentials $V_{\mathcal{I}_{1,J}}$ . The conjugation of the Hamiltonian $K_{\Lambda}$ by the unitary operator $e^{Z_{\mathcal{I}_{1,J}}}$, $1<J<N\cdot\sqrt{t}+1$, has the effect to not only “hook up” to bare interaction potentials, for example $\sqrt{t}\,V_{\mathcal{I}_{1,J-1}}$ and $\sqrt{t}\,V_{\mathcal{I}_{1,J+1}}$, but to also “hook up” to terms of the unperturbed Hamiltonians $H^{0}_{\mathcal{I}_{1,J-1}}\,,\,H^{0}_{\mathcal{I}_{1,J+1}}$, namely to the two projections

where $i_{-}$ and $i_{+}$ are the sites of the microscopic lattice corresponding to the endpoints of the interval $\mathcal{I}_{1,J}$; hence, in the conjugation, $\mathcal{P}^{(2)}_{i_{-}-1,i_{-}}$ and $\mathcal{P}^{(2)}_{i_{+},i_{+}+1}$ (that do not belong to the local Hamiltonian $H^{0}_{\mathcal{I}_{1,J}}$) get “hooked up" to other terms. Indeed, following the strategy of [FP], we should define an anti-symmetric matrix $Z_{\mathcal{I}_{1,J}}$ in order to block-diagonalize the interaction potential $V_{\mathcal{I}_{1,J}}$ and observe that, in the course of the conjugation generated by $Z_{\mathcal{I}_{1,J}}$, new terms of the type

are created whenever

We refer to this process in the conjugation as a “hooking” of $\mathcal{P}^{(2)}_{i,i+1}$ terms.

We warn the reader that the conjugation used in the block-diagonalization step $(1,J)$ is generated by an anti-symmetric matrix $Z_{\mathcal{I}^{*}_{1,J}}$ supported in a somewhat larger interval $\mathcal{I}_{1,J}^{*}\supset\mathcal{I}_{1,J}$. In this informal description, we attempt to explain the problems that would arise in the block-diagonalization if the matrix $Z_{\mathcal{I}_{1,J}}$ supported in the interval $\mathcal{I}_{1,J}$ were used.

The new interaction terms (2.16) show some important differences as compared to the operators in (2.12):

1. 1.

   Although the support of the new term displayed in (2.16) coincides with $\mathcal{I}_{1,J}$, up to a single site, the control of its norm is quite difficult, since the counterpart of (2.12) is

   $$\sum_{n=1}^{\infty}\frac{1}{n!}\,ad^{n}Z_{\mathcal{I}_{1,J}}(\frac{\mathcal{P}^{(2)}_{i,i+1}}{\sqrt{t}})\,,$$

   (2.18)

   which cannot be estimated, in a manner similar to (2.13), in terms of

   $$\mathcal{O}(\sqrt{t}\cdot\|V_{\mathcal{I}_{1,J}}^{\,(1,J)_{-1}}\|\cdot\|\frac{\mathcal{P}^{(2)}_{i,i+1}}{\sqrt{t}}\|\,)\,;$$

   indeed this might appear to make it impossible to prove an inductive estimate as in (2.14).
   

2. 2.

   Unless a term supported in $\mathcal{I}_{K,Q}\cup\{i,i+1\}$ is already block-diagonal, it should be lumped to the effective potential $V^{(K,Q)}_{\mathcal{I}_{K^{\prime},Q^{\prime}}}$, for some interval $\mathcal{I}_{K^{\prime},Q^{\prime}}$ with

   $$\mathcal{I}_{K^{\prime},Q^{\prime}}\supset\mathcal{I}_{K,Q}\cup\{i,i+1\}\,.$$

The complications described here force us to modify the method proposed in [FP]: in the present paper, the strict locality of the on-site operators studied in that paper is given up and replaced by a locality property expressed in terms of decay properties of the Green functions of the local Hamiltonians $H^{0}_{\mathcal{I}}$, or, equivalently, by Lieb-Robinson bounds associated with the one-parameter groups generated by the operators $H^{0}_{\mathcal{I}}$. Locality is exploited in a careful study of the operator in (2.16), but associated with an enlarged interval $\mathcal{I}^{*}_{1,J}\supset\mathcal{I}_{1,J}$ introduced in Definition 2.8, below. Thus, in order to block-diagonalize an effective potential supported in $\mathcal{I}_{K,Q}$, we shall use a local unperturbed Hamiltonian with support in a larger interval $\mathcal{I}^{*}_{K,Q}\supset\mathcal{I}_{K,Q}$. The Lieb-Robinson bound in (1.10) will be used to show that the off-diagonal part of the operator

w.r.t. to spectral projections associated with the enlarged interval $\overline{\mathcal{I}^{*}_{1,J}}$ (introduced in Definition 2.10),

has a norm that decays in $t$ at least as fast as

Here one uses that the distance between $\mathcal{I}_{K,Q}$ and the endpoints of $\mathcal{I}^{*}_{K,Q}$ is of order $\sqrt{t^{-1}}$. This enables us to lump this term, as well as the terms corresponding to $n\geq 2$ in the expression

together with a potential term supported in a larger interval containing $\mathcal{I}^{*}_{1,J}$, which will be block-diagonalized in a subsequent step. As for the diagonal part of the operator in (2.19), no extra power of $t$ is gained from the argument based on the Lieb-Robinson bounds; but this is not a problem, because this term does not need to be block-diagonalized anymore.

Thanks to the property in (1.8), the use of enlarged intervals also solves a problem⁶⁶6In the following we only try to convey the main ideas underlying our modification of the block-diagonalization procedure. related to the degeneracy of the ground-state eigenvalue of Hamiltonians of the type $H^{0}_{\mathcal{I}}$: the new potential, which we will denote by $V^{(K,Q)}_{\overline{\mathcal{I}^{*}_{K,Q}}}$, is supported in the enlarged interval $\overline{\mathcal{I}^{*}_{K,Q}}$ after the block-diagonalization of the potential $V^{(K,Q)_{-1}}_{\mathcal{I}_{K,Q}}$ and is not given by the full-fledged Lie-Schwinger series (see (2.32)) associated with the conjugation $e^{Z_{\mathcal{I}^{*}_{K,Q}}}$. To be more explicit, the potential $V^{(K,Q)}_{\overline{\mathcal{I}^{*}_{K,Q}}}$ will contain the following contributions:

• i)

  The expression

  $$\omega(V^{(K,Q)_{-1}}_{\mathcal{I}_{K,Q}})+P^{(+)}_{\overline{\mathcal{I}^{*}_{K,Q}}}\,P^{(+)}_{\mathcal{I}^{*}_{K,Q}}\,\,\Big{[}V^{(K,Q)_{-1}}_{\mathcal{I}_{K,Q}}-\omega(V^{(K,Q)_{-1}}_{\mathcal{I}_{K,Q}})\Big{]}\,P^{(+)}_{\mathcal{I}^{*}_{K,Q}}\,P^{(+)}_{\overline{\mathcal{I}^{*}_{K,Q}}}\,,$$

  (2.22)

  which originates in the zero-order term in the Lie Schwinger series, i.e., in

  $$P^{(-)}_{\mathcal{I}^{*}_{K,Q}}\,\,V^{(K,Q)_{-1}}_{\mathcal{I}_{K,Q}}\,P^{(-)}_{\mathcal{I}^{*}_{K,Q}}+P^{(+)}_{\mathcal{I}^{*}_{K,Q}}\,\,V^{(K,Q)_{-1}}_{\mathcal{I}_{K,Q}}\,P^{(+)}_{\mathcal{I}^{*}_{K,Q}}\,,$$

  (2.23)

  from which we extract

  $$\omega(V^{(K,Q)_{-1}}_{\mathcal{I}_{K,Q}})\,\Big{(}1-P^{(+)}_{\mathcal{I}^{*}_{K,Q}}\Big{)}+P^{(+)}_{\mathcal{I}^{*}_{K,Q}}\,\,V^{(K,Q)_{-1}}_{\mathcal{I}_{K,Q}}\,P^{(+)}_{\mathcal{I}^{*}_{K,Q}}$$

  (2.24)

  and neglect a remainder whose norm decays exponentially in the distance, $\mathcal{O}(\sqrt{t^{-1}})$, between $\mathcal{I}_{K,Q}$ and the endpoints of $\mathcal{I}^{*}_{K,Q}$; see (1.8). This remainder term and the higher-order terms in the Lie-Schwinger series are treated as perturbations and lumped together with a potential, supported in a larger interval, that will be block-diagonalized in a later step.

• ii)

  The diagonal part w.r.t. $P^{(-)}_{\overline{\mathcal{I}^{*}_{K,Q}}},P^{(+)}_{\overline{\mathcal{I}^{*}_{K,Q}}}$ proportional to the projections $\mathcal{P}^{(2)}_{i,i+1}$, hence of terms of the type in (2.19).

In replacing (2.23) by (2.24) we must exclude from the block-diagonalization steps all intervals touching the endpoints of the lattice $\Lambda$. Therefore, we must henceforth distinguish bulk- from boundary-potentials, as explained in Section 2.4 below. The boundary potentials will get block-diagonalized only in the last step that corresponds to the interval $\Lambda$ (given by the entire chain).

We begin this subsection by introducing enlarged intervals that will be needed in our procedure, as explained in Section 2.3; see also Figures 2 and 3.

In order to implement the block-diagonalization steps, we define two subsets of the set $\mathfrak{I}$ of intervals introduced in Definition 2.2 of Section 1.1.3:

Using successive unitary conjugations, we shall derive a transformed Hamiltonian that, in step $(K,Q)$, will coincide with the operator

where the two types of potentials, “V” and “W,” are specified below: