Abstract
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.
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Notes
The initial step, \((0,N)\rightarrow (1,1)\), is of this type; see the definitions in (3.10) corresponding to a Hamiltonian \(K_N\) with nearest-neighbor interactions.
Recall that \(V_{I_{0,i}}^{(0,N)}:=H_i\) and \(V_{I_{0,i}}^{(k,q)}\) will coincide with \(V_{I_{0,i}}^{(0,N)}\) for all (k, q).
Recall the special steps of type (k, 1) with preceding step \((k-1, N-k+1)\).
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Acknowledgements
A.P. thanks the Pauli Center, Zürich, for hospitality in Spring 2017 when this project got started. A.P. also acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.
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Appendix
Appendix
Lemma A.1
For any \(1\le n \le N\)
where \(P^{\perp }_{\Omega _i}=\mathbb {1}-P_{\Omega _i}\).
Proof
We call \(P_{vac}:=\bigotimes _{i=1}^{n} P_{\Omega _i}\) acting on \(\mathcal {H}^{(n)}:=\bigotimes _{i=1}^{n} \mathcal {H}_i \). We define
Notice that all operators \(P_{\Omega _j}^{\perp }\) and \(P_{vac}\) commute with each other and are orthogonal projectors. Therefore we deduce that
We will show that
If (A.4) holds then \(0\notin \text {spec}(A_n)\). By (A.3) it then follows that
Thus, we are left with proving (A.4).
- (i)
Assume that \(\psi \) is perpendicular to the range of \(A_{n}\), and let \(P_{\Omega _j}^{\perp }\psi =:\phi _j\). Then, since \(\psi \perp \text {Range}\, A_n\), we have that
$$\begin{aligned} 0=\langle \psi , A_n \psi \rangle =\sum _{i=1\,;\,i\ne j}^{n}\langle \psi \,,\, P_{\Omega _i}^{\perp }\psi \rangle +\langle \psi \,\, P_{vac}\psi \rangle +\langle \psi \,,\, P_{\Omega _j}^{\perp }\psi \rangle \ge \langle \psi \,,\, P_{\Omega _j}^{\perp }\psi \rangle \nonumber \\ \end{aligned}$$(A.6)but
$$\begin{aligned} \langle \psi \,,\, P_{\Omega _j}^{\perp }\psi \rangle =\langle P_{\Omega _j}^{\perp }\psi \,,\, P_{\Omega _j}^{\perp }\psi \rangle =\langle \phi _j\,,\,\phi _j\rangle \end{aligned}$$(A.7)where we have used that \(P_{\Omega _j}^{\perp }\) is an orthogonal projector. We conclude that \(\phi _j=0\) for all j.
- (ii)
Let \(\psi \perp \text {Range}\,A_n\). Then, by (i),
$$\begin{aligned} \psi =\Big (\bigotimes _{j=1}^{n}\,(P_{\Omega _j}^{\perp }+P_{\Omega _j})\Big )\,\psi = (\bigotimes _{j=1}^{n}P_{\Omega _j})\,\psi =P_{vac}\psi \end{aligned}$$(A.8)and
$$\begin{aligned} 0=\langle \psi \,,\, A_n\,\psi \rangle =\langle \psi \,,\, P_{vac}\,\psi \rangle =\langle \psi \,,\,\psi \rangle \quad \Rightarrow \quad \psi =0\,. \end{aligned}$$(A.9)
Thus, \(\text {Range}\, A_n=\mathcal {H}^{(n)}\), and (A.4) is proven . \(\quad \square \)
From Lemma A.1 we derive the following bound.
Corollary A.2
For \(i+r\le N\), we define
Then, for \(1\le l \le L \le N-r\),
Proof
Lemma A.1 says that
By summing over i, from \(i=l\) up to L, the l-h-s of (A.12), for each j we get no more than \(r+1\) terms of the type \(P^{\perp }_{\Omega _{j}}\) and the inequality in (A.11) follows . \(\quad \square \)
Lemma A.3
Assume that \(t>0\) is sufficiently small, \(\Vert V^{(k,q-1)}_{I_{r,i}}\Vert \le \frac{8}{(r+1)^2}\,t^{\frac{r-1}{3}}\), and \(\Delta _{I_{k,q}}\ge \frac{1}{2}\). Then, for arbitrary N, \(k\ge 1\), and \(q\ge 2\), the inequalities
hold true for a universal constant C. For \(q=1\), \( V^{(k,q-1)}_{I_{k,q}}\) is replaced by \(V^{(k-1,N-k+1)}_{I_{k,1}}\) on the right side of (A.13) and (A.14).
Proof
In the following we assume \(q\ge 2\); if \(q=1\) an analogous proof holds. We recall that
and
with
and, for \(j\ge 2\),
and, for \(j\ge 1\),
From the lines above we derive
We recall definition (A.20) and we observe that
where we use the inductive hypothesis \(\Delta _{I_{k;q}}\ge \frac{1}{2}\). (Recall that \(\Delta _{I_{k;q}}\) is the gap of \(G_{I_{k,q}}\) above its groundstate energy \(E_{I_{k,q}}\).) Then formula (A.17) yields
From now on, we closely follow the proof of Theorem 3.2 in [DFFR]; that is, assuming \(\Vert V^{(k,q-1)}_{I_{k,q}}\Vert \ne 0\), we recursively define numbers \(B_j\), \(j\ge 1\), by the equations
with \(a>0\) satisfying the relation
Using (A.25), (A.26), (A.24), and an induction, it is not difficult to prove that (see Theorem 3.2 in [DFFR]) for \(j\ge 2\)
From (A.25) and (A.26) it also follows that
which, when combined with (A.28) and (A.27), yield
The numbers \(B_j\) are the Taylor’s coefficients of the function
(see [DFFR]). Therefore the radius of analyticity, \(t_0\), of
is bounded below by the radius of analyticity of \(\sum _{j=1}^{\infty }x^jB_j\), i.e.,
where we have used the assumption that \(\Vert V^{(k,q-1)}_{I_{r;i}}\Vert \le \frac{8}{(r+1)^2}\,t^{\frac{r-1}{3}}\) and \((0<)t<1\). Thanks to the inequality in (A.23) the same bound holds true for the radius of convergence of the series \(S_{I_{k;q}}:=\sum _{j=1}^{\infty }t^j(S_{I_{k;q}})_j\,\) . For \(0<t<1\) and in the interval \((0,\frac{a}{16})\), by using (A.25) and (A.30) we can estimate
for some a-dependent constant \(C_a>0\). Hence the inequality in (A.13) holds true, provided that \(t>0\) is sufficiently small but independent of N, k, and q. In a similar way we derive (A.14). \(\quad \square \)
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Fröhlich, J., Pizzo, A. Lie–Schwinger Block-Diagonalization and Gapped Quantum Chains. Commun. Math. Phys. 375, 2039–2069 (2020). https://doi.org/10.1007/s00220-019-03613-2
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DOI: https://doi.org/10.1007/s00220-019-03613-2