A simple mechanism for unstable degeneracies in local Hamiltonians
- Open Access
- 01-11-2025
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Abstract
The stabilizing mechanisms of ground spaces of local gapped Hamiltonians are key in quantum many-body physics, as global symmetries [1] and topological order [2], and thereof understanding unstable degeneracies. Ref.[3] showed that the W-state is not the unique ground state of any local Hamiltonian –even gapless and no frustration free (FF)– and that such degeneracy is unstable. Here, we show a simple mechanism that enforces degenerate yet unstable energy levels covering the W-state result for FF Hamiltonians.
Let be \(\vert \psi \rangle \) and \(\vert \phi \rangle \) orthogonal states that are mapped by a set of finite-range operators \(\{ O_\alpha \}_{\alpha \in \mathcal {S}}\) asSuppose that \(\vert \psi \rangle \) is an eigenvector of each term in a finite-range Hamiltonian \(H=\sum h_j\), i.e., \(h_j\vert \psi \rangle = e_j \vert \psi \rangle \) for all j, and that for each \(h_j\) there exists an operator \(O_\alpha \) such that \([h_j, O_\alpha ] = 0\). Then, \(\vert \phi \rangle \) is also an eigenvector of \(h_j\) with the same eigenvalue \(e_j\), since \(h_j \vert \phi \rangle = h_j O_\alpha \vert \psi \rangle = O_\alpha h_j \vert \psi \rangle = e_j O_\alpha \vert \psi \rangle = e_j \vert \phi \rangle \).
$$\begin{aligned} O_\alpha \vert \psi \rangle =\vert \phi \rangle , \ \mathrm{for \ all} \ \alpha \in \mathcal {S}\ . \end{aligned}$$
(1)
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Some comments are in order: (1) The states \(\vert \psi \rangle , \vert \phi \rangle \) are neither symmetry broken nor topological ordered states since the operators \(O_\alpha \) are finite-range, excluding global symmetry operators and logical ones. (2) The permutation of \(\vert \psi \rangle \leftrightarrow \vert \phi \rangle \) is only guaranteed to be local for \(\vert \psi \rangle \rightarrow \vert \phi \rangle \): \(\{O_\alpha \}\) may be non-invertible so \(\vert \phi \rangle \rightarrow \vert \psi \rangle \) would require non-LOCC operations. (3) For FF Hamiltonians satisfying \(h_j\vert \psi \rangle = 0\), this implies that \(\vert \psi \rangle \) cannot be the unique ground state. In particular, this is the case of the W-state \(\vert W \rangle = \frac{1}{\sqrt{N}} \sum _i \sigma ^+_i \vert 0 \rangle ^{\otimes N}\), and any of its Dicke-state generalizations, sinceso that either \(\sigma ^-_i\) or \(\sigma ^-_{i+\ell }\) commutes with all \(\{h_j\}\) if \(\ell \ge \textrm{range}\{h_j\}\). So that \(\vert W \rangle \) always appears together with \(\vert 0 \rangle ^{\otimes N}\) as ground states of FF Hamiltonians. (4) The W-example also shows that \([h_j, O_\alpha ] = 0\) is satisfied easily when the supports of \(h_j\) and \(O_\alpha \) do not overlap.
$$(N^{1/2} \sigma ^-_i) \vert W \rangle = \vert 0 \rangle ^{\otimes N} \ \mathrm{for \ any \ site} \ i,$$
Hastings showed [4] that a finite-range gapped Hamiltonian \(H=\sum _i \tilde{h}_i\) with ground state \(\vert \psi \rangle \) is equivalent to another one \(H=\sum _i {h}_i\) which (i) is almost FF: \(|{h}_j \vert \psi \rangle |\le |h_j| e^{-\ell }\), where \(\ell \) can be picked as big as desired and (ii) is exponentially decaying: \(|[h_j, O]| \le |h_j| {\cdot } |O| e^{-d(h_j,O)/c}\), where c depends on the microscopic details of \(h_j\). Then, \(\vert \phi \rangle \) in Eq. (1) is also close to a ground state:provided that \(d(j,O_\alpha )\) can be chosen arbitrary big. This extent our result to ground state degeneracies of non-necessarily FF Hamiltonians.
$$\begin{aligned} | h_j \vert \phi \rangle | = |h_j O_\alpha \vert \psi \rangle | \le |O_\alpha |{\cdot }|h_j|( e^{-\ell } + e^{-d(j,O_\alpha )/c}) \ , \end{aligned}$$
(2)
The degeneracies created in ground states of Hamiltonian given by Eq. (1) are unstable. For a local perturbation \(H_\lambda = H_0 - \lambda \sum _j V_j\) that does not close the gap when \(\lambda \) is small, a quasiadiabatic continuation follows [5]: there is a unitary \(U_\lambda \) that evolves the ground states \(\vert \psi \rangle _\lambda =U_\lambda \vert \psi \rangle \) and preserves locality. To show the instability, we compute:where \(\tilde{V}_j = U^\dagger _\lambda V_j U^\dagger _\lambda \) is local and we have assumed that for every j there is \(\alpha _j \in \mathcal {S}\) such that \([\tilde{V}_j, O_{\alpha _j}] = 0\). Besides the first term –(3)– is expected to contribute exponentially small in \(\lambda \), the second term –(4)– is linear in \(\lambda \) times a contribution \( \sum _j\langle \tilde{V}_j \rangle (1-| O^\dagger _{\alpha } O_{\alpha } | f(d))\) which is O(1) in \(\lambda \), where the function f encodes the correlation decay of \(\psi \) and \(d= max_j\{d(j,O_{\alpha _j})\}\). This linear growth in \(\lambda \) discard a stable degeneracy, i.e. exponentially small separation in \(\lambda \) between the ground state energies.
$$\begin{aligned} \langle H_\lambda \rangle _{\psi _\lambda } - \langle H_\lambda \rangle _{\phi _\lambda }&= \langle H_0 \rangle _{\psi _\lambda } - \langle H_0 \rangle _{\phi _\lambda } \end{aligned}$$
(3)
$$\begin{aligned}&\quad -\lambda \sum \limits _{j}\langle \tilde{V}_j-O^\dagger _{\alpha _j} O_{\alpha _j} \otimes \tilde{V}_j \rangle _{\psi }\ \end{aligned}$$
(4)
$$\begin{aligned}&= O(\lambda ), \end{aligned}$$
(5)
We emphasize that the results shown in this work do not require translation invariance and are valid for any physical dimension. Similar results can also be derived for the steady (mixed) states of frustration-free local Lindbladians mapped by local quantum channels.
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Acknowledgements
I thank A. Molnar and A. Turzillo, A. Franco-Rubio and A.M. Kubicki for enlightening discussions and the Erwin Schrödinger Institute (ESI), where this paper took shape. This research was funded in whole or in part by the Austrian Science Fund FWF [10.55776/J4796]. For the purpose of Open Access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.
Declarations
Conflict of interest
The authors declare no Conflict of interest.
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