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arXiv:2607.01446

Parent Hamiltonians of Ergodic Matrix Product States

Owen Ekblad, Eloy Moreno-Nadales, Eric B. Roon, Jeffrey H. Schenker

math-phquant-phSubmitted Jul 1, 2026• Announced Jul 3, 2026• 0 citations
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Abstract

Matrix product states (MPS) are quintessential examples of frustration-free gapped ground states of local interactions called parent Hamiltonians. In this work, we investigate parent Hamiltonians for a class of ergodic matrix product states (EMPS), which are MPS defined by site-dependent random tensors $\{X_j^{[k]}\}_{j=1}^D$ which are homogeneously distributed at every site $k$ in the spin chain. Here, the EMPS are not translation-invariant but rather statistically translation-invariant. Under a mild injectivity assumption, we show the thermodynamic limit of an EMPS is the unique frustration-free ground state of a parent Hamiltonian on the whole spin chain, which, depending on the statistical properties of the EMPS, may or may not be finite-range. In contrast to the translation-invariant regime, these Hamiltonians need not be gapped. Nevertheless, applying the martingale method while keeping track of local statistics gives conditions for a gap, in addition to pointing towards why there need not be a gap in general. We include examples of EMPS both with and without spectral gaps to illustrate our results.

Comments
40 pages, 1 figure, comments welcome :)

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Matrix product states (MPS) are quintessential examples of frustration-free gapped ground states of local interactions called parent Hamiltonians. In this work, we investigate parent Hamiltonians for a class of ergodic matrix product states (EMPS), which are MPS defined by site-dependent random tensors {Xj[k]}j=1D\{X_j^{[k]}\}_{j=1}^D{Xj[k]​}j=1D​ which are homogeneously distributed at every site kkk in the spin chain. Here, the EMPS are not translation-invariant but rather statistically translation-invariant. Under a mild injectivity assumption, we show the thermodynamic limit of an EMPS is the unique frustration-free ground state of a parent Hamiltonian on the whole spin chain, which, depending on the statistical properties of the EMPS, may or may not be finite-range. In contrast to the translation-invariant regime, these Hamiltonians need not be gapped. Nevertheless, applying the martingale method while keeping track of local statistics gives conditions for a gap, in addition to pointing towards why there need not be a gap in general. We include examples of EMPS both with and without spectral gaps to illustrate our results.
We study frustration-free Hamiltonians of fractional quantum Hall (FQH) states from the point of view of the matrix product state (MPS) representation of their ground and excited states. There is a wealth of solvable models relating to FQH physics, which, however, is mostly derived and analyzed from the vantage point of first-quantized ``analytic clustering properties.'' In contrast, one obtains long-ranged frustration-free lattice models when these Hamiltonians are studied in an orbital basis, which is the natural basis for the MPS representation of FQH states. The connection between MPS-like states and frustration-free parent Hamiltonians is the central guiding principle in the construction of solvable lattice models, but thus far, only for short-range Hamiltonians and MPSs of finite bond dimension. The situation in the FQH context is fundamentally different. Here we expose the direct link between the infinite-bond-dimension MPS structure of Laughlin--conformal field theory (CFT) states and their parent Hamiltonians. While focusing on the Laughlin state, generalizations to other CFT-MPSs will become transparent.
Matrix-product state (MPS) skeletons are connected networks of Hamiltonians with exact MPS ground states that underlie a phase diagram. Such skeletons have previously been found in classes of free-fermion models. For the translation-invariant BDI and AIII free-fermion classes, it has been shown that the underlying skeleton is dense, giving an analytic approach to MPS approximation of ground states anywhere in the class. In this paper, we partially expose the skeleton in certain interacting spin chains: the NNN-state Onsager-integrable chiral clock families. We construct MPS that form a dense MPS skeleton in the gapped regions surrounding a sequence of fixed-point Hamiltonians (the generators of the Onsager algebra). Outside these gapped regions, these MPS remain eigenstates, but no longer give the many-body ground state. Rather, they are ground states in particular sectors of the spectrum. Our methods also allow us to find further MPS eigenstates; these correspond to low-lying excited states within the aforementioned gapped regions. This set of MPS excited states goes beyond the previous analysis of ground states on the N=2N=2N=2 free-fermion MPS skeleton. As an application of our results, we find a closed form for the disorder parameter in a family of interacting models. Finally, we remark that many of our results use only the Onsager algebra and are not specific to the chiral clock model representation.
Proving spectral gaps above the ground space is a central problem in quantum many-body physics. Yet, finding lower bounds on the gap is notoriously difficult. We revisit the martingale method, originally developed by Fannes, Nachtergaele, and Werner [Fannes '92, Nachtergaele '96] to prove the existence of spectral gaps of parent Hamiltonians of Matrix Product States (MPS), and provide improvements to the different steps of the method. Most importantly, we devise a new technique which allows to compute the key quantity in the martingale method -- the overlap of local ground spaces -- exactly and efficiently. This enables a clear improvement of the method, allowing it to outperform other existing techniques to lower bound gaps, which we demonstrate by benchmarking on several models. Remarkably, our -- numerically motivated -- approach at the same time also yields a significantly simplified proof of the fact that parent Hamiltonians of any (well-behaved) MPS are always gapped.
We consider random translation-invariant frustration-free quantum spin Hamiltonians on ZD in which the nearest-neighbor interaction in every direction is randomly sampled and then distributed across the lattice. Our main result is that, under a small rank constraint, the Hamiltonians are automatically frustration-free and they are gapped with a positive probability. This extends previous results on 1D spin chains to all dimensions. The argument additionally controls the local gap. As an application, we obtain a 2D area law for a cut-dependent ground state via recent AGSP methods of Anshu-Arad-Gosset.
We introduce a general model of stochastically generated matrix product states (MPS) in which the local tensors share a common distribution and form a strictly stationary sequence, without requiring spatial independence. Under natural conditions on the associated transfer operators, we prove the existence of thermodynamic limits of expectations of local observables and establish almost-sure exponential decay of two-point correlations. In the homogeneous (random translation-invariant) case, for any error tolerance in probability, the two-point function decays exponentially in the distance between the two sites, with a deterministic rate. In the i.i.d. case, the exponential decay still holds with a deterministic rate, with the probability approaching one exponentially fast in the distance. For strictly stationary ensembles with decaying spatial dependence, the correlation decay quantitatively reflects the mixing profile: (ρρρ)-mixing yields polynomial bounds with high probability, while stretched-exponential (resp. exponential) decay in (ρρρ) (resp. (βββ)) yields stretched-exponential (resp. exponential) decay of the two-point function, again with correspondingly strong high-probability guarantees. Altogether, the framework unifies and extends recent progress on stationary ergodic and Gaussian translation-invariant ensembles, providing a transfer-operator route to typical correlation decay in random MPS.
Let (Ω,¶)(Ω, ¶)(Ω,¶) be a standard probability space and let ϑ:Ω→Ω\vartheta:Ω\to Ωϑ:Ω→Ω be a measure preserving ergodic homeomorphism. Let A\mathcal{A}A be a C∗C^*C∗-algebra with a unit and let AZ\mathcal{A}_{\mathbb{Z}}AZ​ be the quasi-local algebra associated to the spin chain with one-site algebra A\mathcal{A}A. Equip AZ\mathcal{A}_{\mathbb{Z}}AZ​ with the group action of translation by kkk-units, τk∈Aut(AZ)τ_k\in Aut(\mathcal{A}_{\mathbb{Z}})τk​∈Aut(AZ​) for k∈Zk\in \mathbb{Z}k∈Z. We study the problem of finding a disordered matrix product state decomposition for disordered states ψ(ω)ψ(ω)ψ(ω) on AZ\mathcal{A}_{\mathbb{Z}}AZ​ with the covariance symmetry condition ψ(ω)∘τk=ψ(ϑkω)ψ(ω) \circ τ_k = ψ(\vartheta^k ω)ψ(ω)∘τk​=ψ(ϑkω). This can be seen as an ergodic generalization of the results of Fannes, Nachtergaele, and Werner [31]. To reify our structure theory, we present a disordered state νων_ωνω​ obtained by sampling the AKLT model [2] in parameter space. We go on to show that νων_ωνω​ has a nearest-neighbor parent Hamiltonian, its bulk spectral gap closes, but it has almost surely exponentially decaying correlations, and finally, that νων_ωνω​ is time-reversal invariant with a Tasaki index of −1-1−1 almost surely.