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arXiv:2603.19475

Disordered Ground States of Ergodic Quantum Spin Systems

Eric B. Roon, Jeffrey H. Schenker

math-phquant-phSubmitted Mar 19, 2026• Announced Mar 23, 2026• 0 citations
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Abstract

In this letter, we fill a hole in the existing literature about disordered quantum spin systems generated by a random local interaction $\{\mathfrak{h}(Z)\}_{Z\Subset \mathbb{Z}^ν}$ satisfying a statistical version of translation invariance. We show such systems always have disordered ground states in the thermodynamic limit with the same symmetry. A key tool we use is a disordered version of the Lieb-Robinson bounds, which hold almost surely under mild conditions on $\mathfrak{h}$. Along the way, we formalize the notion of a random state on a $C^*$-algebra and prove a weak-$\ast$ version of the Riesz-Markov-Kakutani theorem, which seems not to have been recorded in the vector measures literature. As a consequence of the existence of the aforementioned disordered ground states, we show that the spectrum of the GNS Hamiltonain associated to the bulk dynamics is deterministic with respect to the disorder.

Comments
21 pages; includes a generalization of part of the companion article arXiv:2507.07287; Comments Welcome

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In this letter, we fill a hole in the existing literature about disordered quantum spin systems generated by a random local interaction {h(Z)}Z⋐Zν\{\mathfrak{h}(Z)\}_{Z\Subset \mathbb{Z}^ν}{h(Z)}Z⋐Zν​ satisfying a statistical version of translation invariance. We show such systems always have disordered ground states in the thermodynamic limit with the same symmetry. A key tool we use is a disordered version of the Lieb-Robinson bounds, which hold almost surely under mild conditions on h\mathfrak{h}h. Along the way, we formalize the notion of a random state on a C∗C^*C∗-algebra and prove a weak-∗\ast∗ version of the Riesz-Markov-Kakutani theorem, which seems not to have been recorded in the vector measures literature. As a consequence of the existence of the aforementioned disordered ground states, we show that the spectrum of the GNS Hamiltonain associated to the bulk dynamics is deterministic with respect to the disorder.
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.
We study Fermionic systems on a lattice with random interactions through their dynamics and the associated KMS states. These systems require a more complex approach compared with the standard spin systems on a lattice, on account of the difference in commutation rules for the local algebras for disjoint regions, between these two systems. It is for this reason that some of the known formulations and proofs in the case of the spin lattice systems with random interactions do not automatically go over to the case of disordered Fermion lattice systems. We extend to the disordered CAR algebra some standard results concerning the spectral properties exhibited by temperature states of disordered quantum spin systems. We investigate the Arveson spectrum, known to physicists as the set of the Bohr frequencies. We also establish its connection with the Connes and Borchers spectra, and with the associated invariants for such W ∗-dynamical systems which determine the type of von Neumann algebras generated by a temperature state. We prove that all such spectra are independent of the disorder. Such results cover infinite-volume limits of finite-volume Gibbs states, that is the quenched disorder for Fermions living on a standard lattice ℤ d , including models exhibiting some standard spin-glass-like behavior. As a natural application, we show that a temperature state can generate only a type III\mathop {\rm {III}}III von Neumann algebra (with the type III0\mathop {\rm {III_{0}}}III0​ component excluded). In the case of the pure thermodynamic phase, the associated von Neumann algebra is of type IIIλ\mathop {\rm {III_{\lambda }}}IIIλ​ for some λ∈(0,1], independent of the disorder. All such results are in accordance with the principle of self-averaging which affirms that the physically relevant quantities do not depend on the disorder. The approach pursued in the present paper can be viewed as a further step towards fully understanding the very complicated structure of the set of temperature states of quantum spin glasses, and its connection with the breakdown of the symmetry for the replicas.
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 prove that generic quantum local Hamiltonians are gapless. In fact, we prove that there is a continuous density of states above the ground state. The Hamiltonian can be on a lattice in any spatial dimension or on a graph with a bounded maximum vertex degree. The type of interactions allowed for include translational invariance in a disorder (i.e., probabilistic) sense with some assumptions on the local distributions. Examples include many-body localization and random spin models. We calculate the scaling of the gap with the system's size when the local terms are distributed according to a Gaussian β orthogonal random matrix ensemble. As a corollary, there exist finite size partitions with respect to which the ground state is arbitrarily close to a product state. When the local eigenvalue distribution is discrete, in addition to the lack of an energy gap in the limit, we prove that the ground state has finite size degeneracies. The proofs are simple and constructive. This work excludes the important class of truly translationally invariant Hamiltonians where the local terms are all equal.
Abstract We rigorously examine, in generality, the ergodic properties of quantum lattice models with short range interactions, in the C∗C^*C∗ C ∗ algebra formulation of statistical mechanics. Ergodicity results, in the context of group actions on C∗C^*C∗ C ∗ algebras, assume that the algebra is asymptotically abelian, which is not generically the case for time evolution. The Lieb-Robinson bound tells us that, in a precise sense, the spatial extent of any time-evolved local operator grows linearly with time. This means that the algebra of observables is asymptotically abelian in a space-like region, and implies a form of ergodicity outside the light-cone. But what happens within it? We show that the long-time limit of the n-th moment of a ray-averaged observable, along space-time rays of almost every speed, converges to the n-th power of its expectation in the state (i.e. its ensemble average). Thus ray averages do not fluctuate in the long time limit. This is a statement of ergodicity, and holds in any state that is invariant under space-time translations and that satisfies weak clustering properties in space. The ray averages can be performed in a way that accounts for oscillations, showing that ray-averaged observables cannot sustainably oscillate in the long time limit. We also show that in the GNS representation of the algebra of observables, for any KMS state with the above properties, the long-time limit of the ray average of any observable converges (in the strong operator topology) to the ensemble average times the identity, again along space-time rays of almost every speed. This is a strong version of ergodicity, and indicates that, as operators, observables get “thinner” almost everywhere within the light-cone. A similar statement holds under oscillatory averaging.
By tightening the conventional Lieb-Robinson bounds to better handle systems that lack translation invariance, we determine the extent to which "weak links" suppress operator growth in disordered onedimensional spin chains.In particular, we prove that ballistic growth is impossible when the distribution of coupling strengths μ(J ) has a sufficiently heavy tail at small J and we identify the correct dynamical exponent to use instead.Furthermore, through a detailed analysis of the special case in which the couplings are genuinely random and independent, we find that the standard formulation of Lieb-Robinson bounds is insufficient to capture the complexity of the dynamics-we must distinguish between bounds that hold for all sites of the chain and bounds that hold for a subsequence of sites and we show by explicit example that these two can have dramatically different behaviors.All the same, our result for the dynamical exponent is tight, in that we prove by counterexample that there cannot exist any Lieb-Robinson bound with a smaller exponent.We close by discussing the implications of our results, both major and minor, for numerous applications ranging from quench dynamics to the structure of ground states.