Condensed Locality: Lieb–Robinson Estimates and Quasi-Local Dynamics on the Moduli Stack of Hamiltonians

1 Introduction

1.1 The locality problem in the condensed program

A gapped quantum lattice system is specified by a lattice $L$ of sites in dimension $d$, a finite-dimensional Hilbert space at each site, and a rule that assigns to each finite cluster of sites a Hermitian operator supported there, an interaction. The set of admissible $G$-symmetric interactions, together with a locality (decay) condition, is the topological space $\mathcal{I}_{d,G}$ of the program this series develops. The organizing proposal is to stop treating $\mathcal{I}_{d,G}$ as a bare topological space and instead pass to its condensation, $$X \;\longmapsto\; \underline{X}, \qquad \underline{X}(S)=\operatorname{Cont}(S,X),$$ with $S$ ranging over profinite sets, and then to build over it the moduli stack $\mathfrak{Ham}_{d,G}$ of Hamiltonians, the uniformly gapped substack $\mathfrak{Gap}_{d,G}$, the stabilized phase $∞$-groupoid $\mathfrak{Phase}_{d,G}$, and finally the invertible condensed phase spectrum $\mathbf{IP}^{\mathrm{cond}}_{d,G}$. In one sentence, the thesis of the program is that

a topological phase is a component of the stabilized condensed stack of gapped systems.

The advantage of this move is not a new Chern number; it is that condensed abelian groups form a Grothendieck abelian category with exact derived functors, so that continuous families, profinite disorder, analytic completions, symmetry, stacking, defects, and transition loci can be manipulated together, where topological groups behave badly.

None of this is available until one controls the analysis at the bottom. Before $\mathfrak{Ham}_{d,G}$ can be a condensed object at all, three things must be true and must be uniform in the probe: the space of interactions must be a Banach space so that its condensation is a condensed $\mathbb{R}$-vector space; the Heisenberg dynamics must depend continuously on the interaction and the time in the Banach topology, so that a family of dynamics is a morphism rather than a set-theoretic assignment; and the Lieb–Robinson estimates that make the dynamics quasi-local must survive the passage to profinite limits with constants that do not degenerate. This paper establishes exactly these facts, and no more. The word “uniform” is doing real work throughout: a family of quasi-local dynamics whose Lieb–Robinson velocity is finite pointwise but unbounded over the base is not a morphism of condensed sets, just as a family whose gap is positive pointwise but has infimum zero is not a member of $\mathfrak{Gap}_{d,G}$.

The tools are classical. Lieb and Robinson proved a finite group velocity for quantum spin systems in 1972 [1]; the modern reformulation through $F$-functions, due to Nachtergaele, Sims, and Young [2,3] and surveyed by Hastings [4], gives exactly the Banach-space packaging we need. The condensed formalism is that of Clausen and Scholze [5–7], with the pyknotic variant of Barwick and Haine [8]. The closest precedent on the physics side is the homotopical, sheaf-theoretic study of parametrized families of spin systems by Beaudry and collaborators [9], which organizes families of gapped systems with ordinary, not condensed, topology; the present paper is the condensed refinement of that viewpoint at the locality layer. Our contribution is to show that these two mature bodies of work fit together on the nose: the analytic estimate that Nachtergaele–Sims–Young prove is precisely the continuity statement that condensation requires, and the profinite probes of condensed mathematics are precisely matched to the inverse limits of finite-volume and disorder data that lattice physics produces.

1.2 What this paper proves, and what it only conjectures

We adopt the editorial stance of the whole series: a statement is labelled Theorem or Proposition only when it has a complete proof from cited present-day results, and everything past that boundary is a numbered Conjecture. Under that discipline the results are as follows. Theorem 3.3 (labelled I-A) records that the $F$-normed interactions form a real Banach space $\mathcal{B}_{F}$ and that the Heisenberg dynamics is a strongly continuous one-parameter group of $*$-automorphisms with a Lieb–Robinson bound; this is a repackaging of [2]. Theorem 4.4 (I-B) upgrades this to condensed mathematics: $\underline{\mathcal{B}_{F}}$ is a condensed $\mathbb{R}$-vector space, and the assignment $(t,\Phi)↦\tau^\Phi_t$ is a morphism of condensed sets, because it is continuous in the Banach topology and condensation is fully faithful on compactly generated spaces. Theorem 5.1 (I-C) is the analytic heart: over a profinite base, an $S$-continuous family of interactions is the same datum as a norm-convergent compatible system over the finite quotients, and a uniform bound on the $F$-norm produces a Lieb–Robinson light cone whose velocity and prefactors are uniform in $s\in S$. Theorem 5.4 draws the uniform exponential clustering consequence that Part III will need, and Theorem 6.2 records that quasi-adiabatic continuation and automorphic equivalence [10,11] furnish the morphism-level generators of the equivalence class $\mathcal{W}$.

The conjectures mark the edge of what today’s estimates deliver. Theorem 7.1 (I-1), that $\mathfrak{Ham}_{d,G}$ is a condensed higher stack (the groupoid of gauge and quasi-local automorphisms glues along finite covers of profinite probes), is not proved here and is not asserted anywhere in the series as a theorem. Theorem 7.2 (I-2) proposes that the quasi-local automorphisms generated by time-dependent $F$-function interactions form a condensed group for which quasi-adiabatic continuation is an internal path-lifting. Theorem 7.3 (I-3) proposes that Lieb–Robinson estimates satisfy descent along finite quotients of a disorder hull $\Omega=Q^{\mathbb{Z}^d}$, so that quasi-local invariants are determined by finite-resolution data. These three are the locality-layer instances of the program’s Master Conjecture.

1.3 Relation to companion papers

This is Part I of six. The series is modular: each part takes the previous ones as input and produces new structure, rather than assembling into a single monolith.

Part II, Positivity, $C^*$-Norms, and Condensed State Spaces of Quasi-Local Algebras [12], takes the quasi-local algebra $\mathcal{A}$ and the automorphism morphism built here and attaches the condensed state space and the solid-$K$-theory invariant of Aoki [13]. The Lieb–Robinson bound of Theorem 3.3 is what makes the Heisenberg dynamics act on $\mathcal{A}$ by genuine $*$-automorphisms, which Part II needs before it can speak of states and positivity. Part III, The Uniformly Gapped Substack [14], cuts out $\mathfrak{Gap}_{d,G}\subset\mathfrak{Ham}_{d,G}$ and studies stability of the gap; its exponential-clustering input is our Theorem 5.4, and its automorphic equivalences are the $\mathcal{W}$ generated by our Theorem 6.2. Part IV, From Lattice Models to Effective Field Theories [15], group-completes the invertible sector into $\mathbf{IP}^{\mathrm{cond}}_{d,G}$; the continuity of dynamics in the interaction (Theorem 4.4) is what lets its parametrized families be morphisms of condensed objects. Part V, Physical Realizability of Bordism and Homotopy Classes [16], asks which abstract classes are realized by uniformly gapped lattice families, and inherits the uniform-over-the-base discipline established here. Part VI [17] composes the five modules and states the Master Conjecture. Where those papers speak of “an $S$-continuous family of Hamiltonians”, the precise meaning is the one fixed by Theorem 5.1.

The rest of the paper is organized as follows. Section 2 fixes the lattice, the quasi-local algebra, the $F$-functions, and the Banach space $\mathcal{B}_{F}$. Section 3 recalls the Heisenberg dynamics and the Lieb–Robinson bound and proves I-A. Section 4 recalls exactly as much condensed mathematics as we use and proves the condensed vector-space and dynamics statements (I-B). Section 5 proves the uniform Lieb–Robinson bound over a profinite base and the compatible-finite-data description (I-C), and draws the clustering corollary. Section 6 treats quasi-adiabatic continuation and the generation of $\mathcal{W}$. Section 7 states the three conjectures. Section 8 reports the transverse-field Ising simulation. Sections 9 and 10 discuss limitations and conclude.

2 Quantum lattice systems and interaction spaces

2.1 Lattice, local Hilbert spaces, and the quasi-local algebra

We work in the standard operator-algebraic setting for quantum spin systems; the reader who wants proofs of the folklore statements below will find them in [2] and the references there.

For a finite $X\subset L$ write $\mathcal H_X=\bigotimes_{x\in X}\mathcal H_x$ and let $\mathcal{A}_X=\mathcal B(\mathcal H_X)$ be the matrix algebra of observables supported on $X$. If $X\subset Y$ then $\mathcal{A}_X\hookrightarrow\mathcal{A}_Y$ by $A↦ A\otimes\mathbf 1_{Y\setminus X}$, and the local algebra is the union $$\mathcal{A}_\mathrm{loc}\;=\; \bigcup_{X\Subset L}\mathcal{A}_X,$$ the colimit over finite subsets $X\Subset L$. Its completion in the operator norm is the quasi-local algebra $$\mathcal{A}\;=\; \overline{\mathcal{A}_\mathrm{loc}}^{\,\lVert ·\rVert},$$ a unital $C^*$-algebra; this is the standard $C^*$-inductive-limit (quasi-local) construction of Bratteli–Robinson [18]. Because $L$ is countable and each $\mathcal{A}_X$ is finite-dimensional, $\mathcal{A}$ is separable. An element $A\in\mathcal{A}$ is local if $A\in\mathcal{A}_X$ for some finite $X$, and its support $\operatorname{supp}A$ is the smallest such $X$. We write $\mathcal{A}$ for what the notation table of the series calls the observable algebra $A$; the calligraphic letter is the standard one in the Lieb–Robinson literature and we keep it to avoid clashing with generic operators.

A symmetry is a group $G$ acting on $\mathcal{A}$ by $*$-automorphisms, on-site in the internal case (a representation $u_x:G\to\mathcal U(\mathcal H_x)$ inducing $\alpha_g=\bigotimes_x \mathrm{Ad}(u_x(g))$) and by isometries of $(L,\operatorname{dist})$ in the spatial case. Everything below is compatible with a fixed such $G$; to keep the analysis in the foreground we suppress $G$ from the notation and reinstate it only where symmetry changes a statement.

2.2 $F$-functions and the interaction Banach space

The decay of interactions is measured against a weight on the metric, an $F$-function.

On $\mathbb{Z}^d$ the functions $F_\varepsilon(r)=(1+r)^{-(d+\varepsilon)}$ with $\varepsilon>0$ are $F$-functions, as are the exponentially weighted $F(r)=e^{-\theta r} (1+r)^{-(d+\varepsilon)}$. The next lemma, which we use constantly, says that one may always sharpen the decay of a given $F$-function by an exponential factor at no cost in the two constants; it is the mechanism by which a finite velocity appears.

To keep the constants concrete—the knowledge base for this program insists that the $F$-norm never be left implicit—we verify the axioms for the standard polynomial weights.

The $F$-norm controls, uniformly in the site, the total weight of interaction terms that link any two sites. A finite-range interaction of bounded strength has finite $F$-norm for every $F$-function; a two-body interaction $\Phi(\{x,y\})$ decaying like $F(\operatorname{dist}(x,y))$ does too. The point of Theorem 2.5 is that everything downstream depends on $\Phi$ only through the single number $\lVert \Phi\rVert_{F}$ (and the constants of the chosen $F$), which is what will let us pass to families.

We record the two elementary structures that condensation will act on. The self-adjointness constraint makes $\mathcal{B}_{F}$ a real (not complex) Banach space; scaling an interaction by a real number and adding interactions termwise are continuous, so $\mathcal{B}_{F}$ is in particular a topological real vector space and an abelian topological group under addition.

3 The Heisenberg dynamics and the Lieb–Robinson bound

3.1 Finite-volume dynamics and the infinite-volume limit

For a finite $\Lambda\Subset L$ the local Hamiltonian of $\Phi\in\mathcal{B}_{F}$ is the self-adjoint operator $$H^\Phi_\Lambda \;=\; \sum_{X\subseteq\Lambda}\Phi(X)\;\in\;\mathcal{A}_\Lambda,$$ a finite sum, and the finite-volume Heisenberg dynamics is the one-parameter group of $*$-automorphisms $$\tau^{\Phi,\Lambda}_t(A)\;=\;e^{itH^\Phi_\Lambda}\,A\,e^{-itH^\Phi_\Lambda},\qquad A\in\mathcal{A}.$$ The Lieb–Robinson bound controls the spatial spread of $\tau^{\Phi,\Lambda}_t(A)$ and is the key to removing the cutoff $\Lambda$.

This is Theorem 3.4 of [2] in the present notation; we take it as given. The velocity is made explicit by combining it with reweighting.

The proof of Theorem 3.3(2) already contains the estimate we need for the condensed statement, because it is really a statement about how $\tau$ depends on $\Phi$. We isolate it.

3.2 The generator as a symmetric derivation

The strongly continuous group $\tau^\Phi$ has an infinitesimal generator, and the $F$-norm controls it on local observables. This is the infinitesimal counterpart of the whole picture: the same single number $\lVert \Phi\rVert_{F}$ that bounds the dynamics also bounds its generator, term by term.

The assignment $\Phi↦\delta_\Phi$ is real-linear and, on each $\mathcal{A}_X$, bounded in the $F$-norm: $\lVert \delta_\Phi(A)-\delta_\Psi(A)\rVert\le 2\lvert X\rvert F(0)\lVert \Phi-\Psi\rVert_{F} \lVert A\rVert$. This is the infinitesimal shadow of Theorem 3.4; once condensed it says that the passage to generators is itself a morphism $\underline{\mathcal{B}_{F}}\to\underline{\operatorname{Der}(\mathcal{A}_\mathrm{loc})}$, continuous in the $F$-norm, of which Theorem 4.4 is the exponentiated form.

4 Condensation of the interaction space

4.1 Condensed sets and profinite probes

We recall only what we use. A profinite set is a cofiltered limit $S=\varprojlim_i S_i$ of finite sets, equivalently a compact, Hausdorff, totally disconnected topological space. The site of profinite sets has covers the finite jointly surjective families; a condensed set is a sheaf on this site (equivalently, on the subcategory of extremally disconnected profinite sets, where sheaf and product-preserving presheaf coincide). Condensed abelian groups form a Grothendieck abelian category, so kernels, cokernels, and derived functors behave as in ordinary homological algebra [5]; the pyknotic formulation is [8]. Write $\mathbf{Cond}$ for the category of condensed sets.

The one structural fact we lean on is full faithfulness. A topological space is compactly generated if its topology is final for the maps out of compact Hausdorff spaces into it; metrizable spaces and, more generally, first-countable spaces are compactly generated.

Two comments fix the size of the formalism we need. First, a Banach space $V$ is metrizable, hence compactly generated, so Theorem 4.2 applies to it; moreover $\underline{V}$ is a module over the condensed ring $\underline{\mathbb{R}}$, i.e. a condensed $\mathbb{R}$-vector space, because addition $V× V\to V$ and scaling $\mathbb{R}× V\to V$ are continuous and condensation preserves finite products. Second, for separable target spaces one may replace profinite probes by their light (metrizable, equivalently second-countable) counterparts and work in the category $\mathbf{Cond}^{\mathrm{light}}$ of light condensed sets [7]; this avoids the set-theoretic subtleties of the full theory (no inaccessible cardinal is needed) and is the correct home for separable spin systems, whose observable algebra $\mathcal{A}$ is separable by Theorem 2.1.

4.2 The condensed interaction space and the condensed dynamics

We now assemble the dynamics. Let $\operatorname{Aut}(\mathcal{A})$ carry the topology of strong convergence: a net $\beta_i\to\beta$ iff $\beta_i(A)\to\beta(A)$ in norm for every $A\in\mathcal{A}$. Because $\mathcal{A}$ is separable, this topology is metrizable and $\operatorname{Aut}(\mathcal{A})$ is a Polish group, in particular compactly generated, so Theorem 4.2 applies to it.

Theorem 4.4 is the sense in which “the dynamics is a morphism, not just an assignment.” The diagram

\begin{tikzcd}[column sep=large, row sep=large]
  \underline{\mathbb{R}}×\underline{\mathcal{B}_{F}} \arrow[r, "\underline{D}"] \arrow[d, "\mathrm{pr}_2"']
    & \underline{\operatorname{Aut}(\mathcal{A})} \\
  \underline{\mathcal{B}_{F}} \arrow[ur, "\Phi\ ↦\ \underline{\tau^{\Phi}_{\bullet}}"']
\end{tikzcd}

5 Uniform Lieb–Robinson bounds over a profinite base

The previous section made the dynamics a morphism out of $\underline{\mathcal{B}_{F}}$. We now show that when one plugs in an actual profinite family, the Lieb–Robinson estimate holds simultaneously and uniformly, and we identify what an $S$-continuous family of interactions is in concrete terms.

5.1 Uniformity over the base

The three parts say, in order: an $S$-continuous family is a compatible tower of finite-resolution families; such a family is automatically norm-bounded; and a norm bound is exactly what makes the whole family share one light cone. The last point is the reason the condensed reformulation is not merely cosmetic. In the topological category one would have a map $S\to\mathcal{I}_{d,G}$ and a velocity function $s↦ v(s)$ with no guarantee of a common bound; the Banach structure of $\mathcal{B}_{F}$ upgrades “finite velocity at each $s$” to “one finite velocity for all $s$”, which is what a morphism into a condensed set of quasi-local dynamics requires.

The uniformity is compatible with change of probe, which is what makes it useful for descent.

\begin{tikzcd}[column sep=huge, row sep=large]
  \underline{\mathcal{B}_{F}}(S) \arrow[r, "\tau_t"] \arrow[d, "\varphi^*"'] & \underline{\operatorname{Aut}(\mathcal{A})}(S) \arrow[d, "\varphi^*"] \\
  \underline{\mathcal{B}_{F}}(S') \arrow[r, "\tau_t"'] & \underline{\operatorname{Aut}(\mathcal{A})}(S')
\end{tikzcd}