A mathematical theory of topological invariants of quantum spin systems

Let $M$ be a manifold and $Open(M)$ be the category whose objects are open subsets of $M$, and the set of morphisms from an open $U$ to an open $V$ is the singleton or the empty set depending on whether $U\subseteq V$ or $U\nsubseteq V$. Composition of morphisms is uniquely defined. A pre-cosheaf ${\mathfrak{F}}$ on $M$ with values in a category ${\mathcal{C}}$ is a functor ${\mathfrak{F}}:Open(M)\rightarrow{\mathcal{C}}$. Thus for every inclusion of opens $U\subseteq V$ one is given a co-restriction morphism $e_{VU}:{\mathfrak{F}}(U)\rightarrow{\mathfrak{F}}(V)$ such that for any three opens $U\subseteq V\subseteq W$ one has $e_{WV}\circ e_{VU}=e_{WU}$. Pre-cosheaves (as well as pre-sheaves, which are functors from the opposite category of $Open(M)$ to ${\mathcal{C}}$) can be used to describe local data on $M$. This form of locality is rather weak, since it does not require ${\mathfrak{F}}(U\cup V)$ to be expressible through ${\mathfrak{F}}(U)$ and ${\mathfrak{F}}(V)$.

As an example, consider the Lie algebra of gauge transformations, i.e. the Lie algebra ${\mathfrak{G}}(M):=C^{\infty}(M,{\mathfrak{g}})$ of smooth functions on $M$ with values in a finite-dimensional Lie algebra ${\mathfrak{g}}$. It is a global object attached to $M$. To “localize” it, for any open $U\subseteq M$ we define the Lie algebra ${\mathfrak{G}}(U)$ to be the space of smooth ${\mathfrak{g}}$-valued functions on $M$ whose closed support is contained in $U$. In particular, ${\mathfrak{G}}(\varnothing)=0$. For any inclusion of opens $U\subseteq V$ we have a homomorphism of Lie algebras $\iota_{VU}:{\mathfrak{G}}(U)\rightarrow{\mathfrak{G}}(V)$ such that the Lie algebras ${\mathfrak{G}}(U)$ assemble into a pre-cosheaf ${\mathfrak{G}}$ of Lie algebras on $M$. This is a coflasque pre-cosheaf, ie. all its structure maps $\iota_{VU}$ are injective.²²2The terminology comes from sheaf theory, where a pre-sheaf ${\mathcal{F}}:Open(M)^{opp}\rightarrow{\mathcal{C}}$ is called flasque if for any $U\subseteq V$ the restriction morphism ${\mathcal{F}}(V)\rightarrow{\mathcal{F}}(U)$ is a surjection.

Continuing with the example, for any two opens $U,V$ the following sequence is exact:

Here the first arrow is $\iota_{U,U\cap V}\oplus(-\iota_{V,U\cap V})$ and the second arrow is $\iota_{U\cup V,U}\oplus\iota_{U\cup V,V}$. Exactness follows from the existence of a partition of unity for the cover ${\mathfrak{U}}=\{U,V\}$ of $U\cup V$. In words, the exactness of the sequence (1) means that any element of ${\mathfrak{G}}(U\cup V)$ can be decomposed as a sum of elements of sub-algebras attached to $U$ and $V$ modulo ambiguities which take values in the sub-algebra attached to $U\cap V$.

More generally, the existence of a partition of unity implies that for any collection of opens $U_{i}$, $i\in I$, the following sequence is exact:

By definition, this means that the pre-cosheaf of vector spaces ${\mathfrak{G}}$ is a cosheaf of vector spaces. The cosheaf property is a compatibility of the pre-cosheaf with the notions of intersection and union of opens and expresses a stronger form of locality.

Note that the maps in the above exact sequences are not Lie algebra homomorphisms. Hence ${\mathfrak{G}}$ is not a cosheaf of Lie algebras. Nevertheless, the following additional property of ${\mathfrak{G}}$ can be regarded as a form of locality of the Lie bracket: for any $U,V,$ $[{\mathfrak{G}}(U),{\mathfrak{G}}(V)]\subseteq{\mathfrak{G}}(U\cap V)$. We will call this ”Property I”, where ”I” stands for ”intersection”. In particular, elements of ${\mathfrak{G}}(M)$ which are supported on non-intersecting opens $U$ and $V$ commute.

Symmetries of gapped states of quantum lattice spin systems are typically local only approximately. To phrase locality of symmetries in lattice systems in a similar language, one needs to replace the set $Open(M)$ of open subsets of $M$ with a more general structure which admits the notions of intersection, union, and cover.

The first thing to note is that the category $Open(M)$ is rather special: its objects form a pre-ordered set (i.e. the set of objects carries a relation $\subseteq$ which is reflexive and transitive), and the category structure is determined by the pre-order. The relation $\subseteq$ is also anti-symmetric: $U\subseteq V$ and $V\subseteq U$ implies $U=V$. In other words, $Open(M)$ is a poset. In general, we will not require the pre-order to be anti-symmetric. From the categorical viewpoint, $U\subseteq V$ and $V\subseteq U$ means that $U$ and $V$ are isomorphic objects of the category $Open(M)$, and as a general rule, it is not advisable to identify isomorphic objects.

For any pre-ordered set $(X,\leq)$ there is a natural notion of intersection and union. The intersection of $U,V\in X$ can be defined as the greatest lower bound (or meet) of both $U$ and $V$, i.e. a $W\in X$ such that $W\leq U$, $W\leq V$, and for any $W^{\prime}\leq U,V$ we have $W^{\prime}\leq W$. The meet of $U$ and $V$ is denoted $U\wedge V$. Similarly, the union of $U$ and $V$ can be defined as the smallest upper bound (or join) of both $U$ and $V$. It is denoted $U\vee V$. For a general pre-ordered set, the meet and join may not exist for all pairs of objects. If they exist, they are unique up to isomorphism. We will assume that $(X,\leq)$ is such that $U\wedge V$ and $U\vee V$ exist for all $U,V\in X$.³³3If we turn $X$ into a poset by identifying isomorphic objects, then this means that the poset is a lattice in the sense of order theory. The existence of all pairwise meets and joins implies the existence of all finite meets and joins. In the case of the pre-ordered set $Open(M)$ arbitrary (i.e. not necessarily finite) joins make sense.

Finally, to define covers of elements of $X$, let us assume that $U\wedge(V\vee W)\leq(U\wedge V)\vee(U\wedge W)$ for all $U,V,W\in X$.⁴⁴4The opposite relation is automatic, so this condition ensures that $U\wedge(V\vee W)\simeq(U\wedge V)\vee(U\wedge W)$ for all $U,V,W\in X$. This is equivalent to saying that the poset corresponding to $X$ is a distributive lattice. We will say that $X$ is a distributive pre-ordered set. This condition is certainly satisfied for $Open(M)$. We say that a collection ${\mathfrak{U}}=\{U_{i}\}_{i\in I}$ of elements of $X$ covers $A\in X$ iff $U_{i}\leq A$ for all $i\in I$ and $A\leq\bigvee_{i\in I}U_{i}$. This definition ensures that if ${\mathfrak{U}}$ covers $A$, then for any $B\leq A$ the collection ${\mathfrak{U}}\wedge B=\{U_{i}\wedge B\}_{i\in I}$ covers $B$.

In the case of the pre-ordered set $Open(M)$, the standard topological definition of a cover allows $I$ to be infinite. In general, if $X$ admits only finite joins, $I$ needs to be finite. Also, we may or may not allow $I$ to be empty. This possibility only arises when $A$ is the smallest element of $X$, i.e. $A\leq U$ for any $U\in X$. In the case of the pre-ordered set $Open(M)$, the smallest element is the empty set $\varnothing$, and the standard choice is to allow the empty cover of the empty set. For a general pre-ordered set, it is up to us whether to allow $I$ to be empty.

From a categorical perspective, this notion of a cover equips any distributive pre-ordered set $(X,\leq)$ admitting pairwise meets and joins with a Grothendieck topology, thus making it into a site [15]. Apart from the option of allowing the labeling set $I$ to be empty, this Grothendieck topology is canonical.

For any $W\in X$ we may consider the subset $X^{W}=\left\{U\in X\mid U\leq W\right\}$ with the pre-order inherited from $(X,\leq)$. It is a distributive pre-ordered set in its own right. When equipped with its canonical Grothendieck topology, it can be regarded as a sub-site of the site associated to $(X,\leq)$.

Given any distributive pre-ordered set $(X,\leq)$, we can define the notion of a pre-cosheaf of vector spaces, a pre-cosheaf of Lie algebras, a cosheaf of vector spaces, and a coflasque pre-cosheaf of Lie algebras with Property I exactly as before, i.e. by mechanically replacing $\cup$ with $\vee$ and $\cap$ with $\wedge$. Motivated by the above example, we introduce the following definition.

The following Lemma will be useful in future sections to check the cosheaf property:

The following proof is essentially a restatement of the proof of Proposition 1.3 in [16], adapted to the site $L$:

Let $X$ be a distributive pre-ordered set. Let $W\in X$ and ${\mathfrak{U}}=\{U_{i}\}_{i\in I}$ be a cover of $W$. Let ${\mathfrak{F}}$ be a pre-cosheaf of vector spaces over $X$. The Čech chain complex $C_{\bullet}({\mathfrak{U}},W;{\mathfrak{F}})$ is defined by

(where some linear order on $I$ has been chosen) with the differential is the Čech differential $\partial:C_{n+1}\rightarrow C_{n}$ given by $\partial=\sum_{j=0}^{n}(-1)^{j}\lambda_{j}$, where $\lambda_{j}$ is the canonical map

Let $C^{aug}({\mathfrak{U}},W;{\mathfrak{F}})=\{C_{\bullet}({\mathfrak{U}},W;{\mathfrak{F}})\rightarrow{\mathfrak{F}}(W)\}$ be the augmented Čech complex.

Covers of $W\in X$ form a category whose morphisms are refinements. A refinement of a cover ${\mathfrak{U}}=\left\{U_{i}\right\}_{i\in I}$ to a cover ${\mathfrak{V}}=\left\{V_{j}\right\}_{j\in J}$ is a map $\phi:J\rightarrow I$ such that $V_{j}\leq U_{\phi(j)}$. Refinements are composed in an obvious way.

We will call $C^{aug}_{\bullet+1}(-,W;-)$ the Čech functor. We will need some variants of the Čech functor. First, we can define a graded local Lie algebra over a distributive pre-ordered set $(X,\leq)$ in an obvious manner. The construction of the acyclic DGLA $C^{aug}_{\bullet+1}({\mathfrak{U}},W;{\mathfrak{F}})$ works in this case as well, except that it may have components in negative degrees.

Second, we define a pointed DGLA as a DGLA equipped with a distinguished central cycle of degree $-2$ (which we call the curvature). A morphism of pointed DGLAs is a DGLA morphism which preserves the distinguished central cycle.⁵⁵5The category of pointed DGLAs as defined here is a full subcategory of the category of curved DGLAs as defined in [17]. There, for a curved DGLA with curvature ${\mathsf{B}}$, ${\mathsf{B}}$ is not required to be central and the derivation $\partial$ satisfies $\partial^{2}={\rm ad}_{\mathsf{B}}$. We say that a graded local Lie algebra ${\mathfrak{F}}$ over $(X,\leq)$ with a terminal object $T\in X$ is pointed if it is equipped with a distinguished central element ${\mathsf{B}}\in{\mathfrak{F}}(T)$ of degree $-2$. Morphisms in the category of pointed graded local Lie algebras are required to preserve the distinguished element. Then the DGLA $C^{aug}_{\bullet+1}({\mathfrak{U}},T;{\mathfrak{F}})$ is an acyclic pointed DGLA, the distinguished central cycle being ${\mathsf{B}}\in C^{aug}_{-1}({\mathfrak{U}},T;{\mathfrak{F}})$. Of course, since the DGLA is acyclic, this central cycle is exact. The construction of a pointed DGLA from a pointed graded local Lie algebra and a cover of $T$ is functorial in both arguments.

We use the $\ell^{\infty}$ metric on ${\mathbb{R}}^{n}$, i. e. $d(x,y):=\max_{i=1,\ldots,n}|x_{i}-y_{i}|$. For any $U,V\subset{\mathbb{R}}^{n}$ we write $\operatorname{diam}(U):=\sup_{x,y\in U}d(x,y)$ and $d(U,V):=\inf_{x\in U,y\in V}d(x,y)$. They take values in extended non-negative reals $[0,\infty]$. Thus $\operatorname{diam}(\varnothing)=0$ and $d(U,\varnothing)=\infty$ for any $U\subset{\mathbb{R}}^{n}$. For a nonempty set $U$ and $r\geq 0$ we define $U^{r}:=\{x\in{\mathbb{R}}^{n}:d(x,U)\leq r\}$ while we set $\varnothing^{r}=\varnothing$.

A quantum lattice system consists of a countable subset $\Lambda\subset{\mathbb{R}}^{n}$ (“the lattice”) and a finite-dimensional complex Hilbert space $V_{j}$ for every $j\in\Lambda$. We make the following assumption on the lattice system⁶⁶6In [11] $\Lambda$ was assumed to be a Delone set, i.e. it was required to be uniformly filling and uniformly discrete. These assumptions were imposed on physical grounds. All the results proved in [11] hold under weaker assumptions adopted in this paper. In [8] $\Lambda$ was taken to be ${\mathbb{Z}}^{n}$ for simplicity. : there is a $C_{\Lambda}>0$ such that the number of points of $\Lambda$ in any hypercube of diameter $d$ is bounded by $C_{\Lambda}(d+1)^{n}$.

For any bounded nonempty $X\subset{\mathbb{R}}^{n}$ let ${\mathscr{A}}(X):=\bigotimes_{j\in X\cap\Lambda}\mathrm{Hom}_{{\mathbb{C}}}(V_{j},V_{j})$. For any $X\subset Y$ there is an inclusion ${\mathscr{A}}(X)\hookrightarrow{\mathscr{A}}(Y)$ and the algebras ${\mathscr{A}}(X)$ form a direct system with respect to these inclusions. We extend this direct system to include the empty set by setting ${\mathscr{A}}(\varnothing)={\mathbb{C}}$ and letting the inclusion ${\mathscr{A}}(\varnothing)\hookrightarrow{\mathscr{A}}(X)$ take $\alpha\mapsto\alpha\boldsymbol{1}$. Each ${\mathscr{A}}(X)$ is a finite-dimensional $C^{*}$-algebra with the operator norm, and the inclusions ${\mathscr{A}}(X)\hookrightarrow{\mathscr{A}}(Y)$ preserve this norm. The normed *-algebra of local observables is

The algebra of quasi-local observables ${\mathscr{A}}$ is the norm-completion of ${\mathscr{A}}_{\ell}$; it is a $C^{*}$-algebra.

For any bounded $X\subset{\mathbb{R}}^{n}$, define the normalized trace ${\overline{\rm tr}}:{\mathscr{A}}(X)\to{\mathbb{C}}$ as ${\overline{\rm tr}}({\mathcal{A}})=\operatorname{tr}({\mathcal{A}})/{\sqrt{\dim({\mathscr{A}}(X))}}$. For any bounded $X\subset Y$ the partial trace ${\overline{\rm tr}}_{X^{c}}:{\mathscr{A}}(Y)\to{\mathscr{A}}(X)$ is uniquely specified by the condition ${\overline{\rm tr}}_{X^{c}}({\mathcal{A}}\otimes{\mathcal{B}})={\overline{\rm tr}}({\mathcal{A}}){\mathcal{B}}$ for any ${\mathcal{A}}\in{\mathscr{A}}(Y\backslash X)$ and ${\mathcal{B}}\in{\mathscr{A}}(X)$. Besides forming a direct system with respect to inclusions, the spaces ${\mathscr{A}}(X)$ are also an inverse system with respect to the partial trace. ${\overline{\rm tr}}$ extends to a normalized positive linear functional on ${\mathscr{A}}$, i.e. a state. We say that ${\mathcal{A}}\in{\mathscr{A}}$ is traceless if ${\overline{\rm tr}}({\mathcal{A}})=0$. The space of traceless anti-hermitian elements of ${\mathscr{A}}(X)$ will be denoted ${{\mathfrak{d}}_{l}}(X)$. ${{\mathfrak{d}}_{l}}(X)$ is a real Lie algebra with respect to the commutator. The Lie algebras ${{\mathfrak{d}}_{l}}(X)$ form a direct system over the directed set of bounded subsets of ${\mathbb{R}}^{n}$, and its limit will be denoted ${{\mathfrak{d}}_{l}}$. Equivalently, ${{\mathfrak{d}}_{l}}$ is the Lie algebra of traceless anti-hermitian elements of ${\mathscr{A}}_{\ell}$. Note that ${\mathscr{A}}_{\ell}={\mathbb{C}}{\mathbf{1}}\oplus({{\mathfrak{d}}_{l}}\otimes{\mathbb{C}})$.

The set of bricks exhausts the collection of bounded subsets of ${\mathbb{R}}^{n}$ in the sense that any bounded subset is contained in a brick. In addition, the set of bricks satisfies the following regularity property:

For any brick $Y$ we define the following subspace of ${{\mathfrak{d}}_{l}}(Y)$:

Each ${{\mathfrak{d}}_{l}}(Y)$ decomposes as a direct sum ${{\mathfrak{d}}_{l}}(Y)=\bigoplus_{X\subseteq Y}{{\mathfrak{d}}_{l}}^{X}$ over bricks contained in $Y$, and for any brick $X\subseteq Y$ the partial trace ${\overline{\rm tr}}_{Y\backslash X^{c}}$ is the projection onto $\bigoplus_{Z\subseteq X}{{\mathfrak{d}}_{l}}^{Z}\subseteq{{\mathfrak{d}}_{l}}(Y)$. Intuitively, ${{\mathfrak{d}}_{l}}^{Y}$ consists of elements of ${{\mathfrak{d}}_{l}}(Y)$ which are not localized on any brick properly contained in $Y$.

Derivations of ${\mathscr{A}}$ which appear in the physical context are typically only densely defined and have the form

where ${\mathsf{F}}^{Y}\in{{\mathfrak{d}}_{l}}^{Y}$. We are now going to define for every $U\subset{\mathbb{R}}^{n}$ a real Lie algebra ${\mathfrak{D}}_{al}(U)$ which consists of derivations approximately localized on $U$ and such that all ${\mathfrak{D}}_{al}(U)$ have a common dense domain. Moreover, they form a pre-cosheaf of Lie algebras over a certain category of subsets of ${\mathbb{R}}^{n}$.

If $U$ is empty, then for $k>0$ $\|{\mathsf{F}}\|_{U,k}<\infty$ if and only if ${\mathsf{F}}^{Y}=0$ for all $Y\in{\mathbb{B}}_{n}$. Thus ${\mathfrak{D}}_{al}(\emptyset)=0$. We also denote ${\mathfrak{D}}_{al}({\mathbb{R}}^{n})={\mathfrak{D}}_{al}$.

It is easy to see that (6) is a norm on ${\mathfrak{D}}_{al}(U)$ for each $k\geq 0$. We endow ${\mathfrak{D}}_{al}(U)$ with the locally convex topology given by the norms (6) ranging over all $k\geq 0$. Recall that a topological vector space is called a Fréchet space if it is Hausdorff, and if its topology can be generated by a countable family of seminorms with respect to which it is complete.

Recall that for a nonempty $U\subset{\mathbb{R}}^{n}$ we write $U^{r}:=\{x\in{\mathbb{R}}^{n}:d(x,U)\leq r\}$. The norms $\|\cdot\|_{U,k}$ obey the following dominance relation.

The above Lemma shows that the space ${\mathfrak{D}}_{al}(U)$ only depends on the asymptotic geometry of the region $U$ in the following sense: if $U\subseteq V^{r}$ and $V\subseteq U^{r}$ for some $r\geq 0$ then ${\mathfrak{D}}_{al}(U)={\mathfrak{D}}_{al}(V)$ (as subsets of $\prod_{Y}{{\mathfrak{d}}_{l}}^{Y})$ and are isomorphic as Fréchet spaces. In particular, for any non-empty bounded $U\subset{\mathbb{R}}^{n}$, the space ${\mathfrak{D}}_{al}(U)$ coincides with ${\mathfrak{D}}_{al}(\{0\})$.

To relate the spaces ${\mathfrak{D}}_{al}(U)$ to the traditional $C^{*}$-algebraic picture we prove the following:

In [11], the algebra ${\mathscr{A}}_{a\ell}$ of almost-local operators was defined as a subspace of ${\mathscr{A}}$ where a countable family of norms similar to (6) takes finite values. Here we equivalently define ${\mathscr{A}}_{a\ell}$ as the set of elements of ${\mathscr{A}}$ whose traceless hermitian and anti-hermitian parts live in ${{\mathfrak{d}}_{al}}$, topologized as ${\mathscr{A}}_{a\ell}={\mathbb{C}}{\mathbf{1}}\oplus({{\mathfrak{d}}_{al}}\otimes{\mathbb{C}})$.

In view of Prop. 3.2 it is natural to make the following definition.

For any two bounded sets $U,V$, a local observable is strictly localized on both $U$ and $V$ iff it is localized on their intersection, ie. ${\mathscr{A}}(U)\cap{\mathscr{A}}(V)={\mathscr{A}}(U\cap V)$. An analogous relation for the spaces ${\mathfrak{D}}_{al}(U)$ does not hold in general⁷⁷7Indeed, for any two bounded $U,V$ the spaces ${\mathfrak{D}}_{al}(U)$ and ${\mathfrak{D}}_{al}(V)$ coincide and are nontrivial but if $U$ and $V$ are disjoint then ${\mathfrak{D}}_{al}(U\cap V)=0$., but it does hold if we assume that $U$ and $V$ satisfy the following transversality condition:

We will say $U$ and $V$ are transverse if they are $C$-transverse for some $C>0$.