37 Parent Hamiltonians of injective and regular \(G\)-injective PEPS
For injective PEPS, two-site parent interactions pulled back along left inverses of the site maps isolate the virtual Bell bonds and reconstruct the global ground space when the covering patches intersect [ 18 , Section IV.C.1 ] . For regular \(G\)-injective PEPS, right-multiplication invariance on shared internal bonds transfers local parent constraints into cycle holonomy and flatness conditions [ 272 , Section 5 ] .
37.1 A counterexample to bare inverse-adjoint parent pullbacks
The exact-kernel assertion following the displayed parent-Hamiltonian formula in [ 18 , Section IV.C.1 ] , local source lines 2031–2037, fails for rectangular injective site maps. The correction and its relation to genuine parent uniqueness are recorded in [ 74 ] .
Consider two vertices joined by a bond of dimension \(D=1\). At either endpoint place the site map \(A:\mathbb C\longrightarrow \mathbb C^2\), \(A(z)=(z,0)\), and choose the left inverse \(\ell (z_0,z_1)=z_0\). The product maps \(F=A\otimes A\) and \(L=\ell \otimes \ell \) satisfy \(LF=I\), so both the site and product maps are injective. The virtual Bell vector is the scalar \(1\). Its complement interaction is \(H=0\), which is positive and has exactly the Bell line as its kernel.
Direct multiplication gives \(\ell A=1\) and \((\ell \otimes \ell )(A\otimes A)=1\). The two-endpoint virtual space is one-dimensional, and the Bell vector is nonzero, so its span is the whole virtual space.
Bare inverse-adjoint conjugation gives \(L^\dagger HL=0\) on \(\mathbb C^2\otimes \mathbb C^2\). The vector \(e_1\otimes e_0\) belongs to its kernel, but lies outside \(\operatorname {ran}F\). The bare kernel has dimension four, whereas the intended product-site range has dimension one. Hence
The example uses a positive bond dimension and genuinely injective tensors. The corrected interaction with its physical reconstruction summand is treated separately.
The conjugated interaction is zero because \(H=0\). Every vector in \(\operatorname {ran}(A\otimes A)\) has zero coefficient at \((1,0)\), whereas \(e_1\otimes e_0\) has coefficient one there. Thus it is an additional physical kernel vector.
Let \(F_R\) be the product of injective site maps and let \(L_R\) be the product of their left inverses. For a positive virtual interaction \(H\), define
The second summand constrains a physical ground vector to the image of the product site map. This is a regional consequence of the inverse-adjoint deformation argument in [ 18 , Section IV.C.1 ] . For rectangular site maps, the bare printed interaction leaves unused physical directions in its kernel; the correction is recorded in [ 74 ] .
The pulled-back interaction is positive and satisfies
Equivalently, \(\widetilde H_R\psi =0\) precisely when \(HL_R\psi =0\) and \(F_RL_R\psi =\psi \). In particular, if \(\ker H\) is the genuine virtual regional bond space, then \(\widetilde H_R\) is a parent interaction with kernel \(\mathcal G_R\). The site maps may be rectangular. The reconstruction penalty acts on the entire region \(R\); the assertion does not establish locality of a sum of edge interactions.
Both summands are positive. A vector in the kernel of their sum therefore lies in the kernel of each summand. Since \(L_RF_R=I\), the adjoint \(L_R^\dagger \) is injective, so the first condition is equivalent to \(HL_R\psi =0\). The kernel of the second summand is the kernel of \(I-F_RL_R\). Thus every physical ground vector is the image of a virtual ground vector, and conversely. The regional factorization \(T_R=F_RB_R\) identifies the image of the virtual bond range with \(\mathcal G_R\).
Give each vertex of \(R\) an independent configuration of its incident virtual labels. Thus each internal edge has two labels, one at each endpoint. Let \(F_R=\bigotimes _{v\in R}A_v\) be the product site map. For injective sites let \(L_R\) be the product of their left inverses. Let \(B_R\) denote the virtual region map that pairs the two endpoint labels of every internal edge and leaves the crossing labels open. Its coefficients are sums of basis vectors indexed by the incident-edge assignments, with the boundary labels fixed.
For arbitrary site tensors the open-region map satisfies \(T_R=F_RB_R\), and \(\mathcal G_R=F_R(\operatorname {ran}B_R)\). If all site maps are injective, then
No nonzero-bond hypothesis is needed for these identities. These are the finite-graph virtual-coordinate identities underlying the parent argument in [ 18 , Section IV.C.1 ] .
The product site map takes a virtual basis vector to the product of its site tensor coefficients. Summing over compatible internal edge labels with the crossing labels fixed gives the open-region contraction, so \(T_R=F_RB_R\). Tensoring the site left inverses gives \(L_RF_R=I\), hence injectivity of \(F_R\) and \(L_RT_R=B_R\). The range and inverse-image formulas follow.
37.2 The virtual Bell-bond ground space
The virtual part of the injective-parent uniqueness argument in [ 18 , Section IV.C.1 ] , local source lines 2017–2044, concerns a product of independent maximally entangled bonds. We first establish this part without physical maps.
Let \(E\) be a finite set of bonds with dimensions \(D_e\ge 0\). Write \(\zeta _e=(a_e,b_e)\) for their endpoint labels and put
The common virtual ground space consists of the vectors whose slice on each bond, for every fixed configuration of all other bonds, belongs to \(\operatorname {span}\{ \omega _e\} \).
The common virtual ground space is \(\operatorname {span}\{ \Omega \} \). The vector \(\Omega \) is nonzero exactly when every \(D_e\) is positive; in this case the common ground space is one-dimensional. If a bond dimension vanishes, the entire virtual configuration space is empty and the vector space is zero. The assertion also includes an empty bond collection, whose product vector is the scalar vacuum.
The product vector satisfies each slice condition by separating its corresponding Bell factor. Conversely, the slice condition forces a coefficient to vanish whenever the two labels of some bond differ. On diagonal configurations, replacing one common endpoint label by another leaves the coefficient unchanged. Finitely many such replacements join any two diagonal configurations. Thus every common ground vector is a scalar multiple of \(\Omega \). Its coefficient at any diagonal configuration is one, proving nonvanishing when all dimensions are positive. A zero-dimensional bond makes the configuration space empty, giving the stated degenerate case.
Let the site tensors on a finite graph be injective. Suppose a collection of regions contains every vertex in some region and contains the two endpoints of every edge together in some region. Then its common parent space is
Every finite positive interaction family with the genuine regional kernels has this same zero-energy space. If the tensor is an injective PEPS tensor, so that all bond dimensions are positive, the closed vector is nonzero and the ground space is one-dimensional. The span formula also holds when a bond dimension vanishes, in which case the span can be zero.
This is the covering-region form of the virtual-pair argument in [ 18 , Section IV.C.1 ] . Vertex coverage is explicit, including for isolated vertices. The physical reconstruction correction is recorded in [ 74 ] .
Vertex coverage provides the genuine site-image conditions, so every physical parent vector is the product-site image of a vector satisfying all virtual regional conditions. Edge coverage imposes the Bell condition on each separate virtual bond. Their common space is precisely the line spanned by the product Bell vector. Its image is the closed PEPS vector. Conversely the closed vector satisfies every regional condition. Positivity gives the Hamiltonian-kernel statement, and positive bond dimensions give nonvanishing and dimension one.
37.3 Nearest-neighbor parents of injective PEPS
For each edge \(e=\{ u,v\} \) of a finite graph, use the endpoint region \(R_e=\{ u,v\} \). The augmented family additionally contains the singleton \(\{ v\} \) for each isolated vertex \(v\). All regional spaces are the images of the open-boundary contractions. Isolated-site conditions are needed to exclude unused physical directions at vertices absent from every edge; see [ 74 ] .
Every edge is contained in its endpoint region. If every vertex has a neighbor, the endpoint regions also cover every vertex. On an arbitrary finite graph, the augmented family covers every vertex and every edge.
A vertex with a neighbor belongs to the region of their edge. An isolated vertex belongs to its singleton region.
Suppose every site tensor is injective. If every vertex has a neighbor, the common space for the edge-pair regions is exactly the span of the closed PEPS vector. For an arbitrary finite graph, the same conclusion holds for the augmented edge and isolated-site family. If the tensor is an injective PEPS tensor, so that every bond dimension is positive, each of these spaces is one-dimensional. These are the nearest-neighbor and isolated-site forms of the virtual-pair argument in [ 18 , Section IV.C.1 ] . Without positive bond dimensions the span equality still holds, and the PEPS span may be zero.
Applying the covering-region theorem to the two coverage statements yields the span equalities; positive bond dimensions make the virtual Bell product nonzero, and injectivity preserves this property in the physical PEPS vector.
Choose any positive semidefinite interaction on each region, with kernel exactly its regional PEPS space, and sum their extensions to the full physical graph. Under the corresponding hypotheses of the preceding theorem, the Hamiltonian kernel is exactly the closed PEPS span. For an injective PEPS tensor, whose bond dimensions are positive, this kernel is one-dimensional. The edge-only conclusion requires that every vertex have a neighbor; the augmented conclusion includes the isolated-site singleton interactions explicitly.
The kernel of a finite sum of positive interactions is the intersection of their kernels, which is precisely the common regional parent space.
37.4 Regional supports of ordinary injective tensors
The region map obtained by summing all graph-edge labels is the open-region map multiplied by the number of assignments to edges wholly outside the region. Injectivity of the former implies injectivity of the latter. If every virtual bond is nonzero dimensional, the two maps have the same range.
Split the edge labels into those incident to the region and those wholly outside it. The exterior labels do not enter any site tensor, so they contribute their cardinality. An equality of open-region images implies an equality of globally summed images. A nonzero scalar factor preserves a linear range.
If the open-region map of \(R^c\) is injective, then the unnormalized reduced density of the closed PEPS has support exactly equal to \(\mathcal G_R\). Neither injectivity of \(R\) nor connectedness of either side is needed.
Write the physical coefficient matrix as \(M=T_RS^\mathsf T\), where \(S\) is the complementary open map after matching the crossing labels. A left inverse of \(S\) gives a right inverse of its transpose. Consequently \(\operatorname {ran}M=\operatorname {ran}T_R\). The reduced density \(MM^\dagger \) has the same range as \(M\).
If the open maps of \(R\) and \(R^c\) are injective, then \(\operatorname {rank}\rho _R=\prod _{e\in \partial R}D_e\). If two coefficient vectors coincide and their complementary open maps are injective, their regional ground spaces on \(R\) coincide. These assertions need neither site injectivity nor connectedness.
The complementary map identifies the reduced support with \(\mathcal G_R\), and injectivity of the regional map gives its exact boundary dimension. Equal coefficient vectors have equal reduced densities, so the corresponding support spaces agree.
Let \(A\) be an injective PEPS tensor, so that every site tensor is injective and every virtual bond has positive dimension. For every finite region \(R\),
The region and its complement may be disconnected. The positive-bond convention is recorded in [ 73 ] . This is the injective Schmidt-span consequence used in [ 257 , Section 3 ] and the virtual-pair argument of [ 18 , Section IV.C.1 ] .
A contraction of injective sites with nonzero virtual bonds is injective. This applies both to \(R\) and to its complement. The complementary support theorem identifies the reduced support with \(\mathcal G_R\). Injectivity of the region map identifies its range dimension with the crossing-bond configuration dimension.
Fix all physical indices in \(R\) except the index at \(v\in R\). The resulting one-site coefficient vector is a linear slice of the regional vector.
Every one-site slice at \(v\in R\) of a vector in \(\mathcal G_R\) lies in the image of the site map \(A_v\). Consequently a collection of parent regions covering every vertex enforces all one-site image conditions. No injectivity or positive-bond hypothesis is needed. This is a consequence of the finite contractions in [ 18 , Section IV.C.1 ] .
Expand a regional vector as a sum over incident-edge assignments, with a coefficient depending on the crossing labels. For each assignment the selected site’s coefficient vector is a column of \(A_v\) multiplied by the product of all other site coefficients. Every summand belongs to the site image, hence so does the slice. Apply this to a parent region containing each vertex.
Suppose the site maps are injective and the parent regions cover every vertex. Their common physical ground space is exactly the image under the product site map of the common genuine virtual regional bond space. The kernel of any positive exact-kernel parent Hamiltonian with finitely many terms on these regions has the same description.
The regional conditions imply the site-image conditions by the preceding theorem. The product site inverse therefore reconstructs each physical vector and sends its regional conditions to the genuine virtual conditions. The product site map sends the virtual conditions back to the physical conditions. Positivity identifies the kernel of the parent sum with the common local space.
37.5 Vertex and bond coordinates
The virtual construction in [ 18 , Section IV.C.1 ] , local source lines 2017–2044, can be described either by independent labels at the vertices or by the two endpoint labels of each edge. These descriptions are equivalent on the original finite graph, including vertices incident to no edge.
Let every edge \(e\) of a finite graph have a dimension \(D_e\ge 0\). Independent virtual configurations at all vertices are in bijection with configurations \(\zeta _e=(a_e,b_e)\) of the two endpoints of each edge. Equality of an edge’s two incident vertex labels is precisely the condition \(a_e=b_e\). Such consistent configurations are exactly those obtained by assigning one common label to each bond.
Read the two endpoint labels of each edge. Conversely, recover each vertex’s incident label from its corresponding endpoint. An edge has distinct endpoints, so these operations are inverse. The consistency assertion then follows coordinate by coordinate.
In the full vertex region, sum the consistent virtual bond assignments in the original contraction. Under the endpoint-pair coordinate equivalence, the resulting coefficient vector is exactly
No injectivity, connectedness, or positivity of the bond dimensions is assumed.
The full region has no crossing edges, and its incident labels are exactly its global bond labels. An endpoint configuration therefore contributes once if it is consistent and zero times otherwise. This is the displayed product of diagonal indicators.
37.6 Internal Bell constraints from regional contractions
The virtual-pair argument of [ 18 , Section IV.C.1 ] , local source lines 2017–2044, uses the Bell condition on each original graph edge. Here this condition is derived from the regional virtual bond maps, before any physical injectivity assumption is imposed.
Let both endpoints of an edge \(e\) belong to a finite region \(R\). Every vector in the regional virtual bond range vanishes when the two half-edge labels of \(e\) differ. Its coefficients are unchanged when the same permutation of the bond alphabet is applied to both labels. The assertion includes arbitrary boundary coefficients and vanishing bond dimensions.
Each incident-edge assignment uses one common label at the two endpoints of \(e\), so no assignment contributes to a coefficient with unequal labels. A permutation of this common internal label is a bijection of the incident assignments, leaves every boundary label fixed, and permutes the two endpoint labels together. The resulting identities extend by linearity to the regional virtual map.
Let \(x\) be a global vector in independent virtual vertex coordinates. Suppose that, for every fixed complementary virtual configuration, its slice on \(R\) belongs to the regional virtual bond range. Reorganize \(x\) into its original edge-pair coordinates, obtaining \(y\). For every internal edge \(e\) of \(R\) and every fixed configuration \(\zeta \) of the other endpoint pairs, one has
Thus the corresponding one-bond slice belongs to the Bell line. In particular, the assertion applies to the two-vertex region formed by the endpoints of \(e\). No bond dimension is assumed positive.
Restrict a global configuration to \(R\) and its complement. The regional support condition gives vanishing off the diagonal. Jointly permuting the two endpoint labels leaves the complementary configuration fixed because both endpoints belong to \(R\). Transpositions therefore identify all diagonal coefficients. A coefficient function that vanishes off the diagonal and is constant on it is a scalar multiple of the Bell vector.
Let \((R_i)\) be a family of finite vertex regions such that the two endpoints of every edge lie together in some \(R_i\). If every complementary slice of \(x\) on every \(R_i\) belongs to its regional virtual bond range, then its edge-pair coefficient vector belongs to the common Bell-bond ground space.
For each edge choose a region containing both endpoints and apply the Bell-slice theorem. This gives every defining condition of the common Bell-bond space.
37.7 The physical image of the Bell product
The global virtual-pair construction of [ 18 , Section IV.C.1 ] , local source lines 2017–2044, sends the independent Bell product through the site tensor maps. The following identities retain the graph contraction and include isolated vertices and zero-dimensional bonds.
Let \(F\) be the product of all site tensor maps, expressed on its domain in the independent endpoint-pair coordinates of the edges. Its codomain is the physical coefficient space on the full vertex region.
Write \(\psi \) for the closed PEPS vector and \(\Omega \) for the product of virtual Bell vectors. Then, in full-region physical coordinates,
Neither identity requires site injectivity or positive bond dimensions.
The full-region virtual contraction is exactly \(\Omega \) in endpoint-pair coordinates. Applying the product site map gives the full-region open coefficient vector, which equals the closed state because no boundary edges remain. The common Bell space is the span of \(\Omega \), and linear maps preserve spans.
If every ordinary site tensor map is injective, then \(F\) is injective and
These assertions concern the global state and the full-region range. They do not yet identify the ground space of a parent built from smaller regions.
Tensor the site left inverses and express them in the endpoint-pair coordinates. This gives an inverse on the image of \(F\). The inverse-image assertion follows from injectivity. By \(F\Omega =\psi \), nonvanishing is equivalent to that of \(\Omega \), which holds precisely when every bond dimension is positive.
37.8 The common one-site image space
Let \(A\) be a tensor on a finite graph, with physical space \(\mathcal P_v=\mathbb C^d\) at each vertex and independent virtual half-edge space \(\mathcal V_v\) at that vertex. Write \(A_v:\mathcal V_v\to \mathcal P_v\) for its site map and \(T_A=\bigotimes _v A_v\) for their product. The common one-site image space \(\mathcal K_A\) consists of those physical vectors whose one-site slices lie in \(\operatorname {ran}A_v\), for every vertex \(v\) and every choice of physical indices at all other vertices. If every \(A_v\) is injective, choose left inverses \(L_v\) and set \(P_v=A_vL_v\). These retractions need not be orthogonal. This is the independent site-image condition preceding the virtual bond constraints in [ 18 , Section IV.C.1 ] .
If every site map \(A_v\) is injective, then
More precisely, every \(\psi \in \mathcal K_A\) satisfies
This assertion holds for an arbitrary finite graph. It identifies the product site image, before imposing virtual bond constraints; it does not assert uniqueness of a PEPS parent ground state.
For each \(v\), the map \(P_v=A_vL_v\) fixes exactly \(\operatorname {ran}A_v\). Therefore its action at the vertex \(v\), with the identity at all other vertices, fixes every vector in \(\mathcal K_A\). Multiplying these commuting actions gives \(\bigotimes _vP_v=T_A(\bigotimes _vL_v)\), proving the reconstruction identity and the inclusion \(\mathcal K_A\subseteq \operatorname {ran}T_A\). Conversely, \(P_vA_v=A_v\) shows that every vector in \(\operatorname {ran}T_A\) satisfies every one-site image condition.
37.9 Regional conditions after the site inverses
Let \(R\) be any region of a finite graph and let \(F_v:\mathbb C^d\to \mathcal Q_v\) be local maps, whose output spaces may depend on \(v\). For an outside output configuration \(\tau \), the \(R\)-slice of \((\bigotimes _vF_v)\psi \) is
where \(\beta \) ranges over outside physical configurations and \(\psi _\beta \) denotes the corresponding physical \(R\)-slice. This is a finite contraction identity; it does not require injectivity or an inner product on the output spaces.
Expanding the product map in physical configurations, splitting each configuration into its coordinates in \(R\) and its coordinates outside \(R\), and factoring the product of the local coefficients, summing first over the coordinates in \(R\) gives the displayed slice.
For any tensor on a finite graph and any family of regions \((R_i)_i\), let \(\mathcal K_{\mathrm{virt}}\) consist of the independent virtual vectors whose \(R_i\)-slices belong to the virtual bond range \(\mathcal B_{R_i}\), for every \(i\) and every outside virtual configuration. This definition requires no injectivity.
Suppose every site map is injective, with chosen left inverses \(L_v\), and write \(L_A=\bigotimes _vL_v\). For a region \(R\), let \(\mathcal B_R\) be the image of the virtual bond contraction map, with the crossing bond coordinates free. If every physical \(R\)-slice of \(\psi \) belongs to the regional PEPS space, then every virtual \(R\)-slice of \(L_A\psi \) belongs to \(\mathcal B_R\). The implication holds simultaneously for any family of regions. No reconstruction identity or virtual support condition is assumed. This is the site-inverse step of the virtual-pair argument in [ 18 , Section IV.C.1 ] ; the identification of the common virtual conditions with the globally paired bond space is a separate assertion.
The regional product inverse sends the regional physical space into \(\mathcal B_R\), since composing it with the regional open-boundary map gives the virtual bond contraction map. Applying the slicing identity with \(F_v=L_v\) shows that every summand lies in the subspace \(\mathcal B_R\), and hence so does their linear combination.
37.10 Exact image of the common virtual conditions
Let \(A\) be a tensor on a finite graph, and let \(T_A=\bigotimes _vA_v\) be its product site map. For an arbitrary family of regions \((R_i)_i\), let \(\mathcal K_{\mathrm{virt}}\) be the common space of independent virtual vectors whose \(R_i\)-slices lie in the regional virtual bond range, for every \(i\) and every outside virtual configuration. Let \(\mathcal K_{\mathrm{phys}}\) be the common physical regional PEPS space, defined by all corresponding physical slice conditions. Then
This implication requires no injectivity of the site maps.
Split an independent virtual configuration into its coordinates inside and outside a region. A physical regional slice of \(T_Ax\) is a linear combination of the regional product site map applied to virtual regional slices of \(x\). The regional product site map sends the virtual bond range onto the physical regional PEPS space, so each summand satisfies the physical regional condition.
If every site map is injective, then
where \(\mathcal K_A\) is the common singleton site-image space. These singleton conditions are explicit, and no reconstruction condition is a hypothesis. This is the physical-to-virtual correspondence in the virtual-pair argument of [ 18 , Section IV.C.1 ] . Identifying \(\mathcal K_{\mathrm{virt}}\) with the global paired-bond space remains a separate assertion.
The forward inclusion follows from the preceding theorem and the singleton-image property of the product site map. Conversely, let \(\psi \in \mathcal K_A\cap \mathcal K_{\mathrm{phys}}\). Its product inverse \(x=L_A\psi \) belongs to \(\mathcal K_{\mathrm{virt}}\) by the regional inverse theorem. The singleton-image theorem gives \(T_Ax=\psi \).
37.10.1 Physical changes on the site images
Fix a finite simple graph with ordered vertices \(V\), site maps \(A_v:\mathcal V_v\to \mathbb C^d\), and linear maps \(F_v:\mathbb C^d\to \mathbb C^e\). Write \(B_v=F_vA_v\), \(\mathcal G_S(A)=\operatorname {im}\mathcal C_S(A)\) for the full open-region range, and \(\mathcal P_{\mathcal R}(A)\) for the common parent space of a family \(\mathcal R=(R_i)_{i\in \mathcal I}\). Set \(F_S=\bigotimes _{v\in S}F_v\) and \(F_{\mathrm{all}}=\bigotimes _{v\in V}F_v\). The following are consequences of the site inverse in [ 272 , Definition 5.1 ] and the regional parent construction. The virtual spaces and representations remain fixed.
If every \(F_v\) is injective on \(\operatorname {im}A_v\), there are linear maps \(L_v:\mathbb C^e\to \mathbb C^d\) such that \(L_vF_vx=x\) for \(x\in \operatorname {im}A_v\). For any such choice, \(L_vB_v=A_v\), and \(F_vL_vy=y\) for \(y\in \operatorname {im}B_v\).
The injection \(F_v:\operatorname {im}A_v\to \mathbb C^e\) has a linear left inverse, and composing it with the inclusion of \(\operatorname {im}A_v\) into \(\mathbb C^d\) yields \(L_v\). Applying \(L_vF_v\) to each column of \(A_v\) gives \(L_vB_v=A_v\), and if \(y=B_vz=F_vA_vz\), then \(F_vL_vy=F_vA_vz=y\).
Suppose \(L_vF_v\) fixes \(\operatorname {im}A_v\) at every vertex. For any region \(S\) and \(\psi \in \mathcal G_S(A)\), \(L_SF_S\psi =\psi \). Consequently site-image injectivity of the \(F_v\) implies injectivity of \(F_S\) on \(\mathcal G_S(A)\) for every \(S\). All virtual boundary conditions are allowed.
Writing \(\psi =\mathcal C_S(A)\theta \) for arbitrary boundary data \(\theta \), physical deformation commutes with contraction, so
Applying \(L_S\) to an equality \(F_S\psi =F_S\phi \) yields injectivity, and when only site-image injectivity is given, \(L_v\) is chosen by Theorem 37.10.1.1.
Let \(P_v:\mathbb C^d\to \mathbb C^d\) fix \(\operatorname {im}A_v\) pointwise. If every one-site slice of \(\psi \) at \(v\) belongs to \(\operatorname {im}A_v\), for all \(v\), then \((\bigotimes _vP_v)\psi =\psi \). In particular, if \(L_vF_v\) fixes \(\operatorname {im}A_v\) and every vertex belongs to some \(R_i\), then \(L_{\mathrm{all}}F_{\mathrm{all}}\psi =\psi \) for every \(\psi \in \mathcal P_{\mathcal R}(A)\).
The operator acting as \(P_v\) at one vertex and as the identity elsewhere fixes \(\psi \) because it acts on its one-site slices, and multiplying these operators gives the first assertion. Vertex coverage puts every one-site slice of a parent vector in the appropriate site image, so taking \(P_v=L_vF_v\) and using \(L_{\mathrm{all}}F_{\mathrm{all}}=\bigotimes _v(L_vF_v)\) gives the second assertion.
Every physical product map satisfies \(F_{\mathrm{all}}\mathcal P_{\mathcal R}(A) \subseteq \mathcal P_{\mathcal R}(B)\), without injectivity or coverage assumptions. If each \(F_v\) is injective on \(\operatorname {im}A_v\) and the regions cover all vertices, its restriction is a linear isomorphism
An inverse is \(L_{\mathrm{all}}\) for any site inverses from Theorem 37.10.1.1.
The identity \(\mathcal G_S(B)=F_S\mathcal G_S(A)\) sends each regional constraint to the corresponding constraint for \(B\). In a fixed exterior slice, the physical maps on exterior vertices only form linear combinations of source slices; hence the forward inclusion holds for arbitrary \(F_v\), and applying it also to \(L_v\) with \(LB=A\) gives the reverse inclusion. Since \(LF\) fixes the source site images and \(FL\) fixes the target site images, Theorem 37.10.1.3 gives both inverse identities on the parent spaces, proving (2) and the dimension equality.
Assume the site-image injectivity and vertex coverage of Theorem 37.10.1.4, and suppose \(\mathcal I\) is finite. Let \(H_i,K_i\) be positive semidefinite regional interactions with \(\ker H_i=\mathcal G_{R_i}(A)\) and \(\ker K_i=\mathcal G_{R_i}(B)\). If \(H\) and \(K\) are the sums of their placements on the full graph, then \(F_{\mathrm{all}}(\ker H)=\ker K\).
Positivity identifies the kernel of each sum with the intersection of its local kernels, so \(\ker H=\mathcal P_{\mathcal R}(A)\) and \(\ker K=\mathcal P_{\mathcal R}(B)\), to which (2) applies.
Let \(G\) be finite, \(\rho \) a complex representation on \(\mathcal W\), and \(T:\mathcal W\to \mathcal P\) and \(S:\mathcal W\to \mathcal Q\) be \(G\)-injective for \(\rho \). There is a linear \(F:\mathcal P\to \mathcal Q\) such that \(FT=S\) and \(F\) is injective on \(\operatorname {im}T\). In particular, two tensor families \(A,B\) on the same virtual bond spaces, with site maps \(G\)-injective for the same \(\rho _v\), admit such maps \(F_v\) with \(B_v=F_vA_v\). Their ambient physical dimensions may differ.
Choosing \(L\) with \(LT=\Pi _\rho \), where \(\Pi _\rho =|G|^{-1}\sum _g\rho (g)\), and setting \(F=SL\), invariance of \(S\) gives \(FT=S\Pi _\rho =S\). Since both \(T\) and \(FT=S\) are \(G\)-injective, Theorem 35.4.1 gives injectivity of \(F\) on \(\operatorname {im}T\). Applying this argument at every vertex and evaluating on virtual basis vectors gives equality of the tensor coefficients.
Let \(A,B\) have the same virtual bond spaces and let their site maps be \(G\)-injective for the same representations \(\rho _v\) of a finite group. If the regions \(R_i\) cover every vertex, there are rectangular physical maps \(F_v\) such that, for every region \(S\),
The second restriction is injective, and the parent spaces have equal dimensions. If \(B=FA\) is already specified and both site families are \(G\)-injective for \(\rho _v\), the second equality holds for that \(F\). For a finite region family and arbitrary positive regional interactions with these exact local kernels, the same comparison gives \(F_{\mathrm{all}}(\ker H)=\ker K\), injectivity on \(\ker H\), and \(\dim \ker K=\dim \ker H\).
Theorem 37.10.1.6 gives \(B=FA\) and site-image injectivity. Regional contraction commutes with \(F\), giving the first equality, and Theorem 37.10.1.4 gives the equality on the common parent spaces. For a prescribed \(F\), its site-image injectivity follows directly from \(G\)-injectivity of \(A\) and \(FA\). Finally, positivity identifies each Hamiltonian kernel with its common parent space.
37.11 Holonomy and flatness from regular parent constraints
On any finite graph region \(R\), choose group elements \(r_e\) on its bonds, with \(r_e=1\) on every crossing bond. Simultaneously multiplying both exposed half-edge labels of each internal bond \(e\) on the right by \(r_e\) leaves every canonical regional column of the regular contraction unchanged. In particular this holds for one internal bond, with both its endpoints in \(R\). The same statement holds in the presence of arbitrary inserted left group multiplications.
Let the physical coefficients be \(a_v\), and suppose that local inverse coefficients \(F_v\) satisfy
where \(\Pi _v\) is the regular averaging projector. Then every original regional ground vector exposed by the product of the \(F_v\) has this internal right invariance. Consequently, every complementary slice of an arbitrary global vector satisfying the original regional parent condition has this invariance after exposure. Such inverse coefficients exist for regular \(G\)-injective tensors.
No \(G\)-isometry, flat-connection reconstruction, or global closure-span representation is assumed: this local identity of the contractions in the accessible coordinate argument of [ 272 ] is an auxiliary regular bond relation and does not establish global closure spanning [ 106 ] .
The regular averaging kernel is unchanged when its two configurations are multiplied on the right, leg by leg, by the same labels: common left translation commutes with these right multiplications. Changing each summed shared bond label \(\eta _e\) to \(\eta _e r_e\) bijects the summation variables while fixing the crossing condition, and associativity gives the same identity in the presence of inserted left group multiplications, yielding the canonical column equality.
Applying the stated inverse coefficients to an original regional column produces its canonical column, which inherits the identity. Linearity extends it to the entire original regional ground space. A regional parent constraint places every complementary physical slice in that space.
Consider a finite graph with site-dependent regular \(G\)-injective tensors. Let \(R\) induce a connected graph, and insert regular bond operators \(u_e\) at their ordered heads. If an internal closed walk has nonidentity holonomy, the original regional ground space and the inserted regional ground space have zero intersection. More generally, a nonidentity spanning-tree cycle residual already forces this separation.
Every regional ground vector, exposed by a local \(G\)-injective inverse, vanishes at half-edge coordinates whose cycle labels are not all the identity. In particular, this holds for every complementary slice of an arbitrary vector satisfying an original regional parent constraint. No inserted-bond representation of that vector is assumed.
Consequently, if a nonzero inserted global PEPS vector is annihilated by an original regional parent term on \(R\), every internal closed-walk holonomy in \(R\) is the identity. If a vertex gauge removes all internal inserted operators, the original and inserted regional ground spaces coincide: crossing operators only relabel the boundary condition. This last assertion requires only coefficient invariance. No \(G\)-isometry is assumed. This is a necessary local implication in the closure argument of [ 272 , Theorem 5.5 ] ; it does not assert that every parent-kernel vector admits an inserted-bond description. This necessary local holonomy condition does not give a global closure representation; see [ 106 ] .
Apply a local \(G\)-injective inverse at every vertex of \(R\). This exposes the canonical regular half-edge coordinates, and applying the original local coefficient maps restores each regional ground vector. In spanning-tree coordinates, the uninserted contraction is supported on identity cycle labels. The inserted contraction is supported on simultaneous conjugates of its cycle residuals. A nonidentity residual makes these supports disjoint. Since the local coordinate map has a left inverse on the regional ground spaces, their physical intersection is also zero.
A common nonzero regional vector therefore forces all residuals to be the identity. Evaluating the closed-walk word at these residuals gives identity holonomy. Finally choose a nonzero physical slice of the inserted global vector. It belongs to the inserted regional ground space by contraction, and to the original one by the original parent constraint.
Consider a finite graph with regular \(G\)-injective tensors, a connected region \(R\), and a closed walk \(p\) within \(R\). There is a physical linear operator \(Q\) on \(R\) which fixes the original regional ground space. On a regional ground vector with inserted bonds \(u\), it acts as the identity if the holonomy around \(p\) is the identity, and as zero otherwise.
Extended by the identity outside \(R\), the same operator fixes every global vector satisfying the original regional parent condition and has the corresponding scalar action on every inserted closed PEPS. No \(G\)-isometry is assumed. This is a linear operator; Hermiticity and positivity are not asserted. It is an auxiliary consequence of the closure and accessible-coordinate calculations in [ 272 , Theorem 5.5 ] . These auxiliary regular filters do not establish global closure spanning; see [ 106 ] .
Choose a spanning tree of \(R\). The exposed cycle labels of an inserted block are simultaneous conjugates of its cycle residuals. Evaluation of the closed-walk word commutes with simultaneous conjugation. Thus the diagonal detector of identity word value acts on each inserted block column by the stated scalar. Passing this detector through a local \(G\)-injective inverse and the original physical maps recovers every regional ground vector and preserves the scalar action. Since an uninserted closed walk has identity holonomy, applying the regional statements to each complementary physical slice yields the global statements.
Let both native torus periods be at least three, and let the tensor be regular \(G\)-injective. Let \(\mathcal C\) be the span of all native pair-conjugacy-class closure vectors, including noncommuting labels, and let \(\mathcal C_{\mathrm{comm}}\) be the commuting closure span. Let \(H\) be a finite sum of positive original parent interactions on connected regions whose closed-cell realizations are simply connected. Assume that this family contains the plaquette at the intersection of the two closure seams. Then
Its dimension is the number of commuting pair-conjugacy classes, equivalently the centralizer sum. This equality applies to arbitrary coherent sums of closure vectors and requires no \(G\)-isometry. It proves the final closure constraint within the finite closure span in the regular case of [ 272 , Theorem 5.5 ] . The separate assertion that every parent-kernel vector lies in \(\mathcal C\) remains unproved. The restriction to the closure-span intersection and the remaining full-kernel identification are recorded in [ 106 ] .
At the seam-intersection plaquette, identity holonomy is equivalent to commutativity of the two closure labels. The identity-holonomy filter therefore fixes every commuting closure and removes every noncommuting closure. Its image of \(\mathcal C\) is contained in \(\mathcal C_{\mathrm{comm}}\). Positivity of the parent sum forces each of its local constraints separately, so every vector in \(\ker H\) is fixed by the corner filter. Such a vector in \(\mathcal C\) thus belongs to \(\mathcal C_{\mathrm{comm}}\). The reverse inclusion is the local parent inclusion for commuting closures.
Let both torus periods be at least three, and let the native tensor be regular \(G\)-injective. The original parent condition on the plaquette at the intersection of the two seams forces the labels \(g,h\) of a closure vector to commute. Conversely, every commuting closure satisfies the original parent condition on every connected region whose closed-cell realization is simply connected. These regions may meet either seam.
For any finite family of such regions, let \(H\) be a sum of positive parent interactions with the original regional kernels. Then the span of the commuting closure vectors is contained in \(\ker H\), and
where \(c_C\) represents \(C\). No \(G\)-isometry is required. This proves the inclusion and the resulting lower bound for the regular case of [ 272 , Theorems 5.5 and 5.9 ] . The assertion that every parent-kernel vector lies in the closure span remains separate. The regular native-torus restriction and the missing reverse global spanning inclusion are recorded in [ 106 ] .
Regular \(G\)-injectivity makes every native closure vector nonzero. The local parent constraint therefore forces trivial plaquette holonomy. At the intersection of the two seams, this is precisely the commutation relation for the closure labels.
A simply connected closed-cell region admits an integer lift. For commuting labels, the lift defines a vertex gauge which removes every internal closure operator. The remaining operators merely relabel the virtual boundary, so the original and inserted regional ground spaces agree. Every physical slice of the closed inserted contraction thus satisfies the original regional parent condition. The kernel of a positive sum is the intersection of the local kernels. Finally use linear independence of the commuting pair-conjugacy classes and their centralizer count.
Let a native four-leg regular \(G\)-injective tensor be placed on a square torus with both periods at least three. Insert arbitrary regular group operators on its ordered bonds, and suppose that the resulting physical PEPS vector is nonzero. If an original plaquette parent term annihilates this vector, the holonomy around that plaquette is the identity.
If the sum of all original plaquette parent terms annihilates it, every elementary plaquette has identity holonomy. Equivalently, writing \(A_v\) for the rightward bond operator and \(B_v\) for the upward bond operator, one has
at every vertex. In particular, a nonzero one-bond flux state with nonidentity flux is not annihilated by the original parent term surrounding its endpoint.
No \(G\)-isometry is assumed. This is a necessary local step in [ 272 , Theorem 5.5 ] ; it does not claim that arbitrary parent-kernel vectors have inserted-bond descriptions, or that flatness alone gives the full torus ground-space classification. The regular native-torus scope and the remaining global representation step are recorded in [ 106 ] .
Each native plaquette is the support of a four-step closed walk, so its induced graph is connected. Apply local regular parent flatness to the induced walk and then identify its holonomy with the original native plaquette holonomy. Positivity of every regional parent term implies that a vector killed by their sum is killed by each term separately.
The ordered plaquette holonomy is \(B_v^{-1}A_{v+(0,1)}^{-1}B_{v+(1,0)}A_v\). It is the identity exactly when the two two-step path products around the square agree. The explicit one-bond insertion has its prescribed group element as holonomy, giving the exclusion of nonidentity flux.