Dependencies

For mutually inverse matrices \(U\) and \(U^{-1}\), define \(\Pi _j=U^{-1}E_{jj}U\), the product of the \(j\)-th column of \(U^{-1}\) with the \(j\)-th row of \(U\). Reorder a stacked bond quadruple into its two middle bits and its two outer bits. On the inner bond choose the adapted basis with first two vectors \((1,0,0,\pm 1)\) and last two vectors the remaining coordinate vectors. The Bell idempotents carry the weighted targets. Their tensor products with target matrices are expressed in the original stacked coordinates by undoing the reordering.

Let \(\tau \) be a bijection between the graded coordinate space of a slot family and the bond index set \(\{ 0,\ldots ,D_B-1\} \), and let \(G\) and \(G^{-1}\) be mutually inverse matrices in \(M_{D_B}(\mathbb {C})\). The explicit gauge attached to \((\tau ,G)\) is the invertible map sending a bond vector \(v\) to the graded vector \(b\mapsto (Gv)(\tau b)\). For an integer matrix \(X\), its entrywise coercion is the complex matrix with the same entries; for integer matrix product operator tensors \(M\) and \(N\), their integer bond-space product is \((M\mathbin {\cdot }N)^{ik}=\sum _jM^{ij}\otimes N^{jk}\).

For a stacked product of two integer matrix product operator tensors, read over the pair alphabet, the letter at \(a=do+i\) is \((M\mathbin {\cdot }N)^{oi}\). A one-slot compression datum for a tensor \(B\) onto a single target \(C\) is the data of Theorem 29.5.1 for the one-element slot set: an explicit gauge \((\tau ,G)\) conjugating \(B\) into a matrix that is block upper triangular for a flag with \(C\) at one diagonal position. Two shapes recur in the examples below: a conjugation \(GB^iG^{-1}=C^i\) for an invertible \(G\) gives such a datum with no zero slots (\(z=0\)); and, for a four-dimensional \(B\) and a one-dimensional \(C\), an integer gauge sending every letter to a matrix with \(C\) at the third diagonal position of a four-step flag and zero at the other three gives such a datum with three zero slots (\(z=3\)).

Let \(A\) be the tensor in (4). The two target tensors are \(C_+=A\) and \(C_-=-A\). The source is the bond-dimension-five tensor with letters

and

The product tensor of two MPO tensors contracts the bra index of the first factor with the ket index of the second and takes the tensor product of the bond spaces:

In the vertical reading of [ 13 , Section 4.5 ] , this joins two vertically read tensors along an original horizontal bond. It is the tensor written \(M_\alpha M_\beta \) in [ 13 , Theorem 4.14(iii) ] .

A (translation-invariant, PBC) MPS tensor with physical dimension \(d\) and bond dimension \(D\) is a collection of matrices \(\{ A^i\} _{i=0}^{d-1}\), where \(A^i \in M_{D}(\mathbb {C})\), indexed by a physical index \(i \in \{ 0, \ldots , d{-}1\} \). Such a tensor defines an MPV family. Diagrammatically,

The black node denotes the tensor \(A\), the horizontal legs are virtual, and the upper leg is the physical index \(i\).

A tensor \(A\) is normal [ 17 ] if there exists a positive integer \(L_0\) such that \(A\) is \(L_0\)-block injective.

Let \(B^0,\ldots ,B^{d-1}\in M_{D}(\mathbb {C})\), \(W\in \mathrm{M}_{k\times D}(\mathbb {C})\) and \(V\in \mathrm{M}_{D\times k}(\mathbb {C})\), and write \(\widetilde B^i=WB^iV\). If \(VWB^i=B^i\) for every \(i\), then every nonempty word \(w\) satisfies \(VWB^w=B^w\),

and \(\operatorname{tr}(\widetilde B^w)=\operatorname{tr}(B^w)\). The same conclusions hold, with \(B^wVW=B^w\) in place of \(VWB^w=B^w\), if \(B^iVW=B^i\) for every \(i\).

Let \(f\colon R\to \mathbb {C}\) be a ring homomorphism and let \(M\) and \(N\) be matrix product operator tensors with entries in \(R\). The direct sum of the entrywise images of \(M\) and \(N\) under \(f\) is the entrywise image of the direct sum of \(M\) and \(N\) formed over \(R\).

Conversely, whenever a tensor \(B\) admits the block triangular form of Theorem 29.5.1, clauses (i)–(iii), for a family \((C_s)_{s\in S}\), its periodic data are those of the direct sum: \(\operatorname{tr}(B^w)=\sum _{s\in S}\operatorname{tr}(C_s^w)\) for every nonempty word \(w\), that is, \(|V^{(N)}(B)\rangle =|V^{(N)}(\bigoplus _sC_s)\rangle \) for every \(N\geq 1\). When the blocks are rescaled tensors \(C_s=\mu _sA_s\), this reads \(\operatorname{tr}(B^w)=\sum _{s\in S}\mu _s^{|w|}\operatorname{tr}(A_s^w)\).

Let the bond space of \(B\) be \(\mathbb {C}^m\otimes \mathbb {C}^{D}\) and let \(\Pi _1,\ldots \) be mutually orthogonal rank-one idempotents of \(\mathbb {C}^m\) cut out by an invertible matrix \(U\) (the idempotents \(U^{-1}E_{jj}U\)). If every letter has the form \(B^i=\sum _{s\in S}\mu _s\, \Pi _{j(s)}\otimes C_s^i\) with distinct positions \(j(s)\), then \(B\) admits the block triangular form of Theorem 29.5.1 onto the weighted blocks \(\mu _sC_s\), with \(z=(m-|S|)D\) zero factors, the remainder \(R^i=0\), and sitewise intertwiners \(B^iV_s=V_s\, \mu _sC_s^i\) and \(W_sB^i=\mu _sC_s^iW_s\) for every slot: the extension splits.

Let \(A=\bigoplus _{s\in S}C_s\) be a finite direct sum of nonzero normal tensors \(C_s=(C_s^i)_i\) of bond dimension \(D_s\), with repetitions allowed, and let \(B=(B^i)_i\) be an arbitrary tensor of bond dimension \(D_B\). Assume that the periodic vectors of \(A\) and \(B\) agree at every positive length, \(|V^{(N)}(B)\rangle =|V^{(N)}(A)\rangle \) for every \(N{\gt}0\); equivalently, \(\operatorname{tr}(B^w)=\operatorname{tr}(A^w)\) for every nonempty word \(w\). Then there are an integer \(z\geq 0\), an integer \(r=|S|+z\), an invertible matrix \(G\in \mathrm{GL}_{D_B}(\mathbb {C})\), and maps \(V_s:\mathbb {C}^{D_s}\to \mathbb {C}^{D_B}\), \(W_s:\mathbb {C}^{D_B}\to \mathbb {C}^{D_s}\), with the following properties.

1. Every \(GB^iG^{-1}\) is block upper triangular for one common decomposition into \(r\) blocks.

2. Exactly \(|S|\) of the diagonal slots are matched with the target slots: writing \(j(s)\) for the position matched with \(s\), there is \(X_s\in \mathrm{GL}_{D_s}(\mathbb {C})\) with \(\bigl(GB^iG^{-1}\bigr)_{j(s),j(s)}=X_sC_s^iX_s^{-1}\).

3. The remaining \(z\) diagonal slots are one-dimensional zero tensors.

4. The compression pairs are mutually biorthogonal: \(W_sV_t= \operatorname{id}_{D_s}\) if \(s=t\) and \(0\) otherwise.

5. For every nonempty word \(w\), \(W_sB^wV_s=C_s^w\).

6. Writing \(\widehat C^i=\sum _{s\in S}V_sC_s^iW_s\) and \(R^i=B^i-\widehat C^i\), every product \(R^{i_1}R^{i_2}\cdots R^{i_r}=0\): the non-unital algebra generated by the \(R^i\) is nilpotent.

7. \(D_B=\sum _{s\in S}D_s+z\).

The idempotents satisfy \(\Pi _j^2=\Pi _j\), \(\Pi _j\Pi _k=0\) for \(j\ne k\), and \(U\Pi _jU^{-1}=E_{jj}\).

Let the letters of a tensor \(B\) be, after a relabelling of the bond coordinates, block diagonal with blocks \(B_k\), \(k\) in a finite set \(K\), and let every \(B_k\) carry a one-slot compression datum onto a target \(C_k\) with \(z_k\) zero slots and remainder \(R_k\). Then the direct sum of the gauges of the blocks is a compression datum of \(B\) onto the family \((C_k)_{k\in K}\) in the sense of Theorem 29.5.1, with \(z=\sum _kz_k\) zero slots, and its remainder is

relabelled in the same way. Consequently a word \(w\) satisfies \(R^w=0\) if and only if \(R_k^w=0\) for every \(k\): the nilpotency order of the remainder of \(B\) is the largest nilpotency order of the remainders of the blocks.

Conjugation by the explicit gauge attached to \((\tau ,G)\) sends a bond matrix \(A\) to the matrix \(GAG^{-1}\) relabelled along \(\tau \). Entrywise coercion is multiplicative, and for integer tensors \(M, N\) and \(c \in \mathbb {C}\),

If a compression datum uses the explicit gauge \((\tau ,G)\), then the compression maps for slot \(s\) are given in coordinates by \(W_s = \pi _s G\) and \(V_s = G^{-1} \iota _s\), where \(\pi _s\) selects the rows and \(\iota _s\) embeds the columns belonging to slot \(s\).

Let \(A \in (M_{D}(\mathbb {C}))^d\) have integer entries \(A_i \in \mathrm{M}_D(\mathbb {Z})\). If there exist index pairs \((a_k, b_k)\), integer coefficients \(T_{xyk} \in \mathbb {Z}\), and a nonzero integer \(m \ne 0\) such that for all matrix units \(E_{xy}\),

then \(\operatorname{span}_{\mathbb {C}}\{ A_i A_j : i,j \in \{ 0,\dots ,d-1\} \} = M_{D}(\mathbb {C})\). Similarly, let \(B \in (\mathrm{M}_D(\mathbb {Z}))^d\) and \(c \in (\mathrm{M}_1(\mathbb {Z}))^d\). If a selection of rows from the matrices \(B_i - c_i \mathbb {1}_D\) forms an integer matrix \(K \in \mathrm{M}_D(\mathbb {Z})\) admitting an integer left inverse \(K' K = m \mathbb {1}_D\) with \(m \ne 0\), then every \(v \in \mathbb {C}^D\) satisfying \(B_i v = c_i v\) for all \(i\) satisfies \(v = 0\). Likewise, if a selection of columns from \(B_i - c_i \mathbb {1}_D\) admits an integer left inverse up to \(m \ne 0\), then every \(u \in \mathbb {C}^D\) with \(u B_i = c_i u\) satisfies \(u = 0\).

Let \(M\) and \(N\) be matrix product operator tensors whose letters are, after relabellings of their bond coordinates, block diagonal with blocks \(M_k\) and \(N_l\). Then every letter of the stacked product \(M\mathbin {\cdot }N\) is, after a relabelling of its bond coordinates, block diagonal over the pairs \((k,l)\), with the block \((M_k\mathbin {\cdot }N_l)^{ij}\) at the pair \((k,l)\). The bond coordinates of the block \((k,l)\) are the pairs of a bond coordinate of \(M_k\) and a bond coordinate of \(N_l\).

Let a tensor be one nonzero scalar times an integer tensor. Suppose that for every bond position \((x,y)\) a letter of the integer tensor is a nonzero integer multiple of \(E_{xy}\). Then its one-site matrices span the full bond matrix algebra, so it is normal at blocking length one in the sense of Definition 2.5.5. This establishes injectivity, not transfer spectral-radius normalization; a normal tensor (NT) in the RFP canonical form additionally requires that normalization.

The remainder of a conjugate one-slot datum vanishes identically: the compression is an exact change of coordinates. For every one-slot datum with target tensor \(C\) and nonempty word \(w \ne \emptyset \),

or \(\operatorname{tr}(B^w) = c^{|w|}\operatorname{tr}(C^w)\) when the target is carried with scalar weight \(c\).

The source of Definition I.3.1 carries the block triangular form of Theorem 29.5.1 for \(S=\{ +,-\} \), with targets \(C_+=A\), \(C_-=-A\), one zero slot \(z=1\), and dimension count \(5=2+2+1\). There are compression maps \(V_s\in \mathbb {C}^{5\times 2}\) and \(W_s\in \mathbb {C}^{2\times 5}\) such that, for every \(s,t\in S\),

For every word \(w\) and each \(s\in S\),

Define the remainder letters by

Every product \(R^w\) with \(|w|\geq 3\) vanishes, whereas \(R^0R^0\neq 0\). Thus the bound \(|S|+z=3\) in clause (vi) of Theorem 29.5.1 is attained.

If \(V\in \mathbb {C}^{5\times 2}\) satisfies \(B^iV=V(-A^i)\) for both letters, then \(V=0\); if \(W\in \mathbb {C}^{2\times 5}\) satisfies \(WB^i=(-A^i)W\) for both letters, then \(W=0\).

The block \(A\) is normal: its length-two words span \(M_{2}(\mathbb {C})\).

For each physical label \(i\in \{ 0,1\} \), the tensor satisfies

Consequently, \(a_N(x)=0\) whenever \(n_1(x)\) is odd, and

Suppose \(x\) contains \(m\geq 1\) zeros. Read the sites cyclically, and let \(r_1,\ldots ,r_m\) be the numbers of ones between successive zeros. Define \(b(r)=1\) for even \(r\) and \(b(r)=0\) for odd \(r\). Then

For the word with no zeros,

For every nonempty word \(w\),

Suppose \(B=c_B\widehat B\) and \(C_s=c_C\widehat C_s\), where the tensors with hats have integer entries and \(s\in \{ 0,1\} \). Suppose a nonzero integer \(n\) and integers \(k_s\) satisfy, for each \(s\) and each physical index \(a\),

where \(\widehat\Pi _s=2\Pi _s\) and the right-hand side of (3) is in stacked bond coordinates. Then \(B^a=\sum _s\mu _s(\Pi _s\otimes C_s^a)\) in those coordinates, the hypothesis of Proposition 29.5.5. Here \(\mu _s\) denotes a scalar fusion weight, not a vertical multiplicity matrix.