Dependencies

Given a word \(w = (i_1, \ldots , i_L) \in \{ 0,\ldots ,d{-}1\} ^L\), the word evaluation is the matrix product

The empty word evaluates to the identity, \(A^\varnothing = \mathbb {1}_D\). In tensor-network notation,

in which each black node denotes the same local tensor \(A\), the virtual legs remain open, and the physical legs are labelled by the word \((i_1,\ldots ,i_L)\).

Two tensors \(A\) and \(B\) are gauge-phase equivalent if there exist \(X \in \mathrm{GL}_D(\mathbb {C})\) and \(\zeta \in \mathbb {C}\setminus \{ 0\} \) such that, for every physical index \(i\),

A tensor \(A\) is injective if its matrices span the full matrix algebra: \(\operatorname{span}_{\mathbb {C}}\{ A^i : i = 0,\ldots ,d{-}1\} = M_{D}(\mathbb {C})\).

An MPS tensor \(A\) is irreducible if it has no nontrivial invariant projection.

The mixed transfer operator (or cross transfer operator) for two MPS tensors \(A\) and \(B\) of the same bond dimension \(D\) is the linear map \(F_{AB}:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) defined by

For MPS tensors \(A\) of bond dimension \(D_1\) and \(B\) of bond dimension \(D_2\), possibly with \(D_1\neq D_2\), the rectangular mixed transfer operator is the map \(F_{AB}^{\mathrm{rect}}: M_{D_1\times D_2}(\mathbb {C})\to M_{D_1\times D_2}(\mathbb {C})\) defined by

The transfer map associated to a tensor \(A\) is the linear map \(\mathcal{E}_A : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) defined by

Diagrammatically, the transfer map is the double-layer contraction

in which the upper node denotes \(A\), the lower node denotes \(A^\dagger \), and the physical index is summed over between them.

If \(A\) and \(B\) are injective normalized tensors that are not gauge-phase equivalent, then \(\operatorname{tr}(F_{AB}^N(X))\to 0\) as \(N\to \infty \) for every \(X\in M_{D}(\mathbb {C})\).

Let \(A\) be a normalized tensor of bond dimension \(D_1\) and \(B\) a normalized tensor of bond dimension \(D_2\). Suppose \(\rho _A,\rho _B\) are fixed points of the respective transfer maps and \(S_A,S_B\) are invertible matrices with \(S_AS_A^\dagger =\rho _A\) and \(S_BS_B^\dagger =\rho _B\) (for instance the positive definite Perron–Frobenius fixed points of Theorem 5.3.2 and their square roots, as in the right-canonical gauge of Theorem  [ LTC26 , “Right-canonical (unital) gauge” ] ). Gauge to unital families \(A'^i=S_A^{-1}A^iS_A\) and \(B'^i=S_B^{-1}B^iS_B\). If \(X\in M_{D_1\times D_2}(\mathbb {C})\setminus \{ 0\} \) satisfies \(F_{AB}^{\mathrm{rect}}(X)=\mu X\) with \(|\mu |=1\), then \(X'=S_A^{-1}X(S_B^\dagger )^{-1}\) is nonzero, the families \(A'\) and \(B'\) are unital, and, for every \(i\),

Let \(A\) and \(B\) be tensors of the same bond dimension. Suppose invertible gauges take them to \(A'\) and \(B'\), and suppose there are an invertible matrix \(X'\) and a scalar \(\mu \) with \(|\mu |=1\) such that \(A'^iX'=\mu X'B'^i\) for every physical index \(i\). Then \(A\) and \(B\) are gauge-phase equivalent.

For every \(X\in M_{D}(\mathbb {C})\) and \(N\geq 0\),

where \(A^\sigma =A^{\sigma _1}\cdots A^{\sigma _N}\) denotes the word evaluation of Definition 2.1.2.

Setting \(B=A\) recovers the ordinary transfer map: \(F_{AA}=\mathcal{E}_A\).

Assume \(D \ge 1\). Let \(A\) be an MPS tensor with \(\sum _i (A^i)^\dagger A^i = \mathbb {1}\), so that \(\mathcal{E}_A\) is trace-preserving. Then there exists \(\rho \ge 0\), \(\rho \neq 0\), with \(\mathcal{E}_A(\rho ) = \rho \).

Assume \(D \ge 1\). Let \(A\) be an injective MPS tensor with \(\sum _i (A^i)^\dagger A^i = \mathbb {1}\). Then the transfer map \(\mathcal{E}_A\) has a unique positive semidefinite fixed point \(\rho \) up to scaling, and \(\rho \) is positive definite.

If \(\rho \) is a fixed point of \(\mathcal{E}_A\), then \(\operatorname{tr}(\mathcal{E}_A^N(\rho ))=\operatorname{tr}(\rho )\) for every \(N\geq 0\). In particular, this applies to the distinguished Perron–Frobenius fixed point from Theorem 5.3.2.

For normalized tensors of bond dimensions \(D_1\) and \(D_2\), \(\rho _{\operatorname{spec}}(F_{AB}^{\mathrm{rect}})\leq 1\).

Let \(A\) be an irreducible normalized MPS tensor whose transfer map \(\mathcal{E}_A\) is primitive. Then there is a nonzero positive semidefinite fixed point \(\rho \) of \(\mathcal{E}_A\) with \(\operatorname{tr}(\rho )\neq 0\) such that, for \(P:=P_\rho \) and \(N:=\mathcal{E}_A-P\), \(\rho _{\operatorname{spec}}(N){\lt}1\).

Let \(A\) and \(B\) be injective normalized tensors that are not gauge-phase equivalent. Then \(F_{AB}^n(X)\to 0\) for every \(X\in M_{D}(\mathbb {C})\).