14 Exponential Decay of Correlations

In the thermodynamic limit, spatial correlation functions of matrix product states are governed by the subleading spectrum of the transfer map $\mathcal{E}_A$. Subtracting the leading fixed-point contribution leaves a connected correlator decaying exponentially (with polynomial factors $n^k$ for non-diagonal Jordan blocks) at a rate defining the spatial correlation length. For tensors generating parent Hamiltonians (Chapter 11), this spectral gap controls ground-state correlations.

While Chapter 5 analyzes the peripheral spectrum and overlap decay, this chapter develops the one-point and two-point functionals, their spectral decomposition, and quantitative decay bounds.

14.1 Connected correlators from the transfer map

Definition 14.1.1 Auxiliary-space one-point functional

✓

Fix an MPS tensor and a right fixed point of its transfer map. Write the tensor as $A$ and the fixed point as $\rho ^R$. Under the hypotheses of Theorem 4.3.2, that fixed point exists and is unique up to scaling; we normalize it so that $\operatorname{tr}(\rho ^R)=1$. For an auxiliary-space insertion $X\in M_{D}(\mathbb {C})$, define

\begin{align} \langle X_0 \rangle & = \operatorname{tr}(X\rho ^R). \notag \end{align}

Definition 14.1.2 Auxiliary-space two-point functional at distance $n$

✓

For auxiliary-space insertions $X,Y\in M_{D}(\mathbb {C})$ and $n\ge 0$, define

\begin{align} \langle X_0 Y_n \rangle & = \operatorname{tr}\! \left(Y\, \mathcal{E}_A^n(X\rho ^R)\right). \notag \end{align}

Equivalently, insert $X$ at site $0$, propagate by $\mathcal{E}_A^n$, then insert $Y$ and take the trace.

$\begin{tenkz}[rows={wire,wire}, cols=3, west={cup=$Y$}, east={cup=$X\rho^R$}] \tn[at=(1,2), skin=ring, wires=2]{\E_A^n} \end{tenkz}$

Definition 14.1.3 Auxiliary-space connected correlator

✓

The connected two-point function is

\begin{align} C(X,Y;n) & = \langle X_0Y_n\rangle - \langle X_0\rangle \langle Y_0\rangle . \notag \end{align}

Definition 14.1.4 Fixed-point-centered auxiliary insertion

✓

For a matrix $\rho \in M_{D}(\mathbb {C})$ and an auxiliary insertion $X\in M_{D}(\mathbb {C})$, define

\begin{align} Z_\rho (X) & = X\rho -\operatorname{tr}(X\rho )\rho . \label{eq:correlator_traceless_part} \end{align}

When $\operatorname{tr}(\rho )=1$, this removes the trace component of $X\rho $.

Lemma 14.1.5 Trace of the centered insertion

✓

If $\operatorname{tr}(\rho )=1$, then

\begin{align} \operatorname{tr}\! \bigl(Z_\rho (X)\bigr) & =0. \end{align}

Proof ▶

By linearity of the trace,

\begin{align} \operatorname{tr}\! \bigl(Z_\rho (X)\bigr) & =\operatorname{tr}(X\rho )-\operatorname{tr}(X\rho )\operatorname{tr}(\rho ) =0. \end{align}

Theorem 14.1.6 Fixed-point centering identity for connected correlators

✓

Suppose that $\rho $ is a right fixed point of the transfer map, $\mathcal{E}_A(\rho )=\rho $. Then, for every $n\geq 0$,

\begin{align} C(X,Y;n) & =\operatorname{tr}\! \left(Y\, \mathcal{E}_A^n\bigl(Z_\rho (X)\bigr)\right). \label{eq:connected_correlator_traceless_reduction} \end{align}

When moreover $\operatorname{tr}(\rho )=1$, the preceding lemma shows that $Z_\rho (X)$ is traceless. The identity isolates the fixed-point contribution before the remaining transfer iterates are analysed.

Proof ▶

Since $\mathcal{E}_A(\rho )=\rho $, one has $\mathcal{E}_A^n(\rho )=\rho $ for every $n$. Linearity therefore gives

\begin{align} \operatorname{tr}\! \left(Y\, \mathcal{E}_A^n\bigl(Z_\rho (X)\bigr)\right) & = \operatorname{tr}\! \left(Y\, \mathcal{E}_A^n(X\rho )\right) -\operatorname{tr}(X\rho )\operatorname{tr}(Y\rho ), \end{align}

which is the definition of $C(X,Y;n)$.

14.2 Physical correlations and exponential decay

Definition 14.2.1 Physical connected fixed-point correlator

✓

Let $\rho \in M_{D}(\mathbb {C})$ have nonzero trace, and set $P(Z)=\operatorname{tr}(Z)\rho /\operatorname{tr}\rho $. For physical operators $X,Y$ on independently chosen blocks of lengths $L_1,L_2$, and $n\geq 0$ unobserved intermediate sites, define

\[ C_{\mathrm{phys}}(X,Y;n)= \operatorname{tr}\! \left(\mathcal{E}_X(\mathcal{E}_A-P)^n \bigl(\mathcal{E}_Y(\rho )-P(\mathcal{E}_Y(\rho ))\bigr)\right). \]

The centering of the inner insertion includes adjacent blocks, $n=0$.

Lemma 14.2.2 Physical connected contraction

✓

Suppose $\mathcal{E}_A$ is trace preserving, $\mathcal{E}_A(\rho )=\rho $, and $\operatorname{tr}\rho =1$. Then, for all $n\geq 0$,

\[ C_{\mathrm{phys}}(X,Y;n)= \operatorname{tr}\! \left(\mathcal{E}_X\mathcal{E}_A^n(\mathcal{E}_Y(\rho ))\right) -\operatorname{tr}(\mathcal{E}_X(\rho ))\operatorname{tr}(\mathcal{E}_Y(\rho )). \]

For $n\geq 1$ this also equals $\operatorname{tr}(\mathcal{E}_X(\mathcal{E}_A-P)^n(\mathcal{E}_Y(\rho )))$, including the equal-support specialization used in preparation estimates.

Proof ▶

Trace preservation and stationarity give $P\mathcal{E}_A=\mathcal{E}_AP=P$ and $P^2=P$. Induction yields $(\mathcal{E}_A-P)^n(Z-PZ)=\mathcal{E}_A^nZ-PZ$ at every $n\geq 0$. Taking $Z=\mathcal{E}_Y(\rho )$ proves the first identity. For $n\geq 1$, $\mathcal{E}_A^n=P+(\mathcal{E}_A-P)^n$, which proves the second.

Lemma 14.2.3 Complementary gap in trace-preserving gauge

✓

Let $A$ be normal in trace-preserving gauge, and let $\rho \succeq 0$ satisfy $\mathcal{E}_A(\rho )=\rho $ and $\operatorname{tr}\rho =1$. Then $\rho _{\operatorname{spec}}(\mathcal{E}_A-P){\lt}1$, where $P(Z)=\operatorname{tr}(Z)\rho $.

Proof ▶

Normality supplies irreducibility and peripheral spectrum $\{ 1\} $. The complementary transfer-map gap for irreducible primitive channels applies to the nonzero positive fixed state [ 216 , “Complementary transfer-map gap for primitive maps” ] .

Theorem 14.2.4 Physical correlation decay in trace-preserving gauge

✓

For any tensor $A$ and matrix $\rho $ of nonzero trace, every rate $r{\gt}\rho _{\operatorname{spec}}(\mathcal{E}_A-P)$ and every pair of physical operators on independent finite supports have a constant $C_{XY}{\gt}0$ such that, for all $n\geq 0$,

\begin{align} |C_{\mathrm{phys}}(X,Y;n)|\leq C_{XY}r^n. \label{eq:correlations_decay_bound} \end{align}

If $A$ is normal in trace-preserving gauge and $\rho \succeq 0$ is a normalized fixed state, one can choose a single rate $\rho _{\operatorname{spec}}(\mathcal{E}_A-P){\lt}r{\lt}1$ for every finite support and every pair of observables. Thus the fixed-point infinite-chain connected correlations decay exponentially, without a diagonalizability assumption. The source’s pure-exponential expansion needs the Jordan-block correction recorded in [ 94 ] .

Proof ▶

Put $T=\mathcal{E}_A-P$ and $Z=\mathcal{E}_Y(\rho )-P(\mathcal{E}_Y(\rho ))$. The Gelfand formula gives a constant $C{\gt}0$ such that $\| T^nZ\| \leq Cr^n\| Z\| $ at every $n\geq 0$. The linear functional $W\mapsto \operatorname{tr}(\mathcal{E}_X(W))$ on the finite-dimensional matrix space has a bound $|\operatorname{tr}(\mathcal{E}_X(W))|\leq M_X\| W\| $ with $M_X{\gt}0$. Hence $C_{XY}=M_XC\| Z\| +1$ proves the estimate. Lemma 14.2.3 permits the choice $r{\lt}1$.

Remark 14.2.5 Transfer-map spectral hypothesis

The rate $r{\lt}1$ in Theorem 14.2.4 gives exponential decay. It is a transfer-map gap condition: all non-leading eigenvalues of $\mathcal{E}_A$ lie strictly inside the unit disk. It is distinct from the Hamiltonian spectral gap of Chapter 12. The complementary transfer-map gap $\rho _{\operatorname{spec}}(\mathcal{E}_A-P){\lt}1$ is established for primitive channels (the corresponding theorem in the companion quantum-channel volume [ 216 , “Complementary transfer-map gap for primitive maps” ] ). The mixed-transfer gap needed for separation of non-equivalent canonical blocks is supplied for irreducible trace-preserving tensors (Theorem 5.6.2), and quantitative single-tensor transfer-map bounds for injective tensors are recorded in Theorem 14.3.3.

These transfer-map gap results give exponential convergence of the complementary transfer map at every rate strictly exceeding its spectral radius.

Definition 14.2.6 Correlation length

✓

For every $\lambda _2\in \mathbb {C}$, define the associated correlation-length quantity by

\begin{align} \xi & = -\frac{1}{\log |\lambda _2|}. \notag \end{align}

For a nonzero subleading eigenvalue in the physical range $0{\lt}|\lambda _2|{\lt}1$, this quantity is positive. If all subleading spectral values vanish and correlations therefore vanish after finitely many transfer steps, the total definition gives the limiting value $\xi =0$.

Lemma 14.2.7 Correlation length is positive

✓

If $|\lambda _2| {\lt} 1$ and $\lambda _2 \ne 0$, then $\xi {\gt} 0$.

Proof ▶

Since $0{\lt}|\lambda _2|{\lt}1$, we have $\log |\lambda _2|{\lt}0$, so $\xi =-1/\log |\lambda _2|{\gt}0$.

14.3 Quantitative transfer-map gap bounds

The qualitative bound $\rho _{\operatorname{spec}}(\mathcal{E}_A-P){\lt}1$ for primitive channels [ 216 , “Complementary transfer-map gap for primitive maps” ] admits quantitative refinements in terms of geometric constants and decay rates for the transfer map $\mathcal{E}_A$.

Theorem 14.3.1 Exponential convergence of injective primitive channels

✓

Fix an injective, trace-preserving MPS tensor $A$ with positive definite fixed point $\rho $. Let

\begin{align} P(X) & = \frac{\operatorname{tr}(X)}{\operatorname{tr}(\rho )}\rho \label{eq:correlations_fixed_point_projection} \end{align}

be the corresponding fixed-point projection. Then there exist $C {\gt} 0$ and $0 {\lt} \delta \le 1$ such that, for all $n\ge 0$ and $X\in M_{D}(\mathbb {C})$,

\begin{align} \| \mathcal{E}_A^n(X)-P(X)\| & \le C(1-\delta )^n\| X\| . \label{eq:correlations_channel_convergence} \end{align}

The transfer-map gap $\delta $ exists by primitivity (the corresponding theorem in the companion quantum-channel volume [ 216 , “Complementary transfer-map gap for primitive maps” ] , the complementary transfer-map gap).

Proof ▶

Injectivity implies primitivity of the transfer map. The complementary spectral-radius bound $\rho _{\operatorname{spec}}(\mathcal{E}_A-P){\lt}1$ then follows from the primitive transfer-map gap theorem. Choose $r$ with $\rho _{\operatorname{spec}}(\mathcal{E}_A-P){\lt}r{\lt}1$. The Gelfand formula yields $\| (\mathcal{E}_A-P)^n\| \le Cr^n$ for a suitable constant $C{\gt}0$. Since $P$ is the fixed-point projection, $\mathcal{E}_A P=P\mathcal{E}_A=P$, and hence $\mathcal{E}_A^n-P=(\mathcal{E}_A-P)^n$ for $n\ge 1$. The case $n=0$ is absorbed by increasing $C$ if necessary. Setting $\delta =1-r$ gives (8).

Theorem 14.3.2 Correlation length bound

✓

For an injective trace-preserving MPS tensor, there exist $C{\gt}0$ and $\xi {\gt}0$ such that, for all $n\ge 0$ and all traceless $X\in M_{D}(\mathbb {C})$, i.e. $\operatorname{tr}(X)=0$,

\begin{align} \| \mathcal{E}_A^n(X)\| & \le C e^{-n/\xi }\| X\| . \label{eq:correlations_traceless_decay} \end{align}

Traceless matrices lie in $\ker P$, where $P$ is the fixed-point projection (7). Since $\mathcal{E}_A-P$ has spectral radius strictly less than $1$, choose any $r$ with $\rho _{\operatorname{spec}}(\mathcal{E}_A-P){\lt}r{\lt}1$. The Gelfand formula then gives exponential decay with $\xi =-1/\log r$. Choosing a rate strictly above the spectral radius is necessary in general because a Jordan block at the spectral radius can prevent a uniform bound proportional to $\rho _{\operatorname{spec}}(\mathcal{E}_A-P)^n$.

Proof ▶

For traceless $X$, we have $P(X)=0$, so $\mathcal{E}_A^n(X)=(\mathcal{E}_A-P)^n(X)$. Applying the Gelfand-formula estimate with $\rho _{\operatorname{spec}}(\mathcal{E}_A-P){\lt}r{\lt}1$ gives $\| \mathcal{E}_A^n(X)\| \le Cr^n\| X\| $. With $\xi =-1/\log r{\gt}0$, the identity $r^n=e^{-n/\xi }$ yields (9).

Theorem 14.3.3 Transfer-map gap from injectivity

✓

For an injective trace-preserving MPS tensor, there exists $\delta {\gt}0$ such that every eigenvalue $\mu \ne 1$ of the transfer map satisfies $|\mu |\le 1-\delta $.

Proof ▶

Injectivity implies eventual full Kraus rank. This yields primitivity, hence a complementary transfer-map gap. Since the transfer map has only finitely many eigenvalues, one may choose a uniform $\delta {\gt}0$ separating $1$ from the remaining spectrum.

14.4 Relation to parent Hamiltonians

Remark 14.4.1

Suppose that the MPS tensor $A$ generates a parent Hamiltonian $H_N(A,L)$ (Definition 11.4.42) whose ground state is the matrix product vector (Lemma 11.4.46), and consider its fixed-point infinite-chain state. If the hypotheses of Theorem 14.2.4 hold, then for every rate $r$ with $\rho _{\operatorname{spec}}(\mathcal{E}_A-P){\lt}r{\lt}1$ the corresponding ground-state correlations decay as $r^n=e^{-n/\xi }$ with finite correlation length $\xi =-1/\log r$, the quantity of Definition 14.2.6 evaluated at $r$. When $\mathcal{E}_A-P$ is diagonalizable, one may take $r=|\lambda _2|$, the largest modulus of a subleading eigenvalue, provided $\lambda _2\neq 0$.

14.5 Physical correlations in an arbitrary bond basis

Definition 14.5.1 Physical connected left-right contraction

✓

For matrices $\rho ^L,\rho ^R\in M_{D}(\mathbb {C})$ and physical operators $X,Y$ on independently chosen blocks of lengths $L_1,L_2$, define

\begin{align} C_{\rho ^L,\rho ^R}^{A}(X,Y;n) & =\operatorname{tr}\! \left(\rho ^L\mathcal{E}_X\mathcal{E}_A^n(\mathcal{E}_Y(\rho ^R))\right) -\operatorname{tr}(\rho ^L\mathcal{E}_X(\rho ^R))\operatorname{tr}(\rho ^L\mathcal{E}_Y(\rho ^R)). \label{eq:correlations_left_right} \end{align}

When $\rho ^L,\rho ^R$ are positive fixed matrices of the dual transfer map and the transfer map, respectively, normalized by $\operatorname{tr}(\rho ^L\rho ^R)=1$, this is the connected infinite-chain physical expectation with $n\geq 0$ unobserved intermediate sites. This is the fixed-point contraction of [ 17 , Section II.B.3 ] .

Theorem 14.5.2 Gauge covariance of physical insertions

✓

Suppose $B^i=UA^iU^{-1}$ with $U$ invertible. For every physical operator $O$ on $L$ sites and every $Z\in M_{D}(\mathbb {C})$,

\begin{align} \mathcal{E}_O^B(UZU^\dagger ) & =U\mathcal{E}_O^A(Z)U^\dagger . \label{eq:correlations_insertion_gauge} \end{align}

This includes $L=0$; no unitarity or normalization is assumed.

Proof ▶

Every word satisfies $B^w=UA^wU^{-1}$. Substitute this into the finite word sum defining the insertion and cancel $U^{-1}U$ and $U^\dagger U^{-\dagger }$.

Theorem 14.5.3 Gauge invariance of physical connected correlations

✓

Suppose $B^i=UA^iU^{-1}$ with $U$ invertible, and put $\widetilde\rho ^L=U^{-\dagger }\rho ^LU^{-1}$ and $\widetilde\rho ^R=U\rho ^RU^\dagger $. Then, for every pair of independently supported physical operators and every $n\geq 0$,

\begin{align} C_{\widetilde\rho ^L,\widetilde\rho ^R}^{B}(X,Y;n) & =C_{\rho ^L,\rho ^R}^{A}(X,Y;n). \label{eq:correlations_gauge_invariance} \end{align}

The identity does not require stationarity or positivity of the boundary matrices.

Proof ▶

Insertion of the identity on $n$ sites is $\mathcal{E}_A^n$, so (11) also gives covariance of every transfer power. Apply this and (11) to the two-point and one-point terms. Trace cyclicity gives $\operatorname{tr}(U^{-\dagger }\rho ^LZU^\dagger )=\operatorname{tr}(\rho ^LZ)$ in each term.

Theorem 14.5.4 Left-right contraction in trace-preserving gauge

✓

Suppose $\mathcal{E}_A$ is trace preserving, $\mathcal{E}_A(\rho )=\rho $, and $\operatorname{tr}\rho =1$. For every pair of independently supported physical operators and every $n\geq 0$, $C_{\mathbb {1},\rho }^{A}(X,Y;n)=C_{\mathrm{phys}}(X,Y;n)$.

Proof ▶

Multiplication by $\mathbb {1}$ leaves each contraction unchanged. Lemma 14.2.2 gives the claimed equality at every separation.

Theorem 14.5.5 Physical correlation decay for a normal tensor

✓

Let $A$ be a normal tensor, with leading transfer eigenvalue normalized to one. There are positive definite matrices $\rho ^L,\rho ^R$ satisfying

\begin{align} \mathcal{E}_A^*(\rho ^L) & =\rho ^L, \\ \mathcal{E}_A(\rho ^R) & =\rho ^R, \\ \operatorname{tr}(\rho ^L\rho ^R) & =1, \end{align}

and a rate $0{\lt}r{\lt}1$ such that every pair $X,Y$ of physical operators on independently chosen finite supports admits a constant $C_{XY}{\gt}0$ with $|C_{\rho ^L,\rho ^R}^{A}(X,Y;n)|\leq C_{XY}r^n$ for every $n\geq 0$. No trace-preserving gauge or diagonalizability hypothesis is imposed. This proves the decay conclusion of [ 17 , Section II.B.3 ] for normal tensors. The more general class whose unique eigenvalue of largest modulus is algebraically simple, with possibly singular fixed matrices, is outside this statement.

Proof ▶

Choose $B^i=UA^iU^{-1}$ in trace-preserving gauge, with a positive definite fixed state $\rho $ of trace one. Gauge invariance of normality implies that $B$ is normal. Set $\rho ^L=U^\dagger U$ and $\rho ^R=U^{-1}\rho U^{-\dagger }$. Trace preservation of $\mathcal{E}_B$ implies $\mathcal{E}_A^*(\rho ^L)=\rho ^L$; transfer covariance gives $\mathcal{E}_A(\rho ^R)=\rho ^R$. Congruence preserves positive definiteness, and trace cyclicity gives $\operatorname{tr}(\rho ^L\rho ^R)=\operatorname{tr}\rho =1$. Theorem 14.2.4 supplies a common rate $r{\lt}1$ in the gauge of $B$. Since its left-right contraction with left matrix $\mathbb {1}$ equals the centered fixed-point correlator, (12) transports the bound to $A$ at every separation, including $n=0$.

14.6 Jordan spectral expansions

The pure-exponential formula in [ 17 , Section II.B.3 ] omits both polynomial Jordan factors and finite nilpotent transients. The following all-separation expansion replaces that formula; see [ 94 ] .

Theorem 14.6.1 Exact binomial expansion of physical correlations

✓

Let $T=\mathcal{E}_A-P$ be the complementary transfer map used in the centered physical correlator. There is a finite set $S$ of eigenvalues of $T$ and coefficients $c_{\mu j}$ such that, at every separation $n\ge 0$,

\begin{align} C_{\mathrm{phys}}(X,Y;n) & =\sum _{\mu \in S}\sum _{j=0}^{k_\mu -1} \binom nj\mu ^{\max (n-j,0)}c_{\mu j}, \label{eq:physical_correlator_binomial} \end{align}

where $k_\mu $ is the least exponent at which the generalized $\mu $-eigenspace stabilizes. Terms with $j{\gt}n$ have zero binomial coefficient; the displayed exponent uses truncated natural subtraction. This formula includes $n=0$ and $\mu =0$, with $0^0=1$.

Proof ▶

The generalized-eigenspace decomposition writes the centered inner insertion as a finite sum of components $v_\mu $. On each component, $(T-\mu I)^{k_\mu }v_\mu =0$. Apply the commuting binomial formula to $T=(T-\mu I)+\mu I$ and then the outer physical insertion and trace. Powers beyond $k_\mu $ vanish; missing terms when $n{\lt}k_\mu $ have zero binomial coefficient. The coefficients are the resulting fixed trace pairings with $(T-\mu I)^jv_\mu $.

Theorem 14.6.2 Nonzero exponentials and the nilpotent transient

✓

The same correlator has the all-separation form

\begin{align} C_{\mathrm{phys}}(X,Y;n) & =t(n)+\sum _{\mu \in S\setminus \{ 0\} }p_\mu (n)\mu ^n, \label{eq:physical_correlator_jordan_transient} \end{align}

with $\deg p_\mu {\lt}k_\mu $ and $t(n)=0$ for $n\ge k_0$.

Proof ▶

For $\mu \ne 0$, use the falling-factorial polynomial $p_\mu (x)=\sum _{j{\lt}k_\mu }c_{\mu j}\mu ^{-j} x(x-1)\cdots (x-j+1)/j!$. Its degree is below $k_\mu $ and its value times $\mu ^n$ gives the nonzero-eigenvalue binomial sum, even when $j{\gt}n$. Keep the zero-eigenvalue sum separately. For $n\ge k_0$ every surviving index satisfies $j{\lt}n$, so its factor $0^{n-j}$ vanishes. Replacing this transient by a polynomial times $0^n$ would be incorrect.

Lemma 14.6.3 A common physical normal gauge

✓

A normal tensor $A$ with leading transfer eigenvalue one admits a trace-preserving gauge representative $B$ with positive trace-one fixed state $\rho _B$, and positive normalized left/right fixed matrices $\rho ^L,\rho ^R$ for $A$. At every separation and for every pair of finite-support physical operators, the left-right connected contraction for $A$ equals that for $B,\mathbb {1},\rho _B$.

Proof ▶

If $B^i=UA^iU^{-1}$, set $\rho ^L=U^\dagger U$ and $\rho ^R=U^{-1}\rho _B U^{-\dagger }$. The existing gauge covariance, trace cyclicity and positive congruence identities prove all assertions. This common construction supplies both the decay theorem and the spectral expansion.

Theorem 14.6.4 Jordan-aware physical spectral expansion for normal tensors

✓

Let $A$ be normal with leading transfer eigenvalue one. There are positive normalized left/right fixed matrices for which every finite-support physical connected correlator has the form

\begin{align} C_{\rho ^L,\rho ^R}^{A}(X,Y;n) & =t(n)+\sum _{\mu \in S}p_\mu (n)\mu ^n, \label{eq:correlations_spectral_expansion} \end{align}

at every $n\ge 0$. Here $0\notin S$, each $\mu \in S$ is an actual transfer eigenvalue of $A$ with $|\mu |{\lt}1$, and $t(n)=0$ for $n\ge D^2$. A single canonical representative $B$ supplies all Jordan indices: $\deg p_\mu $ is strictly below the stabilization index of the generalized $\mu $-eigenspace of its complementary transfer map. No diagonalizability, assumed expansion, or trace-preserving condition on the original $A$ is required.

Proof ▶

In trace-preserving gauge, every nonzero complementary eigenvalue is a transfer eigenvalue: its eigenvector has zero trace. Normality and the existing irreducible-channel peripheral theorem place all complementary eigenvalues strictly inside the unit disk. Generalized-eigenspace stabilization occurs by the dimension $D^2$, bounding the transient. Apply the preceding expansion in the common gauge and transport the physical contraction back to $A$. Gauge similarity preserves the transfer eigenvalues, so the frequencies in the resulting physical formula are eigenvalues of the original tensor.