16 Exponential Decay of Correlations

This chapter records an auxiliary-space transfer-map interface for thermodynamic-limit correlation functions and the resulting exponential decay estimates. The key mechanism is spectral: after removing the leading eigenvalue contribution, the connected part is governed by subleading eigenvalues of the transfer map. Passing from physical local observables to the auxiliary insertions used here requires the corresponding observable-inserted transfer map and is not supplied by the present definitions.

When the MPS tensor generates a parent Hamiltonian (Chapter 13), these auxiliary formulas control spatial ground-state correlations once physical observable insertions have been identified with the corresponding transfer-map insertions. The exponential decay rate is then set by the spectral properties of the transfer map, i.e. by the modulus of its subleading eigenvalues.

The spectral analysis of the transfer map and the mixed transfer operator, including the eigenvalue bound $\rho (F_{AB}) \le 1$, the strict transfer-operator gap $\rho (F_{AB}) {\lt} 1$ for non-equivalent blocks, and the resulting overlap decay, is developed in Chapter 7. This chapter focuses on the auxiliary correlator interface: one-point and two-point functionals, their spectral decomposition, and quantitative bounds on decay rates.

16.1 Connected correlators from the transfer map

Definition 16.1.1 Auxiliary-space one-point functional

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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 6.4.2, 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 16.1.2 Auxiliary-space two-point functional at distance $n$

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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.

$\begin{tenkz}[ sandwich, west={cup=$Y$}, east={cup=$X\rho^R$} ] \tnghost{} & \tnX[wires=2]{\E_A^n} & \tnghost{} \\ \tnghost{} & & \tnghost{} \end{tenkz}$

Definition 16.1.3 Auxiliary-space connected correlator

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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}

16.2 Spectral expansion and exponential decay

Theorem 16.2.1 Sum of exponentials (spectral expansion interface)

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If coefficients $c_j\in \mathbb {C}$ and eigenvalues $\lambda _j\in \mathbb {C}$, for $j=1,\ldots ,D^2-1$, satisfy, for all $n\ge 0$,

\begin{align} C(X,Y;n) & = \sum _{j=1}^{D^2-1} c_j\, \lambda _j^n, \label{eq:correlations_spectral_expansion} \end{align}

then the connected correlator equals the stated sum of exponentials. The source [ CPGSV21 , Section II.B.3 ] states that the connected correlator is a sum of $D^2-1$ pure exponentials, $C(X,Y;n)=\sum _{j\ge 2}^{D^2}c_{XY}(j)\lambda _j^n$. This pure-exponential representation follows when the relevant subleading spectral data of the transfer map are diagonalizable. For a non-diagonalizable transfer map, nontrivial Jordan blocks can instead produce terms of the form $n^k\lambda ^n$. Accordingly, the pure-exponential formula is retained here as an explicit interface hypothesis rather than asserted for every normal MPS.

Proof ▶

Given the expansion hypothesis (??), the equality is immediate. Under the corresponding diagonalizability hypothesis, the expansion follows from the spectral decomposition of $\mathcal{E}_A^n$ restricted to the complement of the fixed-point eigenspace. The leading eigenvalue $\lambda _1=1$ contributes $\langle X_0\rangle \langle Y_0\rangle $, which is subtracted in the connected correlator, leaving the sum over subleading eigenvalues. This is the spectral mechanism described in [ CPGSV21 , Section II.B.3 ] .

Theorem 16.2.2 Exponential decay bound (interface)

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If a constant $C_{XY}\in \mathbb {R}$ and $\lambda _2\in \mathbb {C}$ satisfy, for all $n\ge 0$,

\begin{align} |C(X,Y;n)| & \le C_{XY}\, |\lambda _2|^n, \label{eq:correlations_decay_bound} \end{align}

then the connected correlator satisfies that exponential decay bound. The source [ CPGSV21 , Section II.B.3 ] obtains this by combining the sum-of-exponentials expansion with the triangle inequality, while under the primitive or injective hypotheses stated in Chapter 7, the transfer-map gap results ensure that the subleading eigenvalues lie strictly inside the unit disk.

Proof ▶

Given the bound hypothesis (??), the estimate is immediate. For a pure-exponential expansion, the source [ CPGSV21 , Section II.B.3 ] obtains such a bound by applying the triangle inequality and using $|\lambda _j|\le |\lambda _2|$ for all subleading eigenvalues; when the transfer-map gap gives $|\lambda _2|{\lt}1$, this is a decaying bound. In the presence of nontrivial Jordan blocks, an exponential estimate still follows at any rate strictly larger than the complementary spectral radius, as made explicit below.

Remark 16.2.3 Transfer-map spectral hypothesis

The condition $|\lambda _2|{\lt}1$ turns the bound in Theorem 16.2.2 into 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 14. The complementary transfer-map gap $\rho (\mathcal{E}_A-P){\lt}1$ is established for primitive channels (Theorem 4.11.3, Chapter 4). The mixed-transfer gap needed for separation of non-equivalent canonical blocks is supplied for irreducible trace-preserving tensors (Theorem 7.7.2), and quantitative single-tensor transfer-map bounds for injective tensors are recorded in Theorem 16.3.3.

These transfer-map gap results suffice for unconditional exponential convergence at every rate strictly larger than the complementary spectral radius. A pure sum of exponentials, or a bound with the exact rate given by the largest subleading eigenvalue modulus, additionally requires control of the relevant Jordan blocks. The remaining formalization task is to extract the spectral coefficients and, where necessary, the generalized spectral terms from $\mathcal{E}_A$ (Issue #1447).

Definition 16.2.4 Correlation length

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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 16.2.5 Correlation length is positive

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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 $\xi =-1/\log |\lambda _2|{\gt}0$.

16.3 Quantitative transfer-map gap bounds

The transfer-map gap theorems in Chapter 4 establish $\rho (\mathcal{E}_A-P){\lt}1$ for primitive channels qualitatively. This section gives quantitative refinements in terms of geometric constants and rates for the single-tensor transfer map $\mathcal{E}_A$.

Theorem 16.3.1 Exponential convergence of injective primitive channels

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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 (Theorem 4.11.3, the complementary transfer-map gap).

Proof ▶

Injectivity implies primitivity of the transfer map. The complementary spectral-radius bound $\rho (\mathcal{E}_A-P){\lt}1$ then follows from the primitive transfer-map gap theorem. Choose $r$ with $\rho (\mathcal{E}_A-P){\lt}r{\lt}1$. 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 (??).

Theorem 16.3.2 Correlation length bound

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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 (??). Since $\mathcal{E}_A-P$ has spectral radius strictly less than $1$, choose any $r$ with $\rho (\mathcal{E}_A-P){\lt}r{\lt}1$. 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 $\rho (\mathcal{E}_A-P)^n$.

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 $\rho (\mathcal{E}_A-P){\lt}r{\lt}1$ 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 (??).

Theorem 16.3.3 Transfer-map gap from injectivity

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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.

16.4 Relation to parent Hamiltonians

Remark 16.4.1

Suppose that the MPS tensor $A$ generates a parent Hamiltonian $H_N(A,L)$ (Definition 13.3.14) whose ground state is the matrix product vector (Lemma 13.3.18), and that physical observable insertions reduce to the auxiliary functionals defined above. If the hypotheses of Theorem 16.2.2 hold, then the corresponding ground-state correlations decay with finite correlation length $\xi $.