Finite temporal bond dimension can cut off critical slowing down

Key claim

A finite temporal bond dimension $\chi$ bounds the Schmidt rank of an influence matrix across a time cut. In a converged calculation it can also impose an operational infrared timescale, but $\chi$ alone neither defines that timescale nor turns numerical truncation into a physical bath.

Near an impurity quantum critical point, this distinction becomes visible. The physical relaxation time diverges, while finite temperature and finite temporal compression eventually stop the numerical growth of the fitted relaxation time.

The two effects may produce the same leading cutoff in one observable. They need not produce the same multi-time process.

Theme

Temporal tensor networks, critical slowing down, and cutoff-limited Kibble–Zurek dynamics.

Guiding question

When a Keldysh influence functional is compressed into a finite-$\chi$ temporal matrix-product state, in what sense does $\chi$ act as an infrared regulator, and where does the analogy with temperature fail?

Setup

Consider the two-impurity Anderson model used by Lotem and collaborators. Two interacting impurity orbitals couple to independent fermionic baths and to each other through an exchange $K$.

At a critical value $K_c$, the model reaches a non-Fermi-liquid fixed point in the two-channel-Kondo universality class. Write the tuning field as

The leading symmetry-allowed perturbation has scaling dimension

Its RG eigenvalue is therefore

and close to the fixed point

Running the flow until $g({\ell }_{\ast })\sim 1$ gives

The corresponding crossover energy and time obey

Here $A$ is a nonuniversal amplitude with the units required by the definition of $g$ and time. The exponent $2$ is the universal datum.

Universality class

The two-impurity critical point is in the 2CK universality class under the stated symmetries. Calling it simply “the 2CK model” would erase the microscopic distinction.

Analysis

1. What temporal compression represents

Trace out a bath after discretizing the forward and backward Keldysh contours. Its complete action on an impurity history is encoded in an influence object

After folding the two contours, time steps become sites of a one-dimensional tensor network. A temporal cut separates an earlier history from a later history, so the exact influence object admits a Schmidt decomposition

Approximating this state by a temporal MPS of bond dimension $\chi$ retains at most $\chi$ Schmidt channels across each cut:

This is the exact kinematic statement. It says how much temporal information can cross the cut in the chosen representation.

It does not say that correlations vanish after a fixed number of time steps. A finite-dimensional transfer operator can support long exponential tails, degeneracies, or oscillatory modes. Finite bond dimension, finite Markov order, and finite memory duration are different properties.

Rank is not range

Bond dimension limits the number of transmitted temporal modes. It does not put a hard upper bound on how long the slowest retained mode can persist.

For a time-translation-invariant bath, a semigroup influence matrix uses a single repeated tensor. Suppose its normalized transfer operator has eigenvalues

If ${\lambda }_1$ is isolated and the observable overlaps with the ${\lambda }_2$ sector, the transfer spectrum defines a correlation time

This formula is meaningful only after specifying the transfer operator, time step, normalization, spectral sector, and observable overlap. A near-degeneracy or a vanishing overlap changes the relevant time.

An operational cutoff ${\tau }_{\chi }$ is therefore best defined through evidence: convergence of chosen observables with $\chi$, convergence of the relevant transfer spectrum, or a scaling collapse controlled by one extracted time. It should not be inferred from $\chi$ alone.

2. What the 2026 calculation actually finds

Lotem et al. combine separate semigroup influence matrices for four fermionic baths with a spatial MPS for the coupled auxiliary spaces and impurity orbitals. They apply the construction to sudden quenches and ramps through the two-impurity critical point.

Away from $K_c$, the late-time interimpurity correlation is fitted by

Both temperature $T$ and finite SGIM bond dimension limit the fitted $\tau$. The authors introduce the empirical leading-rate ansatz

Because the zero-temperature physical rate vanishes at criticality, they estimate

and recover

Their low-$T$, sufficiently large-$\chi$ data collapse supports this subtraction for that fitted observable and parameter range. The paper presents additivity as an assumption tested numerically, not as a theorem about temporal MPS truncation.

A Matthiessen-type ansatz

Independent rates often add when distinct slow processes produce simple poles. Operator mixing, $K$-dependent truncation error, or nonexponential tails can invalidate that picture.

At $K=K_c$, the same work reports relaxation consistent with $t^{-3/2}$. It also states that no established theoretical prediction for this exponent is known. The observation is a numerical result of the study, not a general 2CK law.

This qualification matters because an algebraic tail has no unique exponential relaxation time. The fitted off-critical $\tau$, the critical power-law window, and the SGIM transfer time are three distinct quantities that may agree only in a controlled scaling regime.

3. Temperature and finite $\chi$ stop the calculation differently

Regulator	What changes	Physical status	What a matched relaxation time proves
$T>0$	The bath state and its correlation functions; KMS periodicity introduces a thermal timescale	Physical control parameter	The selected observable has the same leading cutoff time
Physical decoherence	The system–environment dynamics or reduced generator	Physical process	The selected decay rate can match
Finite $\chi$	The variational representation of the multi-time influence object	Numerical approximation	The calculation resolves the selected observable only up to a comparable time
Finite observation time	The available fitting window and frequency resolution	Measurement or analysis limitation	Only that the inferred scale cannot exceed the window reliably

Temperature acts on every correlation constrained by the thermal state and KMS relations. A finite-$\chi$ approximation discards particular temporal Schmidt channels selected by the compression procedure.

Matching one scalar time does not imply equality of response functions, counting statistics, out-of-time-order correlations, or interventions at several times. Those are properties of the full process tensor.

The safe analogy is structural:

Spatial finite-entanglement scaling supplies a useful precedent: a finite-$\chi$ MPS often generates an effective correlation length near a critical state. Transferring that idea to a temporal influence matrix is a hypothesis to test through the temporal transfer spectrum and observable collapse, not an automatic identity.

4. Cutoff-limited Kibble–Zurek scaling

Ramp the tuning field linearly through the critical point:

Write the physical critical relaxation law in general form as

The Kibble–Zurek time follows by equating the remaining time to the critical point with the instantaneous relaxation time:

Therefore

and

For $\mathrm{\Delta }=1/2$,

Now suppose temperature, compression, or observation time limits the resolved relaxation to an operational ceiling ${\tau }_{\mathrm{I}\mathrm{R}}$. The zero-cutoff KZ construction applies only while

Setting $t_Q(v_{\mathrm{I}\mathrm{R}})={\tau }_{\mathrm{I}\mathrm{R}}$ gives

Thus, for the 2CK value,

The exponent $3/2$ follows from critical scaling. The prefactor depends on the normalization of $g$, the relaxation amplitude, and the operational definition of the cutoff.

Lotem et al. define the impurity dissipated work for a $K$ ramp by

For this protocol, their scaling ansatz and simulations give three velocity regimes:

1. Fast ramps are dominated by microscopic transients and are nonuniversal.

2. Intermediate ramps approach the impurity KZ law

   $$⟨W_d⟩\sim v^{1/(2-\mathrm{\Delta })}=v^{2/3}.$$Copy LaTeX

3. At the slowest ramps, finite temperature and finite-$\chi$ resolution cut off the critical growth. The reported response crosses to

   $$⟨W_d⟩\propto v.$$Copy LaTeX

The last statement belongs to this definition of integrated dissipated work and this ramp protocol. Linear response can produce different powers for a local rate, a total work, or a protocol whose duration is held fixed.

Slower can be worse

Reducing $v$ increases the ideal freeze-out time. Once that time exceeds the numerical or thermal window, a slower ramp probes the regulator more strongly than the critical fixed point.

False claim to diagnose

If finite-$T$ and finite-$\chi$ relaxation curves collapse after subtracting one constant rate, then finite bond dimension is physically equivalent to raising the bath temperature.

The collapse establishes a narrower result: the dominant cutoff in one fitted relaxation rate can be parameterized by the same scalar correction over the tested range.

Physical equivalence would require agreement of the complete multi-time process, including thermal consistency conditions and several independent observables. The reported collapse does not supply that evidence.

What follows — and what does not

Statement	Status
A temporal MPS with bond dimension $\chi$ has Schmidt rank at most $\chi$ across a temporal cut.	Exact representation statement
Every finite-$\chi$ temporal MPS has a hard memory range.	False
A gapped, normalized uniform transfer operator defines exponential correlation times.	True under spectral and overlap assumptions
${\tau }_{\chi }$ is a universal function of $\chi$ alone.	Unsupported
The 2IAM critical point used here has $\mathrm{\Delta }=1/2$ and $\tau\sim	K-K_c
Finite $T$ and finite $\chi$ support an additive correction to the fitted rate in the reported data.	Numerical result plus an empirical ansatz
Additive rates hold for every observable and temporal truncation.	False
The observed $t^{-3/2}$ critical decay is an established universal theorem.	False; the paper labels it an unexplained numerical observation
The KZ time scales as $v^{-1/(2-\mathrm{\Delta })}$.	Follows from the stated relaxation law and linear ramp
The amplitude-free formula $v_{\mathrm{I}\mathrm{R}}={\tau }_{\mathrm{I}\mathrm{R}}^{-(2-\mathrm{\Delta })}$ is exact.	False unless units set $A=1$
Matching ${\tau }_{\mathrm{I}\mathrm{R}}$ makes finite temperature and finite $\chi$ the same physical perturbation.	False

Exercise

Assume

until an operational ceiling ${\tau }_{\mathrm{I}\mathrm{R}}$ is reached, and ramp $g(t)=vt$.

1. Derive $t_Q(v)$ without setting $A=1$.
2. Derive the crossover velocity $v_{\mathrm{I}\mathrm{R}}$ from $t_Q(v_{\mathrm{I}\mathrm{R}})={\tau }_{\mathrm{I}\mathrm{R}}$.
3. Evaluate both exponents at $\mathrm{\Delta }=1/2$.
4. Suppose two simulations have the same ${\tau }_{\mathrm{I}\mathrm{R}}$, one from finite $T$ and one from finite $\chi$. Name one one-time observable and one multi-time diagnostic you would compare before calling the regulators equivalent.
5. A uniform SGIM transfer operator has ${\lambda }_1=1$ and ${\lambda }_2=0.99$ at time step $\delta t$. Compute its leading transfer time. State one reason this eigenvalue might not control the measured impurity observable.

Hint 1

With $a=1/(1-\mathrm{\Delta })$, solve $t_Q=A(v{t}_Q{)}^{-a}$ by collecting all powers of $t_Q$ on the left.

Hint 2

An eigenmode contributes only if the boundary conditions and the observable have nonzero overlap with its left and right eigenvectors.

Oral check 1. What does finite $\chi$ bound exactly?

Oral check 2. Why is a critical power law incompatible with assigning a unique exponential relaxation pole?

Solution

Let $a=1/(1-\mathrm{\Delta })$. Then

$$t_Q^{1+a}=A{v}^{-a}.$$Copy LaTeX

Because

$$\frac{a}{1+a}=\frac{1}{2-\mathrm{\Delta }},\frac{1}{1+a}=\frac{1-\mathrm{\Delta }}{2-\mathrm{\Delta }},$$Copy LaTeX

we obtain

$$t_Q=A^{\frac{1-\mathrm{\Delta }}{2-\mathrm{\Delta }}}{v}^{-\frac{1}{2-\mathrm{\Delta }}}.$$Copy LaTeX

At crossover,

$${\tau }_{\mathrm{I}\mathrm{R}}=A^{\frac{1-\mathrm{\Delta }}{2-\mathrm{\Delta }}}{v}_{\mathrm{I}\mathrm{R}}^{-\frac{1}{2-\mathrm{\Delta }}},$$Copy LaTeX

so

$$v_{\mathrm{I}\mathrm{R}}=A^{1-\mathrm{\Delta }}{\tau }_{\mathrm{I}\mathrm{R}}^{-(2-\mathrm{\Delta })}.$$Copy LaTeX

For $\mathrm{\Delta }=1/2$,

$$t_Q=A^{1/3}{v}^{-2/3},v_{\mathrm{I}\mathrm{R}}=A^{1/2}{\tau }_{\mathrm{I}\mathrm{R}}^{-3/2}.$$Copy LaTeX

A useful one-time comparison is the full relaxation curve of $⟨{\mathbf{S}}_1\cdot {\mathbf{S}}_2⟩(t)$ at several $K$. A stronger multi-time comparison could use a two-time response function, full counting statistics, or interventions reconstructed as a process tensor. Agreement of the fitted $\tau$ alone is insufficient.

For the transfer spectrum,

$${\tau }_{\mathrm{t}\mathrm{r}}=-\frac{\delta t}{\log 0.99}\approx 99.5\delta t.$$Copy LaTeX

This mode will not control an observable if the observable has zero overlap with its symmetry sector. Another sector, a nonnormal transient, or a power-law window may then set the apparent timescale.

Check your understanding

You increase $\chi$ and find that the fitted relaxation time doubles, while the second transfer-matrix eigenvalue and a two-time response function have not converged.

Which statement is justified: “the physical memory doubled,” “the accessible relaxation window increased,” or “the critical exponent changed”? Explain what further convergence test would distinguish them.

You may also reply with “deeper,” “too easy,” “too hard,” or your derivation.

Further Reading

• Multibath Influence Matrices: Universal Scaling from Real-Time Dynamics — the 2026 preprint whose quench collapse, $t^{-3/2}$ observation, and impurity KZ regimes are analyzed here.
• Semigroup Influence Matrices for Nonequilibrium Quantum Impurity Models — the uniform temporal-MPS construction used as the SGIM foundation.
• Non-Markovian quantum processes: Complete framework and efficient characterization — the process-tensor formulation of multi-time quantum dynamics.
• Influence functional of many-body systems: Temporal entanglement and matrix-product state representation — temporal entanglement and MPS representations of influence functionals.
• Scaling of entanglement support for Matrix Product States — the spatial finite-entanglement analogy and its effective correlation length.
• Exact Crossover Green Function in the Two-Channel and Two-Impurity Kondo Models — crossover structure near the related 2CK and two-impurity fixed points.

Connections and next step

• RG layer: the relevant field with $\mathrm{\Delta }=1/2$ gives $T^{\ast }\sim g^2$ and $\tau \sim g^{-2}$.
• Keldysh layer: tracing out the baths produces a multi-time influence object without a Born–Markov approximation.
• Tensor-network layer: $\chi$ bounds temporal Schmidt rank; the transfer spectrum, not $\chi$ alone, supplies candidate correlation times.
• Numerical layer: rate subtraction is a tested collapse ansatz for one observable, not a universal law of truncation.
• Dynamical layer: an infrared ceiling removes the slow end of the KZ window and can expose cutoff-limited linear response.
• Next step: extract the relevant ${\tau }_{\chi }$ from symmetry-resolved SGIM transfer eigenvalues and compare it with observable-by-observable convergence.
• Revisit: contrast this numerical regulator with the physical KMS and fluctuation–dissipation constraints in the July 31 entry.