Echoes from the Flumen's Bed: Probing Many-Body Localization via Quantum Quench Dynamics

For researchers probing the elusive many-body localized phase, quantum quench dynamics have become the tool of choice. Unlike static probes that struggle to distinguish MBL from a glassy Anderson insulator, sudden changes in Hamiltonian parameters reveal how a system explores its Hilbert space over time. The central question is direct: after a quench, does the system retain memory of its initial state, or does it thermalize? This guide is for experimentalists and numerical practitioners who already understand the basics of MBL and want to design quench protocols that yield unambiguous signatures. We avoid textbook introductions and instead focus on the decisions that determine whether your measurement captures localization or an artifact of slow dynamics.

Why Quench Dynamics Expose MBL Where Static Probes Fail

The core advantage of a quantum quench lies in its ability to excite all energy scales simultaneously. In a static measurement—say, a conductivity or spectral function—MBL systems can mimic trivial insulators, especially at finite temperature. A quench, by contrast, drives the system out of equilibrium and lets you watch the relaxation in real time. The hallmark of MBL is the failure of the system to act as its own bath: local observables do not relax to thermal values, and entanglement grows only logarithmically in time.

This logarithmic entanglement growth is the single most robust diagnostic. In a thermalizing system, entanglement entropy grows ballistically (linearly in time) after a quench. In an MBL system, it grows as S(t) ~ log(t). Measuring this requires careful control of the initial state and the quench protocol. A deep quench (large parameter change) can obscure the logarithmic slope, while a shallow quench may not excite enough modes. The sweet spot is a quench that changes the disorder strength or interaction energy by about 10–20% of the bandwidth.

Another static probe that fails is the level spacing statistics. While the Poissonian distribution of energy levels is a signature of MBL, it is only accessible via exact diagonalization of small systems. For larger systems or experiments, quench dynamics provide the only practical route. The key is to measure not just the survival probability (Loschmidt echo) but also the full distribution of local observables across many disorder realizations.

Prerequisites: What You Need Before Designing a Quench Protocol

Before you write a single line of simulation code or align a laser, three prerequisites must be settled. First, your Hamiltonian must support a well-defined MBL transition. This means tunable disorder (e.g., quasiperiodic potential or speckle pattern) and interactions that are neither too strong (which can induce a thermal phase via pair formation) nor too weak (which leaves you in the Anderson regime). The standard model is the disordered Heisenberg chain with on-site potential disorder of strength W. For W greater than the critical value W_c ≈ 3.5–4.0 (in units of the hopping), the system enters the MBL phase at infinite temperature.

Second, you need a method for preparing a reproducible initial state. In cold-atom experiments, this often means a charge density wave (CDW) or a Néel state. In numerical simulations, you can choose any product state, but the most informative are those that break translational symmetry, as the relaxation of the staggered magnetization is a direct probe of localization. Avoid translationally invariant initial states—they can mask the MBL signature because the system may appear localized due to symmetry protection.

Third, establish your measurement resolution. For entanglement entropy, you need access to the full reduced density matrix of a subsystem. In experiments, this requires quantum state tomography or, more practically, measuring the second Rényi entropy via parity oscillations. In simulations, you can compute the von Neumann entropy directly, but finite bond dimension in tensor network methods can artificially suppress entanglement growth. Ensure your bond dimension is at least χ = 512 for a 100-site chain to avoid truncation errors that mimic the logarithmic plateau.

One often overlooked prerequisite is the number of disorder realizations. MBL is a phase that exists in the thermodynamic limit, but finite-size effects are severe. You need at least 10^4 disorder realizations for system sizes up to 16 sites, and 10^3 for larger systems (32–48 sites) to achieve statistically meaningful averages. Fewer realizations can produce apparent logarithmic growth from rare thermal inclusions.

Designing the Quench: Step-by-Step Workflow

A well-designed quench protocol follows a sequence of decisions that trade off between signal clarity and experimental constraints. Here we outline the steps, from initial state preparation to data analysis.

Step 1: Choose the quench type

Global quenches (changing a parameter uniformly across the entire system) are the standard choice. They are simple to implement and produce the cleanest entanglement growth signal. However, they also couple to all modes, which can mask the slow dynamics of rare regions. Local quenches (flipping a single spin or changing the potential on one site) excite only a few modes and are better for probing the spatial structure of the MBL phase—they reveal how far the excitation propagates. For most purposes, start with a global quench of the disorder strength: prepare the system in a strongly disordered state (W > 6) and suddenly reduce it to just above the critical point (W ≈ 4.5). This maximizes the contrast between the initial localized state and the final critical regime.

Step 2: Define the observable

The most reliable single observable is the imbalance, defined as the difference between the occupation of even and odd sites normalized by the total occupation. For a CDW initial state, the imbalance decays to zero in a thermal system but saturates at a finite value in the MBL phase. A complementary observable is the return probability (Loschmidt echo), which measures the overlap of the time-evolved state with the initial state. In MBL systems, it decays as a power law with an exponent related to the fractal dimension of the eigenstates. Avoid measuring only the global magnetization—it is too coarse and often thermalizes even in MBL systems due to global symmetries.

Step 3: Set the time window

The logarithmic entanglement growth only becomes visible after a characteristic time t* ~ exp(1/ξ), where ξ is the localization length. For typical parameters (ξ ≈ 2–4 lattice sites), t* ranges from 10 to 100 hopping times. Your measurement must extend to at least 1000 hopping times to see the logarithmic regime clearly. In cold-atom experiments, this means coherence times of several seconds—challenging but achievable with optical lattices. For numerical simulations, time-evolving block decimation (TEBD) can reach t = 1000 with moderate bond dimension, but you must check convergence by increasing the bond dimension until the entanglement entropy stabilizes.

Step 4: Average over disorder

As mentioned, disorder averaging is essential. But how you average matters. Do not average the logarithm of the entanglement entropy—instead, average the entropy itself and then fit the logarithmic growth. Averaging the log can bias the result toward rare thermal regions. Also, compute the variance of the entropy across realizations; a large variance indicates that the system is not truly MBL but rather in a Griffiths phase where rare thermal inclusions dominate. In the true MBL phase, the variance should decrease with system size.

Step 5: Analyze the data

Fit the late-time entanglement entropy to S(t) = a + b log(t). The coefficient b is the key: for MBL, it is typically between 0.1 and 0.5 (in units of the hopping). A b value less than 0.05 suggests that the system is actually localized in a non-interacting sense (Anderson localization). Also check the imbalance: if it decays to zero within the experimental window, the system is thermalizing, even if the entanglement appears logarithmic. The combination of finite imbalance and logarithmic entanglement is the gold standard.

Tools and Environmental Realities

The choice of platform—experiment versus simulation—determines which tools are available and which constraints dominate.

Experimental platforms

Ultracold atoms in optical lattices remain the workhorse for MBL quenches. The key parameters are the lattice depth (which controls the hopping), the interaction strength (tunable via Feshbach resonances), and the disorder (speckle pattern or quasiperiodic lattice). A typical setup: a 3D optical lattice with a superlattice to create a quasiperiodic potential. The quench is performed by suddenly changing the depth of the primary lattice, which changes the hopping by a factor of 2–10. The main environmental challenge is the harmonic trap: the inhomogeneous density across the trap can mask the MBL signal. Use a flat-bottomed trap or post-select on a central region of uniform density. Another reality is the finite temperature of the initial state. MBL is an eigenstate property at infinite temperature, but experiments start at finite temperature (typically 1–10 nK, corresponding to 0.1–0.5 of the hopping). This prethermalizes the system and can suppress the logarithmic growth. Work at the lowest achievable entropy per particle, and measure the temperature independently via the equation of state.

Numerical tools

For exact diagonalization (ED), you are limited to about 20 sites. ED is useful for benchmarking the MBL transition via level statistics, but it cannot access the long-time dynamics needed for entanglement growth. For larger systems (up to 100 sites), use TEBD or time-dependent DMRG with bond dimensions up to 1024. The main pitfall is the entanglement growth itself: as the simulation progresses, the bond dimension needed to maintain accuracy grows exponentially in time for a thermalizing system. For MBL, the bond dimension grows only logarithmically, so TEBD is efficient. However, you must check convergence by comparing results with different bond dimensions. A rule of thumb: if the entanglement entropy at the final time is more than 90% of the maximum possible for the given bond dimension, your results are unreliable. Also, use a Trotter step of δt ≤ 0.05 to avoid discretization errors that can mimic localization.

Data analysis software

Standard scientific Python (NumPy, SciPy) is sufficient for fitting and plotting. For entanglement entropy, use the built-in singular value decomposition (SVD) from the TEBD output. For the Loschmidt echo, compute the overlap of the time-evolved state with the initial state. Avoid using custom fitting routines without cross-validation—use the curve_fit function with a bootstrap estimate of the fit parameters to gauge uncertainty.

Variations for Different Constraints

Not every lab or numerical group has the same resources. Here we adapt the quench protocol for three common constraints: short coherence times, limited system size, and no access to entanglement entropy.

Short coherence times (t_max < 100 hopping times)

In many solid-state platforms (e.g., doped semiconductors or quantum dot arrays), decoherence sets in before logarithmic growth is visible. In this case, focus on the short-time behavior of the imbalance. Even at early times, the imbalance decays as a power law: I(t) ~ t^{-α}. For a thermalizing system, α > 1; for MBL, α < 0.5. This is a less robust signature but still useful. Another option is to measure the two-point correlation function at the initial time and after a fixed short time. The decay of correlations with distance in MBL is exponential, with a localization length that does not change with time. In a thermalizing system, correlations spread ballistically. This measurement can be done in a single snapshot (no time-resolved dynamics) by preparing the system at different times and measuring the correlation function.

Limited system size (N < 20 sites)

For exact diagonalization studies, you are stuck with small systems. Here, the quench dynamics can still be useful, but you must average over a large number of disorder realizations (at least 10^5) to suppress finite-size fluctuations. The entanglement entropy will saturate at a value set by the system size, not by the MBL phase. Instead, measure the level spacing ratio r = min(δ_n, δ_{n+1}) / max(δ_n, δ_{n+1}), where δ_n is the gap between consecutive energy levels. After a quench, the distribution of r for the time-evolved state can be computed from the eigenstate expansion. For MBL, the distribution is Poissonian (average r ≈ 0.39); for thermal, it is Wigner-Dyson (average r ≈ 0.53). This method works even for small systems and is robust to the quench amplitude.

No access to entanglement entropy

Some experiments cannot measure entanglement directly. In that case, use the number entropy S_N = -∑ p_n log p_n, where p_n is the probability of finding n particles in a subsystem. The number entropy also grows logarithmically in MBL systems, but it is easier to measure because it only requires counting particles. In cold-atom experiments, this can be done via in situ imaging with single-site resolution. The number entropy has the advantage that it is less sensitive to temperature than the full entanglement entropy. Another alternative is the mutual information between two distant sites, which in MBL remains finite at long times, while in thermal systems it decays to zero.

Pitfalls, Debugging, and When the Signal Lies

Even with a carefully designed protocol, several common pitfalls can produce false positives for MBL. Here we list the most frequent ones and how to diagnose them.

Finite-size effects mimicking MBL

Small systems (N < 20) can appear localized because the level spacing exceeds the interaction energy, leading to a 'Fock-space localization' that is not the true MBL phase. This is especially dangerous in numerical studies. To check, increase the system size: if the imbalance decay slows down with increasing N, the system is thermalizing and the apparent localization is a finite-size artifact. Another test: vary the interaction strength. True MBL requires interactions; if you turn off interactions and the system remains localized, you are seeing Anderson localization, not MBL.

Rare thermal regions (Griffiths effects)

In the vicinity of the MBL transition, rare regions of low disorder can thermalize locally and then spread thermalization to the whole system over very long times. This produces a slow power-law decay of the imbalance that can be mistaken for MBL. The signature of Griffiths effects is that the decay exponent α decreases with system size. In true MBL, the imbalance saturates to a finite value, not a power law. To distinguish, fit the imbalance to a stretched exponential: I(t) ~ exp(-(t/τ)^β). For MBL, β = 0 (saturation); for Griffiths, 0 < β < 1; for thermal, β = 1. Also, compute the variance of the entanglement entropy across disorder realizations: in the Griffiths phase, the variance is large and does not decay with system size.

Prethermalization plateau

Even in a thermalizing system, the entanglement entropy can show a temporary plateau at intermediate times due to the presence of a prethermal regime. This is common in systems with a large separation of energy scales (e.g., strong interactions compared to hopping). The plateau can last for hundreds of hopping times and look logarithmic. The fix: extend the measurement window. If the entropy eventually crosses over to linear growth, it is prethermalization, not MBL. Another clue: the imbalance decays to zero during the plateau, while in MBL it remains finite. Always measure both imbalance and entanglement together.

Experimental artifacts: trap inhomogeneity and off-resonant scattering

In cold-atom experiments, the harmonic trap creates a position-dependent chemical potential. This can produce a shell structure where the center is in the MBL phase while the edges are thermal. The measured imbalance then reflects a weighted average. To mitigate, use a flat-bottom trap or post-select a central region. Off-resonant scattering from the imaging laser can cause heating and decoherence, which artificially suppresses entanglement growth. Use low-intensity imaging and subtract the heating rate from the measured entropy growth by performing a control experiment without a quench.

Numerical artifacts: Trotter errors and bond truncation

In TEBD, a Trotter step that is too large (δt > 0.1) introduces energy non-conservation that can mimic localization. Reduce δt until the energy variance is below 10^{-6}. Bond truncation is the other major source: if the bond dimension is too low, the entanglement entropy is artificially capped, producing a plateau that looks like MBL. Increase the bond dimension until the entropy at the final time changes by less than 1% when doubling the bond dimension. Also, check the truncation error (sum of discarded singular values); it should be below 10^{-8} at each time step.

When you encounter a suspicious signal, the first debugging step is to repeat the quench with a different initial state (e.g., from a CDW to a random product state). If the apparent MBL signature persists, it is more likely real. If it disappears, the signal was an artifact of the initial state symmetry. Second, vary the disorder strength across the transition. The MBL phase should show a sharp change in the imbalance saturation value at the critical disorder. A gradual change suggests a crossover, not a phase transition.

Finally, compare your results with the predictions of the eigenstate thermalization hypothesis (ETH). For a thermalizing system, the diagonal ensemble (the time-averaged density matrix) should agree with the microcanonical ensemble. Compute the diagonal entropy and compare it to the thermal entropy. If they disagree, your system is not thermalizing, and MBL is a plausible explanation.

In summary, quantum quench dynamics remain the most direct probe of many-body localization, but the devil is in the details. By carefully designing the quench, choosing the right observables, and systematically ruling out artifacts, you can confidently identify the MBL phase and measure its key properties. The next step is to extend these protocols to higher dimensions or to systems with long-range interactions, where the MBL phase may be fragile. Start with the guidelines here, and always validate your results with multiple independent diagnostics.