Dynamical Phase Transitions and Loschmidt Diagnostics

A dynamical quantum phase transition of Loschmidt type is a nonanalyticity, after the thermodynamic limit, in a return-probability rate function. It is a precise statement about global unitary overlap. Its relation to equilibrium criticality, topology, or a local dynamical order parameter is model dependent and must be demonstrated rather than inferred from the word “transition.”

Required background. Quenches and relaxation fixes the initial-state and order-of-limits problem. Helpful background. Large deviations and phase coexistence supplies the rate-function analogy.

Loschmidt amplitude and rate

For a pure initial state $\lvert\psi_0\rangle$ evolved by $H_f$,

$$\mathcal G(t)=\langle\psi_0\rvert e^{-iH_ft}\lvert\psi_0\rangle, \qquad \mathcal L(t)=\lvert\mathcal G(t)\rvert^2.$$

For volume $V$, define

$$g(t)=-\lim_{V\to\infty}\frac1V\ln\mathcal L(t).$$

The formal resemblance of $\mathcal G(z)=\langle\psi_0\rvert e^{-zH_f}\lvert\psi_0\rangle$ to a boundary partition function motivates a Fisher-zero description. Zeros lie at complex $z$ for finite $V$. Euclidean evolution occupies positive real $z$, whereas real time is the imaginary $z$-axis $z=it$. A nonanalytic real-time rate arises when a line or area of zeros pinches that imaginary $z$-axis in the thermodynamic limit.

This Loschmidt-rate construction was introduced for the transverse-field Ising model by Heyl, Polkovnikov, and Kehrein 2013; its extensions and non-equivalences are reviewed by Heyl 2018.

At finite size, $\mathcal G(t)$ is an entire finite sum of exponentials and the measured rate is rounded. A cusp-like feature in a short trace is therefore evidence for an approach to a singular limit only when its position, width, and height have a systematic size trend.

Solvable two-band quench

Consider independent momenta with

$$h_a(\mathbf k)=\mathbf d_a(\mathbf k)\cdot\boldsymbol\sigma, \qquad \varepsilon_a(\mathbf k)=\lvert\mathbf d_a(\mathbf k)\rvert,$$

where $a=i,f$. Prepare the lower band of $h_i$. Its excitation probability in the upper band of $h_f$ is

$$p_{\mathbf k} =\frac{1-\widehat{\mathbf d}_i(\mathbf k)\cdot \widehat{\mathbf d}_f(\mathbf k)}{2}.$$

The mode-resolved return amplitude is

$$\mathcal G_{\mathbf k}(t) =(1-p_{\mathbf k})e^{i\varepsilon_f t} +p_{\mathbf k}e^{-i\varepsilon_f t},$$

and hence

$$\lvert\mathcal G_{\mathbf k}(t)\rvert^2 =1-4p_{\mathbf k}(1-p_{\mathbf k}) \sin^2[\varepsilon_f(\mathbf k)t].$$

A real-time zero requires a momentum $\mathbf k_*$ with $p_{\mathbf k_*}=1/2$, equivalently $\widehat{\mathbf d}_i\cdot\widehat{\mathbf d}_f=0$, at times

$$t_n^*=\frac{(n+1/2)\pi}{\varepsilon_f(\mathbf k_*)}, \qquad n=0,1,2,\ldots.$$

In the continuum momentum integral, the logarithm of the vanishing factor can produce a nonanalyticity in $g(t)$. Its precise form depends on dimension and on how the zero manifold crosses real time.

This derivation gives useful negative tests. A quench parameter crossing an equilibrium phase boundary does not suffice unless the post-quench occupations pass through $1/2$. Conversely, a suitable occupation texture can yield critical times without an equilibrium transition between $h_i$ and $h_f$. Topology can force such momenta in certain symmetry classes, but that is an additional theorem with hypotheses, not the definition of a dynamical transition.

Local observables are a separate question

The Loschmidt amplitude is a many-body interferometric object. A local magnetization, current, or correlator need not be singular at $t_n^*$. In some models, a dynamical order parameter changes sign or a momentum-space phase vortex crosses a contour at the same time. Establishing that relation requires computing both observables with one convention and showing the coincidence persists with size and perturbations.

The distinction also matters experimentally. Full return probability can be exponentially small in $V$ and sensitive to preparation fidelity. Interferometric protocols may reconstruct a mode-resolved phase in free or weakly interacting systems, while randomized measurements estimate many-body overlaps in small devices. Detector normalization and postselection can alter the apparent rate function.

For a mixed initial state there is no unique extension: interferometric amplitude, fidelity, and purification-based definitions need not have the same zeros. A finite-temperature “dynamical transition” is incomplete unless it states which generalized return object is used and why that object connects to the measured protocol.

Evidence checks and interpretation ceiling

A controlled claim should report:

• the initial density operator and quench Hamiltonian;
• whether $g(t)$ is measured globally, reconstructed from independent modes, or inferred indirectly;
• scaling with $V$ and temporal resolution;
• the Fisher-zero or occupation mechanism producing each critical time;
• local observables evaluated without aligning their features by hand; and
• robustness to preparation errors and weak interactions omitted by an integrable description.

Experiments with trapped ions and optical lattices have observed return-rate features and momentum-space dynamical structures consistent with model predictions Jurcevic et al. 2017, Fläschner et al. 2018. They establish finite-system dynamical critical behavior. The thermodynamic singularity and its classification remain deductions from a tested model plus scaling, not directly observed infinities.

Evidence status

This assessment was checked through 10 August 2026. Loschmidt-rate nonanalyticities are theoretically sharp and have multiple finite-platform realizations, while their universal relation to equilibrium phase boundaries or local order remains absent in general. Dated extensions, null results, and alternative return-probability definitions belong in the Quantum Matter and Emergence Research dossier.

Exercises

1. Critical momentum. Starting from the two-band expression, show that a real-time zero is impossible if $p_{\mathbf k}<1/2$ for every momentum.

Solution

The smallest possible value of $\lvert\mathcal G_{\mathbf k}\rvert^2$ over time is $1-4p_{\mathbf k}(1-p_{\mathbf k})=(1-2p_{\mathbf k})^2$. It is strictly positive if $p_{\mathbf k}<1/2$. Since the total free-system amplitude is a product of nonzero mode factors at finite size, no mode reaches a real-time zero.

2. Preparation infidelity. Suppose an experiment reports $\mathcal L_{\mathrm{meas}}(t)=q\mathcal L(t)+(1-q)B(t)$ with a smooth background $B>0$. What happens to a zero and why must $q$ be calibrated?

Solution

At an ideal zero, the measured return probability is $(1-q)B(t)>0$, so the logarithmic singularity is rounded even before finite-size effects. Without an independently constrained $q$ and background model, rounding cannot be assigned uniquely to system size, interactions, or imperfect preparation.

References

• Fläschner, Nick, Dominik Vogel, Matthias Tarnowski, Benno S. Rem, Dirk-Sören Lühmann, Markus Heyl, Jan Carl Budich, Ludwig Mathey, Klaus Sengstock, and Christof Weitenberg. “Observation of Dynamical Vortices after Quenches in a System with Topology.” Nature Physics 14, 265–268 (2018). DOI.
• Heyl, Markus. “Dynamical Quantum Phase Transitions: A Review.” Reports on Progress in Physics 81, 054001 (2018). DOI.
• Heyl, Markus, Anatoli Polkovnikov, and Stefan Kehrein. “Dynamical Quantum Phase Transitions in the Transverse-Field Ising Model.” Physical Review Letters 110, 135704 (2013). DOI.
• Jurcevic, P., H. Shen, P. Hauke, C. Maier, T. Brydges, C. Hempel, B. P. Lanyon, M. Heyl, R. Blatt, and C. F. Roos. “Direct Observation of Dynamical Quantum Phase Transitions in an Interacting Many-Body System.” Physical Review Letters 119, 080501 (2017). DOI.