Loschmidt Echo and Dynamical Phase Transitions Preview

Prepare a normalized pure state $|\psi_0\rangle$, evolve it with a time-independent many-body Hamiltonian $H_f$, and ask for the amplitude to return to the initial state:

$$\mathcal G_V(t) = \langle\psi_0| e^{-iH_ft/\hbar} |\psi_0\rangle.$$

Its squared magnitude,

$$\mathcal L_V(t) = |\mathcal G_V(t)|^2,$$

is the return probability. In quench studies it is often called a Loschmidt echo. If $V$ is the number of sites, particles at fixed density, or another extensive size variable, the corresponding return-rate density is

$$\lambda_V(t) = -\frac{1}{V} \ln \mathcal L_V(t).$$

A Loschmidt-rate dynamical quantum phase transition occurs at a real critical time $t_c$ when the thermodynamic-limit rate

$$\lambda(t) = \lim_{V\to\infty} \lambda_V(t)$$

is nonanalytic at $t_c$. The order of operations is part of the definition: take the logarithm, divide by the extensive size, take the thermodynamic limit, and only then test analyticity in time.

This is a precise notion of nonequilibrium criticality. It is not a claim that the state becomes thermal, that an equilibrium phase boundary has been crossed, that a local order parameter must be singular, or that a finite system literally changes phase.

Helpful background. Quantum quenches supplies the preparation and final-energy probabilities; the thermodynamic limit explains why the order of limits matters.

The main setting is a pure initial state, a finite-range or otherwise thermodynamically well-scaled lattice Hamiltonian, and closed unitary evolution. Mixed-state and open-system returns require a separately specified experiment and definition.

Return Probabilities and Reversal Echoes

The phrase “Loschmidt echo” is used for several inequivalent quantities.

Name	Formula	Question
Return amplitude	$\mathcal G(t)=\langle\psi_0\rvert e^{-iH_ft/\hbar}\lvert\psi_0\rangle$	What is the complex amplitude to return after one evolution?
Return probability	$\mathcal L(t)=\lvert\mathcal G(t)\rvert^2$	What is the probability of projecting back onto the initial state?
Imperfect-reversal echo	$M_\delta(t)=\lvert\langle\psi_0\rvert e^{i(H_f+\delta H)t/\hbar}e^{-iH_ft/\hbar}\lvert\psi_0\rangle\rvert^2$	How sensitive is reversal to a perturbation?
State fidelity	$F(\rho,\sigma)$	How close are two states under a declared fidelity convention?
Ground-manifold return	$P_0(t)=\sum_\alpha\lvert\langle\psi_\alpha\rvert\psi(t)\rangle\rvert^2$	What is the probability to return to a degenerate reference manifold?

The first two are central here. The imperfect-reversal echo is central in quantum-chaos, decoherence, and control studies. These quantities coincide only in special protocols.

The phrase “dynamical phase transition” also has several meanings. A common modern distinction is:

• DPT-I: a nonanalytic change, as a Hamiltonian or energy parameter is varied, in a long-time order parameter or stationary dynamical regime;
• DPT-II: a nonanalyticity in a Loschmidt return-rate density as real time is varied.

This page uses DQPT for the DPT-II, Loschmidt-rate notion unless explicitly stated otherwise. Usage varies between papers, so read the definition as well as the label.

Spectral Representation

Let

$$H_f|n_f\rangle = E_n^f|n_f\rangle$$

and expand

$$|\psi_0\rangle = \sum_n c_n|n_f\rangle, \qquad p_n = |c_n|^2.$$

Then

$$\mathcal G_V(t) = \sum_n p_n e^{-iE_n^ft/\hbar}.$$

Thus the return amplitude is the characteristic function of the final-energy probability distribution. It depends on the diagonal weights $p_n$, not on relative phases among the $c_n$ in this energy basis. By contrast, a generic observable

$$\langle O(t)\rangle = \sum_{m,n} c_m^*c_n e^{i(E_m^f-E_n^f)t/\hbar} O_{mn}$$

also depends on energy-basis coherences and on the matrix elements of $O$. This already explains why a return-rate singularity need not appear in every local observable.

Energy-origin invariance

Changing the energy zero by

$$H_f \longmapsto H_f+C_V\mathbf 1$$

gives

$$\mathcal G_V(t) \longmapsto e^{-iC_Vt/\hbar} \mathcal G_V(t).$$

Therefore $\mathcal L_V(t)$ and $\lambda_V(t)$ are unchanged. The complex phase of $\mathcal G_V$ depends on the energy origin, but its zeros and squared magnitude do not.

Short-time baseline

For finite-dimensional systems, the cumulant expansion begins as

$$\ln\mathcal G_V(t) = -\frac{i\langle H_f\rangle_0t}{\hbar} -\frac{(\Delta H_f)^2_0t^2}{2\hbar^2} +O(t^3).$$

Consequently,

$$\lambda_V(t) = \frac{(\Delta H_f)^2_0}{V\hbar^2} t^2 +O(t^4).$$

For unbounded Hamiltonians, this expansion requires the corresponding finite energy moments; the stated fourth-order remainder needs a finite fourth moment. For a state with an extensive energy variance, the intensive rate has a finite quadratic onset. A fitted cusp at times comparable to the sampling interval must first be distinguished from this analytic short-time law.

Why an Intensive Logarithm Is Needed

For many global quenches,

$$\mathcal L_V(t) \asymp e^{-V\lambda(t)}$$

over a suitable thermodynamic time window. The raw overlap is then exponentially small even when $\lambda(t)$ is ordinary and smooth. Dividing its logarithm by $V$ extracts an intensive object, just as an equilibrium free-energy density extracts the extensive logarithm of a partition function.

The analogy has limits:

• $\mathcal G(t)$ is generally complex, whereas an equilibrium partition function at real positive inverse temperature is positive;
• the initial state acts as a temporal boundary condition;
• time is not temperature;
• $\lambda(t)$ is a global large-deviation quantity, not an equilibrium thermodynamic potential;
• its singularities need not organize local static correlations in the same way as an equilibrium critical point.

The analogy is mathematically productive precisely when these differences remain visible.

Complex Time and the Boundary Partition Function

Introduce a complex variable $z$ with units of inverse energy:

$$\mathcal Z_V(z) = \langle\psi_0| e^{-zH_f} |\psi_0\rangle = \sum_n p_n e^{-zE_n^f}.$$

Physical real-time evolution lies on the imaginary-$z$ axis:

$$z = \frac{it}{\hbar}.$$

For a finite-dimensional lattice system, $\mathcal Z_V(z)$ is an entire function because it is a finite sum of exponentials. Define a complex boundary free-energy density on any zero-free patch by

$$f_V(z) = -\frac{1}{V} \operatorname{Log}\mathcal Z_V(z).$$

The logarithm requires a branch choice, but its real part on the physical line is unambiguous:

$$\lambda_V(t) = 2\operatorname{Re} f_V\!\left(\frac{it}{\hbar}\right).$$

For unbounded continuum Hamiltonians, convergence and domain questions must be checked before calling $\mathcal Z(z)$ entire. The finite-lattice construction is the clean reference case.

Fisher zeros

The zeros

$$\mathcal Z_V(z_j)=0$$

are called Fisher zeros by analogy with zeros of equilibrium partition functions in a complex temperature-like variable. At finite $V$, they are isolated, possibly with multiplicity: an entire function with $\mathcal Z_V(0)=1$ cannot have an accumulation of zeros at a finite complex point. As $V$ grows, they can become dense on curves or regions.

A DQPT can occur when the thermodynamic accumulation set reaches the physical axis. Schematically,

$$\operatorname{Re}z_j(V) \longrightarrow 0, \qquad \hbar\operatorname{Im}z_j(V) \longrightarrow t_c.$$

The zeros must contribute at the extensive scale of $\ln\mathcal Z_V$: an approaching zero carried by only a subextensive subsystem does not by itself produce a nonzero singularity in the volume-normalized rate. Analyze the limiting zero density or competing extensive branches, as in the examples below.

What finite-size zeros establish

Several cautions matter:

1. An exact zero can occur in a finite quantum system. That fact alone does not establish thermodynamic criticality.
2. A finite-size rate may be smooth because the nearest zero remains off the physical axis; its sharpening with size can still anticipate a DQPT.
3. A sharp analytic peak can arise when a zero line passes close to, but does not reach, the physical axis.
4. The thermodynamic singularity can be expressed as a crossing of dominant transfer-matrix branches even when directly locating every zero is impractical.
5. Experimental noise can prevent an observed probability from reaching zero while leaving a resolvable finite-size precursor.

The question is whether the singularity survives in the intensive thermodynamic limit.

Order of Limits

For fixed finite $V$, unitary dynamics is quasiperiodic when the spectrum is discrete. Recurrences are expected, and $\mathcal G_V(t)$ is analytic in $t$ even though it may have isolated real zeros. A thermodynamic DQPT instead concerns

$$\lambda(t) = -\lim_{V\to\infty} \frac{1}{V} \ln |\mathcal G_V(t)|^2$$

at fixed physical time.

The following operations need not commute:

$$\begin{aligned} &V\to\infty, &&t\to\infty, \\ &\delta t\to0, &&\text{noise strength}\to0. \end{aligned}$$

Taking $t\to\infty$ first at finite size probes recurrences and time averages. Taking $V\to\infty$ first can produce branch competition and real-time nonanalyticity. Convolving with finite time resolution before taking a derivative rounds a cusp. Removing noise after taking a logarithm can be singular if the inferred probability has crossed zero or a background floor.

A One-Qubit Zero Is Not a Phase Transition

Take $\Omega>0$ and

$$H = \frac{\hbar\Omega}{2} \sigma_z, \qquad |\psi_0\rangle = |{+x}\rangle.$$

Then

$$\begin{aligned} \mathcal G_1(t) &= \cos\!\left(\frac{\Omega t}{2}\right), \\ \mathcal L_1(t) &= \cos^2\!\left(\frac{\Omega t}{2}\right). \end{aligned}$$

The return probability vanishes at

$$t_n = \frac{(2n+1)\pi}{\Omega}.$$

This is exact destructive interference in a two-dimensional Hilbert space. There is no thermodynamic limit and therefore no many-body phase transition.

Now take $V$ independent copies:

$$H_V = \frac{\hbar\Omega}{2} \sum_{j=1}^{V} \sigma_j^z, \qquad |\Psi_0\rangle = |{+x}\rangle^{\otimes V}.$$

Factorization gives

$$\mathcal G_V(t) = \left[ \cos\!\left(\frac{\Omega t}{2}\right) \right]^V$$

and

$$\lambda(t) = -2\ln \left| \cos\!\left(\frac{\Omega t}{2}\right) \right|.$$

This thermodynamic rate is singular at the same times, even though the system is noninteracting and never entangles. The example is deliberately austere: a Loschmidt-rate DQPT does not by itself imply interactions, chaos, transport, thermalization, or spatial entanglement.

Exact Benchmark: Transverse-Field Ising Chain

Consider the periodic one-dimensional model

$$\begin{aligned} H(g) &= -J\sum_{j=1}^{V} \sigma_j^z\sigma_{j+1}^z -Jg\sum_{j=1}^{V} \sigma_j^x, \\ &\text{with } J>0. \end{aligned}$$

The infinite chain has equilibrium quantum critical points at $g=\pm1$. For positive field, the familiar ordered-to-paramagnetic transition is at $g=1$. The canonical model conventions, parity sectors, and finite-size subtleties are developed in Transverse-Field Ising Model.

Take even $V$ and prepare the even spin-parity ground state at $g_i$, then quench to $g_f$. The even sector has antiperiodic fermion momenta $k=(2m+1)\pi/V$; use its positive half $0<k<\pi$ below. Choosing a symmetry-broken state or another parity sector requires the corresponding boundary conditions and return definition. A Jordan–Wigner transformation, Fourier transform, and Bogoliubov rotation reduce the even-parity problem to independent momentum pairs. The positive quasiparticle energy is

$$\epsilon_k(g) = 2J \sqrt{ 1+g^2-2g\cos k }.$$

It is useful to define the normalized pseudospin direction

$$\widehat{\mathbf d}_k(g) = \frac{ (\sin k,\,0,\,g-\cos k) }{ \sqrt{ \sin^2k+(g-\cos k)^2 } }.$$

The probability that the final $(k,-k)$ sector is in its excited pair state is

$$p_k = \frac{ 1- \widehat{\mathbf d}_k(g_i) \boldsymbol{\cdot} \widehat{\mathbf d}_k(g_f) }{2}.$$

Up to a nonzero analytic phase, the many-body return amplitude factorizes as

$$\mathcal G_V(t) \doteq \prod_{0<k<\pi} \left[ (1-p_k) +p_k e^{-2i\epsilon_k(g_f)t/\hbar} \right].$$

The symbol $\doteq$ indicates equality up to an overall phase that does not affect zeros or the return probability.

Mode-resolved return probability

For each momentum pair,

$$\begin{aligned} \mathcal L_k(t) ={}& \left| (1-p_k) +p_k e^{-2i\epsilon_k^ft/\hbar} \right|^2 \\ ={}& 1 -4p_k(1-p_k) \sin^2\!\left( \frac{\epsilon_k^ft}{\hbar} \right), \end{aligned}$$

where $\epsilon_k^f\equiv\epsilon_k(g_f)$. The thermodynamic rate is therefore

$$\begin{aligned} \lambda(t) &= -\int_0^\pi \frac{dk}{2\pi} \ln Q_k(t), \\ Q_k(t) &= 1 -4p_k(1-p_k) \sin^2\!\left( \frac{\epsilon_k^ft}{\hbar} \right). \end{aligned}$$

This formula displays the mechanism directly. A mode can become orthogonal to its initial state only if both conditions hold:

$$p_{k_*} = \frac12, \qquad \sin^2\!\left( \frac{\epsilon_{k_*}^ft}{\hbar} \right) = 1.$$

The first condition says that the quench populates the final two-level sector equally. The second says that its two amplitudes have accumulated a relative phase of $\pi$.

Critical momentum

The equal-population condition is

$$\widehat{\mathbf d}_{k_*}(g_i) \boldsymbol{\cdot} \widehat{\mathbf d}_{k_*}(g_f) = 0.$$

For the Ising chain this becomes

$$1+g_ig_f -(g_i+g_f)\cos k_* = 0,$$

When $g_i+g_f\ne0$ and both pseudospin directions are defined,

$$\cos k_* = \frac{1+g_ig_f}{g_i+g_f}.$$

For $g_i,g_f>0$, a real $k_*\in(0,\pi)$ exists for a generic quench across $g=1$. This is the model-specific origin of the close relation between equilibrium and dynamical criticality in the original Ising-chain construction.

Critical times

At $k_*$ the critical times are

$$t_n^* = \left( n+\frac12 \right) \frac{\pi\hbar} {\epsilon_{k_*}^f}, \qquad n=0,1,2,\ldots.$$

The spacing is set by the post-quench energy of the critical mode, not directly by the equilibrium gap at $g=1$.

Fisher-zero lines

For complex $z$, a mode factor vanishes when

$$(1-p_k) +p_k e^{-2\epsilon_k^fz} = 0.$$

One convenient labeling of the zeros is

$$\begin{aligned} z_n(k) &= \frac{ \ln[p_k/(1-p_k)] +i\pi(2n+1) }{ 2\epsilon_k^f }, \\ n&\in\mathbb Z. \end{aligned}$$

This expression assumes $0<p_k<1$ and $\epsilon_k^f>0$; at $p_k=0$ or $1$ the mode factor has no finite zero. As $k$ becomes continuous, these points form Fisher-zero lines. They cross the imaginary axis exactly when $p_{k_*}=1/2$. Substituting $z=it/\hbar$ then reproduces $t_n^*$.

Why a cusp appears

Near a simple critical mode and critical time, let

$$q = k-k_*, \qquad \tau = t-t_n^*.$$

After shifting $q$ to remove a mixed term, the argument of the logarithm has the generic local form

$$\mathcal L_k(t) \simeq Aq^2+B\tau^2, \qquad A,B>0.$$

The singular part of the one-dimensional momentum integral is therefore controlled by

$$-\int_{-\Lambda}^{\Lambda} dq\, \ln(Aq^2+B\tau^2).$$

Relative to its value at $\tau=0$, this contributes

$$\lambda_{\mathrm{sing}}(t) = -C|\tau| +o(|\tau|), \qquad C>0.$$

Thus a simple Fisher-line crossing produces a linear kink in one dimension. Other dimensions, dispersions, zero manifolds, and branch structures can produce different singular forms. The word “cusp” should not be promoted to a universal exponent without deriving the local zero geometry.

Finite-size momentum grids

At finite $V$, allowed momenta depend on fermion parity and spin boundary conditions. The continuum value $k_*$ is generally absent from the grid. Then $\mathcal L_V(t)$ remains nonzero near $t_n^*$ and $\lambda_V(t)$ is rounded. Its width narrows as the nearest allowed momentum approaches $k_*$.

Occasionally $k_*$ is exactly allowed, producing a real zero even at finite size. That arithmetic accident does not remove the need to compare size sequences and sectors. Boundary-condition averaging can help diagnose momentum-grid effects, but it must not be used to manufacture a smoother extrapolation without reporting the individual sectors.

What the Ising Result Does Not Generalize

The Ising-chain criterion is exact and illuminating, but several features are not universal:

• a quench across an equilibrium critical point need not produce a DQPT;
• a DQPT can occur for a quench whose endpoints lie in the same equilibrium phase;
• interacting and nonintegrable systems need not admit momentum-factorized amplitudes;
• critical times need not be periodic;
• a singularity need not come from one isolated quasiparticle momentum;
• a local order parameter need not vanish at a DQPT;
• finite temperature and dissipation do not have a unique extension of the pure-state definition.

Analyses of the XY and XXZ chains provide explicit counterexamples to a one-to-one equilibrium–dynamical correspondence. In transfer-matrix language, a nonanalyticity can appear when two leading boundary-transfer eigenvalues exchange dominance, regardless of whether the quench path crossed an equilibrium phase boundary.

Competing Exponential Branches

Choose an orthonormal set of reference states $|\psi_\alpha\rangle$ spanning the ordered-state manifold of interest. In a finite system, symmetry-broken states need not be exactly degenerate energy eigenstates; specify the chosen reference subspace explicitly. A physically natural return probability is

$$P_0(t) = \sum_{\alpha} \left| \langle\psi_\alpha| e^{-iH_ft/\hbar} |\psi_0\rangle \right|^2.$$

If each contribution has large-deviation form

$$P_\alpha(t) \asymp e^{-V\lambda_\alpha(t)},$$

and the number of contributing branches is finite (or grows subexponentially, with uniform control of their rates), then

$$\begin{aligned} \lambda_0(t) &= -\lim_{V\to\infty} \frac1V \ln P_0(t) \\ &= \min_\alpha \lambda_\alpha(t). \end{aligned}$$

If exponentially many branches contribute, their multiplicity can change the rate and cannot be discarded. For a finite family, the minimum of two analytic branches can have a cusp where their values cross. In a $\mathbb Z_2$-broken system, that crossing may coincide with a switch between positive- and negative-magnetization return sectors.

This mechanism was used in trapped-ion experiments: the two ferromagnetic reference probabilities were measured separately, and their branch exchange sharpened with size. The relation to magnetization is physically meaningful in that symmetry setting, but it is not a universal theorem for arbitrary DQPTs.

Branch crossings and finite systems

For finite $V$,

$$-\frac1V \ln \left( e^{-V\lambda_1} +e^{-V\lambda_2} \right)$$

is smooth when both probabilities are positive. Near the crossing, the rounding width is typically of order $1/V$ if the branch slopes differ. Replacing the sum by $\min(\lambda_1,\lambda_2)$ can expose the limiting kink in a small system, but this replacement is an asymptotic inference, not the measured finite-size rate itself.

Worked example: two product-state returns

Return to the independent-spin Hamiltonian above and set $\theta=\Omega t/2$. Measure the subspace spanned by $|{+x}\rangle^{\otimes V}$ and $|{-x}\rangle^{\otimes V}$. The two probabilities and their total are

$$P_+=\cos^{2V}\theta,\qquad P_-=\sin^{2V}\theta,\qquad P_0=P_++P_-.$$

For $V\ge2$, the finite-size rate is smooth on $0<\theta<\pi/2$. Its limit is

$$\lambda_0(\theta)=-2\ln\max(|\cos\theta|,|\sin\theta|).$$

The cusp at $\theta=\pi/4$ has height $\ln2$, whereas the finite-size value there is $(1-1/V)\ln2$. This is a return to a two-state subspace, distinct from returning to the initial vector alone. The example requires no interactions or entanglement.

For independent spins, the branches $-2\ln|\cos\theta|$ and $-2\ln|\sin\theta|$ cross at $\pi/4$. Summing the two probabilities before taking the logarithm rounds the crossing at finite $V$. The lower panel compares the actual rates for $V=2,8,32$ with their thermodynamic limit.

Boundary Transfer Matrices

In one-dimensional tensor-network formulations, the return amplitude often takes the asymptotic form

$$\mathcal G_V(t) \simeq \sum_a c_a(t) \mu_a(t)^V,$$

where $\mu_a(t)$ are eigenvalues of a boundary transfer operator for one site. If the largest-magnitude eigenvalue is unique and its boundary coefficient is nonzero and not exponentially small in $V$, it controls the rate:

$$\lambda(t) = -2\ln |\mu_{\max}(t)|.$$

If

$$|\mu_1(t_c)| = |\mu_2(t_c)|$$

and the dominant branch changes at $t_c$, the limiting rate can become nonanalytic. This is the dynamical counterpart of competing free-energy branches.

For a transfer operator advancing $b$ sites, use powers $V/b$ and divide the displayed logarithm by $b$. At a tie between complex leading eigenvalues, cancellation can occur for particular size sequences; check that the thermodynamic limit exists instead of replacing a coherent sum by its largest magnitude without qualification.

The transfer-matrix picture is especially useful for interacting chains because it does not require free quasiparticles. It also supplies a practical diagnostic: compare the leading complex eigenvalues, their magnitudes, and their convergence with bond dimension and time step.

Relation to Quantum Work

Use the positive-sign characteristic function $\chi(u)=\langle e^{iuW/\hbar}\rangle$, where $u$ has units of time. For a sudden quench with no intervening evolution and an initial state diagonal in $H_i$, the two-projective-measurement protocol gives

$$\chi(u)=\operatorname{Tr}\!\left[\rho_0 e^{-iH_i u/\hbar}e^{iH_f u/\hbar}\right].$$

For $H_i|\psi_0\rangle=E_i|\psi_0\rangle$,

$$\chi(u)=e^{-iE_i u/\hbar}\mathcal G(-u),\qquad |\chi(u)|^2=\mathcal L(u).$$

Indeed, each final-energy probability $p_n$ now carries the phase $e^{iu(E_n^f-E_i)/\hbar}$, so the inverse transform is

$$P(W)=\frac{1}{2\pi\hbar}\int_{-\infty}^{\infty}du\, e^{-iuW/\hbar}\chi(u).$$

The shift converts final energy to work, $W=E_n^f-E_i$. The sign reversal reflects complex-time zeros; it leaves real-time zero locations paired at $\pm t$ and preserves the return probability. A convention with $\langle e^{-iuW/\hbar}\rangle$ is equally valid but requires the opposite inverse-transform sign.

For a mixed initial state, the initial energy distribution must remain in the two-measurement formula. For an initially coherent state, the first measurement changes that state. Neither situation justifies replacing the work characteristic function with $\operatorname{Tr}(\rho_0e^{-iH_ft/\hbar})$. See work distributions for the complete operational definition.

Equilibrium Criticality: Relation without Equivalence

The resemblance between

$$Z(\beta) = \operatorname{Tr}e^{-\beta H}$$

and

$$\mathcal Z(z) = \langle\psi_0|e^{-zH_f}|\psi_0\rangle$$

motivates the Fisher-zero construction, but the two objects have different boundary data. A useful comparison is:

Equilibrium transition	Loschmidt-rate DQPT
vary temperature or coupling	vary real time after a declared preparation
thermal trace or ground-state energy	boundary amplitude fixed by $\lvert\psi_0\rangle$
nonanalytic thermodynamic potential	nonanalytic return-rate density
thermodynamic state changes across control value	one state follows one unitary orbit
local order and correlations may have universal scaling	local signatures are model- and protocol-dependent

In the nearest-neighbor Ising-chain ground-state quench, crossing the equilibrium critical point produces an equal-population mode and hence DQPTs. In other models, the geometry of initial-state overlaps with final eigenvectors can permit or forbid Fisher-axis crossings independently of equilibrium phase labels.

The reliable statement is therefore conditional:

Equilibrium critical structure can constrain DQPTs in specific symmetry classes and protocols, but a DQPT is not a generic detector of equilibrium phase boundaries.

Order Parameters and Local Observables

A Loschmidt amplitude compares two vectors in the full Hilbert space. A local observable probes a reduced amount of information. Their thermodynamic scalings are different.

For a local operator $O_X$ with bounded support $X$, it is possible that

$$\lim_{V\to\infty} \langle O_X(t)\rangle$$

is smooth at $t_c$ while $\lambda(t)$ has a cusp. Conversely, a long-time order parameter can change nonanalytically as a Hamiltonian parameter is varied even if the Loschmidt rate has no real-time singularity.

Connections do occur in controlled settings:

• symmetry-manifold branch crossings can coincide with an order-parameter sign change;
• scaling near some DQPT fixed points can appear in correlation functions;
• momentum-resolved zeros can organize dynamical vortices in two-band systems;
• anomalous and regular cusp sequences in long-range Ising models can correlate with distinct order-parameter regimes.

Each connection requires its own assumptions. A magnetization zero near a return-rate peak is a useful comparison, but the rate-function limit must be checked separately.

Dynamical Topological Order Parameters

For a translationally invariant two-band or Bogoliubov–de Gennes system, the return amplitude can factorize over momenta:

$$\mathcal G(t) = \prod_k \mathcal G_k(t), \qquad \mathcal G_k(t) = r_k(t)e^{i\phi_k(t)}.$$

Separate the phase into dynamical and geometric parts:

$$\phi_k^{\mathrm G}(t) = \phi_k(t) -\phi_k^{\mathrm{dyn}}(t),$$

with

$$\phi_k^{\mathrm{dyn}}(t) = -\frac1\hbar \int_0^t ds\, \langle\psi_k(s)| H_f(k) |\psi_k(s)\rangle.$$

When the endpoint geometric phases agree modulo $2\pi$, and the amplitude is nonzero along the momentum interval at the time considered, one can define a winding

$$\nu_D(t) = \frac{1}{2\pi} \int_0^\pi dk\, \partial_k \phi_k^{\mathrm G}(t).$$

At a critical mode where $\mathcal G_{k_*}(t_c)=0$, its phase is undefined and $\nu_D$ can jump by an integer. This is a dynamical topological order parameter.

The construction is powerful but not universal:

• it assumes a momentum-resolved factorization or an appropriate generalization;
• quantization depends on symmetries and endpoint conditions;
• it is distinct from the conserved topology of the instantaneously evolved many-body wavefunction;
• a DTOP jump does not imply that every equilibrium topological invariant changed;
• disorder, interactions, degeneracies, and mixed states require additional definitions.

Momentum-space tomography in cold-atom two-band systems has visualized phase vortices associated with such critical modes. Those vortices reveal the mode-resolved phase structure in that setting; a generic interacting spin system requires a different measurement protocol.

Interactions, Integrability, and Chaos

DQPTs occur in free, integrable, nonintegrable, and even product systems. Therefore:

$$\text{DQPT} \not\Rightarrow \text{integrability},$$

and

$$\text{DQPT} \not\Rightarrow \text{many-body chaos}.$$

Likewise, neither ETH nor level repulsion guarantees a Loschmidt-rate cusp. The return amplitude depends on a particular initial-state energy distribution and on coherent phase cancellation. Spectral statistics alone do not determine that distribution.

Weak integrability-breaking perturbations can shift or deform critical times and may preserve early DQPTs over accessible windows. At longer times, new branch crossings or rounded structures can appear. A claim of robustness should specify:

• the perturbation family;
• how its norm scales with $V$;
• the time interval tested;
• whether the thermodynamic limit was direct or extrapolated;
• which cusp property remained stable.

The same caution applies to long-range interactions. Kac normalization, interaction exponent, finite-size geometry, and order of limits can alter both equilibrium extensivity and dynamical branch structure.

Orthogonality Catastrophe and Local Quenches

The Anderson orthogonality catastrophe concerns the vanishing overlap of many-body ground states under a local perturbation as system size grows. A Loschmidt amplitude is instead time dependent and compares an initial state with its evolved image. The two ideas meet in local-quench problems, where boundary field theory and work-edge singularities can govern power-law decay.

For a local quench, however, the natural scaling may be subextensive:

$$-\ln\mathcal L_V(t) = o(V).$$

Then the volume-normalized rate satisfies

$$\lambda(t)=0$$

even though the overlap has nontrivial time dependence. One should choose the normalization dictated by the perturbation geometry rather than forcing every echo into a bulk DQPT framework.

Recurrences and Long Times

Finite isolated systems recur. A near-unity return at a late time does not invalidate earlier dephasing, and a sequence of small returns does not prove irreversibility.

At fixed physical time, compare sizes for which recurrences, boundary reflections, numerical truncation, and experimental decoherence do not dominate. A DQPT may occur on a microscopic time scale; no general condition $t\gg t_{\mathrm{micro}}$ belongs in its definition. DQPT critical times should be shown to stabilize inside a growing pre-recurrence window as $V$ increases.

Periodic critical times in an exactly factorized model are not generic recurrences of the full state. They arise from repeated destructive interference of a critical sector. Conversely, a finite-size revival peak can occur without any thermodynamic nonanalyticity.

Mixed States and Finite Temperature

For an initial density operator $\rho_0$, several plausible quantities coexist.

An interferometric amplitude is

$$\mathcal G_{\mathrm{int}}(t) = \operatorname{Tr} \left[ \rho_0e^{-iH_ft/\hbar} \right].$$

A fidelity return is

$$\mathcal L_F(t) = F\!\left( \rho_0, e^{-iH_ft/\hbar} \rho_0 e^{iH_ft/\hbar} \right),$$

with the squared Uhlmann-fidelity convention. Purification-based amplitudes, interferometric phases, and work-characteristic functions provide still other choices.

They agree in selected pure-state limits but differ at finite temperature. As a result, statements such as “temperature destroys the DQPT” or “the DQPT survives at finite temperature” are incomplete unless the return definition, preparation, and measured protocol are stated.

For example, $\rho_0=I/2$ remains unchanged under $H_f=\hbar\Omega\sigma_z/2$, so its fidelity return is always one. Yet $\mathcal G_{\mathrm{int}}(t)=\cos(\Omega t/2)$ has zeros. Their disagreement is a difference between two experiments, not an ambiguity in computing the same quantity.

Open-System Extensions

Under a quantum channel $\mathcal E_t$, one might study

$$F\!\left( \rho_0, \mathcal E_t(\rho_0) \right),$$

a survival probability, a purification overlap, a trajectory generating function, or the spectrum of a tilted generator. These probe different physics.

Open-system dynamical phase transitions can also refer to:

• nonanalytic steady states of a Lindbladian;
• closing of a Liouvillian spectral gap;
• trajectory-space large-deviation transitions;
• exceptional points;
• transient singularities in a chosen generalized return.

None is automatically the closed pure-state DQPT defined at the start of this page. Cross-pollination is useful, but shared vocabulary is not equivalence.

Experimental Access

No single measurement architecture is optimal for every system.

Direct projective return

Prepare $|\psi_0\rangle$, evolve, and measure the projector

$$\Pi_0 = |\psi_0\rangle\langle\psi_0|.$$

Then

$$\langle\Pi_0\rangle_t = \mathcal L_V(t).$$

For a computational-basis product state, this may be one bit-string probability. For a degenerate reference manifold, sum the probabilities of the declared basis states.

The difficulty is sampling. If

$$\mathcal L_V(t) \sim e^{-V\lambda(t)},$$

then a Bernoulli estimator from $N_{\mathrm{shot}}$ repetitions has relative uncertainty approximately

$$\frac{\delta\mathcal L} {\mathcal L} \sim \frac{1} {\sqrt{ N_{\mathrm{shot}}\mathcal L }}$$

when $\mathcal L\ll1$. Fixed relative precision therefore requires exponentially many shots in the worst case.

Ancilla interferometry

A controlled evolution can encode

$$\operatorname{Re}\mathcal G(t) \quad\text{and}\quad \operatorname{Im}\mathcal G(t)$$

in an ancilla’s coherences. This gives phase information and can connect directly to work-characteristic measurements. Its cost is a controlled many-body evolution, plus careful calibration of ancilla dephasing and control phases.

Momentum-resolved tomography

For factorized two-band systems, Bloch-vector tomography gives the state ray and the magnitude of its overlap with the initial ray. An instantaneous Bloch vector does not determine the overall complex amplitude phase. Extracting geometric phases or dynamical vortices also needs a time-resolved path and the appropriate phase reference or known Hamiltonian evolution. This can locate critical modes without measuring an exponentially small full many-body probability. It depends strongly on translational invariance and an effectively Gaussian mode description.

Tensor-network and quantum-circuit transfer matrices

Infinite matrix product states can represent the thermodynamic state directly. Classical or quantum circuits can estimate a dominant transfer eigenvalue rather than a finite-chain overlap. This trades finite-size extrapolation for ansatz, bond-dimension, optimizer, sampling, and device-noise checks.

The experiments of Jurcevic and collaborators (2017) measured the two ferromagnetic return probabilities in an interacting trapped-ion chain. Fläschner and collaborators (2018) used momentum-resolved tomography in a two-band optical-lattice system. Dborin and collaborators (2022) estimated a transfer-matrix quantity on a superconducting processor. These illustrate three different return measurements; each needs its own finite-size or approximation analysis.

Noise, Resolution, and Background Floors

Suppose the measured probability is

$$\mathcal L_{\mathrm{meas}}(t) = (1-\eta) \mathcal L_{\mathrm{ideal}}(t) +\eta B(t),$$

where $B(t)$ is a background and $\eta$ summarizes preparation, measurement, or decoherence error. Near a deep minimum, even small $\eta$ can dominate. Taking a logarithm produces

$$\lambda_{\mathrm{meas}}(t) = -\frac1V \ln\mathcal L_{\mathrm{meas}}(t),$$

so subtracting a background after the logarithm is not equivalent to correcting probabilities before it.

Finite time resolution convolves the probability or signal with an instrument response. A cusp in the limiting rate can become a smooth peak. A credible analysis reports:

• the raw count model;
• SPAM calibration and uncertainty;
• the time-response function;
• whether corrections were applied before or after logarithms;
• sensitivity to background priors;
• bootstrap or likelihood intervals for $t_c$;
• the size range over which the inferred sharpening persists.

Testing the Thermodynamic Limit

A sharp curve at one size does not determine the thermodynamic limit.

Rate-function sequence

Plot

$$\lambda_V(t) = -\frac1V \ln\mathcal L_V(t)$$

for several sizes with the same Hamiltonian normalization, boundary family, and time units. Track:

• pseudocritical time $t_c(V)$;
• peak or kink width $w(V)$;
• left and right local slopes;
• minimum return probability;
• sensitivity to interpolation and smoothing;
• recurrence and boundary-reflection times.

The limiting claim is strengthened when $t_c(V)$ converges, $w(V)$ shrinks, and the slope discontinuity stabilizes before recurrences.

Complex-zero sequence

When complex-time evaluation is available, locate the nearest zero $z_0(V)$ and test

$$\operatorname{Re}z_0(V) \longrightarrow 0, \qquad \hbar\operatorname{Im}z_0(V) \longrightarrow t_c.$$

The distance to the physical axis is often a cleaner rounding diagnostic than the height of a real-time peak. Numerical analytic continuation is ill-conditioned, however; directly evaluating $\mathcal Z(z)$ is preferable to continuing noisy real-time data.

Branch-spectrum sequence

For transfer-matrix methods, compare

$$\begin{gathered} |\mu_1(t)|, \qquad |\mu_2(t)|, \\ \Delta_\mu(t) = \ln \frac{|\mu_1(t)|}{|\mu_2(t)|}. \end{gathered}$$

Track the same two branches continuously rather than relabeling them by magnitude at each time. A stable sign change of $\Delta_\mu$ at increasing bond dimension supports branch exchange. Convergence of the rate alone can hide a poorly resolved subleading spectrum.

Choosing a finite-size fit

A fit such as

$$t_c(V) = t_c+aV^{-\omega}$$

is a hypothesis, not a definition. The exponent $\omega$ depends on geometry, boundaries, and the singular mechanism. Report alternative fit windows and corrections rather than assigning an equilibrium critical exponent by analogy.

Numerical Methods

Exact diagonalization

Given final eigenpairs,

$$\mathcal G_V(t) = \sum_n p_n e^{-iE_n^ft/\hbar}.$$

Use symmetry blocks to organize the calculation while preserving the prepared state. An initial state can physically occupy several sectors. If $w_s$ is its weight in sector $s$ and $\mathcal G_s$ the amplitude of its normalized sector component, then $\mathcal G=\sum_s w_s\mathcal G_s$: add amplitudes before squaring. Projecting onto one sector or summing sector return probabilities instead changes the experiment.

For larger sparse systems, Krylov propagation can compute

$$|\psi(t)\rangle = e^{-iH_ft/\hbar}|\psi_0\rangle$$

and then its overlap with $|\psi_0\rangle$ without diagonalizing the full Hamiltonian.

Free-fermion determinants

Gaussian states permit determinant, Pfaffian, or mode-product formulas. These are efficient but convention-sensitive. Check:

• Nambu doubling;
• occupied versus empty mode conventions;
• the factor of two in pair energies;
• fermion-parity sectors;
• branch choices for determinant logarithms;
• momenta at $k=0$ and $k=\pi$.

Compute $\ln\mathcal L$ as a sum of mode logarithms to avoid underflow.

Tensor-network evolution

Matrix product states can reach the thermodynamic limit directly or evolve long finite chains. Report:

• bond dimension and truncation threshold;
• time step and integrator;
• energy and norm drift;
• convergence of $\lambda(t)$ and transfer eigenvalues;
• whether entanglement growth ends the reliable window before or after $t_c$.

If a sharp feature appears only when the bond dimension saturates, increase that dimension and verify that the feature remains before interpreting it physically.

Complex-time calculations

Imaginary components of $z$ generate unitary evolution; real components generate nonunitary filtering. Tensor norms can become extremely ill-conditioned. Stabilized transfer matrices, arbitrary precision, or direct root tracking may be required.

Never infer Fisher zeros by applying an unconstrained high-order analytic continuation to noisy sampled data without stability tests. Many distinct analytic functions agree within real-axis error bars and disagree dramatically off axis.

Exercises

1. Invariance under an energy shift

Let

$$H_f' = H_f+C\mathbf1.$$

Show that $\mathcal G'(t)$ differs from $\mathcal G(t)$ only by a phase. Which DQPT data are invariant?

Solution

Because the identity commutes with $H_f$,

$$e^{-iH_f't/\hbar} = e^{-iCt/\hbar} e^{-iH_ft/\hbar}.$$

Hence

$$\mathcal G'(t) = e^{-iCt/\hbar} \mathcal G(t).$$

The return probability, rate function, real critical times, and complex zeros are unchanged because the multiplying exponential is never zero. The absolute phase of the amplitude changes and therefore requires a declared energy origin in an interferometric measurement.

2. Independent-spin return rate

For

$$H_V = \frac{\hbar\Omega}{2} \sum_{j=1}^{V}\sigma_j^z$$

and $|\Psi_0\rangle=|{+x}\rangle^{\otimes V}$, derive $\mathcal G_V(t)$ and $\lambda(t)$. Explain why its singularities do not diagnose interactions.

Solution

Each spin contributes

$$\langle +x| e^{-i\Omega t\sigma_z/2} |+x\rangle = \cos\!\left(\frac{\Omega t}{2}\right).$$

The product state and sum Hamiltonian factorize, so

$$\mathcal G_V(t) = \cos^V\!\left(\frac{\Omega t}{2}\right).$$

Therefore

$$\begin{aligned} \lambda(t) &= -\lim_{V\to\infty} \frac1V \ln|\mathcal G_V(t)|^2 \\ &= -2\ln \left| \cos\!\left(\frac{\Omega t}{2}\right) \right|. \end{aligned}$$

It diverges when $\Omega t/2=(n+1/2)\pi$. Yet the Hamiltonian contains no interactions and the state remains a tensor product at all times. The singularity records an extensive product of identical one-spin destructive-interference events.

3. Critical momentum in the Ising chain

Using

$$\widehat{\mathbf d}_k(g) \propto (\sin k,0,g-\cos k),$$

derive

$$\cos k_* = \frac{1+g_ig_f}{g_i+g_f}.$$

Show that for positive $g_i$ and $g_f$, a generic quench across $g=1$ gives $|\cos k_*|<1$.

Solution

The equal-population condition is orthogonality of the two normalized pseudospins. Normalization does not affect whether their dot product vanishes:

$$\begin{aligned} 0 ={}& \sin^2k +(g_i-\cos k)(g_f-\cos k) \\ ={}& 1+g_ig_f -(g_i+g_f)\cos k. \end{aligned}$$

Solving gives the stated expression. Suppose $0<g_i<1<g_f$. Define the difference $D$ between the two squared expressions. Then

$$\begin{aligned} D &= (g_i+g_f)^2 -(1+g_ig_f)^2, \\ D &= (1-g_i^2)(g_f^2-1) >0. \end{aligned}$$

Both numerator and denominator are positive, so the ratio lies strictly between zero and one. Interchanging $g_i$ and $g_f$ gives the same result.

4. Fisher zeros and critical times

For one Ising momentum pair with $0<p<1$ and $\epsilon>0$, solve

$$(1-p) +p e^{-2\epsilon z} = 0$$

for its zeros. Determine when a zero lies on $z=it/\hbar$.

Solution

Rearranging,

$$e^{-2\epsilon z} = -\frac{1-p}{p}.$$

Taking all logarithm branches gives

$$-2\epsilon z = \ln\!\left(\frac{1-p}{p}\right) +i\pi(2m+1).$$

Relabeling the integer yields

$$z_n = \frac{ \ln[p/(1-p)] +i\pi(2n+1) }{ 2\epsilon }.$$

Its real part vanishes exactly when $p=1/2$. Equating its imaginary part to $t/\hbar$ gives

$$t_n = \left(n+\frac12\right) \frac{\pi\hbar}{\epsilon},$$

up to the equivalent labeling of positive and negative times.

5. Linear cusp from a critical mode

Show that

$$I(\tau) = \int_{-\Lambda}^{\Lambda} dq\, \left[ \ln(q^2+c^2\tau^2) -\ln q^2 \right]$$

has leading small-$|\tau|$ behavior $2\pi c|\tau|$.

Solution

Differentiate with respect to $a=c|\tau|>0$:

$$\frac{dI}{da} = \int_{-\Lambda}^{\Lambda} dq\, \frac{2a}{q^2+a^2} = 4\arctan\!\left(\frac{\Lambda}{a}\right).$$

As $a\to0^+$,

$$\frac{dI}{da} \longrightarrow 2\pi.$$

Since $I(0)=0$,

$$I(\tau) = 2\pi c|\tau| +O(\tau^2/\Lambda).$$

The return rate carries a minus sign in front of the momentum logarithm, so its singular contribution is proportional to $-|\tau|$ plus analytic background.

6. Return to a broken-symmetry manifold

Suppose

$$P_0(t) = e^{-V\lambda_+(t)} +e^{-V\lambda_-(t)}.$$

Show that the limiting rate is $\min(\lambda_+,\lambda_-)$. Estimate the rounding width near a transverse crossing where

$$\lambda_+(t)-\lambda_-(t) \simeq s(t-t_c), \qquad s\ne0.$$

Solution

Factor out the smaller exponential. If $\lambda_+<\lambda_-$,

$$\begin{aligned} \delta\lambda &= \lambda_--\lambda_+ >0, \\ -\frac1V\ln P_0 &= \lambda_+ -r_V, \\ r_V &= \frac1V \ln\!\left[ 1+e^{-V\delta\lambda} \right]. \end{aligned}$$

The correction $r_V$ vanishes as $V\to\infty$, and the other ordering is analogous. Thus the limit is the minimum.

Both branches contribute appreciably when

$$V|\lambda_+-\lambda_-| \lesssim1.$$

Using the linear crossing gives

$$|t-t_c| \lesssim \frac{1}{V|s|}.$$

Hence the finite-size crossover narrows as $1/V$ in this simple branch model.

7. Work characteristic function

Let $|\psi_0\rangle$ satisfy

$$H_i|\psi_0\rangle = E_i|\psi_0\rangle.$$

Starting from

$$\chi(u) = \langle\psi_0| e^{-iH_i u/\hbar} e^{iH_f u/\hbar} |\psi_0\rangle,$$

derive its relation to $\mathcal G(-u)$. Use $\chi(u)=\langle e^{iuW/\hbar}\rangle$. What information changes?

Solution

Acting to the left with the initial-energy exponential gives

$$\begin{aligned} \chi(u) &= e^{-iE_i u/\hbar} \mathcal G(-u), \\ \mathcal G(u) &= \langle\psi_0| e^{-iH_fu/\hbar} |\psi_0\rangle. \end{aligned}$$

The work peaks are at $W=E_n^f-E_i$. The phase factor shifts the energy origin, and the reversed argument reflects the zeros of $\mathcal G$; $|\chi(u)|^2=\mathcal L(u)$ because $\mathcal G(-u)=\mathcal G(u)^*$ for real $u$.

8. Analyze a finite-size return minimum

You are given noisy return-probability data for $V=8,10,12,14$. List a minimal analysis that can distinguish a candidate thermodynamic cusp from a sharp finite-size minimum.

Solution

A useful analysis would:

1. compute $\lambda_V(t)=-V^{-1}\ln\mathcal L_V(t)$ from a probability-level noise model;
2. fit the candidate time, width, and left/right slopes with uncertainty for every $V$;
3. verify convergence under time-grid refinement and alternative local fit windows;
4. test whether the candidate time stabilizes while the width shrinks;
5. locate recurrence and boundary-reflection times and exclude contaminated data;
6. compare parity or boundary sectors if they alter the allowed momentum grid;
7. propagate SPAM and background-floor uncertainty before taking logarithms;
8. where possible, compute the nearest complex zero or leading transfer branches independently;
9. repeat after controlled small Hamiltonian perturbations and numerical-cutoff changes.

Four sizes may still be insufficient for an asymptotic exponent. The appropriate conclusion can be “finite-size precursor consistent with a DQPT” rather than a claimed thermodynamic singularity.

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