Loschmidt Matrix in Quantum Quench Dynamics

• Loschmidt Matrix is a representation in quantum quench dynamics that captures the many-body overlap via its determinant and eigenvalue spectra.
• It appears in formulations such as boundary transfer matrices for 1D systems, overlap matrices in Fermi systems, and Toeplitz matrices in XX-type chains.
• The matrix’s spectral properties reveal Fisher zeros, yield nonanalytic return rates, and signal dynamical quantum phase transitions.

Loschmidt matrix denotes a matrix representation of quench dynamics whose spectrum or determinant controls the Loschmidt amplitude and, through it, the Loschmidt echo. In the literature, the term is used for distinct but structurally related objects: the boundary transfer matrix $\overline T(z)$ for one-dimensional quantum systems, the overlap matrix $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$ for quenched Slater determinants in non-interacting Fermi systems, and the Toeplitz matrix $M_{jk}(w)$ governing complex-time overlaps in XX-type spin chains. Across these formulations, the matrix serves as the carrier of Fisher zeros, return-rate singularities, and dynamical quantum phase transitions (DQPTs) (Andraschko et al., 2013, Vanhala et al., 2022, Santilli et al., 2019).

1. Core definitions

A common starting point is the Loschmidt amplitude, also called a boundary partition function in the transfer-matrix formulation,

$$Z(z)=\langle \Psi_0|e^{-zH}|\Psi_0\rangle,\qquad z\in\mathbb C.$$

Real-time dynamics is obtained by setting $z=i\,t$, so that

$$G(t)\equiv Z(i\,t)=\langle \Psi_0|e^{-iHt}|\Psi_0\rangle,\qquad L(t)=|G(t)|^2.$$

The corresponding return rate, often called the dynamical free-energy rate function, is

$g(t)\equiv -\lim_{L\to\infty}\frac1L\ln L(t).$

In the free-fermion overlap-matrix formulation, if $\{\lambda_i(t)\}_{i=1}^{N_p}$ are the eigenvalues of $\mathcal L(t)$, then

$$Z(t)=\det \mathcal L(t)=\prod_{i=1}^{N_p}\lambda_i(t),\qquad g(t)=-\frac{1}{L}\ln|Z(t)|^2=-\frac{2}{L}\sum_{i=1}^{N_p}\ln|\lambda_i(t)|.$$

In XX-type chains, the complex-time quantity is written as $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$0 with $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$1, and $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$2 (Andraschko et al., 2013, Vanhala et al., 2022, Santilli et al., 2019).

These definitions indicate a family of equivalent roles rather than a single canonical matrix object. A plausible implication is that “Loschmidt matrix” is best understood operationally: it is the matrix whose determinant or dominant eigenvalue governs the many-body overlap.

2. Boundary transfer matrix in one-dimensional quantum systems

For one-dimensional quantum systems, a Trotter–Suzuki decomposition in the “imaginary-time” direction $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$3 maps $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$4 to the partition function of a two-dimensional classical lattice on a cylinder whose ends are fixed by $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$5. One introduces a one-step transfer matrix $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$6 that propagates two sites in the spatial direction and incorporates the boundary weights imposed by the initial state: $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$7 where $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$8 is the even chain length, $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$9 is the Trotter number, and $M_{jk}(w)$0. Ordering the eigenvalues as

$M_{jk}(w)$1

the thermodynamic limit is dominated by $M_{jk}(w)$2, yielding the boundary free-energy density

$M_{jk}(w)$3

At real time,

$M_{jk}(w)$4

Within this framework, Fisher zeros are tied to crossings of leading eigenvalues in the spectrum of $M_{jk}(w)$5. The crossing condition is

$M_{jk}(w)$6

for some $M_{jk}(w)$7. At such points $M_{jk}(w)$8 is nonanalytic, and for $M_{jk}(w)$9 the crossing generates cusps in $$Z(z)=\langle \Psi_0|e^{-zH}|\Psi_0\rangle,\qquad z\in\mathbb C.$$0. Numerically, the approach builds an MPO for $$Z(z)=\langle \Psi_0|e^{-zH}|\Psi_0\rangle,\qquad z\in\mathbb C.$$1 from a Trotter–Suzuki split of $$Z(z)=\langle \Psi_0|e^{-zH}|\Psi_0\rangle,\qquad z\in\mathbb C.$$2, includes fixed boundary weights associated with $$Z(z)=\langle \Psi_0|e^{-zH}|\Psi_0\rangle,\qquad z\in\mathbb C.$$3, and extracts the leading eigenvalues by iterative Lanczos or Arnoldi methods in an MPS/MPO setup working directly in the thermodynamic limit, in LCRG or iTEBD style. The method was benchmarked against the analytically solvable transverse-Ising quench with perfect agreement (Andraschko et al., 2013).

3. Overlap-matrix formulation in disordered Fermi systems

For non-interacting quenched Fermi systems, the Loschmidt matrix is the $$Z(z)=\langle \Psi_0|e^{-zH}|\Psi_0\rangle,\qquad z\in\mathbb C.$$4 overlap matrix of occupied single-particle orbitals. Let $$Z(z)=\langle \Psi_0|e^{-zH}|\Psi_0\rangle,\qquad z\in\mathbb C.$$5 and $$Z(z)=\langle \Psi_0|e^{-zH}|\Psi_0\rangle,\qquad z\in\mathbb C.$$6 be quadratic fermion Hamiltonians, let $$Z(z)=\langle \Psi_0|e^{-zH}|\Psi_0\rangle,\qquad z\in\mathbb C.$$7 be an $$Z(z)=\langle \Psi_0|e^{-zH}|\Psi_0\rangle,\qquad z\in\mathbb C.$$8-particle Slater-determinant eigenstate of $$Z(z)=\langle \Psi_0|e^{-zH}|\Psi_0\rangle,\qquad z\in\mathbb C.$$9, and let $z=i\,t$0 be the $z=i\,t$1 matrix whose columns are the occupied single-particle orbitals of $z=i\,t$2. Under evolution with $z=i\,t$3, the orbital matrix becomes

$z=i\,t$4

The many-body Loschmidt amplitude is then

$z=i\,t$5

where the Loschmidt matrix $z=i\,t$6 has entries

$z=i\,t$7

A zero of the many-body overlap occurs precisely when at least one eigenvalue of $z=i\,t$8 vanishes. After analytic continuation $z=i\,t$9, one studies the complex Loschmidt zeros defined by $$G(t)\equiv Z(i\,t)=\langle \Psi_0|e^{-iHt}|\Psi_0\rangle,\qquad L(t)=|G(t)|^2.$$0. In finite systems, each disorder realization has a generically discrete set of complex zeros. In the thermodynamic limit, these zeros coalesce into a two-dimensional region in the complex-$$G(t)\equiv Z(i\,t)=\langle \Psi_0|e^{-iHt}|\Psi_0\rangle,\qquad L(t)=|G(t)|^2.$$1 plane whose boundary intersects the real-time axis at a sharply defined critical time $$G(t)\equiv Z(i\,t)=\langle \Psi_0|e^{-iHt}|\Psi_0\rangle,\qquad L(t)=|G(t)|^2.$$2. At $$G(t)\equiv Z(i\,t)=\langle \Psi_0|e^{-iHt}|\Psi_0\rangle,\qquad L(t)=|G(t)|^2.$$3, the first zero eigenvalue of $$G(t)\equiv Z(i\,t)=\langle \Psi_0|e^{-iHt}|\Psi_0\rangle,\qquad L(t)=|G(t)|^2.$$4 hits the origin, $$G(t)\equiv Z(i\,t)=\langle \Psi_0|e^{-iHt}|\Psi_0\rangle,\qquad L(t)=|G(t)|^2.$$5, and for all $$G(t)\equiv Z(i\,t)=\langle \Psi_0|e^{-iHt}|\Psi_0\rangle,\qquad L(t)=|G(t)|^2.$$6 the origin remains engulfed by the eigenvalue support so that $$G(t)\equiv Z(i\,t)=\langle \Psi_0|e^{-iHt}|\Psi_0\rangle,\qquad L(t)=|G(t)|^2.$$7 strictly (Vanhala et al., 2022).

4. Toeplitz Loschmidt matrices in XX-type spin chains

In XX spin chains and generalizations with additional interactions to more neighbours, the Loschmidt matrix appears as a Toeplitz matrix for complex-time overlaps of domain-wall states. For the single-domain-wall amplitude,

$$G(t)\equiv Z(i\,t)=\langle \Psi_0|e^{-iHt}|\Psi_0\rangle,\qquad L(t)=|G(t)|^2.$$8

one obtains

$$G(t)\equiv Z(i\,t)=\langle \Psi_0|e^{-iHt}|\Psi_0\rangle,\qquad L(t)=|G(t)|^2.$$9

with

$g(t)\equiv -\lim_{L\to\infty}\frac1L\ln L(t).$0

Equivalently,

$g(t)\equiv -\lim_{L\to\infty}\frac1L\ln L(t).$1

The Toeplitz symbol has the form

$g(t)\equiv -\lim_{L\to\infty}\frac1L\ln L(t).$2

with

$g(t)\equiv -\lim_{L\to\infty}\frac1L\ln L(t).$3

If

$g(t)\equiv -\lim_{L\to\infty}\frac1L\ln L(t).$4

then

$g(t)\equiv -\lim_{L\to\infty}\frac1L\ln L(t).$5

Examples of $g(t)\equiv -\lim_{L\to\infty}\frac1L\ln L(t).$6 given in this framework are the nearest-neighbour case $g(t)\equiv -\lim_{L\to\infty}\frac1L\ln L(t).$7, the exponentially decaying case

$g(t)\equiv -\lim_{L\to\infty}\frac1L\ln L(t).$8

the power-law polylogarithmic case

$g(t)\equiv -\lim_{L\to\infty}\frac1L\ln L(t).$9

and the pure Fisher–Hartwig case

$\{\lambda_i(t)\}_{i=1}^{N_p}$0

This determinantal formulation also makes explicit the factorization of multi-domain-wall correlators into Toeplitz minors and the associated unitary matrix-integral representation (Santilli et al., 2019).

5. Spectral mechanisms of zeros, singularities, and scaling

The three formulations isolate different spectral mechanisms for dynamical singularities. In the boundary transfer-matrix approach, nonanalyticities in the Loschmidt echo and Fisher zeros in the complex plane are caused by a crossing of eigenvalues in the spectrum of $\{\lambda_i(t)\}_{i=1}^{N_p}$1; the dominant eigenvalue changes identity, and $\{\lambda_i(t)\}_{i=1}^{N_p}$2 develops cusps (Andraschko et al., 2013).

In disordered Fermi systems, the determinant representation shifts attention from eigenvalue crossings to the statistics of the smallest eigenvalue. Defining

$\{\lambda_i(t)\}_{i=1}^{N_p}$3

and the disorder-averaged density

$\{\lambda_i(t)\}_{i=1}^{N_p}$4

the critical time $\{\lambda_i(t)\}_{i=1}^{N_p}$5 is identified with a transition in $\{\lambda_i(t)\}_{i=1}^{N_p}$6. For $\{\lambda_i(t)\}_{i=1}^{N_p}$7, the support of $\{\lambda_i(t)\}_{i=1}^{N_p}$8 stays bounded away from $\{\lambda_i(t)\}_{i=1}^{N_p}$9, so $\mathcal L(t)$0 as $\mathcal L(t)$1. For $\mathcal L(t)$2, the distribution collapses towards $\mathcal L(t)$3, and for large $\mathcal L(t)$4

$\mathcal L(t)$5

with local eigenvalue density $\mathcal L(t)$6 near the origin. Both the mean $\mathcal L(t)$7 and standard deviation $\mathcal L(t)$8 then shrink as $\mathcal L(t)$9. Near $$Z(t)=\det \mathcal L(t)=\prod_{i=1}^{N_p}\lambda_i(t),\qquad g(t)=-\frac{1}{L}\ln|Z(t)|^2=-\frac{2}{L}\sum_{i=1}^{N_p}\ln|\lambda_i(t)|.$$0, the finite-size scaling ansatz is

$$Z(t)=\det \mathcal L(t)=\prod_{i=1}^{N_p}\lambda_i(t),\qquad g(t)=-\frac{1}{L}\ln|Z(t)|^2=-\frac{2}{L}\sum_{i=1}^{N_p}\ln|\lambda_i(t)|.$$1

with $$Z(t)=\det \mathcal L(t)=\prod_{i=1}^{N_p}\lambda_i(t),\qquad g(t)=-\frac{1}{L}\ln|Z(t)|^2=-\frac{2}{L}\sum_{i=1}^{N_p}\ln|\lambda_i(t)|.$$2 set by the Rayleigh scaling, and $$Z(t)=\det \mathcal L(t)=\prod_{i=1}^{N_p}\lambda_i(t),\qquad g(t)=-\frac{1}{L}\ln|Z(t)|^2=-\frac{2}{L}\sum_{i=1}^{N_p}\ln|\lambda_i(t)|.$$3 if one identifies $$Z(t)=\det \mathcal L(t)=\prod_{i=1}^{N_p}\lambda_i(t),\qquad g(t)=-\frac{1}{L}\ln|Z(t)|^2=-\frac{2}{L}\sum_{i=1}^{N_p}\ln|\lambda_i(t)|.$$4 (Vanhala et al., 2022).

In XX-type chains, the corresponding singularities appear in a double-scaling limit $$Z(t)=\det \mathcal L(t)=\prod_{i=1}^{N_p}\lambda_i(t),\qquad g(t)=-\frac{1}{L}\ln|Z(t)|^2=-\frac{2}{L}\sum_{i=1}^{N_p}\ln|\lambda_i(t)|.$$5, $$Z(t)=\det \mathcal L(t)=\prod_{i=1}^{N_p}\lambda_i(t),\qquad g(t)=-\frac{1}{L}\ln|Z(t)|^2=-\frac{2}{L}\sum_{i=1}^{N_p}\ln|\lambda_i(t)|.$$6, keeping $$Z(t)=\det \mathcal L(t)=\prod_{i=1}^{N_p}\lambda_i(t),\qquad g(t)=-\frac{1}{L}\ln|Z(t)|^2=-\frac{2}{L}\sum_{i=1}^{N_p}\ln|\lambda_i(t)|.$$7 fixed. The saddle-point equation for the eigenvalue density $$Z(t)=\det \mathcal L(t)=\prod_{i=1}^{N_p}\lambda_i(t),\qquad g(t)=-\frac{1}{L}\ln|Z(t)|^2=-\frac{2}{L}\sum_{i=1}^{N_p}\ln|\lambda_i(t)|.$$8 yields a weak-coupling phase with full-circle support and a strong-coupling phase with one-cut or multi-cut support. For $$Z(t)=\det \mathcal L(t)=\prod_{i=1}^{N_p}\lambda_i(t),\qquad g(t)=-\frac{1}{L}\ln|Z(t)|^2=-\frac{2}{L}\sum_{i=1}^{N_p}\ln|\lambda_i(t)|.$$9,

$\mathcal L(t)=V^\dagger e^{-iH_1 t}V$00

and

$\mathcal L(t)=V^\dagger e^{-iH_1 t}V$01

For the classical XX chain, $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$02 and

$\mathcal L(t)=V^\dagger e^{-iH_1 t}V$03

The third derivative of $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$04 is discontinuous at $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$05, and analogous third-order transitions are reported for exponentially decaying and power-law interactions (Santilli et al., 2019).

6. Analytic controls: bosonization, free-fermion structure, and exact determinants

For quenches within the Luttinger-liquid phase of the XXZ chain, low-energy dynamics is described by the bosonized Hamiltonian

$\mathcal L(t)=V^\dagger e^{-iH_1 t}V$06

plus a $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$07-term. After a quench $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$08, hence $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$09 and $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$10, a Gaussian/Bogoliubov calculation gives

$\mathcal L(t)=V^\dagger e^{-iH_1 t}V$11

From this factorized form, no zeros occur for finite $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$12, because the product stays finite for all $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$13. The corresponding return rate is

$\mathcal L(t)=V^\dagger e^{-iH_1 t}V$14

with ultraviolet cutoff $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$15 of order the inverse lattice spacing. Comparison with full numerics shows qualitative agreement for small quenches, while quantitative differences, including cutoff-dependence and damping rates, quickly set in for larger quenches. The same work also shows analytically that, for a quench to the free fermion point, the Fisher zeros depend sensitively on the initial state and can lie exactly on the real axis already for finite system size (Andraschko et al., 2013).

For XX-type chains with a pure Fisher–Hartwig singularity, the finite-$\mathcal L(t)=V^\dagger e^{-iH_1 t}V$16 determinant is available in closed form: $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$17 where $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$18 is the Barnes double-Gamma. Its large-$\mathcal L(t)=V^\dagger e^{-iH_1 t}V$19 asymptotics separates into the usual Bernoulli-number power series and exponentially small Stokes terms. Along the imaginary-time axis, $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$20, these Stokes terms are absent; along the real-time axis, $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$21, some arguments cross the Stokes lines $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$22, and exponentially small quench-induced corrections appear. Berry’s smoothing describes their smooth onset across the Stokes lines (Santilli et al., 2019).

7. Relation to equilibrium phase structure

The Loschmidt matrix formalism does not support a simple identification of dynamical nonanalyticities with equilibrium phase boundaries. In the transfer-matrix treatment of the XXZ model with uniform and staggered magnetic fields, there are examples in both the integrable and non-integrable case where the Loschmidt echo does not show non-analyticities although the quench leads across an equilibrium phase transition, and examples where non-analyticities appear for quenches within the same phase. This establishes that, in that setting, crossing an equilibrium transition is neither a sufficient nor a necessary condition for a cusp in the return rate (Andraschko et al., 2013).

An analogous decoupling appears in disordered Fermi systems. The DQPT at $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$23 occurs for quenches both within the localized phase, with $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$24, and across $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$25, and it also appears in the parallel GUE random-matrix toy model, which has no spatial structure at all. The singular many-body dynamics encoded by the zero-manifold of $\mathcal L(t)=V^\dagger e^{-iH_1 t}V$26 is therefore not a probe of the equilibrium Anderson localization transition (Vanhala et al., 2022).

Taken together, these results indicate that the Loschmidt matrix is principally a dynamical object. Its determinant, dominant eigenvalue, or eigenvalue distribution controls real-time singularities, but the presence, order, and location of those singularities depend on the quench protocol, the initial state, and the spectral geometry of the matrix representation rather than on a universal correspondence with equilibrium criticality.