Loop Quenches in Symmetry-Protected Topology

• Loop quenches are defined as symmetric pulse protocols (H1 → H2 → H1) that preserve protecting symmetries in nonequilibrium dynamics of SPT systems.
• They restore chiral symmetry at a special midpoint, enabling the measurement of dynamic observables like the Loschmidt chirality amplitude to reveal the winding number.
• Experimental implementations, such as the loop-quench-probe setup, demonstrate quantized conductance responses that robustly reflect equilibrium topological invariants even under weak noise.

Searching arXiv for papers specifically on “loop quenches” and closely related usage to ground the article in the current literature. Loop quenches are a class of nonequilibrium protocols introduced for the study of symmetry-protected topological (SPT) systems in settings where ordinary quench dynamics would dynamically violate the protecting symmetry. In the sense formalized in "Survival and Detection of Symmetry-Protected Topology in Loop Quenches" (Forcellini et al., 12 Aug 2025), a loop quench is a two-step pulse protocol in which the system starts from a target Hamiltonian $H_1$, is quenched to a pulse Hamiltonian $H_2$ for a duration $T_2$, and is then returned to $H_1$, so that the evolution follows $H_1\to H_2\to H_1$. The defining idea is that a time-symmetric pulse can restore the relevant symmetry at a special point of the protocol and can encode equilibrium topology into measurable dynamical observables. In the broader quench literature, closely related but distinct "loop-like" usages also occur, notably annulus-to-torus local quenches and topology-changing splitting or joining quenches; these usages are conceptually adjacent but are not identical to the $H_1\to H_2\to H_1$ protocol (Bhattacharyya et al., 2019, Shimaji et al., 2018).

1. Definition and protocol

In the named sense of the recent SPT literature, a loop quench is a symmetric pulse protocol tailored to evade the usual dynamical loss of symmetry under nonequilibrium evolution. The protocol consists of three stages: start from the target Hamiltonian $H_1$, quench to a pulse Hamiltonian $H_2$ for a duration $T_2$, and return to $H_1$. The protocol is therefore written

$H_2$0

Its central structural feature is that the pulse is arranged symmetrically in time and centered at the midpoint of the evolution (Forcellini et al., 12 Aug 2025).

This construction was introduced for SPT phases protected by symmetries such as time-reversal symmetry, chiral symmetry, and particle-hole symmetry. The stated motivation is that ordinary single-step quenches $H_2$1 typically violate the protecting symmetry dynamically, rendering the equilibrium topological classification inaccessible out of equilibrium. Loop quenches were proposed precisely as a loophole: the symmetry can be preserved at a special symmetric point of the protocol, and the equilibrium topological invariant of $H_2$2 can remain encoded in measurable dynamical quantities (Forcellini et al., 12 Aug 2025).

The paper that introduced this terminology focuses on chiral-SPT phases and on chiral-symmetric one-dimensional two-band insulators. A plausible implication is that the term "loop quench" is most precise when reserved for this time-symmetric return protocol, rather than for every quench with a geometrically or topologically loop-like representation.

2. Dynamical symmetry restoration

The formal mechanism underlying the protocol is a dynamical symmetry condition centered at a restoration time $H_2$3. For chiral symmetry generated by $H_2$4, the condition is

$H_2$5

with $H_2$6. For the evolution operator

$H_2$7

choosing symmetric times $H_2$8 and $H_2$9 gives

$T_2$0

This is the stated sense in which the loop quench preserves chiral symmetry at the midpoint of the protocol (Forcellini et al., 12 Aug 2025).

For the one-dimensional chiral-symmetric two-band insulator studied in detail, the Hamiltonian is

$T_2$1

with

$T_2$2

and chiral symmetry imposes

$T_2$3

The loop-quench evolution operator is written as

$T_2$4

with $T_2$5. The symmetry-breaking $T_2$6 component is proportional to

$T_2$7

The explicit consequence drawn in the paper is that at $T_2$8 the chiral-symmetry-breaking part vanishes, and for generic $T_2$9 this is the only time when the symmetry is restored for all $H_1$0 (Forcellini et al., 12 Aug 2025).

The same work also emphasizes that the system may be studied beyond the symmetry-restoration time according to

$H_1$1

so the protocol is not restricted to the exact restoration point. This suggests that the loop geometry of the quench is being used not merely to enforce a symmetry instantaneously, but to implant topological information into the subsequent dynamics.

3. Loschmidt chirality amplitude and topological encoding

The central dynamical observable introduced for loop quenches is the Loschmidt chirality amplitude (LCA),

$H_1$2

where

$H_1$3

Here $H_1$4 are eigenstates of the initial Hamiltonian $H_1$5, and the equivalence follows from

$H_1$6

The LCA is therefore a chiral-symmetry-resolved overlap that measures the amplitude for the evolved state to occupy the chiral partner of the initial state (Forcellini et al., 12 Aug 2025).

For the one-dimensional chiral two-band model, the equilibrium topology of $H_1$7 is characterized by the winding number

$H_1$8

To connect dynamics to this invariant, the pulse Hamiltonian is chosen as a weak magnetic-flux pulse on a ring: $H_1$9 The required conditions are that $H_1\to H_2\to H_1$0 be weak, that the ring be large enough that the vector potential is uniform, and that the pulse be short,

$H_1\to H_2\to H_1$1

Under these conditions, the LCA becomes to lowest order in $H_1\to H_2\to H_1$2

$H_1\to H_2\to H_1$3

This is the key formula showing that the dynamical response is proportional to the winding density (Forcellini et al., 12 Aug 2025).

The frequency-space form is obtained from the one-sided retarded Fourier transform

$H_1\to H_2\to H_1$4

At zero frequency,

$H_1\to H_2\to H_1$5

In the short-pulse limit $H_1\to H_2\to H_1$6, this yields

$H_1\to H_2\to H_1$7

The equilibrium invariant $H_1\to H_2\to H_1$8 is thus encoded in the zero-frequency imaginary part of the Fourier-transformed LCA (Forcellini et al., 12 Aug 2025).

4. Experimental readout and robustness

To access the LCA experimentally, the same work proposes a loop-quench-probe (LQP) setup involving two weakly coupled identical systems. System A is the target SPT insulator subjected to the loop quench, while system B is an identical copy that is not directly quenched. After the quench, the two are weakly tunnel-coupled through

$H_1\to H_2\to H_1$9

System B is initialized in $H_1\to H_2\to H_1$0, and the measured quantity is its induced chirality,

$H_1\to H_2\to H_1$1

The integrated response satisfies

$H_1\to H_2\to H_1$2

so the chirality imprinted on B directly reveals the LCA of A (Forcellini et al., 12 Aug 2025).

The corresponding LQP conductance is defined such that, in the weak-pulse and weak-coupling limit,

$H_1\to H_2\to H_1$3

The intended interpretation is a quantized conductance-like response whose value is fixed by the equilibrium winding number (Forcellini et al., 12 Aug 2025).

The protocol was also tested against noisy perturbations of the form

$H_1\to H_2\to H_1$4

To quantify symmetry breaking, the paper defines a chiral-symmetry-violation measure

$H_1\to H_2\to H_1$5

When chiral symmetry is preserved, $H_1\to H_2\to H_1$6. The numerical result reported is that $H_1\to H_2\to H_1$7 remains close to the winding number for weak noise, with stronger degradation when the noise directly points along $H_1\to H_2\to H_1$8, since that explicitly breaks chiral symmetry (Forcellini et al., 12 Aug 2025).

5. Scope and generalizations

The loop-quench construction was presented as more than a single model calculation. The central claims include that symmetry-protected topology can survive in a special quench protocol even though generic quenches destroy it, that loop quenches evade dynamical violation of the protecting symmetry by using a time-symmetric pulse, that the equilibrium topological invariant of $H_1\to H_2\to H_1$9 is encoded in the Loschmidt chirality amplitude, and that a suitable pump-probe