Bloch oscillations and the lack of the decay of the false vacuum in a one-dimensional quantum spin chain

1) The manuscript address a timely and challenging topic which is the suppression of the false vacuum decay in many-body quantum systems displaying confinement of excitations.

2) The numerical analysis presented is detailed.

3) The manuscript is clearly written and it can be understood also by non-experts in the field.

1) A discussion and a quantification of the timescales required for the decay of the false vacuum is missing (see also the report).

The manuscript addresses the problem of the suppression of the false vacuum decay in the Ising model with both transverse $h$ and longitudinal $g$ field, which is a paradigmatic model displaying confinement of the quasi-particle excitations. The false vacuum decay consists in the transition of an unstable-false vacuum state to the stable-true vacuum by means of the nucleation of bubbles of the true vacuum inside the false one. This problem is timely and challenging as the false vacuum decay is a non-perturbative effect which is very hard to treat analytically and to access numerically due to the large timescales involved.

Within this framework, the analysis of the paper addresses the suppression of the false vacuum decay in quantum quenches from the false vacuum (ground state for $h>0$) to the anti-confining ($h<0$) regime of the quantum Ising chain. In particular, the false vacuum is observed not to decay for the time scales the authors numerically access. The system, instead, displays oscillations in the (suppressed) entanglement growth and in the (suppressed) correlation spreading . The oscillatory motion is ascribed to Bloch oscillations. These statements are supported by a detailed numerical analysis based on the iTEBD method. The analysis presented, in particular, clearly distinguishes the different physics of the oscillations happening in the confining regime ($h>0$-meson spectrum) from the one of the Bloch oscillations happening in the anti-confining regime ($h<0$ with frequency of oscillations $\propto h$) .

Based on these facts, I think the manuscript deserves publication in Scipost Physics. However, some aspects of the discussion must be improved . Once the authors thoroughly address the Requested changes below, I will be happy to recommend the manuscript for publication in Scipost Physics.

1) The authors do not discuss at all the relevant timescales of the false vacuum decay. I am aware of the fact that this is a very difficult problem and I do not demand the authors to find a formula for this aspect. However, the authors should mention and discuss the result of Phys. Rev. B 60, 14525, which gives an analytical prediction for the timescale $\tau$ of the decay of the false vacuum because of bubbles nucleation in the Ising chain as for $h \to 0^{-}$ and $g$ not too close to $1$. This is, to my knowledge, the only theoretical prediction for this challenging problem and it is therefore extremely relevant for the analysis of the paper. I therefore expect the authors to discuss the relation between such a result and the outcome of their numerical simulations. I would expect that the value of $\tau$ is far too large to be able to observe numerically the false vacuum decay for the values of the post-quench parameters $(g,h)$ used by the authors (when the formula for $\tau$ applies).


2) In the abstract and in the introduction-conclusions the claim is made that "the suppressed decay is a lattice effect". I find this claim in general not correct since the suppression of the false vacuum decay is present also in the continuum/field theory description. The mechanism underlying the decay of the false vacuum by bubbles formation is analogous to the Schwinger mechanism of QED [Phys. Rev. 82 (1951) 664], where particle/anti-particle pairs are produced out of the false vacuum from a quench of the background electric field. As a matter of fact, the decay rate of the false vacuum predicted by the Schwinger effect presents an exponential form similar to the one computed in the aforementioned Reference Phys. Rev. B 60, 14525. As a consequence I would not classify the suppression of the false vacuum decay as a lattice effect since it can be suppressed also in field theory. For the specific case of the lattice, the mechanism underlying the suppression of the false vacuum decay is given by the Bloch oscillations, but this does not imply that the same process cannot happen also in the continuum. The authors should therefore change the discussion in the abstract and in the other parts of the manuscript, such as the conclusions, accordingly.

3)The prediction of the authors for the spatial extention of the vacuum bubbles is based on the semiclassical two-fermion method, first proposed for the Ising chain in [Journal of Statistical Physics volume 131, pages917–939 (2008)]. The authors should cite this reference and accordingly discuss the physical meaning and the regime of parameters $(g,h)$ where this approximation is expected to work. This aspect must be clearly discussed before Eq.(2.7)

4) At the beginning of page 7 the authors say that "kink-antikink pairs collide once during every oscillation period". I think this is not true in general. If the initial length $r_0$ of the bubble is larger than the amplitude of the oscillations of the kink-antikink (set by $h$ and $g$) at the extremes of the bubble, then the extremes just perform independent Bloch oscillations.

Some minor points
5) On page 2 the authors should specify the boundary conditions used for the quantum Ising chain.

6) Regarding Eq.(2.2) for the entanglement entropy, the authors should specify where the bipartition is done.

7) Regarding the iTEBD simulations, the authors should mention the maximum bond dimension used in the simulations.

- Broad appeal
- Very accessible and easily readable

- The results lack novelty to some extent

The authors study the spreading of correlation and entanglement in quantum quenches in the one-dimensional Ising spin chain, in the presence of both a transverse and longitudinal field.
When considered in a pure transverse field, the model is equivalent to free fermions and thus easily solvable, but a weak longitudinal field dramatically changes the setup.
Indeed, excitations over the ferromagnetic phase (i.e. domain walls) experience a pairwise linear potential proportional to the longitudinal field and hence experience confinement.
In particular, the authors closely follow the investigation of [Nat. Phys. 13 (2017) 246-249], but with a longitudinal field of opposite sign. In this case, the system initially lies in the false vacuum of the confining theory and the quench excites bubbles of true vacuum in the false-vacuum sea. In this setup, domain walls feel a repulsive force and, in the case of a continuum theory, this would cause the domain walls to indefinitely drift far apart.
On the lattice, the finiteness of the kinetic energy changes this picture and the domain walls undergo Bloch oscillations, resulting in false vacuum bubbles with a finite maximum extent.

The study of confinement in condensed matter setups attracted a lot of interest in recent times, therefore I believe this work is timely and of broad appeal.
However, I have a main criticism concerning the originality of the results here presented. The authors often use adjectives like "quite counter-intuitively" and "unexpectedly" while referring to bubbles of finite length despite the presence of a repulsive interaction, but the crucial role of the lattice in inducing Bloch oscillations has already been pointed out in Refs. [34] and [38] of the manuscript.
In Ref. [34] a domain wall in the confining Ising has been studied: in a continuum theory, the domain wall would accelerate towards the false-vacuum region, but due to lattice-induced Bloch oscillations the domain wall oscillates around its original position.
Then, in Ref. [38] suppression of transport is studied in the confining Ising in the case of dilute domain walls. These domain walls are essentially experiencing Stark localization in a linear potential induced by the surrounding excitations, hence they undergo Bloch oscillations around their original position. Of course, this effect would be absent in a continuum theory.
The setup studied by the authors is different from that of the above references, nevertheless the general mechanism inducing finite-sized vacuum bubbles has already been presented there: I invite the authors to give proper credit to Refs. [34] and [38] and reconsider the choice of wording concerning the novelty of the phenomenon.
For this reason, I am not sure if the acceptance criteria of SciPost Physics are met, but I support publication in SciPost Core after my detailed comments have been properly addressed.

Detailed comments:

- pg 2, "The resulting confining force [41] inhibits the propagation of the domain walls to large distances, and prevents thermalization of the system within all time scales accessible to numerical simulations."

I think Ref. [38] should be properly mentioned here in connection with the absence of the Schwinger mechanism on very long time scales.

- Fig. 1.2.
The (approximately linear) entanglement growth signals the presence of excitations of finite velocity in the system. Two possible effects come to my mind. The first one is the Schwinger effect: new mesons are dynamically produced due to string breaking.
The second mechanism is: mesons are too dense and they collide during the expansion, causing scattering and becoming traveling excitations.
Can the authors comment on which of the two mechanisms (or maybe another one I could not think of) is responsible for the drift? I think that by reducing (in modulus) the longitudinal field, the meson becomes shallower (i.e. it favors the second mechanism) and string breaking becomes harder (i.e. it mitigates the first effect). Checking whether the linear growth increases or diminishes after this operation should help in characterizing the underlying mechanism.

- pg 7.
"Kink-antikink pairs collide once during every oscillation period. They could, in principle, annihilate into mesons then; however our numerics shows no such effect in the accessible time frame, which is consistent with recent findings [52] that the inelastic scattering is very ineffective"

I believe Ref. [38] should be mentioned here, in particular the result that the Schwinger effect takes place on exponentially long time scales.

- pg 7.
"This requires the longitudinal field to be larger than some minimum value, which can be computed as a function of g and turns out to be smaller than the fields used in our simulations."

I think it could be worth giving the typical value of $2\epsilon(\pi)\rho_{bubble }\chi $

- pg 7.
"The energy gap of these mesonic excitations is always larger than twice the kink gap, grows with increasing |h|, and can also be computed theoretically using a semi-classical method [53, 31]. For small quench fields, we do not observe frequencies corresponding to Bloch oscillations"

I am familiar with the method, but I think a short summary should be provided here as well. This will make the paper more self-contained and easily accessible for a broader audience.

- pg 7 (and below)
"For small quench fields, we do not observe frequencies corresponding to Bloch oscillations." and the following discussion.

I think the authors should be more specific in this discussion and use better wording. Indeed, in the limit of a weak transverse field the size of the bubble is governed by the ratio h/g. If h/g is large, the bubble is small. Maybe the authors should be more specific rather than writing "small quench field", since one can still have g and h both small, but h/g large (thus small bubbles). I invite the authors to clarify the quoted sentence and the following discussion.