# Topological aspects of worldline configurations in hardcore Bose-Hubbard models

### Submission summary

 As Contributors: Fabio Lingua Preprint link: scipost_202007_00013v1 Date submitted: 2020-07-10 02:00 Submitted by: Lingua, Fabio Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Quantum Physics Approaches: Theoretical, Computational

### Abstract

In this manuscript, we present a proposal to relate topological structure of worldline configurations to ground-states of hardcore Bose-Hubbard models. Configurations result from the path-integral formulation of the density matrix in the limit of zero temperature. We consider hard-core bosons for which configurations, i.e. collections of particle paths, can be seen as geometric braids with a certain topological structure. We show that certain properties of worldline configurations allow to differentiate ground-states of different models. Within the path-integral picture, worldline configurations can provide a visualization of the generation of quantum fluctuations, which essentially give rise to the entanglement of the ground-state. We propose that this approach may provide new insights in the study and deciphering of multipartite entanglement. By means of Monte Carlo calculations, we study checkerboard, stripe, valence-bond solids, $\mathbb{Z}_2$ topologically ordered spin liquid, and superfluid phase. We find that each ground-state is characterized by a certain `topological spectrum' which can be used to differentiate among different ground-states.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission scipost_202007_00013v1 on 10 July 2020

## Reports on this Submission

### Report

This manuscript reports on a proposal to relate ground state properties of hard-core boson models to the topological properties of their worldline configurations in an imaginary-time path-integral formulation. The authors define appropriate means to quantify and structure the topological properties of the worldlines in terms of the statistics of braiding numbers and finally present results from QMC simulations of various characteristic ground state phases that exhibit distinct fingerprints in the topological invariant histograms.

As a general approach, I find this approach interesting and it might have the potential to become a new pathway to identifying ground states from such an analysis also in more general cases. If that would be the case, I would indeed range this idea beyond a mere curiosity.

However, I find that the present paper does not allow to draw helpful conclusions regarding its more general applicability. For this to be the case, the authors must perform a more extended investigation with respect to (i) how well does the distinction of different states work if one approaches parameter regions near quantum phase transitions between different ground states, (ii) how do the invariant distributions look like for low but significantly finite temperatures. In short: how useful is this approach if one is not looking at parameter regimes in which the nature of the ground state readily follows from cartoon pictures.

In addition to the above general request, I also would ask the authors to improve on the following points:

-The second identity regarding the unions of configurations right before Eq. (6) is a purely set-theoretical identify, and thus holds true irrespectively of the fact that C_tau can be seen as the union of the C_phi. This should be clarified, as I find the current formulation confusing in that respect.

-The meaning of the bold numbers in the worldline configs. in Fig. 3 should be explained.

-The manuscript should be read by a native speaker, as there are still a few language issues with the current version.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

Author Fabio Lingua on 2020-10-15
(in reply to Report 1 on 2020-09-21)
Category:

REFEREE: "As a general approach, I find this approach interesting and it might have the potential to become a new pathway to identifying ground states from such an analysis also in more general cases. If that would be the case, I would indeed range this idea beyond a mere curiosity."
We thank the referee for showing interest in our work and recognizing its potential. Indeed, the main goal of the manuscript is presenting to the scientific community a novel approach which we believe may be useful in advancing our understanding of certain properties of many-body systems, in particular, multipartite entanglement. We hope our work will stimulate a fruitful conversation within the scientific community, a conversation that will contribute to this advancement.

REFEREE: "However, I find that the present paper does not allow to draw helpful conclusions regarding its more general applicability."
In the present paper, we introduce a novel approach and show its potential by calculating certain topological invariants of worldline configurations which can fingerprint different ground states. We hope that these results will convince the readers that the approach introduced has the potential of characterizing properties of many-body systems. At the same time, we would like to emphasize that more complex topological invariants needs to be introduced to fully achieve this characterization and its generalization to other ground states. In order to do so, algorithms which fully identify and characterize the braids present in configurations, and more complex topological invariants need to be introduced. These algorithms are non-trivial as it is well-known in knot theory and their development is ongoing work of the authors. Nonetheless, we believe that our current findings already show the potential of this approach as, for example, they are able to discern spatially short-ranged fluctuations typical of conventional solids like the checkerboard or stripe solids from more complex fluctuations presents in Z2 topological order and superfluid phase.

REFEREE: "...the authors must perform a more extended investigation with respect to (i) how well does the distinction of different states work if one approaches parameter regions near quantum phase transitions between different ground states,"
In our work, we have mainly focused on topological signatures deep in well-known quantum phases, for different system sizes. We have shown that it is indeed possible to differentiate the various ground states considered (in some cases the Monte Carlo simulations did require very long CPU time). The study of phase transitions along with other results has been addressed in [L. Shpani et. al. Phys. Rev. B 101, 195103 (2020)]. The current manuscript introduces a new approach and emphasizes its potential as a novel way to ultimately characterize ground states entanglement.

REFEREE: "(ii) how do the invariant distributions look like for low but significantly finite temperatures."
We plan to address the finite temperature case in the future, once further algorithms (as mentioned above) have been developed. This is because the finite temperature quantum state is mixed and as such we expect a more complex relationship between topological invariants and the mixture of thermal and quantum fluctuations. This relationship will be understood only once the tools/algorithms to further characterize braids have been developed.

REFEREE: "In short: how useful is this approach if one is not looking at parameter regimes in which the nature of the ground state readily follows from cartoon pictures."
The authors are not sure about which cartoon pictures the referee refers to. If the referee refers to the states in Appendix D, there, we use the simple pictures of checkerboard and stripe phases far away from the transition point to illustrate the lack of more complex braids in the configurations of these two ground states. Otherwise, if the referee is asking how our proposed approach can be used to probe phases with no known ground state structure, we answer as follows.
In the manuscript, we propose, and our results demonstrate, that topological properties of worldline configurations have the potential to be used as a novel way to characterize quantum phases. Based on our results, we believe that a more systematic study on the general relationship between topological properties of worldlines and entanglement in ground states can be developed and will be useful to probe known quantum phases and detect novel ones.
Some general characteristics of the relationship between topological properties of worldline configurations and ground state entanglement can already be inferred from the numerical results of this manuscript. For example, the superfluid phase is expected to be described by coherent state (which is a superposition of all Fock states). Likewise, we noticed that worldline configurations with all kinds of braiding events among worldlines participate in the path integral formulation of the ground state. For gapped insulating phases, instead, our results show that worldline configurations are characterized by simple intertwining among worldlines. This is no longer valid for ground state with long-range properties of entanglement. In this case, the topological structure of the configurations becomes more complex and further algorithms are needed in order to fully unveil the dominant topological structures. Based on these observations, we expect that the method proposed here will provide a new path to understand the entanglement for nontrivial quantum phases, and, more generally, the path to develop a more general description of quantum phases.

REFEREE: "In addition to the above general request, I also would ask the authors to improve on the following points:
-The second identity regarding the unions of configurations right before Eq. (6) is a purely set-theoretical identify, and thus holds true irrespectively of the fact that C_tau can be seen as the union of the C_phi. This should be clarified, as I find the current formulation confusing in that respect."
It is true, it is a general theoretical identity. Here, we use it to split the integral of equation (5) as sum of integrals (eq. 6). We will clarify it better in the next version of the manuscript.

REFEREE: "-The meaning of the bold numbers in the worldline configs. in Fig. 3 should be explained."