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Charting the space of ground states with tensor networks
by Marvin Qi, David T. Stephen, Xueda Wen, Daniel Spiegel, Markus J. Pflaum, Agnès Beaudry, Michael Hermele
Submission summary
Authors (as registered SciPost users):  Marvin Qi 
Submission information  

Preprint Link:  scipost_202306_00044v1 (pdf) 
Date submitted:  20230630 20:03 
Submitted by:  Qi, Marvin 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We employ matrix product states (MPS) and tensor networks to study topological properties of the space of ground states of gapped manybody systems. We focus on families of states in one spatial dimension, where each state can be represented as an injective MPS of finite bond dimension. Such states are shortrange entangled ground states of gapped local Hamiltonians. To such parametrized families over X we associate a gerbe, which generalizes the line bundle of ground states in zerodimensional families (i.e. in fewbody quantum mechanics). The nontriviality of the gerbe is measured by a class in H3(X, Z), which is believed to classify onedimensional parametrized systems. We show that when the gerbe is nontrivial, there is an obstruction to representing the family of ground states with an MPS tensor that is continuous everywhere on X. We illustrate our construction with two examples of nontrivial parametrized systems over X = S3 and X = RP2 ×S1. Finally, we sketch using tensor network methods how the construction extends to higher dimensional parametrized systems, with an example of a twodimensional parametrized system that gives rise to a nontrivial 2gerbe over X = S4.
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This paper studies topological properties of families Hamiltonians defined on onedimensional spin chains, where the family depends on some continuous parameter. A paradigmatic 0d example why this is interesting is the Chern number of a family of 0d Hamiltonians (describing bands of free fermions).
In the current work, the authors introduce a classification of such phases for families of 1d systems. A key tool are tensor network states, which are well established as tools to characterize *individual* 1D systems. In order to generalize this to a setting of families of 1D systems, the authors build upon and extend mathematical results from the representation theory of 1D tensor networks. From there, they can extract the necessary mathematical structure, arising from reparametrizations of the MPS family, which allows them to construct a gerbe associated to the corresponding family. This allows them to classify the phases of the parametrized 1D system through the classification of gerbes in terms of cohomology. Finally, the authors also discuss possible generalization to higher dimensions, which, due to the lack of similarly strong statements about 2D tensor networks, are more speculative or examplebased.
I think this work makes an important contribution to the classification of phases of parametrized systems using tensor networks, a topic which is both important and yet unexplored. This work is thus extremely timely, and opens up a new avenue in the field (together with Ref. 58 which appeared briefly before). It introduces new mathematical ideas which connect tensor networks and classification of phases in terms of cohomology, and should thus also form a starting point for relevant followup work. In addition, the paper is very well written, and reads very nicely, in particular given its heavy mathematical nature, which makes it possible to get an understanding of the results without having to dive into all details immediately.
Overall, I strongly recommend publication of the paper in its present form.
Very few minor corrections:
 eq. 95: There should be no physical indices on the rhs.
 pg 18, right col opposite of (95): The sentence "The semiinjective PEPS are also suited [...]" does not seem to make sense.
 6 lines above (100): Should there be a "that" between "such [that] the columns"?