Gapped spinful phases obtained via Gutzwiller projections of Euler states

Abstract: Gutzwiller projections of non-interacting chiral topological phases are known to give rise to fractional, topologically ordered chiral phases. Here, we carry out a similar construction using two copies of non-interacting Euler insulators to produce a class of spinful interacting Euler models. To that end, we take advantage of the recently discovered exact representation of certain Euler insulators by a projected entangled pair state (PEPS) of bond dimension $D = 2$. The Gutzwiller projection can be implemented within the tensor network formalism, giving rise to a new PEPS of bond dimension $D = 4$. We, moreover, apply very recent state-of-the-art tensor network tools to evaluate these phases. In particular, we analyze its entanglement entropy scaling and find no topological correction to the area law, indicating that the state is not intrinsically topologically ordered. Its entanglement spectrum shows a clear cusp at momentum zero, similar to non-interacting Euler insulators, and the spectrum of the transfer operator indicates that the state is gapped, which could imply non-intrinsic topological features. Finally, the static structure factor displays Bragg peaks, indicating the simultaneous presence of local order.

• The paper introduces a novel method using Gutzwiller projections on Euler insulator PEPS to generate spinful models with a doubled bond dimension.
• It employs tensor network analysis to quantify entanglement entropy and spectrum, with linear scaling suggesting a lack of intrinsic topological order.
• Structure factor calculations reveal clear Bragg peaks, indicating localized order and correspondence with gapped local Hamiltonians.

Gapped Spinful Phases from Gutzwiller Projections of Euler States

Introduction

This paper explores the construction of spinful interacting Euler models derived from non-interacting Euler insulators, employing Gutzwiller projections. It leverages an exact representation of certain Euler insulators as PEPS with bond dimension $D = 2$. This technique transforms the state, effectively doubling the bond dimension to $D = 4$. Utilizing tensor network formalism, the authors evaluate these phases, focusing on their entanglement entropy and spectrum, and the structure factor to assess order and topological features.

Construction of PEPS for Euler States

The construction begins with spinless fermions configured in a kagome lattice encapsulated in PEPS form, which uses virtual fermions in $W$ states. Mapping these virtual components to physical entities involves localized mappings via $\hat{M}_j$. The application of Gutzwiller projection on two such PEPS copies results in a new spinful PEPS state, as articulated:

Figure 1: Construction of the exact free fermionic Euler state [source].

Following this formulation, two copies of the stated $\hat{M}_j$ mappings undergo Gutzwiller projections to form spin systems. This conversion uses tensor network methodology, adapting calculations for PEPS tensor assessments, heading towards spin model implementations:

Figure 2: Construction of the PEPS tensor $T_G$ of $|\psi_{\text{spin}\rangle. Obtaining a square lattice PEPS.</p></p> <h3 class='paper-heading' id='results-analysis'>Results Analysis</h3> <p>Key metrics analyzed include entanglement entropy and spectrum. Calculations on cylinder geometries reveal pertinent traits of the engaged system. Specifically, the entanglement entropy&#39;s linear scaling indicates negligible topological correction, suggesting the absence of intrinsic topological order.</p> <p>Another critical aspect explored is the entanglement spectrum on cylindrical configurations. With visible spectrum cusps at momentum $K = 0$$, these characteristics affirm notable interactions that resemble non-interacting Euler insulators: <img src="https://emergentmind-storage-cdn-c7atfsgud9cecchk.z01.azurefd.net/paper-images/2508-10957/entanglement_entropy.png" alt="Figure 3" title="" class="markdown-image" loading="lazy"> <p class="figure-caption">Figure 3: Entanglement entropy as a function of cylinder circumference (red dots). The dashed line indicates a linear fit, demonstrating the absence of topological corrections.</p> <img src="https://emergentmind-storage-cdn-c7atfsgud9cecchk.z01.azurefd.net/paper-images/2508-10957/entanglement_spectrum.png" alt="Figure 4" title="" class="markdown-image" loading="lazy"> <p class="figure-caption">Figure 4: Entanglement spectrum for a cylinder circumference$$N_y = 7$$. K denotes the many-body momentum and entanglement energies.</p></p> <h3 class='paper-heading' id='transfer-operator-gap-analysis'>Transfer Operator Gap Analysis</h3> <p>Another pivotal exploration addresses the gap found in the transfer operator on the cylinder. The <a href="https://www.emergentmind.com/topics/transfer-gap" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">transfer gap</a> implies potential correspondence with gapped local Hamiltonians. Compensation for even-odd fluctuations in computations depicts consistency with a finite gap in simulated thermodynamic limits. <img src="https://emergentmind-storage-cdn-c7atfsgud9cecchk.z01.azurefd.net/paper-images/2508-10957/transfer_gap.png" alt="Figure 5" title="" class="markdown-image" loading="lazy"></p> <p><img src="https://emergentmind-storage-cdn-c7atfsgud9cecchk.z01.azurefd.net/paper-images/2508-10957/transfer_gap_peps.png" alt="Figure 5" title="" class="markdown-image" loading="lazy"> <p class="figure-caption">Figure 5: Gap of the transfer operator as a function of the inverse cylinder circumference$$N_y.

Structure Factor and Local Order

The structure factor calculations further assess local order presence. Results exhibit clear Bragg peaks, demonstrating localized order within the constructed states. With configurations projected upon varying cylinder dimensions, the implications reaffirm conditional stability across geometrically varied implementations:

Figure 6: Magnitude of the structure factor |S(k)| manifesting local order presence with distinct Bragg peaks.

Conclusion

The paper successfully highlights that Gutzwiller projections on Euler insulators yield spinful models with particular entanglement signatures, analogous to non-interacting systems, without intrinsic topological order. However, these states maintain local order, reflected in calculated structure factors. This establishes foundational advancements in understanding spinful representations within Euler phase frameworks, steering further studies to discern interactions across dynamic settings.

The research provides an overview of potential symmetry and topological implications of emergent phase states, opening avenues for more extended exploration within such intricate intensive physics formulations.