A new rung on the ladder: exploring topological frustration towards two dimensions

The gist: This work addresses topological frustration in two dimensions by studying a three-legged antiferromagnetic quantum Ising ladder on a torus using tensor network methods, revealing that topological frustration shifts the position of the quantum critical point and that the ground state can be characterized as hosting three delocalized quasiparticles.

Topological Frustration and its Manifestation

Topological frustration arises when boundary conditions impose geometric frustration in a quantum system, creating delocalized defects in the ground states and profoundly altering the low-energy properties The authors study this by inducing topological frustration by an odd number of spins along both spatial directions on a three-legged ladder using tensor network methods This geometry induces an extensive amount of frustration with respect to the total system size, which is not the case for a chain or a fully two-dimensional system

Quasiparticle Picture in 1D Systems

The simplest systems that allow us to understand why topological frustration induces quasiparticle excitations in spin systems are one-dimensional quantum spin chains with nearest neighbors interactions In the context of a fully anisotropic Heisenberg chain, applying frustrated boundary conditions induces topological frustration At the classical point h = Jy = Jz = 0, the ground state manifold contains 2L degenerate classical configurations corresponding to L possible locations of the ferromagnetic defect Adding a small perturbation lifts this degeneracy and results in a state where the ferromagnetic defect is completely delocalized along the chain, which corresponds to the injection of a quasiparticle excitation The entanglement entropy can be expressed as S(x) = S QP (x) + Sloc(h, J⃗) + log2 2 in the thermodynamic limit.

Topological Frustrated Ising Model on a Ladder

The study focuses on a quantum Ising model in a transverse field on an L×3 ladder with periodic boundary conditions along both directions, i.e., a thin torus Topological frustration is induced as a combination of odd lattice sites number (L = 2m + 1 and m ∈ N), periodic boundary conditions (PBC), and antiferromagnetic (AFM) interactions Numerical studies using the density matrix renormalization group (DMRG) algorithm show that the presence of TF modifies the phase diagram by shifting the critical point position The unfrustrated system exhibits a quantum phase transition between an ordered phase and a disordered phase when h/J ≈ 3.04438.

Entanglement Entropy and Quasiparticle Characterization

The bipartite entanglement entropy of the frustrated case differs from the non-frustrated case, as the unfrustrated system obeys an area law while the states in the frustrated case are highly entangled The quantum phase transition in a frustrated system manifests as a peak in the first derivative of entanglement entropy (Fig. 2d) rather than a peak in entanglement entropy itself The numerical results are well reproduced by the ansatz S(x, h, J⃗) = S QP (x) + Sloc(h, J⃗) + c. The term b(x), which is the entanglement entropy of the ground state of a system of two free bosons hopping on a ring of L sites, arises because the reduced density matrix possesses only three nonzero eigenvalues.

Critical Point Shift and Scaling Behavior

The authors observe a clear shift of the quantum critical point position hc in the presence of TF when switching from AFM to FM interactions This shift is attributed to the fact that for an L×3 system, the number of ferromagnetic defects, L + 3, is comparable to the total number of spins 3L. The scaling exponent α approaches 2 in the limit of small h in the topologically frustrated phase, i.e., h << hc region. Furthermore, the energy gap ∆E at h = 0.2 as a function of lattice side size L is plotted with a fitted scaling exponent α = 1.976. The compatibility of this scaling with the 1D topologically frustrated Ising chain suggests that the energy gap scaling is a direct consequence of the single-particle dispersion relation and the quantization of lattice momenta.

Conclusion on Quasiparticle Interpretation

The analysis confirms that the entanglement entropy scaling is consistent with the presence of three bosonic quasiparticles in the ground state of the system The constant c turns out to be different between the frustrated and non-frustrated case due to a different degeneracy of the entanglement spectrum. The results suggest that topological frustration has a quantitative effect on the phase diagram, shifting the critical point relative to its unfrustrated counterpart. This work provides a concrete starting point for future studies on topological frustration in fully two-dimensional models.

Data and Code Availability

The datasets are available upon the reasonable request to the authors The simulations were partially performed using the open source Quantum TEA library. The work acknowledges computational resources from Cineca on the Leonardo machine.

Acknowledgments

The authors thank Philippe Lecheminant, Federico Becca and Siddhartha Lal for their insights on the general behavior of quantum spin ladders The research leading to these results has received funding from various organizations including European Union via UNIPhD programme (Horizon 2020 under Marie Skłodowska-Curie grant agreement No. 101034319 and NextGenerationEU) The Italian Ministry of University and Research (MUR) provided funding through Quantum Frontiers (the Departments of Excellence 2023-2027). The work also acknowledges support from the Croatian Science Foundation (HRZZ) Projects DOK-2020-01-9938.

Appendix A Summary

A ground state can be seen as an Ly×Lx matrix whose rows is made by kink states, i.e., antiferromagnetically aligned states with one ferromagnetic defect at some lattice site The number of broken bonds between the two layers along the vertical direction must be equal to Lx, denoted as b1 = Lx. The total number of ground states will be 4Lx / (Lx + Ly - 1) / (2LxLy).

Appendix B Summary

In Fig. 3, the fitting coefficients for log(hc − h∗c) = a log(L) + b are given as a = -1.740 ± 0.009 and b = 0.96 ± 0.01 for FM. In Fig. 6, the fitting coefficients for log(1 − R) = a log(3L) + b are a = -0.902 ± 0.005 and b = 1.556 ± 0.024 for x = 1/2.

Appendix C Summary

The ansatz that best reproduces the numerical results is the bosonic one presented in Eq. (8). The comparison shows that the ansatz for three bosons best reproduces our results.

Appendix D Summary

To find the ground states in the matrix product state (MPS) form, a variational DMRG algorithm is run The simulations are performed by grouping the three spins on the same rung into a single site of larger local dimension d = 8. The ground states for different external fields h are found sequentially using the obtained state ΨMPS(h) as an initial guess for optimizing ΨMPS(h−∆h).

1. This study provides a quasiparticle picture of topological frustration, which can be used to develop novel error correction codes for quantum computation by characterizing delocalized excitations in frustrated ground states. The improved AI system can accurately predict the low-energy properties and entanglement structure of complex quantum many-body systems exhibiting topological features.

2. The understanding that the bipartite entanglement entropy of this system is compatible with the presence of three bosonic excitations in the ground state allows for designing more efficient quantum simulation algorithms capable of efficiently calculating entanglement in quasi-2D systems. This enables the AI to perform more accurate simulations for materials science problems involving strongly correlated electrons or spin systems.

3. The numerical analysis shows that the presence of TF modifies the phase diagram of the model by shifting the critical point position, which can be leveraged to create machine learning models that predict phase transitions in complex quantum materials based on topological invariants and boundary conditions. This allows for rapid screening of material candidates for desired quantum phases.

4. The paper's construction rules for ground states, such as the sum of the indices of the translation operators appearing in the product must be equal to an integer multiple of Lx, can inspire new variational methods within AI systems to search efficiently through large, degenerate Hilbert spaces characteristic of topologically frustrated systems. This improves the speed and accuracy of finding ground states for quantum optimization problems.