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Order Parameter Discovery for Quantum Many-Body Systems

Nicola Mariella, Tara Murphy, Francesco Di Marcantonio, Khadijeh Najafi, Sofia Vallecorsa, Sergiy Zhuk, Enrique Rico

TL;DR

The paper addresses identifying quantum phase transitions without relying on predefined order parameters by introducing a two-dimensional reduced fidelity susceptibility (RFS) vector field derived from bipartite ground-state reduced density matrices. It then formulates a QCQP to extract Hermitian observables that act as order parameters, enabling quantitative phase characterization. The approach yields accurate phase diagrams and recovered order parameters for the ANNNI model, the cluster Hamiltonian, and a Rydberg spin chain, with finite-size scaling confirming expected critical behavior in at least one case. By linking phase transitions to sources in the fidelity vector field and providing a tomography-free diagnostic, the method offers a versatile tool for exploring complex quantum phases, including potential extensions to topological orders through nonlinear reduced-state information. The framework also supports practical quantum hardware certification by reducing reliance on full wavefunction tomography.

Abstract

Quantum phase transitions reveal deep insights into the behavior of many-body quantum systems, but identifying these transitions without well-defined order parameters remains a significant challenge. In this work, we introduce a novel approach to constructing phase diagrams using the vector field of the reduced fidelity susceptibility (RFS). This method maps quantum phases and formulates an optimization problem to discover observables corresponding to order parameters. We demonstrate the effectiveness of our approach by applying it to well-established models, including the Axial Next Nearest Neighbour Interaction (ANNNI) model, a cluster state model, and a chain of Rydberg atoms. By analyzing observable decompositions into eigen-projectors and finite-size scaling, our method successfully identifies order parameters and characterizes quantum phase transitions with high precision. Our results provide a powerful tool for exploring quantum phases in systems where conventional order parameters are not readily available.

Order Parameter Discovery for Quantum Many-Body Systems

TL;DR

The paper addresses identifying quantum phase transitions without relying on predefined order parameters by introducing a two-dimensional reduced fidelity susceptibility (RFS) vector field derived from bipartite ground-state reduced density matrices. It then formulates a QCQP to extract Hermitian observables that act as order parameters, enabling quantitative phase characterization. The approach yields accurate phase diagrams and recovered order parameters for the ANNNI model, the cluster Hamiltonian, and a Rydberg spin chain, with finite-size scaling confirming expected critical behavior in at least one case. By linking phase transitions to sources in the fidelity vector field and providing a tomography-free diagnostic, the method offers a versatile tool for exploring complex quantum phases, including potential extensions to topological orders through nonlinear reduced-state information. The framework also supports practical quantum hardware certification by reducing reliance on full wavefunction tomography.

Abstract

Quantum phase transitions reveal deep insights into the behavior of many-body quantum systems, but identifying these transitions without well-defined order parameters remains a significant challenge. In this work, we introduce a novel approach to constructing phase diagrams using the vector field of the reduced fidelity susceptibility (RFS). This method maps quantum phases and formulates an optimization problem to discover observables corresponding to order parameters. We demonstrate the effectiveness of our approach by applying it to well-established models, including the Axial Next Nearest Neighbour Interaction (ANNNI) model, a cluster state model, and a chain of Rydberg atoms. By analyzing observable decompositions into eigen-projectors and finite-size scaling, our method successfully identifies order parameters and characterizes quantum phase transitions with high precision. Our results provide a powerful tool for exploring quantum phases in systems where conventional order parameters are not readily available.
Paper Structure (24 sections, 1 theorem, 69 equations, 17 figures)

This paper contains 24 sections, 1 theorem, 69 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Phase Diagram Construction
  4. Order parameter discovery
  5. Experiments
  6. The ANNNI model
  7. Order parameters discovery
  8. Finite Size Scaling
  9. The cluster Hamiltonian
  10. The Rydberg model spin chain
  11. Model description
  12. Phase diagram construction
  13. Order parameters
  14. Conclusion
  15. Acknowledgment
  16. ...and 9 more sections

Key Result

Theorem C.1

Let $\Xi: \mathcal{S}_{n} \to \mathcal{S}_{n}$ be the linear operator defined in eq:xi-op-def. Assume that for the density matrices $\rho_+, \rho_-$, defining $\Xi$, it holds that $\langle\widehat{\rho_+}, \widehat{\rho_-}\rangle\ne 1$. Then, is the reduced-SVD of $\Xi$, with non-zero singular values and unnormalized right $V_i$ and left $U_i$ singular vectors

Figures (17)

  • Figure 1: Visualisation of the reduced fidelity susceptibility (RFS) uses. Here, we show that it can be used for both phase diagram construction of quantum systems and order parameter discovery.
  • Figure 2: (a) Color-mapped visualization of the fidelity vector field angle $\theta(\boldsymbol{\lambda})$. (b) Quiver plot of the vector field $P(\boldsymbol{\lambda})$ at a phase transition line. We note that the angle (thus the associated color) drastically changes at the phase transition in (a), which corresponds to a source (arrows radiate outward) in the vector field in (b).
  • Figure 3: (a) Phase Diagram of one-dimensional ANNNI Model. Here the pink represents the Ferromagnetic Phase (FM), where all spins are aligned along the x direction. The grey area corresponds to the paramagnetic phase (PM), in which the magnetic field dominates and all spins align along the z direction. The antiphase (AP) corresponds to the green region where the ground state takes the form of a staggered magnetization pattern with period four. Both the PM and AP are separated by the floating phase (FP). Here the spin chain can be seen as a ladder of two spin chains as sketched in the cartoon spin configurations.
  • Figure 4: Phase Diagram obtained using the reduced fidelity susceptibility of the one-dimensional ANNNI Model. Here a chain length of $L = 50$ spin sites was used and a two-site RDM was used when calculating the gradient of the reduced fidelity susceptibility. (a) The angle of the vector field given in \ref{['eq:angle-grad']} is plotted (b) the vector field given in \ref{['eq:vfield-def']} is plotted.
  • Figure 5: Experiments for the order parameter discovery on the ANNNI model. Here a chain length of $L = 50$ spin sites was used and a two-site RDM was used when calculating the gradient of the reduced fidelity susceptibility. (a) Expectation of the optimal observable $M$ for the paramagnetic phase. The expectations for the projectors $M^{(1)}$ and $M^{(4)}$ of \ref{['eq:annni-optim-obs-eigd']}, are respectively in (c) and (b).
  • ...and 12 more figures

Theorems & Definitions (2)

  • Theorem C.1
  • proof