Discovering Order Parameters for Quantum Many-Body Systems Using Reduced Fidelity Susceptibility

Core Concepts

This paper introduces a novel method for identifying quantum phase transitions and discovering order parameters in quantum many-body systems using the reduced fidelity susceptibility (RFS) vector field.

Abstract

• Bibliographic Information: Mariella, N., Murphy, T., Di Marcantonio, F., Najafi, K., Vallecorsa, S., Zhuk, S., & Rico, E. (2024). Order Parameter Discovery for Quantum Many-Body Systems. arXiv:2408.01400v3 [quant-ph].

• Research Objective: To develop a new approach for constructing phase diagrams and discovering order parameters in quantum many-body systems where conventional methods are challenging.

• Methodology: The authors introduce the concept of a reduced fidelity susceptibility (RFS) vector field, derived from the fidelity between reduced density matrices of the ground state at neighboring points in parameter space. By analyzing the behavior of this vector field, specifically the presence of sources and sinks, the authors identify phase transitions. Furthermore, they formulate an optimization problem based on the RFS vector field to discover observables that serve as order parameters for the identified transitions. The method is demonstrated on three well-established models: the Axial Next Nearest Neighbor Interaction (ANNNI) model, a cluster state model, and a chain of Rydberg atoms.

• Key Findings:

  • The RFS vector field effectively identifies phase transitions in all three models, accurately reproducing known phase diagrams.
  • The optimization problem successfully discovers order parameters that capture the different phases in each model.
  • The method reveals intricate details in the phase diagrams, such as the structure of the floating phase in the ANNNI model.
  • Finite-size scaling analysis confirms the validity of the discovered order parameters.

• Main Conclusions: The RFS vector field provides a powerful and versatile tool for exploring quantum phases and discovering order parameters in complex quantum systems. This method offers advantages over traditional approaches, particularly in situations where defining order parameters is not straightforward.

• Significance: This research contributes significantly to the field of quantum many-body physics by providing a new framework for understanding and characterizing quantum phase transitions. The ability to discover order parameters has important implications for the development of quantum simulation platforms and the exploration of novel quantum phases.

• Limitations and Future Research: While the method proves successful for the tested models, further investigation is needed to assess its performance on a wider range of quantum systems. Future research could explore the application of this technique to higher-dimensional systems and systems with more complex interactions. Additionally, investigating the scalability of the method for larger system sizes is crucial for its practical implementation in studying real-world quantum materials.

Stats

The 1D quantum Ising model has a critical exponent ν = 1.
For the Ising universality class the expected value of β = 1/8.
The irrelevant parameter exponent θ = 6.8 × 10−5.

Quotes

"In this paper, we extend the use of the fidelity susceptibility from previous studies (e.g. [14, 15]) and construct a vector field using a reduced fidelity susceptibility (RFS)."
"In summary, our mathematical framework establishes the phases diagram and enables the discovery of the order parameters for a Hamiltonian model, facilitating the understanding of different phases."
"We stress that our method can provide efficient certification and verification processes for quantum simulations and phase transition detection [18] and circumvents the need for full tomography of the wave function, relying instead on reduced density matrix to capture the relevant characteristic thermodynamic information."

Key Insights Distilled From

Order Parameter Discovery for Quantum Many-Body Systems

by Nicola Marie... at arxiv.org 11-19-2024

https://arxiv.org/pdf/2408.01400.pdf

Deeper Inquiries

How might this method be adapted to study open quantum systems, where interactions with the environment are non-negligible?

Adapting the Reduced Fidelity Susceptibility (RFS) vector field method to open quantum systems presents a significant challenge due to the inherent complexity introduced by interactions with the environment. Here's a breakdown of the challenges and potential adaptations:
Challenges:

Mixed States: Open quantum systems exist in mixed states, described by density matrices, rather than pure states. The fidelity and fidelity susceptibility concepts need careful extension to accommodate mixed states.
Non-Unitary Dynamics: The evolution of open quantum systems is non-unitary, meaning the system's coherence can decay over time due to entanglement with the environment. This non-unitary evolution complicates the calculation of fidelity and its derivatives.
Environment Characterization: Accurately modeling the environment and its coupling to the system is crucial. Often, simplifying assumptions are made about the environment (e.g., Markovian dynamics), but these might not always hold.
Potential Adaptations:

Generalized Fidelity Measures: Explore generalizations of fidelity suitable for mixed states, such as the Uhlmann-Jozsa fidelity used in the paper. These measures can quantify the "closeness" of two density matrices.
Quantum Trajectories: Employ quantum trajectory methods to simulate the open system dynamics. These methods unravel the system's evolution into stochastic trajectories, allowing for the calculation of fidelity along each trajectory and subsequent averaging.
Effective Hamiltonians: In some cases, it might be possible to derive an effective Hamiltonian that captures the essential features of the system-environment interaction. This effective Hamiltonian can then be used within the existing RFS framework.
Focus on Steady States: Instead of ground states, shift the focus to analyzing the system's steady states, which are often of interest in open system dynamics. Adapt the RFS method to identify transitions between different steady states.
Key Considerations:

The computational cost of simulating open quantum systems is generally higher than for closed systems. Efficient numerical techniques will be crucial for practical implementations.
The interpretation of the RFS vector field in the context of open systems might differ from closed systems. Careful analysis is needed to understand how the environment influences the vector field's structure.

Could the reliance on computationally intensive methods like DMRG limit the applicability of this approach to larger systems, and are there alternative ways to determine ground states that could be explored?

Yes, the reliance on computationally intensive methods like Density Matrix Renormalization Group (DMRG) can indeed limit the applicability of the RFS vector field approach to larger systems, especially those with high entanglement.
Limitations of DMRG:

Entanglement Growth: DMRG's efficiency stems from its ability to represent low-entanglement states effectively. However, near critical points or in systems with long-range interactions, entanglement can grow rapidly, making DMRG computationally expensive or even intractable for large system sizes.
System Geometry: DMRG is most efficient for one-dimensional systems or systems with a moderate number of spatial dimensions. Its performance degrades for higher-dimensional systems.
Alternative Ground State Determination Methods:

Variational Methods:

Variational Quantum Eigensolver (VQE): A hybrid quantum-classical algorithm that uses a parameterized quantum circuit to approximate the ground state and a classical optimizer to minimize the energy. VQE is well-suited for near-term quantum computers.
Density Matrix Embedding Theory (DMET): A method that breaks down the system into smaller fragments and solves them self-consistently. DMET can handle larger systems than DMRG in some cases.

Tensor Network Methods:

Projected Entangled Pair States (PEPS): A generalization of DMRG to higher dimensions. PEPS can represent certain entangled states more efficiently than DMRG in higher dimensions.
Multi-scale Entanglement Renormalization Ansatz (MERA): A tensor network approach that captures entanglement at different length scales. MERA can be more efficient than DMRG for critical systems.

Quantum Monte Carlo (QMC): A family of stochastic methods that sample the system's wavefunction or density matrix. QMC can handle larger systems than DMRG, but it suffers from the sign problem for certain Hamiltonians.
Key Considerations:

Accuracy vs. Efficiency: The choice of method involves a trade-off between accuracy and computational efficiency. DMRG is highly accurate for systems with manageable entanglement, while alternative methods might sacrifice some accuracy for improved scalability.
Quantum Hardware Suitability: Some methods, like VQE, are specifically designed for near-term quantum computers, while others are purely classical. The choice depends on the available computational resources.

If we view the evolution of a quantum system as traversing a landscape of different phases, what insights does the RFS vector field offer about the "topology" of this landscape and the possible pathways between different phases?

The RFS vector field provides a compelling visualization of the "quantum phase landscape" and offers insights into its topology and the pathways a system might take during its evolution:
Topological Insights:

Sources as Phase Transitions: The emergence of sources in the RFS vector field directly corresponds to quantum phase transitions. These sources act as "peaks" or "hills" in the landscape, indicating points of instability where the system's properties change abruptly.
Sinks as Stable Phases: Sinks in the vector field represent regions of local minima in fidelity susceptibility, corresponding to stable quantum phases. These sinks are akin to "valleys" in the landscape, where the system tends to settle.
Flow Lines as Adiabatic Pathways: The flow lines of the vector field, tracing the direction of steepest descent in fidelity susceptibility, can be interpreted as potential adiabatic pathways for the system's evolution. By slowly changing the system's parameters along these flow lines, one could, in principle, guide the system from one phase to another while minimizing excitations.
Pathways and Dynamics:

Phase Transition Traversal: The RFS vector field can reveal the "easiest" or most "natural" paths for a system to undergo a phase transition. The system is more likely to follow paths along the flow lines, as these correspond to directions of lower fidelity susceptibility and hence less drastic changes in the system's state.
Critical Slowing Down: The density of flow lines near a critical point can provide information about the system's dynamics. A high density of flow lines converging towards a source suggests critical slowing down, where the system's relaxation time diverges near the phase transition.
Limitations:

Ground State Focus: The RFS vector field, as presented in the paper, primarily focuses on the system's ground state. It doesn't directly capture the full complexity of the system's evolution, which might involve excited states and non-adiabatic processes.
Parameter Space Dependence: The topology of the RFS vector field depends on the chosen parameter space. Exploring different parameter spaces can reveal different aspects of the phase landscape.
Future Directions:

Dynamical Extensions: Extend the RFS vector field concept to incorporate time evolution and explore how the landscape changes over time, especially in driven or out-of-equilibrium systems.
Excited State Analysis: Investigate how the RFS vector field can be used to study transitions between excited states and their role in the system's dynamics.
Experimental Verification: Design experiments to verify the predictions of the RFS vector field regarding adiabatic pathways and critical slowing down.