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Chinese Physics B

Quantum phase transitions with eigen microstate approach in one-dimensional transverse-field Ising model

Z. B. Su, Yuan Jiang, Gaoke Hu, Yuehua Su, Liangsheng Li, Wen‐Long You, Maoxin Liu, Xiaosong Chen

Published Apr 28, 2025• 0 citations
Publisher

Abstract

Abstract We propose an eigen microstate approach (EMA) for analyzing quantum phase transitions in quantum many-body systems, introducing a novel framework that does not require prior knowledge of an order parameter. Using the transverse-field Ising model (TFIM) as a case study, we demonstrate the effectiveness of EMA by identifying key features of the phase transition through the scaling behavior of eigenvalues and the structure of associated eigen microstates. Our results reveal substantial changes in the ground state of the TFIM as it undergoes a phase transition, as reflected in the behavior of specific components ξ i ( k ) within the eigen microstates. This method is expected to be applicable to other quantum systems where predefining an order parameter is challenging.

DOI
10.1088/1674-1056/add1c0

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Abstract We propose an eigen microstate approach (EMA) for analyzing quantum phase transitions in quantum many-body systems, introducing a novel framework that does not require prior knowledge of an order parameter. Using the transverse-field Ising model (TFIM) as a case study, we demonstrate the effectiveness of EMA by identifying key features of the phase transition through the scaling behavior of eigenvalues and the structure of associated eigen microstates. Our results reveal substantial changes in the ground state of the TFIM as it undergoes a phase transition, as reflected in the behavior of specific components ξ i ( k ) within the eigen microstates. This method is expected to be applicable to other quantum systems where predefining an order parameter is challenging.
We study the one-dimensional quantum Ising model in a transverse magnetic field where the exchange couplings are ordered according to the Thue-Morse (TM) sequence. At zero temperature, this model is equivalent to a two-dimensional classical Ising model in a magnetic field with TM aperiodicity along one direction. We compute the order parameter (magnetization) of the chain and the scaling behavior of the energy spectrum when the system undergoes a phase transition. Analogous to the quasiperiodic (QP) quantum Ising chain, the onset of long-range order is signaled by a nonanaliticity in the exponent \ensuremath{\delta} which describes the scaling of the total bandwidth with the size of the chain. The critical spin-coupling can be computed analytically and it is found to be lower than the QP case. Furthermore, the energy bands are found to be narrower than the corresponding QP chain. The former and latter results are consistent with the fact that the present structure has a degree of ordering intermediate between QP and random.
The Transverse Field Ising Model (TFIM) is an important quantum many-body model used to study the competition between spin-spin interactions and quantum fluctuations generated by a transverse field. Due to the exponential growth of the Hilbert space with increasing system size, numerical methods quickly become computationally infeasible motivating the use of approximate approaches. In this work Neural Quantum States were used as a variational method to approximate the ground-state properties of the one-dimensional TFIM. The NQS was represented using a feedforward Neural Network and optimized through Monte Carlo sampling together with stochastic reconfiguration. The implementation was done through the NetKet framework. The obtained results showed stable convergence of the variational energy and good agreement with exact diagonalization for the smaller system. The quantum phase transition was observed through the order parameter and spin-spin correlations. Finite-size effects were also observed when comparing the two system sizes. The smaller system more closely approaches the symmetric cat-like ground-state. In contrast, the larger system tends to remain near one symmetry-broken configuration. As a result, the smaller system showed stronger spin expectation fluctuations and a smoother behavior near the phase transition. The result demonstrates that the NQS approach is a good approximate method in studying the ground-state physics of the one-dimensional TFIM.
The one-dimensional transverse field Ising model with boundary fields is studied analytically and numerically. The phase diagram in the ordered state is obtained. We find that there exist two types of phase transitions. The longitudinal boundary fields applied to the left and right ends of the Ising chain are hL{h}_{L}hL​ and hR{h}_{R}hR​, respectively. For $|{h}_{L}|,|{h}_{R}|<\sqrt{1\ensuremath{-}g}$, where ggg is the transverse field and hLhR<0{h}_{L}{h}_{R}<0hL​hR​<0, a first-order phase transition occurs with the changing hL{h}_{L}hL​ or hR{h}_{R}hR​. The energy gap and boundary magnetization are solved exactly. The analytical expressions of the finite-size scaling for the first-order phase transition are obtained. For $|{h}_{R}|>\sqrt{1\ensuremath{-}g}$ and hLhR<0{h}_{L}{h}_{R}<0hL​hR​<0, there exists a continuous phase transition with changing hL{h}_{L}hL​ and vice versa. This transition is identified as a quantum wetting transition. The singularity of the boundary magnetization in this phase transition is explicitly shown. A simple computational procedure with high accuracy and efficiency is proposed to calculate the magnetization. The magnetization profile, correlation functions, and wetting layer thickness are studied numerically.
In a many-body quantum system, the interplay of various energies and entropies leads to different phases. The system undergoes a phase transition in the case that certain feature becomes dominant over all the others by varying either adjustable interaction constants or applied external fields in the Hamiltonian. In this work, we investigate the quantum phase transition from the perspective of coherence through a significant model. It is shown that when an extra two-level system is injected into the transverse field Ising model, its coherence will suffer an explicit sudden change in dynamics around the phase transition point of the surrounding model. With such dynamic sensitivity, coherence can be exploited to characterize quantum phase transitions and reveal the location of critical point.
In this thesis I have successfully recreated the results that describe the behavior of a transverse-field Ising chain. There is a phase transition when the parameter that controls the influence of the transverse field is one (that is: the factor in front of the transverse-field term and the interaction term are equally big). However, I could not determine exactly what the order parameter should be.\nThen for the transverse-field Ising ladder I have used perturbation theory in the parameter that controls the interaction between the chains. I have found that the phase transition doesn't change position, but that the ground state is a different state compared to the transverse-field Ising chain, when the transverse-field term is smaller than the interaction.
Identifying phases of matter presents considerable challenges, particularly within the domain of quantum theory, where the complexity of ground states appears to increase exponentially with system size. Quantum many-body systems exhibit an array of complex entanglement structures spanning distinct phases. Although extensive research has explored the relationship between quantum phase transitions and quantum entanglement, establishing a direct, pragmatic connection between them remains a critical challenge. In this work, we present a novel and efficient quantum phase transition classifier, utilizing disentanglement with reinforcement learning-optimized variational quantum circuits. We demonstrate the effectiveness of this method on quantum phase transitions in the transverse field Ising model (TFIM) and the XXZ model. Moreover, we observe the algorithm's ability to learn the Kramers-Wannier duality pertaining to entanglement structures in the TFIM. Our approach not only identifies phase transitions based on the performance of the disentangling circuits but also exhibits impressive scalability, facilitating its application in larger and more complex quantum systems. This study sheds light on the characterization of quantum phases through the entanglement structures inherent in quantum many-body systems.