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Springer proceedings in physics

The Random Transverse Field Ising Ferromagnet: The Simplest Disordered Model with a Quantum Phase Transition

A. P. Young, Christian Pich

Published Jan 1, 1999• 1 citation
Publisher
DOI
10.1007/978-3-642-60095-1_8

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From Classical to Quantum: Extending Prometheus to Uncover Quantum Critical Behavior in Disordered Transverse Field Ising Chains

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Abstract We extend the Prometheus framework from classical to quantum phase transitions, demonstrat- ing unsupervised discovery of quantum critical phenomena in the disordered transverse field Ising model (DTFIM). Building upon our previous work on classical spin systems [1, 2], we develop a quantum-aware variational autoencoder (Q-VAE) architecture that operates directly on quantum ground state wavefunctions obtained via exact diagonalization. The Q-VAE learns latent represen- tations that capture the essential physics of the quantum phase transition between paramagnetic and ferromagnetic phases, achieving detection of the quantum critical point hc = 1.00 ± 0.02 in the clean limit, consistent with the theoretical value hc/J = 1. We introduce disorder through random transverse fields h i ∼ Uniform[h−W, h+W ] and investigate the disorder-driven infinite-randomness fixed point characteristic of the random TFIM. Our framework successfully identifies the shift in critical behavior with increasing disorder strength and extracts effective critical exponents through finite-size scaling analysis. We formulate and test the hypothesis that the Q-VAE latent space structure reflects the activated dynamical scaling ln ξ ∼ |h − hc| −ψ with ψ ≈ 0.5 expected at the infinite-randomness fixed point. The Prometheus quantum discovery pipeline provides a systematic approach to exploring quantum phase diagrams in systems where analytical solutions are unavail- able, with potential applications to quantum materials, cold atom systems, and quantum computing platforms.

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We present some exact results for the effect of disorder on the critical properties of an anisotropic XY\mathrm{XY}XY spin chain in a transverse field. The continuum limit of the corresponding fermion model is taken and in various cases results in a Dirac equation with a random mass. Exact analytic techniques can then be used to evaluate the density of states and the localization length. In the presence of disorder the ferromagnetic-paramagnetic or Ising transition of the model is in the same universality class as the random transverse field Ising model solved by Fisher using a real-space renormalization-group decimation technique (RSRGDT). If there is only randomness in the anisotropy of the magnetic exchange then the anisotropy transition (from a ferromagnet in the x direction to a ferromagnet in the y direction) is also in this universality class. However, if there is randomness in the isotropic part of the exchange or in the transverse field then in a nonzero transverse field the anisotropy transition is destroyed by the disorder. We show that in the Griffiths' phase near the Ising transition that the ground-state energy has an essential singularity. The results obtained for the dynamical critical exponent, typical correlation length, and for the temperature dependence of the specific heat near the Ising transition agree with the results of the RSRGDT and numerical work.
Quantum multicritical points (QMCPs) emerge at the junction of two or more quantum phase transitions due to the interplay of disparate fluctuations, leading to novel universality classes. While quantum critical points have been well characterized, our understanding of QMCPs is much more limited, even though they might be less elusive to study experimentally than quantum critical points. Here, we characterize the QMCP of an interacting heterogeneous quantum system in two and three dimensions, the ferromagnetic random transverse-field Ising model (RTIM). The QMCP of the RTIM emerges due to both geometric and quantum fluctuations, studied here numerically by the strong disorder renormalization group method. The QMCP of the RTIM is found to exhibit ultraslow, activated dynamic scaling, governed by an infinite disorder fixed point. This ensures that the obtained multicritical exponents tend to the exact values at large scales, while also being universal---i.e., independent of the form of disorder---providing a solid theoretical basis for future experiments.
Abstract We extend the Prometheus framework from classical to quantum phase transitions, demonstrat- ing unsupervised discovery of quantum critical phenomena in the disordered transverse field Ising model (DTFIM). Building upon our previous work on classical spin systems [1, 2], we develop a quantum-aware variational autoencoder (Q-VAE) architecture that operates directly on quantum ground state wavefunctions obtained via exact diagonalization. The Q-VAE learns latent represen- tations that capture the essential physics of the quantum phase transition between paramagnetic and ferromagnetic phases, achieving detection of the quantum critical point hc = 1.00 ± 0.02 in the clean limit, consistent with the theoretical value hc/J = 1. We introduce disorder through random transverse fields h i ∼ Uniform[h−W, h+W ] and investigate the disorder-driven infinite-randomness fixed point characteristic of the random TFIM. Our framework successfully identifies the shift in critical behavior with increasing disorder strength and extracts effective critical exponents through finite-size scaling analysis. We formulate and test the hypothesis that the Q-VAE latent space structure reflects the activated dynamical scaling ln ξ ∼ |h − hc| −ψ with ψ ≈ 0.5 expected at the infinite-randomness fixed point. The Prometheus quantum discovery pipeline provides a systematic approach to exploring quantum phase diagrams in systems where analytical solutions are unavail- able, with potential applications to quantum materials, cold atom systems, and quantum computing platforms.