Unsupervised Phase Discovery with Deep Anomaly Detection

Korbinian Kottmann, Patrick Huembeli, Maciej Lewenstein, and Antonio Acín

Phys. Rev. Lett. 125, 170603 – Published 21 October 2020

Abstract

We demonstrate how to explore phase diagrams with automated and unsupervised machine learning to find regions of interest for possible new phases. In contrast to supervised learning, where data is classified using predetermined labels, we here perform anomaly detection, where the task is to differentiate a normal dataset, composed of one or several classes, from anomalous data. As a paradigmatic example, we explore the phase diagram of the extended Bose Hubbard model in one dimension at exact integer filling and employ deep neural networks to determine the entire phase diagram in a completely unsupervised and automated fashion. As input data for learning, we first use the entanglement spectra and central tensors derived from tensor-networks algorithms for ground-state computation and later we extend our method and use experimentally accessible data such as low-order correlation functions as inputs. Our method allows us to reveal a phase-separated region between supersolid and superfluid parts with unexpected properties, which appears in the system in addition to the standard superfluid, Mott insulator, Haldane-insulating, and density wave phases.

Received 30 March 2020
Revised 22 July 2020
Accepted 24 September 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.170603

Physics Subject Headings (PhySH)

Biological neural networks Machine learning Phase diagrams Quantum entanglement Quantum phase transitions

Supersolids

Tensor network methods

Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Korbinian Kottmann ^1,*, Patrick Huembeli ¹, Maciej Lewenstein^1,2, and Antonio Acín ^1,2

¹ICFO—Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain
²ICREA, Pg. Llus Companys 23, 08010 Barcelona, Spain

^*Korbinian.Kottmann@gmail.com

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Issue

Vol. 125, Iss. 17 — 23 October 2020

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Images

Figure 1
Schematic one-dimensional convolutional neural network autoencoder with symmetric shortcut connections (SSC) that connect latent layers of the same dimension directly via addition, thereby improving the model performance [34]. Shaded boxes in the background indicate schematically the convolutional nature of the layers. We illustrate how the input data $x$ gets encoded into the lower dimensional latent vector $z$ and decoded again to $\bar{x}$ .
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Figure 2
Extended BH phase diagram with five distinct phases obtained by the correlators Eqs. (3)–(5). The dashed lines indicate the transition points observed from diverging correlation lengths between MI-HI-DW and nonzero $S$ in Eq. (6) between SF and $SF + SS$ .
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Figure 3
2D loss map of the AE after training near the parameter space origin (blue square frame). The insets (a),(b), and (c) show the loss along the dashed lines. Vertical green dashed line in inset (a) indicates critical $U_{c} = 3.33$ [42]. Vertical gray dashed lines in inset (b) and (c) are the transitions from Fig. 2. The phase boundaries are determined by a rise in loss [insets (a) and (c)]. The anomalous regions are already well separated by decreasing losses because of the critical behavior at the phase boundaries [inset (b)], which share similarities with the critical SF phase. Higher loss indicates that this region is more different from the training region in the blue square; lower loss indicates similarity.
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Figure 4
2D loss map of the AE after training in the two blue square frames in the SF and the MI phase. The insets (a) and (b) show the loss along the dashed lines. Instead of the ES, we use the physically accessible correlator $C_{SF}$ as input data. The HI is not recognized as this correlator does not contain information about the topological order of this phase.
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Figure 5
Transition from SF to phase separated $SF + SS$ at fixed $U = 0.5$ . The solid long-range order emerges while SF correlations sustain. Inset (a) shows the phase separation in the density $⟨ n_{j} ⟩$ for a state at $(U, V) = (0.5, 4)$ with $L_{\infty} = 200$ , $d = 6$ , and $χ_{\max} = 500$ . Inset (b) shows the power-law decay of $C_{SF} (0, j)$ in the solid part via the doubly logarithmic plot of every second value, confirming supersolidity.
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Figure 6
Finite size scaling of the vanishing second derivative of the ground state energy per site $E$ with respect to the filling $f$ . This shows that the system becomes mechanically unstable, leading to phase separation as depicted in Fig. 5 for $f = 1$ .
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