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# Isospin Pomeranchuk effect in twisted bilayer graphene

## Abstract

In condensed-matter systems, higher temperatures typically disfavour ordered phases, leading to an upper critical temperature for magnetism, superconductivity and other phenomena. An exception is the Pomeranchuk effect in 3He, in which the liquid ground state freezes upon increasing the temperature1, owing to the large entropy of the paramagnetic solid phase. Here we show that a similar mechanism describes the finite-temperature dynamics of spin and valley isospins in magic-angle twisted bilayer graphene2. Notably, a resistivity peak appears at high temperatures near a superlattice filling factor of −1, despite no signs of a commensurate correlated phase appearing in the low-temperature limit. Tilted-field magnetotransport and thermodynamic measurements of the in-plane magnetic moment show that the resistivity peak is connected to a finite-field magnetic phase transition3 at which the system develops finite isospin polarization. These data are suggestive of a Pomeranchuk-type mechanism, in which the entropy of disordered isospin moments in the ferromagnetic phase stabilizes the phase relative to an isospin-unpolarized Fermi liquid phase at higher temperatures. We find the entropy, in units of Boltzmann’s constant, to be of the order of unity per unit cell area, with a measurable fraction that is suppressed by an in-plane magnetic field consistent with a contribution from disordered spins. In contrast to 3He, however, no discontinuities are observed in the thermodynamic quantities across this transition. Our findings imply a small isospin stiffness4,5, with implications for the nature of finite-temperature electron transport6,7,8, as well as for the mechanisms underlying isospin ordering and superconductivity9,10 in twisted bilayer graphene and related systems.

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## Data availability

All data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. Source data are provided with this paper.

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## Acknowledgements

We acknowledge discussions with S. Kivelson, A. Macdonald, B. Spivak and M. Zaletel, as well as experimental assistance from H. Polshyn and C. Tschirhart. Y.S. acknowledges the support of the Elings Prize Fellowship from the California NanoSystems Institute at University of California, Santa Barbara. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan and the CREST (JPMJCR15F3), JST. Transport and fabrication experiments at UCSB were supported by the ARO under MURI W911NF-16-1-0361. Thermodynamic measurements were supported by the National Science Foundation under DMR-1654186. A.F.Y. acknowledges the support of the David and Lucille Packard Foundation under award 2016-65145. E.B. was supported by the European Research Council (ERC) under grant HQMAT (grant number 817799), and by the US–Israel Binational Science Foundation (BSF) under the NSF–BSF DMR programme (grant number DMR-1608055). X.L. and J.I.A.L. are supported by Brown University.

## Author information

Authors

### Contributions

Y.S., J.G. and X.L. fabricated tBLG devices. Y.S. and F.Y. performed the measurements. Y.S., F.Y. and A.F.Y. analysed the data. A.F.Y. and E.B. constructed the thermodynamic model. Y.S., F.Y., E.B. and A.F.Y. wrote the paper with input from J.I.A.L. T.T. and K.W. grew the hBN crystals.

### Corresponding author

Correspondence to Andrea F. Young.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks Xuan Gao, Emanuel Tutuc and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 tBLG devices.

a, b, Top, optical images of device 1 (a) and device 2 (b). Scale bars correspond to 5 μm. Bottom, schematic of the tBLG heterostructures for device 1 (a) and device 2 (b).

### Extended Data Fig. 2 Temperature dependence of the resistivity ρxx.

a, b, d, e, ρxx as a function of the nominal filling factor ν0 at various temperatures up to 30 K in device 1 (a, linear scale; b, logarithmic scale) and device 2 (d, linear scale; e, logarithmic scale). The traces in a, b are measured at T = 1.5, 5, 8, 12, 17, 22 and 30 K and the traces in d, e are measured at T = 1.7, 5, 10, 15, 20, 25 and 30 K. c, f, Two-dimensional map of ρxx as a function of ν0 and T in device 1 (c) and device 2 (f).

### Extended Data Fig. 3 Temperature dependence of the Hall density behaviour.

a, c, The Hall density νH expressed in electrons per superlattice unit cell as a function of ν0 up to 20 K at a fixed B = 0.5 T. The data of device 1 are measured with T = 0.5, 2.5, 4.5, 8 and 20 K (a) and the data of device 2 are measured with T = 1.7, 4.3, 6, 10 and 20 K (c). b, d, Subtracted Hall density νH − ν0 as a function of ν0 at each temperature in device 1 (b) and device 2 (d). Insets, d(νH − ν0)/dν0 as a function of ν0 at each temperature around ν0 = −1. d(νH − ν0)/dν0 is calculated from νH − ν0 using a 20-point moving average (b) and a 40-point moving average (d) in ν0.

### Extended Data Fig. 4 In-plane magnetic field dependence of the Hall density in device 1.

a, b, Hall density νH (a) and subtracted Hall density νH − ν0 (b) expressed in electrons per superlattice unit cell, and measured with Btot = 0.5, 3, 6, 9 and 12 T and fixed B = 0.5 T. c, νH − ν0 as a function of Btot and ν0 with the magnetic field applied at an angle θB = 20.5°, measured at a nominal T = 20 mK. Blue and pink circles correspond to the positions of peaks of ρxx and the points of maximum descent in νH − ν0, respectively, and denote phase boundaries between symmetry-breaking isospin ferromagnets (IF1, IF2 and IF3) and an isospin unpolarized state (IU).

### Extended Data Fig. 5 Landau fan diagram at the hole side in a tilted magnetic field in device 1.

a, ρxx as a function of ν0 and total magnetic field Btot oriented at an angle with respect to the plane θB of 4.1°(a), 9.6°(b) and 20.5°(c). df, Schematics of the Landau fan diagram based on ac, respectively. The numbered labels denote the Bloch band filling index, which encodes the number of electrons bound to each lattice unit cell.

### Extended Data Fig. 6 Thermodynamic measurements in device 2.

a, Chemical potential μ as a function of ν0 at T = 4.2 K and B = 0, 3, 6 and 9 T. b, μ as a function of ν0 at B = 0 T and T = 4.2, 12 and 20 K. c, Inverse compressibility dμ/dν as a function of ν0 at T = 4.2 K and B = 0, 3, 6 and 9 T. d, dμ/dν as a function of ν0 at B = 0 T and T = 4.2, 12 and 20 K. e, dμ/dB as a function of ν0 at T = 4.2 K, calculated from (μ(9 T) − μ(6 T))/3 T, (μ(6 T) − μ(3 T))/3 T and (μ(3 T) − μ(0 T))/3 T. f, dμ/dT as a function of ν0, calculated from (μ(12 K) − μ(4.2 K))/7.8 K at B = 0, 3 and 6 T.

### Extended Data Fig. 7 High-temperature transport in a tilted magnetic field in device 1.

ad, ρxx (top) and d(νH − ν0)/dν0 (bottom) as a function of ν0 at T = 5 K (a), 10 K (b), 15 K (c) and 20 K (d) at Btot = 3, 6, 9, 12 and 15 T, oriented at an angle of 9.1° relative to the plane. d(νH − ν0)/dν0 is calculated from νH − ν0 using a 20-point moving average in ν0.

### Extended Data Fig. 8 Temperature and in-plane magnetic field dependence of resistive peak around ν0 = +1 in device 2.

a, ρxx as a function of nominal filling factor ν0 around ν0 = +1 between 1.7 and 30 K in device 2. b, The ρxx peak position as a function of ν0 and T. c, ρxx as a function of nominal filling factor ν0 around ν0 = +1 at B = 0, 3, 6, 9 and 12 T. d, The ρxx peak position as a function of ν0 and B.

### Extended Data Fig. 9 Temperature-dependent chemical potential and resistance in device 2.

a, Chemical potential μ as a function of ν0 at 4.2, 8.0, 12, 16, 20, 26, 32, 40, 59, 76 and 96 K. dμ/dν0 in Fig. 4 is calculated by the derivative of these data. b, Inverse electronic compressibility dμ/dν as a function of ν0 at 4.2, 8, 12, 16, 20, 26, 32, 40, 59, 76 and 96 K. c, ρxx as a function of ν0 at 4.3, 8.0, 12, 16, 20, 26, 31, 40, 59, 76 and 96 K.

### Extended Data Fig. 10 Entropy change per superlattice unit cell ΔS/kB from the transport data.

White triangles and black circles are phase boundaries for the Zeeman-tuned transition ($${E}_{{\rm{Z}}}^{* }/{k}_{{\rm{B}}}$$) and temperature-tuned transition (T*), respectively, determined by the ρxx peak near ν = −1. The pink curve is Δs/kB as a function of ν0, determined by equation (7).

## Supplementary information

### Supplementary Information

This file contains Supplementary Information on error bar eestimation and Supplementary Figures 1-7.

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Saito, Y., Yang, F., Ge, J. et al. Isospin Pomeranchuk effect in twisted bilayer graphene. Nature 592, 220–224 (2021). https://doi.org/10.1038/s41586-021-03409-2

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