Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Observation of a marginal Fermi glass


A long-standing open problem in condensed-matter physics is whether or not a strongly disordered interacting insulator can be mapped to a system of effectively non-interacting localized excitations. Using terahertz two-dimensional coherent spectroscopy, we investigate this issue in phosphorus-doped silicon, a classic example of a correlated disordered electron system in three dimensions. Despite the intrinsically disordered nature of these materials, we observe coherent excitations and strong photon echoes that provide us with a powerful method for the study of their decay processes. We extract the energy relaxation and decoherence rates close to the metal–insulator transition. We observe that both rates are linear in excitation frequency with a slope close to unity. The energy relaxation timescale counterintuitively increases with increasing temperature, and the coherence relaxation timescale has little temperature dependence below 25 K, but increases as the material is doped towards the metal–insulator transition. Here we argue that these features imply that the system behaves as a well-isolated electronic system on the timescales of interest, and relaxation is controlled by electron–electron interactions. Our observations constitute a distinct phenomenology, driven by the interplay of strong disorder and strong electron–electron interactions, which we dub the marginal Fermi glass.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Prices vary by article type



Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Linear and nonlinear optical response of phosphorus-doped silicon.
Fig. 2: Two-dimensional terahertz spectra at different phosphorus concentrations.
Fig. 3: Frequency and doping dependence of the pump–probe and rephasing relaxation rates.
Fig. 4: Temperature dependence of the pump–probe and rephasing anti-diagonal spectra.

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.


  1. Nozieres, P. Theory of Quantum Liquids (CRC Press, 2018).

  2. Anderson, P. W. Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958).

    Article  ADS  Google Scholar 

  3. Abrahams, E. 50 Years of Anderson Localization (World Scientific, 2010).

  4. Anderson, P. W. The Fermi Glass: Theory and Experiment (World Scientific, 2004).

  5. Efros, A. L. & Shklovskii, B. I. Coulomb gap and low temperature conductivity of disordered systems. J. Phys. C 8, L49–L51 (1975).

    Article  ADS  Google Scholar 

  6. Fleishman, L. & Anderson, P. Interactions and the Anderson transition. Phys. Rev. B 21, 2366–2377 (1980).

    Article  ADS  Google Scholar 

  7. Freedman, R. & Hertz, J. Theory of a Fermi glass. Phys. Rev. B 15, 2384–2398 (1977).

    Article  ADS  Google Scholar 

  8. Gornyi, I. V., Mirlin, A. D. & Polyakov, D. G. Interacting electrons in disordered wires: Anderson localization and low-T transport. Phys. Rev. Lett. 95, 206603 (2005).

    Article  ADS  Google Scholar 

  9. Basko, D. M., Aleiner, I. L. & Altshuler, B. L. Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states. Ann. Phys. 321, 1126–1205 (2006).

    Article  ADS  MATH  Google Scholar 

  10. Nandkishore, R. & Huse, D. A. Many-body localization and thermalization in quantum statistical mechanics. Annu. Rev. Condens. Matter Phys. 6, 15–38 (2015).

    Article  ADS  Google Scholar 

  11. Abanin, D. A., Altman, E., Bloch, I. & Serbyn, M. Colloquium: many-body localization, thermalization and entanglement. Rev. Mod. Phys. 91, 021001 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  12. Burin, A. L., Kagan, Y., Maksimov, L. A. & Polishchuk, I. Y. Dephasing rate in dielectric glasses at ultralow temperatures. Phys. Rev. Lett. 80, 2945–2948 (1998).

    Article  ADS  Google Scholar 

  13. Burin, A. L. Energy delocalization in strongly disordered systems induced by the long-range many-body interaction. Preprint at (2006).

  14. Yao, N. Y. et al. Many-body localization in dipolar systems. Phys. Rev. Lett. 113, 243002 (2014).

    Article  ADS  Google Scholar 

  15. Gutman, D. B. et al. Energy transport in the Anderson insulator. Phys. Rev. B 93, 245427 (2016).

    Article  ADS  Google Scholar 

  16. Nandkishore, R. M. & Sondhi, S. L. Many-body localization with long-range interactions. Phys. Rev. X 7, 041021 (2017).

    Google Scholar 

  17. Woerner, M., Kuehn, W., Bowlan, P., Reimann, K. & Elsaesser, T. Ultrafast two-dimensional terahertz spectroscopy of elementary excitations in solids. New J. Phys. 15, 025039 (2013).

    Article  ADS  Google Scholar 

  18. Lu, J. et al. Coherent two-dimensional terahertz magnetic resonance spectroscopy of collective spin waves. Phys. Rev. Lett. 118, 207204 (2017).

    Article  ADS  Google Scholar 

  19. Wan, Y. & Armitage, N. Resolving continua of fractional excitations by spinon echo in THz 2D coherent spectroscopy. Phys. Rev. Lett. 122, 257401 (2019).

    Article  ADS  Google Scholar 

  20. Rosenbaum, T. et al. Metal–insulator transition in a doped semiconductor. Phys. Rev. B 27, 7509–7523 (1983).

    Article  ADS  Google Scholar 

  21. Paalanen, M., Rosenbaum, T., Thomas, G. & Bhatt, R. N. Critical scaling of the conductance in a disordered insulator. Phys. Rev. Lett. 51, 1896–1899 (1983).

    Article  ADS  Google Scholar 

  22. Helgren, E., Armitage, N. P. & Grüner, G. Frequency-dependent conductivity of electron glasses. Phys. Rev. B 69, 014201 (2004).

    Article  ADS  Google Scholar 

  23. Helgren, E., Armitage, N. P. & Grüner, G. Electrodynamics of a Coulomb glass in n-type silicon. Phys. Rev. Lett. 89, 246601 (2002).

    Article  ADS  Google Scholar 

  24. Lee, M. & Stutzmann, M. L. Microwave ac conductivity spectrum of a Coulomb glass. Phys. Rev. Lett. 87, 056402 (2001).

    Article  ADS  Google Scholar 

  25. Shklovskii, B. & Efros, A. Phononless hopping conduction in disordered systems. Zh. Eksp. Theor. Fiz. 81, 406–415 (1981).

    Google Scholar 

  26. Mukamel, S. Principles of Nonlinear Optical Spectroscopy (Oxford Univ. Press, 1995).

  27. Hamm, P. & Zanni, M. Concepts and Methods of 2D Infrared Spectroscopy (Cambridge Univ. Press, 2011).

  28. Aue, W., Bartholdi, E. & Ernst, R. R. Two-dimensional spectroscopy. Application to nuclear magnetic resonance. J. Chem. Phys. 64, 2229–2246 (1976).

    Article  ADS  Google Scholar 

  29. Cundiff, S. T. & Mukamel, S. Optical multidimensional coherent spectroscopy. Phys. Today 66, 44–49 (2013).

    Article  Google Scholar 

  30. Kuehn, W., Reimann, K., Woerner, M., Elsaesser, T. & Hey, R. Two-dimensional terahertz correlation spectra of electronic excitations in semiconductor quantum wells. J. Phys. Chem. B 115, 5448–5455 (2011).

    Article  Google Scholar 

  31. Lu, J. et al. Nonlinear two-dimensional terahertz photon echo and rotational spectroscopy in the gas phase. Proc. Natl Acad. Sci. USA 113, 11800–11805 (2016).

    Article  ADS  Google Scholar 

  32. Choi, W., Lee, K. H. & Kim, Y. B. Theory of two-dimensional nonlinear spectroscopy for the Kitaev spin liquid. Phys. Rev. Lett. 124, 117205 (2020).

    Article  ADS  Google Scholar 

  33. Tomakoff, A. Nonlinear and Two-Dimensional Spectroscopy (Univ. Chicago, 2011);

  34. Galperin, Y., Gurevich, V. & Parshin, D. in Hopping Transport in Solids (eds Pollak, M. & Shklovskii, B) Ch. 3, 81–123 (Modern Problems in Condensed Matter Sciences Vol. 28, Elsevier, 1991).

  35. Bhatt, R. N. & Lee, P. A. Scaling studies of highly disordered spin-1/2 antiferromagnetic systems. Phys. Rev. Lett. 48, 344–347 (1982).

    Article  ADS  Google Scholar 

  36. Fisher, M. P. A. & Zwerger, W. Quantum Brownian motion in a periodic potential. Phys. Rev. B 32, 6190–6206 (1985).

    Article  ADS  Google Scholar 

  37. Varma, C., Nussinov, Z. & van Saarloos, W. Singular or non-Fermi liquids. Phys. Rep. 361, 267–417 (2002).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Varma, C., Littlewood, P. B., Schmitt-Rink, S., Abrahams, E. & Ruckenstein, A. Phenomenology of the normal state of Cu–O high-temperature superconductors. Phys. Rev. Lett. 63, 1996–1999 (1989).

    Article  ADS  Google Scholar 

  39. Burin, A. & Kagan, Y. Low-energy collective excitations in glasses. New relaxation mechanism for ultralow temperatures. Zh. Eksp. Teor. Fiz. 106, 633–647 (1994).

    Google Scholar 

  40. Levitov, L. S. Absence of localization of vibrational modes due to dipole–dipole interaction. Europhys. Lett. 9, 83–86 (1989).

    Article  ADS  Google Scholar 

  41. Parameswaran, S. A. & Gopalakrishnan, S. Non-Fermi glasses: localized descendants of fractionalized metals. Phys. Rev. Lett. 119, 146601 (2017).

    Article  ADS  Google Scholar 

  42. Thorsmølle, V. & Armitage, N. Ultrafast (but many-body) relaxation in a low-density electron glass. Phys. Rev. Lett. 105, 086601 (2010).

    Article  ADS  Google Scholar 

  43. Greenland, P. et al. Coherent control of Rydberg states in silicon. Nature 465, 1057–1061 (2010).

    Article  ADS  Google Scholar 

  44. Lynch, S. A. et al. First observation of a THz photon echo. In 35th International Conference on Infrared, Millimeter and Terahertz Waves 1–2 (IEEE, 2010).

  45. Thurber, W. R., Mattis, R. L., Liu, Y. M. & Filliben, J. J. Resistivity-dopant density relationship for phosphorus-doped silicon. J. Electrochem. Soc. 127, 1807–1812 (1980).

    Article  ADS  Google Scholar 

Download references


We thank A. Burin, Y. Galperin, Y.-B. Kim, A. Legros, I. Martin, A. Millis, V. Oganesyan, S. Paramesweran, B. Shklovskii and Y. Yuan for helpful discussions. This project was supported by a now-cancelled DARPA DRINQS programme grant and by the Gordon and Betty Moore Foundation, EPiQS initiative, grant number GBMF-9454. S.G. acknowledges support from NSF grant no. DMR-1653271.

Author information

Authors and Affiliations



F.M. and D.C. built the 2D terahertz set-up and carried out experiments and analysis. S.G. and R.N. provided theoretical support. N.P.A. directed the project. All authors contributed to the writing and editing of the manuscript.

Corresponding authors

Correspondence to Fahad Mahmood or N. P. Armitage.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.

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

Supplementary information

Supplementary Information

Supplementary Sections I–IV and Figs. 1–10.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mahmood, F., Chaudhuri, D., Gopalakrishnan, S. et al. Observation of a marginal Fermi glass. Nat. Phys. 17, 627–631 (2021).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing