# How to Recognize the Universal Aspects of Mott Criticality?

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. In Search of Mott Criticality

## 3. Experiments

#### 3.1. Dilute 2DEG in Semiconductors

#### 3.2. Organic Compounds

#### 3.3. Moiré Materials

#### 3.4. Universal Criticality

## 4. Competing Theoretical Pictures

#### 4.1. Spin Liquid Picture of the Mott Point

#### 4.2. Dynamical Mean Field Theory Picture of the Mott Point

#### 4.3. Percolative Phase Coexistence Picture

## 5. Interpreting Resistivity Maxima

#### 5.1. Resistivity Maxima from Thermally Destroying Coherent Quasiparticles

#### 5.2. Percolation Scenario Due to Phase Coexistence

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Phillips, P. Mottness. Ann. Phys.
**2006**, 321, 1634–1650. [Google Scholar] [CrossRef] - Mott, N. Metal-Insulator Transitions; Taylor & Francis: Abingdon, UK, 1990. [Google Scholar]
- Imada, M.; Fujimori, A.; Tokura, Y. Metal-insulator transitions. Rev. Mod. Phys.
**1998**, 70, 1039–1263. [Google Scholar] [CrossRef] [Green Version] - Dobrosavljević, V.; Trivedi, N.; Valles Jr, J.M. Conductor Insulator Quantum Phase Transitions; Oxford University Press: Oxford, UK, 2012. [Google Scholar]
- Cao, Y.; Fatemi, V.; Demir, A.; Fang, S.; Tomarken, S.L.; Luo, J.Y.; Sanchez-Yamagishi, J.D.; Watanabe, K.; Taniguchi, T.; Kaxiras, E.; et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature
**2018**, 556, 80–84. [Google Scholar] [CrossRef] [PubMed] - Szentpéteri, B.; Rickhaus, P.; Vries, F.K.d.; Márffy, A.; Fülöp, B.; Tóvári, E.; Watanabe, K.; Taniguchi, T.; Kormányos, A.; Csonka, S.; et al. Tailoring the Band Structure of Twisted Double Bilayer Graphene with Pressure. Nano Lett.
**2021**, 21, 8777–8784. [Google Scholar] [CrossRef] [PubMed] - Mooij, J.H. Electrical conduction in concentrated disordered transition metal alloys. Phys. Status Solidi A
**1973**, 17, 521–530. [Google Scholar] [CrossRef] - Ciuchi, S.; Sante, D.D.; Dobrosavljević, V.; Fratini, S. The origin of Mooij correlations in disordered metals. NPJ Quantum Mater.
**2018**, 3, 1–6. [Google Scholar] [CrossRef] - Evers, F.; Mirlin, A.D. Anderson transitions. Rev. Mod. Phys.
**2008**, 80, 1355–1417. [Google Scholar] [CrossRef] [Green Version] - Lee, P.A.; Nagaosa, N.; Wen, X.G. Doping a Mott insulator: Physics of high-temperature superconductivity. Rev. Mod. Phys.
**2006**, 78, 17–85. [Google Scholar] [CrossRef] - Balents, L. Spin liquids in frustrated magnets. Nature
**2010**, 464, 199–208. [Google Scholar] [CrossRef] - Sondhi, S.L.; Girvin, S.M.; Carini, J.P.; Shahar, D. Continuous quantum phase transitions. Rev. Mod. Phys.
**1997**, 69, 315–333. [Google Scholar] [CrossRef] [Green Version] - Saito, Y.; Nojima, T.; Iwasa, Y. Highly crystalline 2D superconductors. Nat. Rev. Mater.
**2016**, 2, 74. [Google Scholar] [CrossRef] [Green Version] - Lee, P.A.; Ramakrishnan, T.V. Disordered electronic systems. Rev. Mod. Phys.
**1985**, 57, 287–337. [Google Scholar] [CrossRef] - Belitz, D.; Kirkpatrick, T.R. The Anderson-Mott transition. Rev. Mod. Phys.
**1994**, 66, 261–380. [Google Scholar] [CrossRef] - Abrahams, E.; Kravchenko, S.V.; Sarachik, M.P. Metallic behavior and related phenomena in two dimensions. Rev. Mod. Phys.
**2001**, 73, 251. [Google Scholar] [CrossRef] [Green Version] - Spivak, B.; Kravchenko, S.V.; Kivelson, S.A.; Gao, X.P.A. Colloquium: Transport in strongly correlated two dimensional electron fluids. Rev. Mod. Phys.
**2010**, 82, 1743–1766. [Google Scholar] [CrossRef] [Green Version] - Kravchenko, S. Strongly Correlated Electrons in Two Dimensions; Jenny Stanford Publishing: New York, NY, USA, 2017. [Google Scholar]
- Melnikov, M.Y.; Shashkin, A.A.; Dolgopolov, V.T.; Zhu, A.Y.X.; Kravchenko, S.V.; Huang, S.H.; Liu, C.W. Quantum phase transition in ultrahigh mobility SiGe/Si/SiGe two-dimensional electron system. Phys. Rev. B
**2019**, 99, 081106. [Google Scholar] [CrossRef] [Green Version] - Pustogow, A.; Rösslhuber, R.; Tan, Y.; Uykur, E.; Böhme, A.; Wenzel, M.; Saito, Y.; Löhle, A.; Hübner, R.; Kawamoto, A.; et al. Low-temperature dielectric anomaly arising from electronic phase separation at the Mott insulator-metal transition. NPJ Quantum Mater.
**2021**, 6, 9. [Google Scholar] [CrossRef] - Li, T.; Jiang, S.; Li, L.; Zhang, Y.; Kang, K.; Zhu, J.; Watanabe, K.; Taniguchi, T.; Chowdhury, D.; Fu, L.; et al. Continuous Mott transition in semiconductor moiré superlattices. Nature
**2021**, 597, 350–354. [Google Scholar] [CrossRef] - Simonian, D.; Kravchenko, S.V.; Sarachik, M.P. Reflection symmetry at a B=0 metal-insulator transition in two dimensions. Phys. Rev. B
**1997**, 55, R13421–R13423. [Google Scholar] [CrossRef] [Green Version] - Dobrosavljević, V.; Abrahams, E.; Miranda, E.; Chakravarty, S. Scaling Theory of Two-Dimensional Metal-Insulator Transitions. Phys. Rev. Lett.
**1997**, 79, 455–458. [Google Scholar] [CrossRef] [Green Version] - Kravchenko, S.V.; Mason, W.E.; Bowker, G.E.; Furneaux, J.E.; Pudalov, V.M.; D’iorio, M. Scaling of an anomalous metal-insulator transition in a two-dimensional system in silicon at B=0. Phys. Rev. B
**1995**, 51, 7038–7045. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Furukawa, T.; Miyagawa, K.; Taniguchi, H.; Kato, R.; Kanoda, K. Quantum criticality of Mott transition in organic materials. Nat. Phys.
**2015**, 11, 221–224. [Google Scholar] [CrossRef] [Green Version] - Ando, T.; Fowler, A.B.; Stern, F. Electronic properties of two-dimensional systems. Rev. Mod. Phys.
**1982**, 54, 437–672. [Google Scholar] [CrossRef] - Camjayi, A.; Haule, K.; Dobrosavljević, V.; Kotliar, G. Coulomb correlations and the Wigner–Mott transition. Nat. Phys.
**2008**, 4, 932–935. [Google Scholar] [CrossRef] [Green Version] - Amaricci, A.; Camjayi, A.; Haule, K.; Kotliar, G.; Tanasković, D.; Dobrosavljević, V. Extended hubbard model: Charge ordering and Wigner–Mott transition. Phys. Rev. B
**2010**, 82, 155102. [Google Scholar] [CrossRef] [Green Version] - Radonjić, M.M.; Tanasković, D.; Dobrosavljević, V.; Haule, K.; Kotliar, G. Wigner–Mott scaling of transport near the two-dimensional metal-insulator transition. Phys. Rev. B
**2012**, 85, 085133. [Google Scholar] [CrossRef] [Green Version] - Shashkin, A.; Melnikov, M.Y.; Dolgopolov, V.; Radonjić, M.; Dobrosavljević, V.; Huang, S.H.; Liu, C.; Zhu, A.Y.; Kravchenko, S. Manifestation of strong correlations in transport in ultraclean SiGe/Si/SiGe quantum wells. Phys. Rev. B
**2020**, 102, 081119. [Google Scholar] [CrossRef] - Shashkin, A.; Melnikov, M.Y.; Dolgopolov, V.; Radonjić, M.; Dobrosavljević, V.; Huang, S.H.; Liu, C.; Zhu, A.Y.; Kravchenko, S. Spin effect on the low-temperature resistivity maximum in a strongly interacting 2D electron system. Sci. Rep.
**2022**, 12, 5080. [Google Scholar] [CrossRef] - Moon, B.H.; Han, G.H.; Radonjić, M.M.; Ji, H.; Dobrosavljević, V. Quantum critical scaling for finite-temperature Mott-like metal-insulator crossover in few-layered MoS 2. Phys. Rev. B
**2020**, 102, 245424. [Google Scholar] [CrossRef] - Popović, D.; Fowler, A.B.; Washburn, S. Metal-Insulator Transition in Two Dimensions: Effects of Disorder and Magnetic Field. Phys. Rev. Lett.
**1997**, 79, 1543–1546. [Google Scholar] [CrossRef] - Bogdanovich, S.c.v.; Popović, D. Onset of Glassy Dynamics in a Two-Dimensional Electron System in Silicon. Phys. Rev. Lett.
**2002**, 88, 236401. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Shashkin, A.A.; Kravchenko, S.V.; Klapwijk, T.M. Metal-Insulator Transition in a 2D Electron Gas: Equivalence of Two Approaches for Determining the Critical Point. Phys. Rev. Lett.
**2001**, 87, 266402. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Shashkin, A.A.; Kravchenko, S.V.; Dolgopolov, V.T.; Klapwijk, T.M. Sharp increase of the effective mass near the critical density in a metallic two-dimensional electron system. Phys. Rev. B
**2002**, 66, 073303. [Google Scholar] [CrossRef] [Green Version] - Smolenski, T.; Dolgirev, P.E.; Kuhlenkamp, C.; Popert, A.; Shimazaki, Y.; Back, P.; Lu, X.; Kroner, M.; Watanabe, K.; Taniguchi, T.; et al. Signatures of Wigner crystal of electrons in a monolayer semiconductor. Nature
**2021**, 595, 53. [Google Scholar] [CrossRef] [PubMed] - Dressel, M.; Tomić, S. Molecular quantum materials: Electronic phases and charge dynamics in two-dimensional organic solids. Adv. Phys.
**2020**, 69, 1–120. [Google Scholar] [CrossRef] - Jerome, D. The physics of organic superconductors. Science
**1991**, 252, 1509–1514. [Google Scholar] [CrossRef] - Kagawa, F.; Sato, T.; Miyagawa, K.; Kanoda, K.; Tokura, Y.; Kobayashi, K.; Kumai, R.; Murakami, Y. Charge-cluster glass in an organic conductor. Nat. Phys.
**2013**, 9, 419–422. [Google Scholar] [CrossRef] - Kurosaki, Y.; Shimizu, Y.; Miyagawa, K.; Kanoda, K.; Saito, G. Mott Transition from a Spin Liquid to a Fermi Liquid in the Spin-Frustrated Organic Conductor κ-(ET)
_{2}Cu_{2}(CN)_{3}. Phys. Rev. Lett.**2005**, 95, 177001. [Google Scholar] [CrossRef] [Green Version] - Pustogow, A.; Bories, M.; Löhle, A.; Rösslhuber, R.; Zhukova, E.; Gorshunov, B.; Tomić, S.; Schlueter, J.A.; Hübner, R.; Hiramatsu, T.; et al. Quantum spin liquids unveil the genuine Mott state. Nat. Mater.
**2018**, 17, 773–777. [Google Scholar] [CrossRef] - Vučičević, J.; Terletska, H.; Tanasković, D.; Dobrosavljević, V. Finite-temperature crossover and the quantum Widom line near the Mott transition. Phys. Rev. B
**2013**, 88, 075143. [Google Scholar] [CrossRef] [Green Version] - Pustogow, A.; Saito, Y.; Löhle, A.; Sanz Alonso, M.; Kawamoto, A.; Dobrosavljević, V.; Dressel, M.; Fratini, S. Rise and fall of Landau’s quasiparticles while approaching the Mott transition. Nat. Commun.
**2021**, 12, 1571. [Google Scholar] [CrossRef] [PubMed] - Pomeranchuk, I. On the thery of He
^{3}. Zh. Eksp. Teor. Fiz.**1950**, 20, 919. [Google Scholar] - Terletska, H.; Vucicevic, J.; Tanasković, D.; Dobrosavljević, V. Quantum Critical Transport near the Mott Transition. Phys. Rev. Lett.
**2011**, 107, 026401. [Google Scholar] [CrossRef] - Deng, X.; Mravlje, J.; Žitko, R.; Ferrero, M.; Kotliar, G.; Georges, A. How Bad Metals Turn Good: Spectroscopic Signatures of Resilient Quasiparticles. Phys. Rev. Lett.
**2013**, 110, 086401. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Emery, V.J.; Kivelson, S.A. Superconductivity in Bad Metals. Phys. Rev. Lett.
**1995**, 74, 3253–3256. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hussey, N.E.; Takenaka, K.; Takagi, H. Universality of the Mott–Ioffe–Regel limit in metals. Philos. Mag.
**2004**, 84, 2847–2864. [Google Scholar] [CrossRef] [Green Version] - Rademaker, L. Spin-Orbit Coupling in Transition Metal Dichalcogenide Heterobilayer Flat Bands. arXiv
**2021**, arXiv:2111.06208. [Google Scholar] [CrossRef] - Ghiotto, A.; Shih, E.M.; Pereira, G.S.S.G.; Rhodes, D.A.; Kim, B.; Zang, J.; Millis, A.J.; Watanabe, K.; Taniguchi, T.; Hone, J.C.; et al. Quantum Criticality in Twisted Transition Metal Dichalcogenides. Nature
**2021**, 597, 345. [Google Scholar] [CrossRef] - Savary, L.; Balents, L. Quantum spin liquids: A review. Rep. Prog. Phys.
**2016**, 80, 016502. [Google Scholar] [CrossRef] - Senthil, T. Theory of a continuous Mott transition in two dimensions. Phys. Rev. B
**2008**, 78, 045109. [Google Scholar] [CrossRef] [Green Version] - Georges, A.; Kotliar, G.; Krauth, W.; Rozenberg, M.J. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys.
**1996**, 68, 13–125. [Google Scholar] [CrossRef] [Green Version] - Spivak, B.; Kivelson, S.A. Phases intermediate between a two-dimensional electron liquid and Wigner crystal. Phys. Rev. B
**2004**, 70, 155114. [Google Scholar] [CrossRef] - Spivak, B.; Kivelson, S.A. Transport in two dimensional electronic micro-emulsions. Ann. Phys.
**2006**, 321, 2071–2115. [Google Scholar] [CrossRef] [Green Version] - Baskaran, G.; Zou, Z.; Anderson, P. The resonating valence bond state and high-Tc superconductivity—A mean field theory. Solid State Commun.
**1987**, 63, 973–976. [Google Scholar] [CrossRef] - Jacko, A.; Fjærestad, J.; Powell, B. A unified explanation of the Kadowaki–Woods ratio in strongly correlated metals. Nat. Phys.
**2009**, 5, 422–425. [Google Scholar] [CrossRef] [Green Version] - Podolsky, D.; Paramekanti, A.; Kim, Y.B.; Senthil, T. Mott Transition between a Spin-Liquid Insulator and a Metal in Three Dimensions. Phys. Rev. Lett.
**2009**, 102, 186401. [Google Scholar] [CrossRef] - Xu, Y.; Wu, X.C.; Ye, M.X.; Luo, Z.X.; Jian, C.M.; Xu, C. Metal-Insulator Transition with Charge Fractionalization. arXiv
**2021**, arXiv:2106.14910. [Google Scholar] - Musser, S.; Senthil, T.; Chowdhury, D. Theory of a Continuous Bandwidth-tuned Wigner–Mott Transition. arXiv
**2021**, arXiv:2111.09894. [Google Scholar] - Vučičević, J.; Tanasković, D.; Rozenberg, M.; Dobrosavljević, V. Bad-metal behavior reveals Mott quantum criticality in doped Hubbard models. Phys. Rev. Lett.
**2015**, 114, 246402. [Google Scholar] - Lee, T.H.; Florens, S.; Dobrosavljević, V. Fate of spinons at the Mott point. Phys. Rev. Lett.
**2016**, 117, 136601. [Google Scholar] - Brinkman, W.F.; Rice, T.M. Application of Gutzwiller’s Variational Method to the Metal-Insulator Transition. Phys. Rev. B
**1970**, 2, 4302–4304. [Google Scholar] [CrossRef] - Eisenlohr, H.; Lee, S.S.B.; Vojta, M. Mott quantum criticality in the one-band Hubbard model: Dynamical mean-field theory, power-law spectra, and scaling. Phys. Rev. B
**2019**, 100, 155152. [Google Scholar] [CrossRef] [Green Version] - Radonjić, M.M.; Tanasković, D.; Dobrosavljević, V.; Haule, K. Influence of disorder on incoherent transport near the Mott transition. Phys. Rev. B
**2010**, 81, 075118. [Google Scholar] [CrossRef] [Green Version] - Jamei, R.; Kivelson, S.; Spivak, B. Universal Aspects of Coulomb-Frustrated Phase Separation. Phys. Rev. Lett.
**2005**, 94, 056805. [Google Scholar] [CrossRef] [Green Version] - Dagotto, E. Complexity in Strongly Correlated Electronic Systems. Science
**2005**, 309, 257–262. [Google Scholar] [CrossRef] [Green Version] - Urai, M.; Furukawa, T.; Seki, Y.; Miyagawa, K.; Sasaki, T.; Taniguchi, H.; Kanoda, K. Disorder unveils Mott quantum criticality behind a first-order transition in the quasi-two-dimensional organic conductor κ-(ET)
_{2}Cu[N(CN)_{2}]Cl. Phys. Rev. B**2019**, 99, 245139. [Google Scholar] [CrossRef] - Yamamoto, R.; Furukawa, T.; Miyagawa, K.; Sasaki, T.; Kanoda, K.; Itou, T. Electronic Griffiths Phase in Disordered Mott-Transition Systems. Phys. Rev. Lett.
**2020**, 124, 046404. [Google Scholar] [CrossRef] - Aharony, A.; Imry, Y.; Ma, S.k. Lowering of Dimensionality in Phase Transitions with Random Fields. Phys. Rev. Lett.
**1976**, 37, 1364–1367. [Google Scholar] [CrossRef] - Kim, S.; Senthil, T.; Chowdhury, D. Continuous Mott transition in moiré semiconductors: Role of long-wavelength inhomogeneities. arXiv
**2022**, arXiv:2204.10865. [Google Scholar] - Economou, E.N. Green’s Functions in Quantum Physics; Springer: Berlin, Germany, 2005. [Google Scholar]
- Efros, A.L.; Shklovskii, B.I. Critical Behaviour of Conductivity and Dielectric Constant near the Metal-Non-Metal Transition Threshold. Phys. Stat. Sol. B
**1976**, 76, 475. [Google Scholar] [CrossRef] - Miranda, E.; Dobrosavljević, V. Disorder-driven non-Fermi liquid behaviour of correlated electrons. Rep. Prog. Phys.
**2005**, 68, 2337. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**The key observable revealing a metal-insulator transition is the resistivity. Here we show $\rho $ vs. T resistivity curves as a function the tuning parameter, for representative examples of the three material systems considered: (

**a**) 2DEG in Si-MOSFET tuned by electronic density (reprinted with permission from Ref. [19] Copyright 2019 American Physical Society), (

**b**) Mott organic material $\kappa {-(\mathrm{BEDT}-\mathrm{TTF})}_{2}{\mathrm{Cu}}_{2}{\left(\mathrm{CN}\right)}_{3}$ tuned by pressure [20], and (

**c**) TMD moiré bilayer MoTe${}_{2}$/WSe${}_{2}$ tuned by displacement field (Data imported from [21]). In all cases, one observes distinct resistivity maxima on the metallic side, at a temperature ${T}_{max}$ that decreases towards the transition.

**Figure 2.**Critical scaling has been observed in all three experimental systems, when the resistivity is plotted versus $T/{T}_{o}$ where ${T}_{o}$ is the characteristic crossover (quantum critical) energy scale. Note that in all cases a strong “mirror” symmetry [22,23] exists between the insulating (upper) and metallic (lower) scaling branch. (

**a**) In a dilute 2DEG, scaling of the bare resistivity $\rho \left(T\right)$ was achieved by simply rescaling T with ${T}_{o}\sim {\left|\delta \right|}^{1.6}$ (Adapted with permission from Ref. [24] Copyright 1995 American Physical Society); (

**b**) In organic compounds, the normalized resistivity $\tilde{\rho}$ is obtained by normalizing the resistivity by the critical resistivity along the Widom line. This leads to excellent scaling collapse with ${T}_{o}\sim {\left|\delta \right|}^{0.60\pm 0.01}$ (Adapted with permission from Ref. [25] Copyright 2015 Springer Nature); (

**c**) A similar approach was followed in TMD moiré bilayer MoTe${}_{2}$/WSe${}_{2}$, with similar ${T}_{o}\sim {\left|\delta \right|}^{0.70\pm 0.05}$ (Data imported from [21]).

**Figure 3.**Characteristic scaling of the resistivity maxima has been reported in several 2DEG electron systems in semiconductors: (

**a**) Si-MOSFETs (adapted with permission from Ref. [29] Copyright 2012 American Physical Society); (

**b**) p-GaAs/AlGaAs quantum wells (adapted with permission from Ref. [29] Copyright 2012 American Physical Society); (

**c**) SiGe/Si/SiGe quantum wells (adapted with permisssion from Ref. [30] Copyright 2020 American Physical Society); (

**d**) few layered-$Mo{S}_{2}$ (Adapted with permission from Ref. [32] Copyright 2020 American Physical Society). All data collapse to the same (theoretical) scaling function [29] obtained from the Hubbard model at half-filling, in the vicinity of the Mott point.

**Figure 4.**Finite temperature phase diagram of the Mott organic materials. (

**a**) first-order phase transition line, as observed in $\kappa $-Cu${}_{2}$(CN)${}_{3}$ (adapted with permission from Ref. [25] Copyright 2015 Springer Nature) at $T<{T}_{c}\sim 20K$, displaying “Pomeranchuk” behavior [45], by “sloping” towards the metallic phase. The corresponding “Quantum Widom Line” [43] arises at $T>{T}_{c}$, which is identified as the center of the quantum critical region [46] with resistivity scaling [25]. (

**b**) Phase diagram [44] for $\kappa $-Cu${}_{2}$(CN)${}_{3}$ over a broader T-range, displaying the convergence of the quantum Widom line (QWL) on the insulating side, and the “Brinkman-Rice” line (${T}_{BR}={T}_{max}$, which intersect at the critical end-point $T={T}_{c}$. The Fermi-Liquid line ${T}_{FL}<{T}_{BR}$ is also shown. (

**c**) The universal phase diagram for a series of spin-liquid Mott organics compounds was established [42] by rescaling the temperature T and the interaction strength U by the respective electronic bandwidth W. The parameters W and U were independently measured [42] for each material using optical conductivity.

**Figure 5.**Fermi liquid behaviour at low temperatures, for (

**a**) Mott organic material $\kappa -{\left[{(\mathrm{BEDT}-\mathrm{TTF})}_{1-x}{(\mathrm{BEDT}-\mathrm{STF})}_{x}\right]}_{2}{\mathrm{Cu}}_{2}{\left(\mathrm{CN}\right)}_{3}$ [44] and (

**b**) MoTe${}_{2}$/WSe${}_{2}$ moiré bilayers (Data imported from [21]). Clear $\rho ={\rho}_{0}+A{T}^{2}$ behavior is observed in both cases, up to a temperature scale ${T}_{FL}$ that seems to decrease linearly towards the metal-insulator transition. The resistivity curves can be collapsed by plotting $\rho (E,T)/{\rho}_{c}\left(T\right)$ vs. $T/{T}_{0}$ where ${T}_{0}\sim {|E-{E}_{c}|}^{0.70\pm 0.05}$, see Figure 2c. Note that this crossover scale seems to follow both the gap size on the insulator, as well as the destruction of the Fermi liquid on the metallic side.

**Figure 6.**Transport behavior vs. dielectric response across the phase diagram of $\kappa $-Cu${}_{2}$(CN)${}_{3}$ [20]. (

**a**) DC transport shows only very gradual change across the BR line (resistivity maxima), and cannot one see any clear indication of the phase coexistence region. (

**b**) In dramatic contrast, the low-frequency dielectric function ${\u03f5}_{1}$ assumes small positive values in the Mott insulator (pale pink), and large negative values in the quasiparticle regime (deep blue); we clearly see the boundaries of these regimes tracing the QWL and the BR line (following ${T}_{max}$), as observed in transport. Remarkably, “resilient” quasiparticles [47] persist past the Fermi Liquid line, at ${T}_{FL}<T<{T}_{BR}={T}_{max}$, where bad metal behavior [48] (metallic transport above the Mott-Ioffe-Regel [49] limit is observed). At low temperature, the Mott point is buried below the phase coexistence dome, which is vividly visualized through colossal dielectric response (${\u03f5}_{1}\sim {10}^{3}$–${10}^{4}$).

**Figure 7.**Predictions of the spinon theory (reprinted with permission from Ref. [59] Copyright 2009 American Physical Society). (

**a**) The phase diagram features a quantum critical point at $T=0$, and two distinct finite-T crossover scales ${T}^{*}$ (above which the system is quantum critical) and ${T}^{**}$ (below which the system is either a metal or a gapless spin liquid). (

**b**) and (

**c**) Resistivity and conductivity along the lines A, B and C in the phase diagram in (

**a**). Critical resistivity is predicted to diverge as ${\rho}_{c}\left(T\right)\sim 1/t$ in $d=3$, leading to resistivity maxima on the metallic side (coductivity minima). In contrast, the same theory predicts finite critical resistivity ${\rho}_{c}\left(T\right)\sim {\rho}^{*}$ in $d=2$ [53], hence monotonic behavior on both sides of the transition and no resistivity maxima.

**Figure 8.**Predictions of DMFT theory. (

**a**) Phase diagram featuring a phase coexistence region at $T<{T}_{c}$, and a Quantum Critical region centered around the Quantum Widom Line (QWL) (adapted with permission from Ref. [46] Copyright 2011 American Physical Society). (

**b**) Resistivity (normalized by the Mott-Ioffe-Regel (MIR) limit) as a function of temperature T across the transition. Note the pronounced resistivity maxima on the metallic side (adapted with permission from Ref. [66] Copyright 2010 American Physical Society. (

**c**) scaling collapse of the resistivity curves, displaying pronounced “mirror symmetry” of the two branches (adapted with permission from Ref. [46] Copyright 2011 American Physical Society).

**Figure 9.**(

**a**) DMFT results for the evolution of the single-particle Density of States (DOS) for several values of the temperature (reprinted with permission from Ref. [66] Copyright 2010 American Physical Society), as well as (

**b**) that of the optical conductivity, in the strongly correlated metallic regime. Different colors correspond to the four distinctive transport regimes (inset in (

**b**)). DOS features a distinct quasiparticle peak at low temperatures, which is thermally destroyed at temperature ${T}_{max}={T}_{BR}\sim {\left({m}^{*}\right)}^{-1}$, where the resistivity (inset of right panel) reaches a maximum. The optical conductivity displays the corresponding suppression of the low-frequency Drude peak around the same temperature.

**Figure 10.**(

**a**) DC resistivity as a function of temperature for several interaction strengths. (

**b**) Scaled resistivity curves. (

**c**) Real part of dielectric function ${\u03f5}_{1}$ at $\omega /D=0.01$, as a function of temperature for several interaction strengths. (

**d**) Scaled dielectric function curves. Results are obtained for a half-filled Hubbard model solved within DMFT.

**Figure 11.**(

**a**) ${T}_{drop}$ as a function of ${T}_{max}$. (

**b**) ${T}_{max}$ as a function of Z. (

**c**) ${T}_{drop}$ as a function of Z.

**Figure 12.**(

**a**) The red line is $x=x\left({T}^{*}\right)$. For T larger than the blue dashed line, $x=0$. We calculate the percolation results along the grey dashed lines. (

**b**) $R/{R}_{M}^{o}$ as a function of T for different ${T}^{*}$. (

**c**) Scaled resistivity curves.

**Figure 13.**(

**a**) The dielectric constant ${\u03f5}_{1}$ as a function of T for different ${T}^{*}$. (

**b**) ${\u03f5}_{1}$ as a function of $\tau $ for different ${T}^{*}$. (

**c**) Scaled dielectric function curves.

**Table 1.**A summary of available experimental results for the three classes of systems considered. The sources (references) are given in the text below. Question-marks indicate the lack of reliable data. Fermi liquid (${T}^{2}$) transport behavior has not been documented in 2DEG systems, in contrast to strong evidence for it in Mott organics and TMD moiré bilayers. Note that the characteristic energy scales $\Delta $, ${\left({m}^{*}\right)}^{-1}$, ${T}_{FL}$, ${T}_{max}$, as well as ${T}_{o}$ display similar continuous decrease towards the transition in all three systems, consistent with general expectations for quantum criticality. One should keep in mind that the error bars on the estimated exponent could be substantial, since the results typically depend strongly on the utilized fitting range.

System | Dilute 2DEG | Mott Organics | TMD Moiré Bilayers |
---|---|---|---|

Transition Type | continuous? | weakly first order (at $T<{T}_{c}\sim 0.01{T}_{F}$) | continuous? |

$\Delta $ | $|n-{n}_{c}|$ | $|P-{P}_{c}{|}^{\nu z}$, $\nu z\approx 0.7-1$ | $|E-{E}_{c}{|}^{\nu z}$, $\nu z\approx 0.6$ |

$\frac{1}{{m}^{*}}$ | $|n-{n}_{c}|$ | ? | ? |

${T}_{o}$ | $|n-{n}_{c}{|}^{\nu z}$, $\nu z\approx 1.6$ | $|P-{P}_{c}{\left(T\right)|}^{\nu z}$, $\nu z\approx 0.5-0.7$ | $|E-{E}_{c}{|}^{\nu z}$, $\nu z\approx 0.7$ |

${T}_{\mathit{FL}}$ | ? | $|P-{P}_{c}|$ | $|E-{E}_{c}{|}^{\nu z}$, $\nu z\approx 0.7$ |

${T}_{max}$ | $|n-{n}_{c}|$ | $|P-{P}_{c}|$ | $|E-{E}_{c}{|}^{\nu z}$, $\nu z\approx 0.7$ |

**Table 2.**A summary of predictions from competing theoretical pictures. The expected transition type differs between the three pictures, with observable differences in the behavior of the mass enhancement ${m}^{*}$, the Kadowaki–Woods ratio $A/{\left({m}^{*}\right)}^{2}$, the destruction of the Fermi liquid at ${T}_{\mathit{FL}}$, and the appearance of a resistivity maxima at ${T}_{max}$. Details are provided in the text below.

Theory Predictions | 2D Spinon Theory | DMFT | Percolation Theory |
---|---|---|---|

Transition Type | continuous | weakly first order (at $T<{T}_{c}\sim 0.01{T}_{F}$) | first order |

$\Delta $ | $|g-{g}_{c}|{S}^{\nu z}$, $\nu z=0.67$ | $|U-{U}_{c1}{|}^{\nu z}$, $\nu z\approx 0.8$ | remains finite |

${m}^{*}$ | weak: $ln\frac{1}{|g-{g}_{c}|}$ | strong: $|U-{U}_{c2}{|}^{-1}$ | no divergence |

$A/{\left({m}^{*}\right)}^{2}$ | ? | constant (KW law obeyed) | diverges: ${({x}_{o}-{x}_{c})}^{-t}$; $t=s/m$ |

${T}_{\mathit{FL}}$ | $|g-{g}_{c}{|}^{2\nu}$ | $|U-{U}_{c2}|$ | ${T}^{*}\sim |{x}_{o}-{x}_{c}|$ |

${T}_{max}$ | ${T}_{max}=\infty $ | $|U-{U}_{c2}|$ | ${T}^{*}\sim |{x}_{o}-{x}_{c}|$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tan, Y.; Dobrosavljević, V.; Rademaker, L.
How to Recognize the Universal Aspects of Mott Criticality? *Crystals* **2022**, *12*, 932.
https://doi.org/10.3390/cryst12070932

**AMA Style**

Tan Y, Dobrosavljević V, Rademaker L.
How to Recognize the Universal Aspects of Mott Criticality? *Crystals*. 2022; 12(7):932.
https://doi.org/10.3390/cryst12070932

**Chicago/Turabian Style**

Tan, Yuting, Vladimir Dobrosavljević, and Louk Rademaker.
2022. "How to Recognize the Universal Aspects of Mott Criticality?" *Crystals* 12, no. 7: 932.
https://doi.org/10.3390/cryst12070932