# How to Recognize the Universal Aspects of Mott Criticality?

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## Abstract

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## 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

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

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**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