**2. Fermion Condensation**

The theory of FC has been described several times, see, e.g., [4,15,19,20]; nonetheless, for the readers' convenience, we briefly present this methodology. The usual approach to describe the ensembles of itinerant Fermi particles is the well-known Landau Fermi liquid theory [34,35]. This theory represents the real properties of a solid with itinerant electrons in terms of a Fermi gas of so-called quasiparticles with weak interaction. In this case, the quasiparticles represent the excited states of a solid or liquid states and are responsible for the low temperature thermodynamic, transport and relaxation properties of common metals. These quasiparticle excitations are characterized by the effective mass *M*<sup>∗</sup>, that is of the order of the bare mass of electron, *M*, and depends weakly on external parameters such as temperature *T*, magnetic field *B*, external pressure *P*, etc. [34,35]. However, the LFL theory cannot explain why the effective mass *M*∗ begins to depend strongly on the stimuli above and, for example, can even be a divergent function of magnetic field *B* or temperature *T*, see, e.g., [4,15,19,36]. Such a dependence is called the NFL behavior and is connected to the growth of the effective mass that occurs when the system approaches the topological fermion condensation quantum phase transition (FCQPT) leading to an FC state with flat bands [1,4,15,19]. Beyond the FCQPT, the system develops a flat band, formed by FC, and characterized by the topological charge that is different from both the topological charges of the Landau Fermi liquid (LFL) and marginal Fermi liquid, representing a new type of Fermi liquid [2,4,15,19,37]. Thus, the stability of FC is ensured by its topological charge, and it can be destroyed only by the first order phase transition, since the topological charge cannot acquire continuous values [2,15,19,37]. As a result of these unique properties of the FC state, a new state of matter is generated, represented by QSL, HF metals, quasicrystals, 2D liquids such as 3He and high-*Tc* superconductors, so that 1D, 2D and 3D strongly correlated Fermi systems exhibit universal scaling behavior irrespective of their microscopic structure [15,19,20,38,39].

The main feature of FC theory is the existence of one more instability channel (additional to those of Pomeranchuk) that cannot be described within the framework of the Landau theory of Fermi liquid [35]. Indeed, under some conditions, the effective mass *M*∗ of LFL quasiparticle diverges, see, e.g., [15,19]. As a result, to keep the finite and positive effective mass at zero and finite temperatures, the Fermi surface changes its topology: the Fermi surface transforms into a Fermi layer, as seen in Figure 1. This topological phase transition generates the effective mass dependence on temperature, magnetic field, etc. We assume, without loss of generality [15,19], that the Fermi liquid is homogeneous. That is, in our model we account for the most important and common features only, neglecting marginal effects related to the crystalline anisotropy of solids [15,19,20]. The Landau equation for the quasiparticle effective mass *M*∗ reads [15,34,35]

$$\frac{1}{M\_{\sigma}^\*(\mathbf{B}, T)} = -\frac{1}{M} + \sum\_{\sigma\_1} \int \frac{\mathbf{P}\_F \mathbf{P}}{p\_F^3} \mathbf{F}\_{\sigma, \sigma\_1}(p\_{F\_\sigma} p)$$

$$\times \quad \frac{\partial n\_{\sigma\_1}(\mathbf{p}\_\prime, T, B)}{\partial p} \frac{dp}{(2\pi)^3} \tag{3}$$

where *<sup>F</sup><sup>σ</sup>*,*σ*<sup>1</sup> (*pF*, *p*) is the interaction function, introduced by Landau. The function *<sup>F</sup><sup>σ</sup>*,*σ*<sup>1</sup> (*pF*, *p*), depending on momentum *p*, Fermi momentum *pF* and spin indices *σ*, *σ*1, has the form of

spherical harmonics with coefficients taken from the best fit to experiment. The fermion occupation number *n* in the Fermi–Dirac statistics reads

$$m\_{\sigma}(p,T) = \left\{ 1 + \exp\left[\frac{\left(\varepsilon\_{\sigma}(\mathbf{p}\_{\prime}T) - \mu\_{\sigma}\right)}{T}\right] \right\}^{-1},\tag{4}$$

where *εσ*(*p*, *T*) is the single-particle spectrum, and *μσ* is a spin-dependent chemical potential: *μσ* = *μ* ± *μBB* where *μB* is the Bohr magneton. The magnetic field dependence occurs due to the Zeeman splitting shifting the system from its topological FCQPT [15].

**Figure 1.** Diagram of flat bands near the FCQPT at zero temperature, *T* = 0. Panel (**a**) shows normal Fermi sphere and corresponding quasiparticles spectrum *ε*(*p*) *p*2/(2*<sup>M</sup>*) and occupation number *n*(*p*) being a step function. Panel (**b**) displays the system in the FC state after the topological FCQPT. The Fermi sphere alters its topology, which is shown schematically as an emergence of a spherical layer of the thickness *pf* − *pi*. In this case, the Fermi momentum *pF* is hidden inside the flat band, defined by the condition *ε*(*p*) = *μ* (7). This condition defines the flat band, shown as a dispersionless part of the spectrum *ε*(*p*) = *μ*, with *μ* being the chemical potential. The function *n*(*p*) decreases gradually from *n*(*pi*) = 1 to *n*(*pf*) = 0 without violating the Pauli exclusion principle.

The standard procedure for obtaining the single-particle spectrum *εσ*(*p*, *T*) in the Landau theory is to vary the system energy *<sup>E</sup>*[*<sup>n</sup>σ*(*p*, *T*)] with regard to the occupation number *n*

$$
\varepsilon\_{\sigma}(p, T) = \frac{\delta E[n(p)]}{\delta n\_{\sigma}(p)}.\tag{5}
$$

We note that the Landau interaction entering Equation (3) is not of a special form since it is fixed by the simple condition that the system is in the FCQPT point [15,19]. The explicit form of the variational Equation (5) reads

$$\frac{\partial \varepsilon\_{\sigma}(p, T)}{\partial p} = \frac{p}{M} - \sum\_{\sigma\_1} \int \frac{\partial F\_{\sigma, \sigma\_1}(p\_\prime, p\_1)}{\partial \mathbf{p}} n\_{\sigma\_1}(p\_1, T) \frac{d^3 p\_1}{(2\pi)^3 \prime} \tag{6}$$

Later on for simplicity, we omit the spin indexes *σ*. In the FC phase (i.e., beyond the FCQPT) at *T* = 0, Equation (5) takes the form [1]

$$
\varepsilon(p, T=0) = \mu, \ p\_i \le p \le p\_f; \ 0 \le n(p) \le 1. \tag{7}
$$

where *pi*, *f* stands for initial and final momenta (not to be confused with Fermi momentum *pF*), where the flat band resides, see Figure 1. Condition (7) defines the flat band since in this case the quasiparticles have no dispersion. By this virtue, quasiparticles have the Fermi velocity *VF* = 0 and at *T* = 0 are condensed with the same energy *<sup>ε</sup>*(*p*, *T* = 0) = *μ*, representing the superconducting state with the finite order parameter *κ*, while the superconducting gap Δ = 0, see Section 7. As this resembles the case of Bose condensation, the corresponding phenomenon is called fermion condensation, being separated from LFL by the first order phase transition [1,2,37]. The system with FC acquires properties, being very different from those of ordinary Fermi liquids, since the Fermi liquid with FC forms a new, topologically-protected (and thus "extremely stable") state of matter. This means that if FC is formed in a substance, it will define its properties at *T* = 0 and at elevated temperatures as well. Figure 1 visualizes (at *T* = 0) the consequences of the FCQPT on the Fermi surface, spectrum and occupation number of a Fermi liquid. The transformation from panel (a) (normal Fermi liquid) to panel (b) is represented by altering the Fermi surface topology so that in the normal Fermi liquid the layer of finite length *pf* − *pi* appears instead of the Fermi surface located at Fermi momentum *p* = *pF*. This immediately implies the emergence of the flat part of the spectrum defined by Equation (7), where all the condensed fermions are located. This, in turn, generates the gradual (instead of abrupt on the panel (a) decay of the occupation numbers *n*(*p*) from *n* = 1 at *p* < *pi* to *n* = 0 at *p* > *pf* .

Equations (3) and (7) allow one to determine the energy spectrum *εσ*(*p*, *T*) and occupation numbers *<sup>n</sup>σ*(*p*, *T*) in a self-consistent way. These quantities, in turn, permit the calculation of the effective mass, *pF*/*M*<sup>∗</sup> = *∂ε*(*p*)/*∂p*|*<sup>p</sup>*=*pF* = *VF*. We emphasize that both magnetic field and temperature dependences of the effective mass *<sup>M</sup>*<sup>∗</sup>(*<sup>B</sup>*, *T*) in the FC phase come from Equation (3) and from the *T*, *B*-dependence of *εσ*(*p*) and *<sup>n</sup>σ*(*p*). Calculated (by Equations (3) and (7)) spectrum and occupation numbers [15] in the FC phase are reported in Figure 2. At (almost) zero temperature, the flat portion of the spectrum is clearly seen at *pi* < *p* < *pf* . This shape of the spectrum defines *n*(*p*) (Figure 2, panel (b)) in the form of "two steps", gradually decaying from one to zero. Simultaneously, at relatively high temperatures (equal to *T*/*EF* = 0.01, which at *EF* ∼ 1eV implies *T* 100 K) this part is rather strongly upward tilted. This shows that finite temperatures erode the FC state, making the effective mass *M*∗ finite, while the system acquires features similar to ordinary Fermi liquid [4,15].

To gain more insights into the physical properties of the FC state, it is helpful to explore the system behavior at *T* → 0. It was shown earlier [1,15,19] that the ground state of a system with FC is highly degenerate. In this case, the occupation numbers *<sup>n</sup>*0(*p*) of the FC state quasiparticles (i.e., having dispersionless spectrum or belonging to the flat band) change gradually from *n* = 1 to *n* = 0 at *T* = 0. This variation occurs at *pi* ≤ *p* ≤ *pf* . It is clear that such a property of the occupation numbers drastically differs from the property of the usual Fermi–Dirac function property at *T* = 0. Indeed, in that case, the Fermi–Dirac function is represented by the step function between *n* = 1 and *n* = 0 at *p* = *pF*, where *pF* stands for Fermi momentum, see Figure 1.

At *T* = 0, the infinite degeneracy of the ground state with FC leads to a *T*-independent entropy term [4,15], remaining finite at *T* = 0 in violation of the Nernst theorem

$$S\_0 = -\sum\_p [n\_0(p)\ln n\_0(p) + (1 - n\_0(p))\ln(1 - n\_0(p))].\tag{8}$$

**Figure 2.** Flat band induced by FC. The calculated single-particle spectrum (**a**) and the quasiparticle occupation number (**b**) at small but finite temperatures versus the dimensionless momentum *k* = *p*/*pF*, where *pF* is the Fermi momentum [15]. Temperature is measured in the units of *EF*. At *T* = 0.01*EF* and *T* = 0.0001*EF*, the vertical lines show the position of the Fermi level *EF* at which *<sup>n</sup>*(*k*, *T*) = 0.5 (see the horizontal line in panel (b)). At *T* = 0.0001*EF* (blue curve), the single-particle spectrum *<sup>ε</sup>*(*k*, *T*) is almost flat (marked "Flat band") in the range *k f* − *ki* (with *ki* = *pi*/*pF* and *k f* = *k f* /*pF* denoting, respectively, the initial and final momenta for FC realization, and *k* = *p*/*pF*). Thus, in the range *k f* − *ki* the density of states *N*0 → <sup>∞</sup>, and outside the range *N*0 is finite. The distribution function *<sup>n</sup>*(*k*, *T*) becomes more asymmetric with respect to the Fermi level *EF*, generating the NFL behavior, and C invariance is broken. To illuminate the asymmetry, the area occupied by holes in panel (b) is labeled h (red) and that occupied by quasiparticles by *p* (maroon).

Thus, the infinite degeneracy of the FC ground state generated by flat bands, see Refs. [19,20] for a comprehensive discussion. We note that for systems where the Nernst theorem is violated due to the ground state degeneracy is a spin glass [40,41]. It is well known that in normal Fermi liquid the function *n*(*p*) at finite temperatures loses its steplike feature at *p* = *pF*, becoming continuous around this point. The same is valid for a Fermi liquid with flat bands; this conclusion follows from Equation (4). This means that at small but finite temperatures *T* = 0 the degeneracy of the above ground state is lifted, consequently the single-particle energy *<sup>ε</sup>*(*p*, *T* = 0) acquires a small dispersion [4]

$$
\varepsilon(p, T \to 0) = T \ln \frac{1 - n\_0(p)}{n\_0(\mathbf{p})}.\tag{9}
$$

From Equation (9), we see that the dispersion is proportional to *T* since the occupation numbers *n*0 approximately remain the same as at *T* = 0. This means that the entropy *S* in this case still remains *S*(*T*) ≥ *S*0. This situation also jeopardizes the Nernst theorem. To avoid this unphysical situation, the nearly flat bands representing the FC state should acquire dispersion in a way that the excess entropy *S*0 should "dissolve" as *T* → 0. This occurs by virtue of some additions to the FCQPT phase transition such as a ferromagnetic and/or a superconductive one, etc. [4,15,19]. Thus, at low temperatures the FC state has to be consumed by a number of phase transitions. This "consumption" can be viewed as a complicated phase diagram of an HF metal at its quantum critical point. In fact, at *T* = 0 the FC state is represented by the superconducting state with the superconducting order parameter *κ* = *n*(*p*)(<sup>1</sup> − *n*(*p*)) that is finite in the region (*pi* − *pf*) [15,33,42], for in the region *n*(*p*) < 1, as shown in Figure 2. Nonetheless, the superconducting gap , Δ = 0, can be absent provided that the superconducting coupling constant *g* = 0. In case of finite *g*, the gap exhibits very specific non-BCS behavior [43] Δ ∝ *g*, see, e.g., [1,4,44,45] and Section 7.

#### **3. Scaling of Physical Properties**

Experimental manifestations of the FC phenomenon correspond to the universal behavior of the physical properties of HF metals [15,19,20,46]. The physical properties of HF metals are formed due to flat bands and are widespread compounds [6]. To reveal the scaling, consider now the approximate solutions of Equation (3) [15,19,20]. At *B* = 0, Equation (3) becomes strongly temperature dependent, which is a typical NFL feature and can be solved analytically [15,19,20,46]:

$$M^\*(T) \simeq a\_T T^{-2/3}.\tag{10}$$

At *T* = 0, the analytical solution is

$$M^\*(B) \cong a\_B B^{-2/3}.\tag{11}$$

Here, *aT* and *aB* are constants. Under the application of a magnetic field, the system transits to the LFL state with the effective mass becoming almost temperature independent and strongly dependent on *B*, as seen from Equation (11).

#### *3.1. Internal Variables Revealing the Scaling Behavior*

Equations (10) and (11) allow us to construct the approximate solution of Equation (3) in the form *<sup>M</sup>*<sup>∗</sup>(*<sup>B</sup>*, *T*) = *<sup>M</sup>*<sup>∗</sup>(*T*/*B*). The introduction of "internal" scales simplifies the problem of constructing the universal scaling of the effective mass *M*<sup>∗</sup>, since in that case we eliminate the microscopic structure of the compound in question [15,19,20]. From the Figure 3a, we see that the effective mass *<sup>M</sup>*<sup>∗</sup>(*<sup>B</sup>*, *T*) reaches a maximum *<sup>M</sup>*<sup>∗</sup>*M* at a certain temperature *TM* ∝ *B* [15]. Accordingly, to measure the effective mass and temperature, it is convenient to introduce the scales *<sup>M</sup>*<sup>∗</sup>*M* and *TM*. In this case, we have new variables *<sup>M</sup>*<sup>∗</sup>*N* = *<sup>M</sup>*<sup>∗</sup>/*M*<sup>∗</sup>*M* that we call normalized effective mass and *TN* = *T*/*TM* that we call normalized temperature. As a result, *<sup>M</sup>*<sup>∗</sup>*N* becomes a function of the only variable *TN* ∝ *T*/*B*, as seen from Figure 3b.

**Figure 3.** Electronic specific heat of YbRh2Si2. Panel (**a**): Specific heat *C*/*T*, versus temperature *T* as a function of magnetic field *B* [36] shown in the legend. Panel (**b**): The normalized effective mass *<sup>M</sup>*<sup>∗</sup>*N* as a function of normalized temperature *TN* ∝ *T*/*B*. *<sup>M</sup>*<sup>∗</sup>*N* is extracted from the measurements of the specific heat *C*/*T* on YbRh2Si2 in magnetic field *B* [36], see panel (**a**), listed in the legend. Approximate constant effective mass *M*∗ at *TN* < 1 is typical for the normal Landau Fermi liquids, and is shown by the arrow.

In the vicinity of the FCQPT, the normalized effective mass *<sup>M</sup>*<sup>∗</sup>*N*(*TN*) can be well approximated by a certain universal function [15,19], interpolating the solutions of Equation (3) between the LFL state, given by Equation (11), and the NFL one, given by Equation (10) [15]

$$M\_N^\*(T\_N) \approx c\_0 \frac{1 + c\_1 T\_N^2}{1 + c\_2 T\_N^{8/3}}.\tag{12}$$

Here, *TN* = *T*/*TM*, *c*0 = (1 + *<sup>c</sup>*2)/(<sup>1</sup> + *<sup>c</sup>*1), where *c*1 and *c*2 are free parameters. Since the magnetic field *B* enters Equation (3) as *μBB*/*T*, the maximum temperature *TM* ∼ *μBB*. Consequently, from Equation (12),

$$T\_M \simeq a\_1 \mu\_B B; \quad T\_N = \frac{T}{T\_M} = \frac{T}{a\_1 \mu\_B B} \propto \frac{T}{B}.\tag{13}$$

where *a*1 is a dimensionless parameter, and *μB* is the Bohr magneton. Equation (13) shows that Equation (12) determines the effective mass as a function of the single variable *TN* ∝ *T*/*B*. That is, the curves *<sup>M</sup>*<sup>∗</sup>*N*(*<sup>T</sup>*, *B*) merge into a single one *<sup>M</sup>*<sup>∗</sup>*N*(*TN*), *TN* = *T*/*TM*, as shown in Figure 4. Since *TM* ∝ *B*, from Equation (13) we conclude that the curves *<sup>M</sup>*<sup>∗</sup>*N*(*<sup>T</sup>*, *B*) coalesce into a single one *<sup>M</sup>*<sup>∗</sup>*N*(*TN* = *<sup>T</sup>*/*B*), *TN* = *T*/*TM* = *T*/*B*, demonstrating the universal scaling in HF metals [15,19,20]. This universal scaling exhibited by *MN* is also shown in Figure 4. We note that Equations (12) and (13) allow one to describe the universal scaling behavior of HF metals, see, e.g., [15,19,20].

**Figure 4.** Scaling of the thermodynamic properties governed by the normalized effective mass *<sup>M</sup>*<sup>∗</sup>*N* in the case of the application of a magnetic field *TN* ∝ *T*/*B*, as follows from Equation (13). The solid curve depicts *<sup>M</sup>*<sup>∗</sup>*N* versus normalized temperature *TN*. It is clearly seen that at finite *TN* < 1, the normal Fermi liquid properties take place. At *TN* ∼ 1, *M*-*N* enters the crossover state, and at growing temperatures it exhibits the NFL behavior.

One more important feature of the FC state is that apart from the fact that the Landau quasiparticle effective mass starts to depend strongly on external stimuli such as *T* and *B*, all relations, inherent in the LFL theory, formally remain the same. Namely, the famous LFL relation [35],

$$M^\*(B, T) \propto \chi(B, T) \propto \frac{\mathcal{C}(B, T)}{T} \propto \gamma\_0. \tag{14}$$

still holds. Here, *γ*0 is the Sommerfeld coefficient. Expression (14) has been related to the FC case, where the specific heat *C*, magnetic susceptibility *χ* and effective mass *M*∗ depend on *T* and *B*. Taking Equation (14) into account, we obtain that the normalized values of *C*/*T* and *χ* are of the form [15,19]

$$M\_N^\*(B, T) = \chi\_N(B, T) = \left(\frac{\mathbb{C}(B, T)}{T}\right)\_N. \tag{15}$$

From Equation (15) we see that the above thermodynamic properties have the same scaling displayed in Figure 4. As a result, we shall see that the observed scaling allows us to construct a general schematic phase diagram, see Section 3.3.
