Entropy and de Haas-van Alphen oscillations of a three-dimensional
marginal Fermi liquid

Magnetic oscillations in metals - the periodic variation of virtually all physical quantities with an external magnetic field - are remarkable macroscopic quantum effects. Such quantum oscillations are periodic in $1/B$, where $B$ is the external magnetic field, and are routinely used in the experimental characterization of metals [1, 2]. Among the foremost examples is the field-induced oscillation of magnetization, known as the de Haas-van Alphen (DHVA) effect, which has played a pivotal role in the study of Fermi surfaces of elemental metals, and even those of strongly correlated systems such as the cuprates [3, 4, 5, 6, 7], iron-pnictides [8, 9], and heavy fermion materials [10, 11].

A quantitative theory of DHVA oscillations was first put forward by Lifshitz and Kosevich (LK) [12], who considered the oscillatory behavior of the thermodynamic potential of a clean, three-dimensional gas of non-interacting electrons with arbitrary dispersion. According to the LK theory, the oscillation frequencies reveal areas of “extremal orbits”, i.e., the Fermi surface orbits having extremal momentum-space areas in the plane perpendicular to B. In addition, from the temperature-dependence of the oscillation amplitude for the $k$-th harmonic (also known as the LK formula)

with $k=1,2,...$, it is possible to extract cyclotron masses associated with extremal orbits through $\omega_{c}=eB/m$. In the presence of impurities, the oscillations are exponentially damped by a finite elastic scattering rate.

Despite the obvious presence of sizable interactions in real metals, nearly all experimental observations of DHVA oscillations to date are interpreted in terms of the LK theory, often with striking success. The reason for such robustness of the predictions LK theory was given by Luttinger [13], and by Bychkov and Gorkov [14], who showed that so long as the metal remains a Fermi liquid, many-body interactions do not alter the LK predictions; instead many-body effects are manifest in renormalized parameters, such as altered extremal orbits and the fully dressed cyclotron masses $m^{*}$ as extracted from the LK formula in Eq.(1). We refer to such an extension of LK theory to interacting systems as the extended LK paradigm, which provides a well-defined theoretical prescription for interpreting DHVA oscillations. In practice, one starts with the expression for the thermodynamic potential, a functional of fully dressed propagators and self-energies. In a magnetic field, all such quantities contain both uniform and oscillatory components. The extended LK prescription involves neglecting both the temperature dependence and oscillatory components of the fermion self-energy. Doing so, one can show that the oscillations satisfy LK theory albeit with renormalized parameters. Specific applications of this theory include coupled electron-phonon systems [15, 16, 17] and disordered two-dimensional Fermi liquids [18, 19], also see [20] for a review.

In a Fermi liquid, the extended LK method is well-justified. For instance, the Fermi liquid self-energy is quadratic in temperature, which can be neglected. Moreover, the oscillatory piece of the self-energy is suppressed by a factor $\left(\omega_{c}/\mu\right)^{3/2}$ compared to the smooth part, where $\mu$ is the Fermi energy. As a consequence, the corresponding contribution to the oscillations of the thermodynamic potential is suppressed by a factor $\left(\omega_{c}/\mu\right)^{1/2}$ compared to the leading term, and thus, can also be neglected. It is far from clear, however, whether such a prescription remains valid when a metal is tuned to a quantum critical point (QCP), where interactions induced by soft order parameter fluctuations become singular, resulting in the breakdown of Fermi liquid theory, and in non-analytic temperature dependence of all self-energies.

Indeed, the naive adoption of the extended LK prescription to the case of a metal with singular interactions was questioned recently[21], from a perspective that it results in the violation of the third law of thermodynamics (Nernst’s theorem), which requires that the entropy density and the specific heat must vanish as $T\rightarrow 0$. A simple manipulation shows that this requirement implies the temperature derivative of the oscillation amplitude, $\partial A_{k}/\partial T$, must vanish in the zero temperature limit. For concreteness, we will consider a 3D metal coupled to some collective bosons which can be either undamped or overdamped. When the system is far away from the QCP it behaves like a conventional Fermi liquid, and the application of the LK prescription still leads to Eq.(1) (with renormalized quantities) from which it is obvious to see that $A_{k}(T)-A_{k}(T=0){\sim}{-}T^{2}$ at $T\ll\omega_{c}$ and therefore $\partial A_{k}/\partial T=0$ as $T\to 0$. However, when the system is close to the QCP, singular fermion interaction mediated by critical bosons results in a marginal Fermi liquid (MFL) behavior. In this regime, working within the LK paradigm leads to $A_{k}(T)-A_{k}(T=0)\sim T\ln T$ at low temperatures, and therefore the apparent divergence of $\partial A_{k}/\partial T$ as $T\to 0$, i.e. the violation of the third law. This observation leads to the breakdown of the LK paradigm, and calls for a more careful analysis of the quantum oscillations for a metal in a quantum critical regime.

In this paper, we aim to resolve this issue by providing a new theory of quantum oscillations for a quantum critical metal at low temperatures beyond the LK paradigm. In principle, one could follow Luttinger’s approach by carefully keeping track of the oscillatory parts of the fermion self-energy, which enters the thermodynamic potential. Then, the remedy of the zero temperature divergence stated above would require an exact cancellation of the divergence in the thermodynamic potential, tracking which would be a rather involved task. Here we show that one can circumvent Luttinger’s approach by utilizing the saddle point conditions for the thermodynamic potential. We present an exact formulation for computing the entropy in a magnetic field of quantum critical metals within the Migdal-Eliashberg (ME) approximation and show that the entropy manifestly satisfies the laws of thermodynamics. As concrete examples, we considered two different types of models with parabolic dispersion and MFL behavior: one with overdamped bosons and a dynamical exponent $z=3$ and the other with undamped bosons and $z=1$. Although the $z=1$ model serves as a simple and elegant demonstration of the essentials of our calculation, the $z=3$ model is more physically relevant. The comparison of the results based on these two models also unveils the important role of different dynamical scaling laws of the bosons in determining the oscillatory entropy and magnetization. Proceeding further, we obtain expressions for the entropy at temperatures and magnetic fields small compared to the bare Fermi energy of the metal, from which the expressions for the oscillation amplitudes $A_{k}(T)$ are obtained via the Maxwell relation. Our approach naturally captures the LK expressions in the non-interacting limit and results in new temperature dependence for the DHVA amplitude near a quantum critical point. Specifically, for $z=1$ bosons, we find that the low temperature behavior of $A_{k}(T)$ is $T^{2}\ln T$, while for $z=3$ case, we find that the leading contribution to $A_{k}(T)$ is $T^{4/3}$. Both of these behaviors obey the thermodynamic laws, and deviate from the standard LK formula in Eq.(1). In addition, we find that the tail of $A_{k}(T)$ at temperatures greater than the cyclotron frequency $T\gg\omega_{c}$ is of the form $T\exp\{-\#T\ln T\}$ for both cases $z=1$ and $z=3$. These asymptotic limits are summarized in Fig. 1. Such apparent deviation from the LK formula could be, in principle, detected in experiments by measuring the quantum oscillations of either magnetization or specific heat, and can be used as a direct evidence for the existence of quantum critical point in various materials.

With hindsight, our theory also provides a clear picture on where the recipe of Luttinger for the DHVA oscillations breaks down. Perhaps not surprisingly, the temperature dependence of the fermion self-energy cannot be neglected near a quantum critical point, since it exhibits non-analytic behavior. Additionally, we show explicitly that the oscillatory parts of the self-energy cannot be neglected; due to their singular temperature dependence, they in fact can contribute even more strongly to the low-temperature entropy. Lastly, we observe that the contribution from overdamped order parameter fluctuations in the metallic environment produces the dominant correction to the entropy, which in fact can contribute significantly to the temperature dependence of the DHVA amplitude. This is in sharp contrast to conventional metals (Fermi liquid regime) where the quantum oscillations are determined only by the fermions.

The outline of the paper is as follows. In Sec. II we summarize the ME theory of a marginal Fermi liquid in three dimensions and in the presence of a magnetic field. In Sec. III we recall the main steps in the derivation of the extended LK formula, and then demonstrate that, if applied to our theory at criticality, it leads to violations of the third law of thermodynamics. Next, in Sec. IV we consider the Luttinger-Ward-Eliashberg thermodynamic potential corresponding to our ME theory in a magnetic field. Starting from this representation, in Sec. V we derive a formally exact formula for the entropy that manifestly satisfies the thermodynamic constraints. This formula is then used to evaluate specific heat and magnetization oscillations in the particular case of the undamped critical boson in Sec. VI, while the role of the Landau damping is discussed in Sec. VII. Discussion and conclusions are presented in Sec. VIII. Some details of calculations are presented in the Appendices.

We will study DHVA oscillations in models exhibiting marginal Fermi liquid behavior in zero magnetic field, as it arises in a three dimensional metal tuned to a continuous quantum phase transition. Broadly stated, there are two classes of symmetry breaking quantum phase transitions in metals in the absence of disorder: those that preserve lattice translations and those that break them. Examples in the former category include ferromagnetic, Ising nematic transitions and more generally, Pomeranchuk instabilities. Examples of the latter describe the onset of density wave order. Within the former category, we can further distinguish between conserved order parameters (such as ferromagnetism) and non-conserved order parameters. We will study transitions that preserve lattice translations associated with non-conserved order parameters.

If such transitions are continuous, their universal properties are governed by a low energy theory consisting of fermions at finite density coupled to gapless bosonic order parameter fluctuations by a Yukawa-like coupling, which is the most relevant coupling in the sense of the renormalization group. A straightforward approach to such problems involves the use of Migdal-Eliashberg (ME) theory to obtain self-consistent descriptions in terms of propagators of the fermions and bosons and their respective self-energies. The bare fermion and boson propagators, $G_{0},D_{0}$ respectively, are

Here $\varepsilon_{n}=2\pi T(n+1/2)$ and $\omega_{n}=2\pi Tn$ are the fermionic and bosonic Matsubara frequencies respectively, $m$ is the fermionic mass, $c$ is the boson’s velocity, and $\mu$ is the chemical potential. The boson mass $m_{b}$ is varied by a non-thermal parameter such as pressure, doping, etc., and the quantum critical point is accessed by tuning the physical boson mass to zero. We also note that Eq. (2) assumes a simple parabolic dispersion. We will comment on possible extensions in Sec. VIII.

The self-consistent relations involving these propagators are[22, 23]

The first two equations are the Dyson equations whereas the last two define the respective self-energies. Originally used in the context of the electron-phonon problem, the ME theory neglects all quantum corrections of the boson-fermion vertex. It arises as a saddle point of recent theories with random-flavor Yukawa couplings, studied in Ref. [24, 25]. While this approximation can be justified by invoking formal large $N$ limits, it is sufficient also to assume, as is done in the electron-phonon context, that the characteristic boson speed is small compared to the Fermi velocity: (i.e. if $\beta:=v_{F}/c\gg 1$ where $v_{F}=\sqrt{2\mu/m}$ is the Fermi velocity, or if the bosons are Landau overdamped), so that a version of the Migdal’s theorem applies. For our purposes, we will take the ME equations above as the starting point for describing the quantum critical metal.

The self-consistent solution of Eqs.(3) has been extensively studied before. Here we only briefly summarize its main properties, and the details of the calculation are presented in Appendix B. First, the boson self-energy contains a Landau damping term[26],

where $\nu=k_{F}^{2}/(2\pi^{2}v_{F})$ is the density of states near the Fermi level. From this result, the $z=3$ dynamical scaling is readily recognized. Inserting this into the equation for $\Sigma(\varepsilon_{n},k)$ and neglecting any weak momentum-dependence of the fermion self-energy near the Fermi surface, i.e. approximating $\Sigma(\varepsilon_{n},k)\approx\Sigma(\varepsilon_{n})$, the solution for $\Sigma(\varepsilon_{n})$ can be found in a straightforward way. In the low-temperature limit where $\varepsilon_{n}$ is treated as a continuous variable, $\Sigma(\varepsilon_{n})$ contains a leading non-analytic temperature dependence $T\ln T$, with a subleading term of the order of $\mathcal{O}(T)$. Keeping only the leading term, we find, as $T\to 0$,

where $\bar{g}^{2}=g^{2}/(4\pi^{2}c^{2}v_{F})$ is the dimensionless coupling constant, and $\Lambda_{D}=2\beta\omega_{D}^{3}/(\pi g^{2}\nu)$ is the energy cutoff expressed in terms of the effective ”Debye frequency”, $\omega_{D}=E_{F}/\beta\ll E_{F}$. The first two terms of Eq.(5) is the marginal Fermi liquid self-energy which was first discussed as a phenomenological model of the normal state of cuprate superconductors[27]. Similar MFL self-energy can also be obtained in the undamped boson model, in which the boson self-energy is neglected and the dynamical exponent is, therefore, $z=1$. Following completely parallel calculations, we find that the fermion self-energy is identical to Eq.(5) with $\bar{g}^{2}/3$ replaced with $\bar{g}^{2}$ and $\Lambda_{D}$ replaced with $\omega_{D}$ (see Appendix B for calculation details). These expressions can also be analytically continued to the real frequency axis according to the following convention

Here the superscript ‘R’ denotes retarded functions which are analytic in the upper half-plane.

Before constructing a theory of DHVA oscillations of marginal Fermi liquids described in the previous section, we recount the standard approach to DHVA oscillations and show how they break down at a quantum critical point. Since the breakdown of the extended LK approach to DHVA is not specific to the ME approximation, we present a more general discussion in what follows.

A clean, three-dimensional spherically symmetric Fermi gas in a magnetic field $B{\widehat{z}}$ has the following thermodynamic potential per unit volume ($\hbar=k_{B}=1$):

where $\epsilon_{np_{z}}=\frac{p_{z}^{2}}{2m}+\omega_{c}(n+1/2)$. Upon summing over the Landau levels via the Poisson formula ¹¹1We refer to the first term in this summation formula as the ‘smooth’ part, whereas the second term is denoted as the ‘oscillatory’ part.,

and performing the $p_{z}$ integration via saddle point methods, one finds the following expression for the oscillatory component of the thermodynamic potential [12, 2]:

where $A_{k}$, the amplitude of the k^th harmonic of the non-interacting system is given by Eq. (1). Throughout the rest of the paper, we are going to use the terms ‘smooth’, ‘non-oscillatory’, or ‘uniform’ interchangeably to denote contributions to various quantities (i.e. observables, self-energies, etc.) that are smooth functions of the magnetic field. Such contributions are typically produced by the first term in the Poisson summation formula, Eq.(19). On the other hand, the remaining contributions that exhibit oscillations as a function of $\mu/\omega_{c}$ will be referred to as ‘oscillatory’ terms which originate from the second term on the r.h.s. of Eq.(19). Upon introducing the scaling variable $\lambda=2\pi^{2}T/\omega_{c}$, it becomes evident that $\Xi_{osc}/\Xi_{n-osc}\sim(\omega_{c}/\mu)^{5/2}$, where $\Xi_{n-osc}$ denotes the non-oscillatory piece of the thermodynamic potential. Next, including interactions, the thermodynamic potential for a system with Hamiltonian $H=H_{0}+H_{int}$ (and $H_{0}$ is the free Fermi gas Hamiltonian), namely

can equivalently be expressed as [30]

where the Green function ${\widehat{G}}$ and electron self-energy ${\widehat{\Sigma}}$ are matrix-valued, the trace operation takes care of momentum integration and Landau level summation, and the regularization factor $e^{i\varepsilon_{m}0^{+}}$ is assumed. The last term $\Phi$ is the so-called Luttinger-Ward functional, which involves an infinite set of closed connected skeleton diagrams (i.e. those that have no self-energy insertions but with fully dressed propagators on each internal line). It can be shown that the expression for $\Xi$ in Eq. (22) is stationary with respect to variation of the exact self-energy.

In a magnetic field, the self-energy will consist of a uniform piece and an oscillatory piece: ${\widehat{\Sigma}}(\varepsilon_{m})={\widehat{\Sigma}}^{(n-osc)}(\varepsilon_{m})+{\widehat{\Sigma}}^{(osc)}(\varepsilon_{m})$. In three dimensions, ${\widehat{\Sigma}}^{(osc)}$ is smaller than its non-oscillatory component by a factor $(\omega_{c}/\mu)^{3/2}$, provided that the interactions responsible for self-energy corrections are sufficiently short-ranged (as an example, in Appendix A we explicitly demonstrate this suppression in case of the electron-phonon interaction). Thus, expanding the thermodynamic potential in powers of ${\widehat{\Sigma}}^{(osc)}$ and making use of the stationary condition, the correction to the thermodynamic potential from the oscillatory self-energy is $\mathcal{O}([{\widehat{\Sigma}}^{(osc)}]^{2})\sim(\omega_{c}/\mu)^{3}$. It is thus parametrically smaller than the bare LK expression above, and can be neglected.

Upon neglecting all contributions from ${\widehat{\Sigma}}^{(osc)}$, it is a simple matter to convince oneself that a cancellation occurs between the second term in Eq. (22) and $\Phi$. We therefore arrive at Luttinger’s starting expression from which to extract DHVA oscillations:

After performing the Landau level summation and the $k_{z}$-integral, we can extract the expression for the oscillation amplitudes,

Here we assumed that ${\widehat{\Sigma}}^{(n-osc)}(\varepsilon_{m})$ is only weakly dependent on the magnetic field, and thus, can be diagonalized in the momentum basis as $\Sigma(\varepsilon_{m},k_{\parallel},k_{z})$ at the leading order in $\omega_{c}/\mu$ [14]. Note that by setting $\Sigma=0$ we immediately arrive at Eq.(1).

For later convenience, we convert the Matsubara sum above into a real frequency integral ²²2We use the following sign convention for analytic continuation: $i\epsilon_{m}\rightarrow\omega+i\delta,i\Sigma(\epsilon_{m})\rightarrow-i\Sigma(-i\omega)=\Sigma^{Ret}(\omega)$. , to obtain the following expression for the amplitude:

where the self-energy appearing above is the non-oscillatory self-energy. In the absence of self-energy corrections, the expression above reproduces the LK form for the amplitude. We next discuss thermodynamic constraints on the expression above in the case of interacting systems.

In this section, we return to the marginal Fermi liquid in the ME approximation. For a generic coupled electron-boson system, the corresponding thermodynamic potential, viewed as a variational free energy functional of propagators and self-energies, yields the self-consistent relations as a saddle point condition [22]. A key advantage of the ME approximation is that the Luttinger-Ward functional $\Phi$ becomes especially simple and could be expressed as a single skeleton diagram (rather than an infinite number of them), such that the resulting saddle point conditions coincide with the ME equations. A thermodynamic potential having these properties can be written as the sum of three different contributions:

where the fermionic part $\Xi_{\rm el}$ is given by

Here the fermionic Green’s function and the self-energy are expressed in the coordinate-frequency representation. The bare Green’s function $G_{0}$ is defined through the standard equation $(i\varepsilon_{m}-\bm{\pi}^{2}/2m+\mu)\delta(\bm{r}-\bm{r}^{\prime})=G_{0}^{-1}(\bm{r},\bm{r}^{\prime},\varepsilon_{m})$.

The bosonic contribution to the thermodynamic potential $\Xi_{\rm bos}$ has the following form

The bare propagator $D_{0}$ is defined in the same way as in Eq. (2). Finally, the Luttinger-Ward functional $\Phi$ in the ME approximation captures the interaction between the bosons and fermions, and is given by

At zero magnetic field, the thermodynamic potential can be evaluated in the momentum basis, yielding the following expression for the fermionic part

and the Luttinger-Ward functional reads as

Viewing $\Xi$ above as a functional of $G,D,\Pi,\Sigma$, it is evident that the first two relations in Eq. 3 follow from $\partial\Xi/\delta\Pi=0$ and $\delta\Xi/\delta\Sigma=0$, whereas the latter two relations in Eq. 3 derive from $\partial\Xi/\delta G=0$ and $\delta\Xi/\delta D=0$.

In a similar way, the thermodynamic potential $\Xi$ in a finite magnetic field results in Eqs. (7,8,14) upon varying it with respect to propagators and self-energies. One could also directly express the thermodynamic potential in the basis of Landau levels. The fermionic part is then given by

and the electron-boson interaction (the Luttiner-Ward functional) takes the form

with the Green’s functions $G_{n}(\varepsilon_{m},k_{z})$ and self-energies $\Sigma_{n}\left(\varepsilon_{m},k_{z}\right)$ defined for a given Landau level, and related to each other via Eq. (12). Variations of $\Xi$ with respect to these propagators directly lead to the Eq. (17) and the second line of Eq. (14).

At this point, there are multiple ways of how one could proceed. For instance, by following the logic outlined in [33], one could note that the second term in the bosonic part of the thermodynamic potential Eq. (35) exactly cancels the LW (interaction) term $\Phi$, Eq. (40). As a result, $\Xi$ can be written in the following form

Next, if $\Sigma_{n}(\varepsilon_{n},k_{z})$ does not have any essential dependence on the Landau level $n$ then the ‘smooth’ contributions to the Poisson summation in the last two terms cancel each other, and we obtain

Note that the bosonic term also contains oscillatory contributions via oscillations of $\Pi(\omega_{m},q)$. The amplitude in the first term is given by

However, Eq. (42) and Eq. (43) are not very convenient for extracting the low-temperature behavior of the oscillation amplitudes since here one has to evaluate the self-energy dependence on temperature $T$. Instead, one can make the temperature dependence more manifest by considering the low-$T$ asymptotic of the entropy instead of the thermodynamic potential itself, which we derive in the next section. The DHVA magnetization can be then obtained via a Maxwell relation. To this end, we analytically continue the expression for the thermodynamic potential and express it in terms of real frequencies. Doing so, we find

where $\Phi$ can be expressed in terms of the fermion and boson spectral functions as

Thus, starting with a thermodynamic potential whose saddle point yields the Migdal-Eliashberg self-consistency relations, we obtained an alternate expression for the thermodynamic potential expressed in the basis of Landau levels, which will prove to be more convenient in analyzing DHVA oscillations. In the next section, we obtain an exact expression for the entropy, from which the low-temperature behavior of the DHVA oscillations can be extracted via a Maxwell relation.

In this section, we derive the exact expression for the entropy that only involves temperature derivatives of the distribution functions. Our derivation extends the result of [34] to the finite magnetic field case. Starting with the expression for the thermodynamic potential in terms of real frequencies derived in the previous section, we make use of the following schematic relation