Pomeranchuk instabilities in holographic metals

Strongly correlated electron systems have been at the heart of most recent research in condensed matter theory sachdev_2000 . The underlying strongly coupled dynamics is thought to be responsible of the richness of the phase diagram of high $T_{c}$ superconductors. It contains normal metallic regions, as well as an exotic phase known as “strange metal” or “non-Fermi liquid” RevModPhys.73.797 which is believed to be described by strongly coupled fermionic degrees of freedom. There are also regions in which rotational symmetry is broken, the so-called “nematic” phases, as well as inhomogeneous “smectic” and “chessboard” phases.

The transition from an isotropic Fermi liquid to a nematic phase is believed to be driven by a Pomeranchuk instability pomeranchuk . Such instability arises when an excitation of the ground state of the Fermi liquid results in a net decrease of the total energy. In Landau theory, such perturbation is represented by a deformation of the Fermi surface. By decomposing the deformation onto an orthonormal basis, Pomeranchuk obtained a set of conditions under which the Fermi liquid is stable. This method can be generalized to lattice systems or anisotropic Fermi surfaces PhysRevB.78.115104 ; PhysRevB.80.075108 and to finite temperature and magnetic field uno ; dos ; RodriguezPonte2013 , and it relies on the weakness of the quasi-particle coupling, or in other words on the validity of the Landau formula. This implies that from the strange metal perspective, since the dynamics is strongly coupled, the detection of fermionic instabilities becomes more difficult.

The holographic description AHARONY2000183 of a homogeneous fermionic phase has been shown to account for some of the interesting properties of the strange metal Lee:2008xf ; Liu:2009dm ; Faulkner:2009wj ; Faulkner:2010da ; Hartnoll:2010gu ; Hartnoll:2011dm . In particular, the resulting spectral function is compatible with a Fermi surface with or without long-lived quasi-particles. Based on that, in this paper, we analyze Pomeranchuk instabilities of the strange metal phase from the holographic perspective. We use a holographic background in which we propagate a Dirac spinor. Being weakly coupled, such spinor can be described by Landau’s theory, and its stability under Fermi surface deformations can be studied. This accounts for a description of the anisotropic instabilities of the dual strange metal.

The generalization of Pomeranchuk method to arbitrary curved spaces is not possible, since it relies on a momentum space representation of the fermionic system. Nevertheless, the special kind of “planar” bulk spacetimes used in holography have the additional feature of a translational symmetry in the spatial directions spanning the boundary. This allows for a labeling of the bulk states by a momentum index $\vec{k}$, complemented by an additional index $m$ labeling the oscillation mode in the holographic direction. Then the $D+1$ dimensional curved space fermionic system can be interpreted as a $(D-1)+1$ dimensional flat space system with multiple fermion species labeled by $m$. Pomeranchuk method can then be straightforwardly applied to such multi-fermion flat space weakly coupled system.

In what follows, we sketch the necessary steps needed to go from a $3+1$ dimensional action for spinors in AdS spacetime to a $2+1$ dimensional Hamiltonian for a multi-fermion system in flat space. Then we construct the corresponding Landau theory, and apply the Pomeranchuk method to it. To improve readability, only the relevant steps of the calculations are shown in the bulk of the paper, leaving the details to the appendices ¹¹1Notice that Pomeranchuk instabilities were studied in the holographic context in Edalati:2012eh from a different perspective, focusing in the distortions of spectral functions. See also Liu:2014mva for a related issue..

We consider a holographic background consisting of a metric and an electromagnetic field with the generic “planar” form

This is a solution of the Einstein-Maxwell equations with a negative cosmological constant, as well as possible additional matter contributions to the energy-momentum tensor. The metric components $f(z)$ and $g(z)$ are functions of the holographic coordinate $z$, and the geometry asymptotes AdS spacetime as $z$ goes to zero, provided $f(z)\sim g(z)\sim 1/z^{2}$. The electric potential $h(z)$ is also a function of $z$, and approaches a constant $h(z)\sim\mu$ at the boundary, which is identified with the chemical potential on the holographic theory.

Specific backgrounds with the form (1) are the Reisner-Nordstrom AdS black-hole Lee:2008xf ; Liu:2009dm ; Faulkner:2010da ; Faulkner:2009wj , the holographic superconductor Hartnoll_2008 , and the electron star at zero Hartnoll:2010gu ; Hartnoll:2011dm ; Cubrovic:2011xm and finite Puletti:2010de ; Hartnoll:2010ik temperatures. Although we keep our formalism as general as possible, in order to get concrete results in Section 4 we apply it to the background of reference Hartnoll:2010gu .

In the background defined above we propagate a spinorial perturbation $\Psi$, whose dynamics is dictated by the free Dirac action

where $/\!\!\!\!{\cal D}$ stands for the covariant derivative containing both curved spacetime and gauge contributions, contracted with the curved space Dirac matrices. Since the energy-momentum tensor and the electric current are quadratic in the spinor perturbation, to linear order in $\Psi$ we can work in the probe limit in which the background (1) is not perturbed.

The general solution to (2) can be decomposed as

in terms of time dependent coefficients $c_{\alpha m\vec{k}}(t)$ and $z$-dependent spinors $\psi_{\alpha m\vec{k}}(z)$. Here the label $\alpha=1,2$ is a spin index, and $\vec{k}\in\mathbb{R}^{2}$ represents the momentum in the $xy$ plane, while $m\in\mathbb{N}$ characterizes the number of oscillations in the $z$ direction. Finally ${\cal N}_{\alpha m\vec{k}}$ is a normalization constant and the factor $z/f^{1/4}(z)$ was introduced for calculational convenience. The spinors $\psi_{\alpha m\vec{k}}(z)$ are written in terms of the solution to an ordinary differential equation in the variable $z$, with a Schrödinger-like form. When physically meaningful boundary conditions are imposed at the extremes of the $z$ range, a (possibly complex) quantized dispersion relation is obtained $\omega_{m}(k)$, with which we can write $c_{\alpha m\vec{k}}(t)\propto\exp(-i\omega_{m}(k)t)$.

Expression (3) allows us to quantize the system by promoting the coefficients $c_{\alpha m\vec{k}}(t)$ to operators $c_{\alpha m\vec{k}}$ and $c^{\dagger}_{\alpha m\vec{k}}$ satisfying fermionic anticommutation relations. Then $c_{\alpha m\vec{k}}^{\dagger}$ creates a fermionic perturbation with momentum $\vec{k}$, spin $\alpha$ and mode index $m$, and $c_{\alpha m\vec{k}}$ annihilates it.

The steps of the derivation leading to decomposition (3) as well as the explicit form of the components of the spinor $\psi_{\alpha mk}(z)$ are reviewed in detail in Appendix A.

Now we introduce interactions among the spinorial perturbations, which represent $1/N$ corrections to the holographic description. We do so by supplementing the action with the additional term

where we made explicit the spinor indices $\sigma\in\{1,2,3,4\}$, and the Lorentz invariant tensor $T^{\bar{\sigma}\bar{\sigma}^{\prime}}_{\sigma\sigma^{\prime}}$ represents the most general four fermion covariant interaction in curved space PhysRevB.59.7140 .

We see that it is written completely in terms of Dirac matrices, and depends linearly on a set of five coupling constants $g_{1},\dots,g_{5}$, which we assume are small.

We can now obtain the Hamiltonian resulting from (2) and (4), it takes the form

Here the frequencies $\omega_{m}(k)$ play the role of dispersion relations for each fermionic mode. On the other hand the new tensor $t^{\alpha_{1}m_{1}{\vec{k}}_{1};\alpha_{2}m_{2}{\vec{k}}_{2}}_{\alpha_{3}m_{3}{\vec{k}}_{3};\alpha_{4}m_{4}{\vec{k}}_{4}}$ contains the information about the interaction strengths among different modes, and it results from contracting the position space interaction tensor $T^{\bar{\sigma}\bar{\sigma}^{\prime}}_{\sigma\sigma^{\prime}}$ with integrals in the $z$ direction of quartic products of the free fermionic eigenstates $\psi_{\alpha m\vec{k}}(z)$.

The explicit forms of the interaction tensors $T^{\bar{\sigma}\bar{\sigma}^{\prime}}_{\sigma\sigma^{\prime}}$ and $t^{\alpha_{1}m_{1}{\vec{k}}_{1};\alpha_{2}m_{2}{\vec{k}}_{2}}_{\alpha_{3}m_{3}{\vec{k}}_{3};\alpha_{4}m_{4}{\vec{k}}_{4}}$. as well as that of the aforementioned quartic integrals, is not relevant for the moment. For the details of the derivation of (3) we refer the reader to Appendix B.

With equation (3), we have succeded in re-writing the bulk dynamics as that of a second quantized Hamiltonian for a two dimensional multi-fermion system in which the index $m$ denotes fermion species. Since the coupling in the bulk is assumed to be weak, we can safely rely on the Landau description of the Fermi liquid.

Assuming that the ground state of the Hamiltonian (3) is characterized by a certain set of occupation numbers $N_{\alpha m\vec{k}}$, then the excitations can be described by their variations $\delta N_{\alpha m\vec{k}}$. The grand canonical energy of an excitation is then written as the Landau formula

where $f_{\alpha m\bar{\alpha}\bar{m}}(\vec{k},\vec{\bar{k}})$ is the so-called “interaction function” which is obtained from the tensor $t^{\alpha_{1}m_{1}{\vec{k}}_{1};\alpha_{2}m_{2}{\vec{k}}_{2}}_{\alpha_{3}m_{3}{\vec{k}}_{3};\alpha_{4}m_{4}{\vec{k}}_{4}}$, and the quasiparticle dispersion relation $\epsilon_{m}(k)$ takes the form

For future use, notice that the quasiparticle dispersion relation equals the frequency plus corrections of first order in the coupling constants.

For the most general covariant quartic perturbation (4), the interaction function takes a particularly simple angular dependence in the momentum plane. Indeed, if we write $\vec{k}=k\,(\cos\theta,\sin\theta)$ then it can be decomposed in only three Fourier modes, as

where the constant, sine and cosine coefficients depend on a reduced subset of the components of the tensor $t^{\alpha_{1}m_{1}{\vec{k}}_{1};\alpha_{2}m_{2}{\vec{k}}_{2}}_{\alpha_{3}m_{3}{\vec{k}}_{3};\alpha_{4}m_{4}{\vec{k}}_{4}}$, or in other words on the position space interaction tensor $T^{\bar{\sigma}\bar{\sigma}^{\prime}}_{\sigma\sigma^{\prime}}$ contracted with integrals of the fermionic states $\psi_{\alpha m\vec{k}}(z)$.

The details of the construction of the Landau description starting with the underlying Hamiltonian (3), as well as the explicit form of the interaction function (10) in terms of the interaction tensor and wavefuntion integrals, are presented in Appendix C.

We can now make use of the Pomeranchuk technique for the above defined two dimensional multicomponent Landau Fermi liquid, in order to diagnose instabilities arising from an anisotropic deformation of its Fermi surface.

We start by decomposing the deformation on the occupation numbers, that charactherize the excitations of the Fermi liquid, in the form

where ${\sf H}(\cdot)$ is the Heavyside unit-step function, and $\delta g_{\alpha m}(\vec{k})$ are arbitrary functions of the momentum characterizing the excitation. When plugged back into (8) the delta functions force the evaluation of the expression on the Fermi momentum $k_{F}^{m}$ defined by $\epsilon_{m}(k_{F}^{m})=0$. The remaining angular integrals result on a grand canonical energy that is a quadratic form in the Fourier components $\delta g_{\alpha m}{}_{n}^{(c,s)}$ of the parameters $\delta g_{\alpha m}(\vec{k})|_{k=k_{F}^{m}}=\sum_{n=0}^{\infty}\left(\delta g_{\alpha m}{}_{n}^{c}\;\cos(n\theta)+\delta g_{\alpha m}{}_{n}^{s}\;\sin(n\theta)\right)$. Notice that, since the interaction function (10) has only three Fourier components, the only modes that depend on the couplings (and hence could lead to negative minors in the quadratic form) are $\delta g_{\alpha m}{}_{0}^{c}$ and $\delta g_{\alpha m}{}_{1}^{(c,s)}$. The first one is isotropic, while the remaining two completely break the rotational symmetry.

If the grand canonical energy of an excitation is negative, the system decreases its energy by creating more such excitations. To avoid such instability, we need to impose that the aforementioned quadratic form is positive definite. This is guaranteed provided all the minors of the quadratic kernel are positive, or in other words

where $v_{F}^{m}=d\epsilon_{m}(k)/dk|_{k=k_{F}^{m}}$ are the Fermi velocities and $f^{(0,c,s)}_{\alpha m\alpha^{\prime}m^{\prime}}$ are the Landau parameters, given by the Fourier components in (10) evaluated at the corresponding Fermi momenta. Notice that, provided the Fermi velocities are positive, the quotient ${k_{F}^{m}}/{v_{F}^{m}}$ can be removed from the prefactors. Moreover, we can perturbatively replace ${k_{F}^{m}}/{v_{F}^{m}}$ in the parenthesis by its free version, defined by $\omega_{m}(k_{f}^{m\,{\sf free}})=0$ and $v_{F}^{m\,{\sf free}}=d\omega_{m}(k)/dk|_{k=k_{F}^{m\,{\sf free}}}$.

When some of the minors in (12) is negative, a linear combination of the modes $\delta g_{\alpha m}{}_{0}^{c}$ causes a net decrease in the energy, resulting in an instability. As mentioned above, such modes are isotropic and therefore the corresponding instability would be a Stoner one, i.e. an instability with respect to adding or removing particles from the ground state. On the other hand, if the negative minor shows up in (13), the unstable direction correspond to a linear combination of the modes $\delta g_{\alpha m}{}_{1}^{(s,c)}$, and the resulting deformation completely breaks rotational invariance. It is tempting to imply that such symmetry breaking pattern is inherited by the phase that stabilizes the system after the phase transition.

A review of Pomeranchuk method, including the details of the calculations resulting in formulas (12)-(13), can be seen in Appendix D.

In summary, in order to test the anisotropic Pomeranchuk instabilities of an holographic Fermi liquid, we need to

1. 1.

   Obtain the fermionic modes $\psi_{\alpha mk}(z)$ in the expansion (3), that we read from (58) and (64)), and their frequencies $\omega_{m}(k)$. This is done by solving a Schrödinger-like equation with potential (44) in the holographic direction for a function $\phi_{mk}(z)$. Notice that we only need the modes with very small frequency, since we will evaluate them at the free Fermi momentum $k_{F}^{m\,{\sf free}}$ where the frequency vanishes.

2. 2.

   Calculate the free Fermi velocities as $v_{F}^{m\,{\sf free}}=d\omega_{F}^{m}/dk|_{k=k_{F}^{m\,{\sf free}}}$. Incidentally, this is why we need modes with non-vanishing small frequency, even if just those with vanishing frequency contribute to the quantities evaluated at the free Fermi momenta.

3. 3.

   Integrate a particular set of quartic products of fermionic modes in the holographic direction given in (LABEL:eq:fermion.integrals.all), in order to go from the coupling tensor in spin indices $T^{\bar{\sigma}\bar{\sigma}^{\prime}}_{\sigma\sigma^{\prime}}$ given by to $t^{\alpha_{1}m_{1}{\vec{k}}_{1};\alpha_{2}m_{2}{\vec{k}}_{2}}_{\alpha_{3}m_{3}{\vec{k}}_{3};\alpha_{4}m_{4}{\vec{k}}_{4}}$ and then to the Landau parameters $f^{(c,s)}_{\alpha m\alpha^{\prime}m^{\prime}}$.

4. 4.

   Check whether all the minors in (12)-(13) are positive. Notice that each minor of $M\times M$ has degree $M$ on the interaction function, which is linear in the coupling constants. This implies that each condition imposes a polynomial restriction of order $M$ in coupling space.

At any point in coupling space at which condition 4 is not satisfied, the quadratic form has a negative eigenmode, implying that the grand canonical energy is decreased by the corresponding excitation. This results in an instability of the system. If the negative minor appears in (12), the resulting instability is isotropic, while if it shows up in (13), it breaks completely the rotational symmetry.

We applied the above steps to the specific example of the zero temperature electron star background of Hartnoll:2010gu . The electron star represents the holographic dual to the ground state of a highly degenerate system of fermions at zero temperature. It is constructed by coupling the gravitational and electromagnetic degrees of freedom in the bulk to a perfect fluid representing the fermions, whose equation of state is obtained in the Thomas-Fermi approximation. A summary of the electron star solution is given in Appendix E.

In order to obtain the fermionic modes and calculate the Fermi momenta and velocities as required by points 1 and 2, we used a Wentzel-Kramers-Brillouin (WKB) approximation on the spinorial perturbation. This is consistent with the use of the Thomas-Fermi approximation for the bulk fluid. The relevant details of the WKB method are presented in Appendix F.

We explored the phase diagram by turning on the coupling constants $g_{1},\dots,g_{5}$ by pairs. Sitting at different values of the non-vanishing couplings on every coupling plane, we evaluated the minors in equations (12) and (13) order by order. For example, on the plane $g_{2}$-$g_{3}$, we obtained the plots at the top of Fig.1. The regions are painted with a color that depends on the order $M$ of the first unstable minor. The plot on the top left correspond to instabilities that preserve rotational invariance, while that on the top right denotes instabilities that break it completely. These plots were combined on the one at the bottom. The resulting unstable regions of the phase diagram can be seen in Fig. 2, in which all different coupling planes are presented. We see that as $M$ grows (and so does the degree of the polynomial in point 4 above) the region in coupling space where the system is stable gets smaller, converging to an isolated wide island around the origin, which is simply connected.

In order to interpret our results from the boundary perspective, we recall that the electron star describes a holographic strongly coupled theory whose ground state has a fermionic condensate Hartnoll:2010gu . Classical spinor perturbations on the bulk then correspond to fermionic excitations on the boundary theory Hartnoll:2011dm .

The interesting point is that, as we decompose the bulk spinor into $z$-modes, the set of Fermi momenta we obtained for them corresponds exactly to those defining the “holographically smeared Fermi surface” in reference Hartnoll:2011dm . It is natural then to interpret collective bulk modes, made by deforming the Fermi surface of each bulk $z$ mode, as corresponding to collective excited states on the boundary theory. These “deformations of the smeared Fermi surface” become unstable when a bulk Pomeranchuk instability shows up, leading to an instability of the boundary state that could easily be interpreted as a boundary Pomeranchuk instability.

We developed a general method to study anisotropic fermionic instabilities on a strongly coupled Fermi liquid via the AdS/CFT duality. It entails to perform a Pomeranchuk analysis on the dual bulk fermion, adapting the formalism to a curved space-time with a planar slice. The key step is to rewrite the single fermion in the bulk in terms of its modes in the holographic direction, resulting on a multi-component systems of lower dimensional fermions which move in the planar slice. By analyzing the minors of a quadratic form, we are able to detect when the fermionic modes become unstable under anisotropic deformations of their Fermi surface. The resulting phase transition breaks the rotational symmetry in the planar slice, and consequently in the boundary theory, potentially leading to a nematic phase.

We kept the formalism as general as possible, being suitable to be applied to many Euclidean-invariant boundary theories. This includes the zero temperature electron star Hartnoll:2010gu ; Hartnoll:2011dm , the finite temperature electron star Puletti:2010de ; Hartnoll:2010ik , as well as other examples²²2Since the method relays on the existence of a bulk Fermi surface, to adapt it to backgrounds such like the holographic metal Lee:2008xf ; Liu:2009dm ; Faulkner:2010da ; Faulkner:2009wj , the holographic superconductor Hartnoll_2008 , the magnetically charged black-hole Albash_2008 , or the Lifshitz geometry Kachru:2008yh , one would need to work in the limit in which a bulk fermion condensate would not backreact. This limit can be subtle, and it is hard to reconcile with the validity conditions for the WKB method, so the fermion wave functions should be solved using a different approximation.. We applied the method to the zero-temperature electron star background, being able to identify the unstable region on a five dimensional coupling space.

The method can be extended to more general situations relevant to condensed matter systems. Some examples of possible future directions are:

• •

  Finite doping: the inclusion of a doping axis should be straitforward, following the lines described in Kiritsis2016 .

• •

  Finite magnetic field: the inclusion of a magnetic field would require a magnetically charged background Albash_2008 ³³3See footnote 2., and it would modify the bulk fermion coupling. Also, the Pomeranchuk analysis needs to be adapted dos .

• •

  Finite temperature fluid: as it stands, the method can be applied to the background of Puletti:2010de ; Hartnoll:2010ik in which the temperature is included via the presence of a horizon, while the bulk fluid is approximated as a finite temperature Fermi liquid. Including the effect of temperature in the bulk matter would modify the background, as well as the Pomeranchuk analysis uno .

• •

  Lifshitz scaling: considering Lifshitz geometries Kachru:2008yh would require a rewriting of the asymptotic conditions on the near boundary fermion, but the rest of the analysis remains mostly unchanged${}^{\ref{foot2}}$.

• •

  Lattice fermions: describing an underlying lattice in the holographic setup would imply the inclusion of a lattice-like metric Horowitz2012 or momentum relaxation Andrade2014 , and a modification of the Pomeranchuk analysis according to the lines of PhysRevB.78.115104 .

These are only some of the situations to which the method can be extended, we think that they deserve further investigation.

As a last remark, one could wonder whether Pomeranchuk instabilities are evident in physical quantities as we approach the edge of the unstable region⁴⁴4We would like to thank the referee for rising this point.. In a weakly coupled Fermi liquid, standard Landau theory arguments show that Landau parameters appear into some response functions. This happens for example for the compressibility or the spin susceptibility. The expressions are such that these functions become infinite when the instability is triggered by the correct Landau parameter.

Recalling that, according to Kubo formulae, response functions are written in terms of two-point correlators of local quantities, we can calculate them in holography. This is done by perturbing the background fields with the right IR boundary conditions and then evaluating the quotient of the leading and subleading parts of the perturbations at the UV. However, it is evident that in such calculation the fermionic couplings $g_{1},\dots,g_{5}$ would only enter at higher loops. Since there is no reason to assume that bulk perturbation theory would break down close to the edge of the unstable region, the natural conclusion is that, in the strongly coupled boundary theory, the response functions are not sensitive to the instability.

The authors thank Pablo Rodríguez Ponte and Ignacio Salazar Landea for relevant contributions during the early stages of this work. We also thank Carlos Lamas for helpful comments. This work has been funded by the CONICET grants PIP-2017-1109, PIP-2015-0688 and PUE 084 “Búsqueda de Nueva Física”, and UNLP grants PID-X791, PID-11/X910.