Exciton-driven quantum phase transitions in holography
We study phase transitions driven by fermionic double-trace deformations in gauge-gravity duality. Both the strength of the double trace deformation and the infrared conformal dimension/self-energy scaling of the quasiparticle can be used to decrease the critical temperature to zero, leading to a line of quantum critical points. The self-energy scaling is controlled indirectly through an applied magnetic field and the quantum phase transition naturally involves the condensation of a fermion bilinear which models the spin density wave in an antiferromagnetic state. The nature of the quantum critical points depends on the parameters and we find either a Berezinskii-Kosterlitz-Thouless-type transition or one of two distinct second order transitions with non-mean field exponents. One of these is an anomalous branch where the order parameter of constituent non-Fermi liquid quasiparticles is enhanced by the magnetic field. Stabilization of ordered non-Fermi liquids by a strong magnetic field is observed in experiments with highly oriented pyrolytic graphite.
Keywords: AdS/CFT, strongly correlated electrons, quantum criticality, graphene
The anti-de Sitter/conformal Field Theory correspondence (AdS/CFT) or gauge/gravity duality is a new proving ground to describe strongly correlated systems, and its application to unresolved questions in condensed matter is an exciting new direction. It is especially compelling, as conventional methods, such as large- Ref.largeN () and -type Ref.epsilon () expansions fail to describe quantum critical behavior in -dimensional systems. The primary examples of such are the strange metal states in the high cuprates and heavy fermion systems. Both systems are characterized by anomalous behavior of transport and thermodynamic quantities. In heavy fermions, the Sommerfeld coefficient grows as the temperature is lowered, meaning that the effective mass of the electrons on the Fermi surface diverges or the Fermi energy of the electrons vanishes Ref.Schofield:2005 (). In the strange metal phase of the high superconductors as well as in heavy fermions near a quantum phase transition, the resistivity is linear with temperature . These anomalous behaviors are partly explained by the phenomenological marginal Fermi liquid model Ref.Varma:1989zz (), and it is an early success of AdS/CFT that the marginal Fermi liquid can be seen to emerge as the low-energy dynamics of a consistent theory.
A particularly simple gravity description for strongly interacting finite density matter is the planar AdS-Reissner-Nordström (AdS-RN) black hole (BH), which is dual to a system at finite chemical potential. While the AdS-RN black hole is a natural starting point to study the universal aspects of finite charge density systems, the universality of a black hole makes it difficult to explain experiments that are keen on the nature of the charge carriers, such as transport properties (e. g. conductivity). In particular the dominance of Pauli blocking for observed physics, requires that at the minimum one needs to add free Dirac fermions to the AdS-RN background. A self-consistent treatment shows that this system is unstable to a quasi-Lifshitz geometry in the bulk Refs.Review:2010 (); Hartnoll:es (); czs2010 (), that encodes for a deconfined Fermi liquid system Refs.Hartnoll:2010xj (); Sachdev:2010um (); Huijse:2011hp (); Sachdev:2011ze (). Here we shall initiate the study of instabilities in the unstable metallic AdS-RN phase that are driven by Fermi bilinears.
The essential low-energy property of the metallic system dual to the AdS-RN black hole background is the emergence of Fermi surfaces Refs.Leiden:2009 (); MIT () where the notion of a quasiparticle needs not be well defined, i.e. stable Ref.Faulkner:2009 (). In Ref.Leiden:2010 (), we used the magnetic field as an external probe to change the characteristics of the Fermi surface excitations and as a consequence the transport properties of the system. It strongly suggested that a quantum phase transition should occur when the underlying quasiparticle becomes (un)stable as a function of the magnetic field. The study in this article of the influence on stability of Fermi bilinears allows us to show that there is a phase transition between the two regimes and that for a specific set of parameters the critical temperature vanishes. Our work is therefore also a fermionic companion to Ref.Faulkner:2011 ().
Continuing the connection of AdS models to actual observations, the results we find resemble other experimental findings in quantum-critical systems. At low temperatures and in high magnetic fields, the resistance of single-layer graphene at the Dirac point undergoes a thousandfold increase within a narrow interval of field strengths Ref.Novoselov-Geim:2007 (). The abruptness of the increase suggests that a transition to a field-induced insulating, ordered state occurs at the critical field Ref.Checkelsky:2009 (). In bilayer graphene, measurements taken at the filling factor point show that, similar to single layer graphene, the bilayer becomes insulating at strong magnetic field Ref.Cadden-Zimansky:2009 (). In these systems, the divergent resistivity in strong magnetic fields was analyzed in terms of Kosterlitz-Thouless localization Ref.Checkelsky:2009 () and the gap opening in the zeroth Landau level Ref.Novoselov:2009 (). However, it remains a theoretical challenge to explain a highly unusual approach to the insulating state. Despite the steep divergence of resistivity, the profile of vs. at fixed saturates to a -independent value at low temperatures, which is consistent with gapless charge-carrying excitations Ref.Checkelsky:2009 (). Moreover, in highly oriented pyrolytic graphite in the magnetic field, the temperature of the metal-insulator phase transition increases with increasing field strength, contrary to the dependence in the classical low field limit Ref.Kopelevich:1999 (). The anomalous behavior has been successfully modeled within a dynamical gap picture Ref.Shovkovy:2d (). The available data suggest that by tuning the magnetic field graphene approaches a quantum critical point, beyond which a new insulating phase develops with anomalous behavior . This picture is in agreement with expectations of quantum critical behavior, where e. g. in heavy fermion metal a new magnetically ordered state (antiferromagnet) emerges when tuned through the quantum critical point Ref.Schofield:2005 ().
We shall see that the same qualitative physics emerges with our use of the the magnetic field as a knob to tune to the IR fixed point to gain some insight into the quantum critical behavior driven by fermion bilinears. In our gauge/gravity dual prescription, the unusual properties characteristic for quantum criticality can be understood as being controlled by the scaling dimension of the fermion operator in the emergent IR fixed point. The novel insight of AdS/CFT is that the low-energy behavior of a strongly coupled quantum critical system is governed by a nontrivial unstable fixed point which exhibits nonanalytic scaling behavior in the temporal direction only (the retarded Green’s function of the IR CFT is ) Ref.Faulkner:2009 (). This fixed point manifests itself as a near-horizon region of the black hole with AdS geometry which is (presumably) dual to a one-dimensional IR CFT. Building on the semilocal description of the quasiparticle characteristics by simple Dyson summation in a Fermi gas coupled to this 1+1-dimensional IR CFT Ref.FaulknerPol () an appealing picture arises that quantum critical fermionic fluctuations in the IR CFT generate relevant order parameter perturbations of the Fermi liquid theory. Whether this is truly what is driving the physics is an open question. Regardless, quantum critical matter is universal in the sense that no information about the microscopic nature of the material enters. Qualitatively our study should apply to any bilinear instability in the strange metal phase of unconventional superconductors, heavy fermions as well as for a critical point in graphene. Universality makes applications of AdS/CFT to quantum critical phenomena justifiable and appealing.
The paper is organized as follows. In Sec. II, we review the AdS-RN black hole solution in AdS-Einstein-Maxwell gravity coupled to charged fermions and the dual interpretation as a quantum critical fermion system at finite density. In Sec. III we use the bilinear formalism put forward in Ref.czs2010 () to explore an instability of a quantum system towards a quantum phase transition using the AdS dual description. We study a quantum phase transition to an insulating phase as a function of the magnetic field. For completeness we test the various phases by a spectral analysis in Sec. IV. We conclude by discussing a phase space in variables for a quantum critical matter at nonzero temperatures.
Ii Holographic fermions in the background of a dyonic black hole
The gravity dual to a -dimensional CFT at finite density in the presence of a magnetic field starts with the Einstein-Maxwell action describing an asymptotically AdS geometry
Here is the gauge field, is an effective dimensionless gauge coupling and the curvature radius of AdS is set to unity. The equations of motion following from eq.(1) are solved by a dyonic AdS black hole, having both electric and magnetic charge
where the redshift factor and the vector field are given by
The AdS boundary is reached for , the black hole horizon is at and the electric and magnetic charge of the black hole and , encoding the chemical potential and magnetic field of the dual CFT, are scaled such that the black hole temperature equals222The independent black hole mass parameter is restored after rescaling and .
In these units, the extremal black hole corresponds to and in this case the red shift factor develops a double zero at the horizon
To include the bulk fermions, we consider a spinor field in the AdS of charge and mass , which is dual to a fermionic operator in the boundary CFT of charge and dimension
with (in units of the AdS radius). The quadratic action for reads
where , and
where is the spin connection, and . Here, and denote the bulk space-time and tangent-space indices respectively, while are indices along the boundary directions, i. e. . The Dirac equation in the dyonic AdS-black hole background becomes
in terms of which the Dirac equation is . In order to separate the variables, we can proceed by finding the matrix such that . The idea is that, although and do not commute, we can find so that commute and can be diagonalized simultaneously Ref.Leiden:2010 ().333Rather the part in not proportional to the identity anticommutes with . This realization shows why the relations in the next sentence are the solution. To this end, must satisfy the relations , , , . A clear solution is .
In a convenient gamma matrix basis (Minkowski signature) Ref.Faulkner:2009 ()
the matrix equals
This choice of the basis allows one to obtain spectral functions in a simple way. In the absence of a magnetic field one can use rotational invariance to rotate to a frame where this is so. The gauge choice for the a magnetic field obviously breaks the isotropy, but the physical isotropy still ensures that the spectral functions simplify in this basis Ref.Leiden:2010 (). The -dependent part of the Dirac equation can be solved analytically in terms of Gaussian-damped Hermite polynomials with eigenvalues quantized in terms of the Landau index Refs.Albash:2009wz (); Albash:2010yr (); Leiden:2010 (). The Dirac equation , where is a diagonal matrix in terms of and whose square is proportional to the identity, then reduces to
We introduce now the projectors that split the four-component bispinors into two two-component spinors where the index is the Dirac index of the boundary theory
The projectors commute with both and (recall that ). At zero magnetic field projectors are given by with unit vector . The projections with therefore decouple from each other and one finds two independent copies of the two-component Dirac equation
where the magnetic momentum is Landau quantized with integer values and . It is identical to the AdS-Dirac equation for an AdS-RN black hole with zero magnetic charge when the discrete eigenvalue is identified with the (size of the) momentum .
Iii Bilinear approach to particle-hole pairing
The objective of this paper is to use the magnetic field as a tool to probe our unstable quantum critical system dual to the dyonic AdS-RN geometry. We show that the instability is manifest in the appearance of ordering in the system: the magnetic field acts as a catalyzer for the particle-hole pairing. In particular, we will find an unusual behavior for the critical temperature of the normal to paired phase transition as the dialing of the magnetic field drives the system to a quantum crtitical point: for a critical magnetic field the critical temperature vanishes indicating a new emergent quantum critical point.
We will identify the bulk quantities in the bilinear approach which are dual to the sought-for quantities on the CFT side. We have given the setup of the bilinear formalism in Ref.czs2010 (). Here, we will first give a concise review with the focus on the transport properties and the influence of magnetic fields, and then derive the bilinear equations relevant for computing the pairing gap.
iii.1 Bulk propagators and currents
A controlled method for calculating the expectation value of some composite operator with the structure of a fermion bilinear () has been put forward in Ref.czs2010 () and it is based on a relation between the bulk and the boundary propagator in the isotropic single-particle approximation. This allows us to identify the familiar quantities at the boundary by matching the resulting expression to known thermodynamic relations. The crucial object was identified in Ref.czs2010 ()
and it is the spatial average of the current four-vector in the bulk444As shown in Ref.czs2010 (), even though the current is defined as spatial average, the only mode that contributes at the leading order (tree level) is the quasinormal mode at .. The metric then assumes the form given in the first section by eq.(2) (so that the horizon is located at and the boundary is at ). Having defined the radial projection of the bulk Dirac equation in eq.(14) we can also define the radial projections of the current as
where and is a Pauli matrix acting in the boundary frame.
The boundary interpretation of this current is, however, subtler than the simple conserved current which it is in the bulk Ref.czs2010 (): it expresses the Migdal theorem, i.e. the density of quasiparticles in the vicinity of the Fermi surface. To see this, express the bulk spinors at an arbitrary value of through the bulk-to-boundary propagators and the boundary spinors as
The meaning of the above expressions is clear: the spinors evolve from their horizon values toward the values in the bulk at some , under the action of the bulk-to-boundary propagator acting upon them (normalized by its value at the boundary). To find the relation with the boundary Green’s function we need to know the asymptotics of the solutions of the Dirac equation (15) at the boundary, see eq.(133) in Appendix A
The bulk-to-boundary Green’s function (in dimensionless units) can be constructed from the solutions to the Dirac equation Ref.Hartman:2010 () as in eq.(131). Using eq.(133) and the expression for the Wronskian, we arrive at the following relation between the boundary asymptotics of the solutions and
It is well known Ref.Landau9 () that the integral of the propagator is related to the charge density. In particular, for and for the horizon boundary conditions chosen so that (Feynman propagator), we obtain
i.e. the bilinear directly expresses the charge density . Notice that to achieve this we need to set , i.e. look at the location of the Fermi surface. By analogy, we can now see that the components correspond to current densities. In particular, the ratio of the spatial components in the external electric field readily gives the expression for the conductivity tensor . Finally, the formalism outlined above allows us to define an arbitrary bilinear and to compute its expectation value. By choosing the matrix appropriately we are able to model particle-hole, particle-particle or any other current. Notice however that all bilinears are proportional on shell, as can be seen from eqs.(22-23), which hold also for any other matrix sandwiched between the two bulk propagators. The proportionality is at fixed parameters (, , etc) so the dependences of the form and will be different for different choices of .
To introduce another crucial current, we will study the form of the action. (We will define our action to model the quantum phase transition and to define the pairing excitonic gap in section III.2.) We pick a gauge, eq.(3), so that the Maxwell field is , meaning that the non-zero components of are , , and their antisymmetric pairs. The total action eqs.(1,7) is now
where . The second integral is the boundary term added to regularize the bulk action, for which the fermion part vanishes on shell. Knowing the metric eq.(2) and the form of , we find that the total action (free energy, from the dictionary) can be expressed as Ref.czs2010 ()
where is the free energy at the horizon, which does not depend on the physical quantities on the boundary as long as the metric is fixed Ref.czs2010 () so we can disregard it here. In eq.(26), and are the leading and subleading terms in the electric and magnetic field
and the fermionic contribution is proportional to
i. e. it is the real part of the boundary propagator555In Ref.czs2010 () this bilinear is denoted by . In the present paper a different bilinear is called .. The bulk fermionic term does not contribute, being proportional to the equation of motion, while the boundary terms include the holographic factors of the form . In accordance with our earlier conclusion that the on-shell bilinears are all proportional, we can reexpress the free energy in eq.(26) as
where the chemical potential reappears in the prefactor and the fermionic term becomes of the form , confirming again that can be associated with the number density.
iii.2 Pairing currents
Now we will put to work our bilinear approach in order to explicitly compute the particle-hole (excitonic) pairing operator. We add a scalar field which interacts with fermions by the Yukawa coupling as done in Ref.Faulkner_photoemission:2009 (). Both scalar and fermion fields are dynamical. The matter action is given by
where the covariant derivatives are , , and . The gamma-matrix structure of the Yukawa interaction is specified further. Matter action is supplemented by the gauge-gravity action
we take the AdS radius and . The gauge field components and are responsible for the chemical potential and magnetic field, respectively, in the boundary theory. As in Ref.Faulkner_photoemission:2009 (), we assume and and the scalar is real . For the particle-hole sector, the scalar field is neutral .
The Yukawa coupling is allowed to be positive and negative. When the coupling is positive , a repulsive interaction makes it harder to form the particle-hole condesate. Therefore it lowers the critical temperature and can be used as a knob to tune to a vanishing critical temperature at a critical value which defines a quantum critical point. When the coupling is negative , an attractive interaction facilitates pairing and helps to form the condensate.
Both situations can be described when the interaction term is viewed as a dynamical mass of either sign due to the fact that it is in channel. For , interaction introduces a new massive pole: massless free fermion field aquires a mass which makes it harder to condense. For , there is a tachyonic instability. The exponentially growing tachyonic mode is resolved by a condensate formation, a new stable ground state. It can be shown that we do not need a nonzero chemical potential to form a condensate in this case. A similar situation was considered in Ref.Faulkner:2011 () for the superconducting instability where the spontaneous symmetry breaking of was achieved by the boundary double-trace deformation. In our case for the electron-hole pairing, symmetry is spontaneously broken by a neutral order parameter. Next we discuss the choice for the gamma-matrix structure of the Yukawa interaction eq.(31) and the corresponding pairing parameter
Now we explain our choice of the pairing operator and give a rigorous justification for this choice.
In principle, any operator that creates a particle and a hole with the same quantum numbers could be taken to define . This translates into the requirements
(Anti) commutation with (time) space gamma matrices is required for the preservation of homogeneity and isotropy, and the last one is there to preserve the particle-hole symmetry. In the basis we have adopted, eq.(11), and , and therefore the charge conjugation is represented as
We will also consider the parity of the order parameter. As defined in Ref.Stefano-Bolognesi (), parity in the presence of the AdS boundary acts as with unchanged, while the transformation of the spinor is given by
We can now expand in the usual basis
where the indices in the commutators run along the six different combinations, and check directly that the conditions eq.(34) can only be satisfied by the matrices , and . This gives three candidate bilinears
For we get the bulk current , i.e. the mass operator in the bulk. As noted in this section and in more detail in Ref.czs2010 (), it can be identified as proportional to the bulk mass term. As such, it describes the free energy per particle, as can be seen from the expression for the free energy eq.(26). The equation of motion for eq.(28) exclusively depends on the current and thus cannot encapsulate the density of the neutral particle-hole pairs: indeed, we directly see that the right-hand side equals zero if the total charge current vanishes.
For , the bulk current is . The crucial difference with respect to the first case is the relative minus sign. It is due to this sign that the current couples to itself, i.e. it is a response to a nonzero parameter , as we will see soon.
For , the resulting bulk current is . It sources the radial gauge field which is believed to be equal to zero in all meaningful holographic setups, as the radial direction corresponds to the renormalization group (RG) scale. Thus, this operator is again not the response to the attractive pairing interaction.
We are therefore left with one possibility only: which is also consistent with the choice of our gauge at nonzero magnetic field. We will therefore work with the channel
As we have discussed earlier, the isotropy in the plane remains unbroken by the radial magnetic field, and hence the expectation value should in fact be ascribed to the current with . We show the equivalence of the and order parameters below. The choice of the channel is motivated by technical simplicity due to the form of the projection operator and the fermion basis we use, eq.(14): with , since with . Finally, we note that the structure of the currents defined in eqs.(17,18) depends on the basis choice and that the currents as such have no physical interpretation in the boundary theory: physical meaning can only be ascribed to the expectation values Ref.czs2010 (). It is exactly the expectation values that encode for the condensation (order) on the field theory side Refs.czs2010 (), Stefano-Bolognesi ().
The AdS/CFT correspondence does not provide a straightforward way to match a double-trace condensate to a boundary operator, though only single-trace fields are easy to identify with the operators at the boundary. Indeed, in holographic superconductors a superconducting condensate is modeled by a charged scalar field (see e.g. Ref.Horowitz:2009 ()). As in Ref.Stefano-Bolognesi (), we argue by matching discrete symmetries on the gravity and field theory sides, that the expectation of the bulk current is gravity dual of the pairing particle-hole gap. Let us consider properties of the corresponding condensates with respect to discrete symmetries, parity and charge conjugation, in the AdS four-dimensional space. According to eq.(36), and are scalars and parity even, while is a pseudoscalar and parity odd. As for the charge conjugation, we easily find that and commute with , while anticommutes. Since the latter is the component of a vector current while the former two are (pseudo)scalars, we find that all operators preserve the particle number, as promised. The magnetic field is odd under both parity and charge conjugation, and therefore it is unaffected by . The condensate is also unaffected by , however and spontaneously break the symmetry.
In the three-dimensional boundary theory, gamma matrices can be deduced from the four-dimensional bulk gamma matrices; and the four component Dirac spinor is dual to a two-component spinor operator . As has been also found in Ref.Stefano-Bolognesi (), the three-dimensional condensate is odd under parity and even under charge conjugation, and therefore it is odd under . We summarize the transformation properties of the four- and three-dimensional condensates together with the magnetic field
which shows that the symmetry properties are matched between and condensates: they spontaneously break the CP symmetry while the magnetic field leaves it intact. Therefore our AdS/CFT dictionary between the bulk and boundary quantities is and , with the corresponding conformal dimensions of boundary operators given by eq.(84) and eq.(83).
The natural bulk extension is now the current
and it is understood that in nonzero magnetic field the integration over degenerates into the sum over Landau levels (this holds for all currents in this section). We will soon show that a complete set of bulk equations of motion for the operator eq.(40) requires a set of currents that we label , and . In the representation eq.(11) we introduce the following bilinears of the fermion field
where the pairing parameter in eq.(40) is , the index for the zeroth component is omitted in , and .
Let us now study the dynamics of the system. We need to know the evolution equations for the currents and the scalar field and to complement them with the Maxwell equations. We will show that the equations of motion for all currents generically have nonzero solutions. This suggests that, due to the coupling with the UV CFT, the pairing can occur spontaneously, without explicitly adding new terms to the action (there is no need to add an interaction for fermions in the bulk). Nevertheless, we will also analyze the situation with nonzero and show what new phenomena it brings as compared to UV CFT-only coupling (i.e. no bulk coupling).
Let us start from the equations of motion. The Dirac and Klein-Gordon equations are to be complemented with the Maxwell equation
which is reduced when the scalar is real, , to
In the background of a dyonic black hole with the metric
the Maxwell equation for the component is
where we have used .
In our setup we ignore the backreaction to , treating it as a fixed external field. The justification comes from the physics on the field theory side: we consider a stationary nonmagnetic system with zero current and magnetization density. In the bulk, this means that the currents sourced by — and backreacting to — the magnetic field arise as corrections of higher order that can be neglected to a good approximation.666To see this, consider the corresponding Maxwell equation (46) and insert the ansatz . The resulting relation for the neutral scalar predicts , compared to the analogous estimate for the electrostatic backreaction . Thus the spatial current is of order of the small correction to the field, . The reason obviously lies in the fact that the magnetic monopole sources a -independent field. Inclusion of the second Maxwell equation for would likely only lead to a renormalization of the magnetic field without quantitative changes of the physics.
The equations of motion for the matter fields read
where we included the connection to the definition . In the dyonic black hole background, the Dirac equation is
where , the scalar is neutral , and
with . In the limit it is written as follows
We write the bilinears in short as
with . Therefore because . We rewrite the Dirac equation for the bilinears
The pairing parameter is obtained by averaging the current
This system should be accompanied by the equation of motion for the neutral scalar field. In the limit of and it is given by
where . In the dyonic black hole background, the equation of motion for the scalar is