We construct a family of holographic duals to anisotropic states in a strongly coupled gauge theory. On the field theory side the anisotropy is generated by giving a vacuum expectation value to a dimension three operator. We obtain our gravity duals by considering the geometry corresponding to the intersection of D3- and D5- branes along 2+1 dimensions. Our backgrounds are supersymmetric and solve the fully backreacted equations of motion of ten-dimensional supergravity with smeared D5-brane sources. In all cases the geometry flows to $AdS_{5}\times {\mathbb S}^5$ in the UV, signaling an isotropic UV fixed point of the dual field theory. In the IR, depending on the parameters of the solution, we find two possible behaviors: an isotropic fixed point or a geometry with anisotropic Lifshitz-like scaling symmetry. We study several properties of the solutions, including the entanglement entropy of strips. We show that any natural extension of existing $c$-functions will display non-monotonic behavior, conforming with the presence of new degrees of freedom only at intermediate energy scales.
We discuss fermions in a spontaneously generated holographic lattice background. The lattice structure at the boundary is generated by introducing a higher-derivative interaction term between a U(1) gauge field and a scalar field. We solve the equations of motion below the critical temperature at which the lattice forms, and analyze the change in the Fermi surface due to the lattice. The fermion band structure is found to exhibit a band gap due to lattice effects.
According to the gauge-gravity duality, we systematically study the Schwinger effect and electric instability with anisotropy in a top-down holographic approach. The anisotropic black brane and bubble (soliton) background in IIB supergravity are employed and the dual theories in these backgrounds are expected to be anisotropic theory at finite temperature and anisotropic theory with confinement respectively. Then performing the potential analysis, we find due to the anisotropy, the potential barrier behaves oppositely with parallel and perpendicular electric fields, and this behavior agrees wi…
Holographic CFTs and holographic RG flows on space-time manifolds which are $d$-dimensional products of spheres are investigated. On the gravity side, this corresponds to Einstein-dilaton gravity on an asymptotically $AdS_{d+1}$ geometry, foliated by a product of spheres. We focus on holographic theories on $S^2\times S^2$, we show that the only regular five-dimensional bulk geometries have an IR endpoint where one of the sphere shrinks to zero size, while the other remains finite. In the $Z_2$-symmetric limit, where the two spheres have the same UV radii, we show the existence of a infinite d…
We study the effect of anisotropy on holographic entanglement entropy (HEE) and holographic mutual information (MI) in the Q-lattice model, by exploring the HEE and MI for infinite strips along arbitrary directions. We find that the lattice always enhances the HEE. The MI, however, is enhanced by lattice for large sub-regions; while for small sub-regions, the MI is suppressed by the lattice. We also discuss how these phenomena result from the deformation of geometry caused by Q-lattices.
The density response of charged liquids contains a collective excitation known as the plasmon. In holographic systems with translational invariance the origin of this collective excitation is traced back to the presence of zero-sound. Using a holographic model in which translational symmetry is broken pseudo-spontaneously, we show the density response is not dominated by a single isolated mode at low momentum and temperature. As a consequence, the density response contains a broad asymmetric peak with an attenuation which does not increase monotonically with momentum and temperature.
We study holographic entanglement entropy in spatially anisotropic field theory. We observe that for the background we consider in this paper, to a good approximation, the holographic entanglement entropy can be decomposed into two terms. One of them is the entanglement entropy of the isotropic field theory at fixed temperature and the other term is only a function of anisotropy parameter. Moreover, for large enough values of anisotropy parameter, our numerical results indicate that the entanglement entropy in the perpendicular direction to anisotropic direction is greater than the parallel ca…
We study a Maxwell-Proca action in an asymptotically AdS black hole background. When moving the temperature of the black hole, we find rich phase diagrams having, that depend strongly on the dimension of the operator dual to the Proca field . We present different solutions in the bulk that correspond to the holographic dual for $s$, $p$ or $s+p$-wave superfluids. In the last case we observe the onset of a spontaneously induced current.
We holographically investigate the scalarization in the Einstein-Scalar-Gauss-Bonnet gravity with a negative cosmological constant. We find that instability exists for both Schwarzschild-AdS and Reissner-Nordstrom-AdS black holes with planar horizons when we have proper interactions between the scalar field and the Gauss-Bonnet curvature corrections. We relate such instability to possible holographic scalarization and construct the corresponding hairy black hole solutions in the presence of the cosmological constant. Employing the holographic principle we expect that such bulk scalarization co…
A real anti-symmetric tensor field was introduced to realize a holographic magnetic ordered phase in our previous works. However, a more careful analysis shows there is a vector ghost in the model. In this paper we present a modified Lagrangian density for the anti-symmetric tensor, which is ghost free and causality is well-defined, and keeps all the significant results in the original model qualitatively. We show this modified Lagrangian density could come from the dimensional compactification of $p$-form field in String/M-theory. For static curved space-time, we also prove that this modified…
We study the spontaneous parity breaking and generating of Hall viscosity and angular momentum in holographic p+ip model, which can describe strongly-coupled chiral superfluid states in many quantum systems. The dual gravity theory, an SU(2) gauge field minimally coupled to Einstein gravity, is parity-invariant but allows a black hole solution with vector hair corresponding to a parity-broken superfluid state. We show that this state possesses a non-vanishing parity-odd transport coefficient -- Hall viscosity -- and an angular momentum density. We first develop an analytic method to solve this…
We present a large class of holographic models where the boundary ${\mathbb R}^{2,1}$ dimensional conformal field theory has a thermal phase with a spontaneously broken global symmetry. The dual black branes in a Poincare patch of asymptotically $AdS_4$ violate the no-hair theorem.
We consider the spontaneous breaking of translational symmetry and identify the associated Goldstone mode -- a longitudinal phonon -- in a holographic model with Bianchi VII helical symmetry. For the first time in holography, we observe the pinning of this mode after introducing a source for explicit breaking compatible with the helical symmetry of our setup. We study the dispersion relation of the resulting pseudo-Goldstone mode, uncovering how its speed and mass gap depend on the amplitude of the source and temperature. In addition, we extract the optical conductivity as a function of freque…
We examine holographic theories where Lifshitz symmetry is broken with spatial anisotropy. In particular, we focus on the conditions imposed by the null energy condition, and demonstrate that it is possible to have unusual anisotropic fixed points where a subset of the spatial dimensions have negative scaling exponents. We also construct interpolating solutions between UV and IR fixed points and show that there is essentially no restriction placed on the endpoints of the flow once anisotropic scaling is allowed. As an example, we demonstrate a flow from AdS to AdS with $c_{IR}>c_{UV}$ that is …
We revisit HHH model in [Phys. Rev. Lett. {\bf 101}, 031601 (2008)] and extend the ansatz of matter fields to being of depending on a spatial dimension except the holographic direction. Despite homogeneous solutions of ground and excited states, especially for the excited states, there also exists solutions where the translational invariance is broken. It is worth mentioning that no periodic sources are assigned to the matter fields, so the translational symmetry is broken spontaneously. We investigate how the new solutions and the condensates of excited states develop with the change of tempe…
We study dynamical condensation process in a holographic superconductor model with anisotropy. The time-dependent numerical solution is constructed for the Einstein-Maxwell-dilaton theory with complex scalar in asymptotic AdS spacetime. The introduction of dilaton field generates the anisotropy in boundary spatial directions. In analogy of isotropic case, we have two black hole solutions below certain critical temperature $T_c$, the anisotropic charged black hole with and without scalar hair, corresponding respectively to the supercooled normal phase and superconducting phase in the boundary t…
We analyze the concomitant spontaneous breaking of translation and conformal symmetries by introducing in a CFT a complex scalar operator that acquires a spatially dependent expectation value. The model, inspired by the holographic Q-lattice, provides a privileged setup to study the emergence of phonons from a spontaneous translational symmetry breaking in a conformal field theory and offers valuable hints for the treatment of phonons in QFT at large. We first analyze the Ward identity structure by means of standard QFT techniques, considering both spontaneous and explicit symmetry breaking. N…
We investigate an anisotropic model of superconductors in the Einstein-Maxwell-dilaton theory with a charged scalar field. It is found that the critical temperature decreases as the anisotropy becomes large. We then estimate the energy gap of the superconductor, and find that the ratio of the energy gap to the critical temperature increases as the anisotropy increases and so it is larger than that in the isotropic case. We also find that peudogap appears due to the anisotropy.
When the system with internal tendency to a spontaneous formation of a spatially periodic state is brought in contact with the external explicit periodic potential, the interesting phenomenon of commensurate lock in can be observed. In case when the explicit potential is strong enough and its period is close to the period of the spontaneous structure, the latter is forced to assume the periodicity of the former and the commensurate state becomes a thermodynamically preferred one. If instead the two periods are significantly different, the incommensurate state is formed. It is characterized by …
The Schwarzschild black two-brane in four-dimensional anti-de Sitter space is dual to a finite temperature state in three-dimensional conformal field theory. We show that the solution acquires a nonzero angular momentum density when a gravitational Chern-Simons coupling is turned on in the bulk, even though the solution is not modified. A similar phenomenon is found for the Reissner-Nordstrom black two-brane with axionic coupling to the gauge field. We discuss interpretation of this phenomenon from the point of view of the boundary three-dimensional conformal field theory.
We consider the spontaneous formation of striped structures in a holographic model which possesses explicit translational symmetry breaking, dual to an ionic lattice with spatially modulated chemical potential. We focus on the perturbative study of the marginal modes which drive the transition to a phase exhibiting spontaneous stripes. We study the wave-vectors of the instabilities with largest critical temperature in a wide range of backgrounds characterized by the period and the amplitude of the chemical potential modulation. We report the first holographic observation of the commensurate lo…
We formulate a theory of dissipative hydrodynamics with spontaneously broken translations, describing charge density waves in a clean isotropic electronic crystal. We identify a novel linear transport coefficient, lattice pressure, capturing the effects of background strain and thermal expansion in a crystal. We argue that lattice pressure is a generic feature of systems with spontaneously broken translations and must be accounted for while building and interpreting holographic models. We also provide the first calculation of the coefficients of thermal and chemical expansion in a holographic …
We present a five-dimensional twice anisotropic holographic model supported by Einstein-dilaton-three-Maxwell action describing light quarks. The first of the Maxwell fields provides finite chemical potential. The second Maxwell field serves for anisotropy, characterizing spatial anisotropy of the QGP produced in heavy-ion collisions (HIC). The third Maxwell field is related to a magnetic field that appears in HIC. The dependence of the 5-dim black hole solution and confinement/deconfinement phase diagram on this magnetic field is considered. The effect of the inverse magnetic catalyses for li…
We first calculate the jet quenching parameter of an anisotropic plasma with a U(1) chemical potential via the AdS/CFT duality. The effects of charge, anisotropy parameter and quark motion direction on the jet quenching parameter are investigated. We then discuss the situation of anisotropic black brane in the IR region. We study both the jet quenching parameters along the longitudinal direction and transverse plane.
We study holographic entanglement entropy in 5-dimensional charged black brane geometry obtained from Einstein-SU(2)Yang-Mills theory defined in asymptotically AdS space. This gravity system undergoes second order phase transition near its critical point affected by a spatial component of the Yang-Mills fields, which is normalizable mode of the solution. This is known as phase transition between isotropic and anisotropic phases. We get analytic solutions of holographic entanglement entropies by utilizing the solution of bulk spacetime geometry given in arXiv:1109.4592, where we consider subsys…
Using the gauge/gravity duality we calculate the heavy quark-antiquark potential in a hot, anisotropic and strongly coupled Yang-Mills plasma in (3+1)-dimensions. As the anisotropic medium we take a deformed version of \mathcal{N} = 4 super Yang-Mills theory at finite temperature following a recent work where the dual type IIB supergravity solution is also proposed. We turn on a small value of the anisotropy parameter, for which the gravity dual is known analytically (perturbatively), and compute the velocity-dependent quark-antiquark interaction potential when the pair is moving through the p…
In this paper a holographic model of $s$-wave superconductor with anisotropic Lifshitz scaling has been considered. In the presence of an external magnetic field our holographic model exhibits both vortex and droplet solutions. Based on analytic methods we have shown that the anisotropy has no effect on the vortex and droplet solutions whereas it may affect the condensation. Our vortex solution closely resembles the Ginzburg-Landau theory and a relation between the upper critical magnetic field and superconducting coherence length has been speculated from this comparison. Using Sturm-Liouville…
It has been shown that holographic massive gravities can effectively realize the states with spontaneous breaking of translational symmetry in a homogenous manner. In this work, we consider a toy model of such category by adding a special gauge-axion coupling to the bulk action. Firstly, we identify the existence of spontaneous breaking of translations by the UV analysis. In the absence of explicit breaking, the black hole solution is simply the same as the Reissner-Nodstrom(RN) black holes, regardless of the non-trivial profile of the bulk axions. The associated Goldstone modes exist only whe…
We present a class of holographic massive gravity models that realize a spontaneous breaking of translational symmetry - they exhibit transverse phonon modes whose speed relates to the elastic shear modulus according to elasticity theory. Massive gravity theories thus emerge as versatile and convenient theories to model generic types of translational symmetry breaking: explicit, spontaneous and a mixture of both. The nature of the breaking is encoded in the radial dependence of the graviton mass. As an application of the model, we compute the temperature dependence of the shear modulus and fin…
In this paper, we study behaviour of Wilson loops in the boost-invariant nonequilibrium anisotropic quark-gluon plasma produced in heavy-ion collisions within the holographic approach. We describe the thermalization studying the evolution of the Vaidya metric in the boost-invariant and spatially anisotropic background. To probe the system during this process we calculate rectangular Wilson loops oriented in different spatial directions. We find that anisotropic effects are more visible for the Wilson loops lying in the transversal plane unlike the Wilson loops with partially longitudinal orien…
The electric conductivity and direct photons emission rate are considered in the holographic theory with two types of anisotropy. The electric conductivity is derived in two different ways, and their equivalence for the twice anisotropic theory is shown. Numerical calculations of the electric conductivity were done for Einstein-dilaton-three-Maxwell holographic model [29]. The dependence of the conductivity on the temperature, the chemical potential, the external magnetic field, and the spatial anisotropy of the heavy-ions collision (HIC) is studied. The electric conductivity jumps near the fi…
We numerically construct asymptotically $AdS$ black brane solutions of $D=4$ Einstein theory coupled to a scalar and two $U(1)$ gauge fields. The solutions are holographically dual to $d=3$ CFTs in a constant external magnetic field along one of the $U(1)$'s. Below a critical temperature the system's magnetisation density becomes inhomogeneous, leading to spontaneous formation of current density waves. We find that the transition can be of second order and that the solutions which minimise the free energy locally in the parameter space of solutions have averaged stressed tensor of a perfect fl…
Most holographic models of superconducting systems neglect the effects of dynamical boundary gauge fields during the process of spontaneous symmetry-breaking. Usually a global symmetry gets broken. This yields a superfluid, which then is gauged "weakly" afterwards. In this work we build (and probe the dynamics of) a holographic model in which a local boundary symmetry is spontaneously broken instead. We compute two-point functions of dynamical non-Abelian gauge fields in the normal and in the broken phase, and find non-trivial gapless modes. Our AdS3 gravity dual realizes a p-wave superconduct…
We employ methods of gauge/string duality to analyze the Brownian motion and the concomitant Langevin equation of a heavy quark in a strongly coupled, thermal, anisotropic Yang-Mills plasma. We consider fluctuations both along and perpendicular to the direction of anisotropy and study the effects of anisotropy on the drag coefficient, the diffusion constant and the Langevin coefficient for both the directions. We also verify the fluctuation-dissipation theorem for Brownian motion in an anisotropic medium.
We study the realisation of higher-form symmetries in the holographic dual of gauge theories coupled to probe matter in the fundamental. We particularly focus on the dual of U(N) gauge theory coupled to fundamental matter. We demonstrate the existence of a continuous 1-form symmetry associated with the conservation of magnetic flux and show that this symmetry is spontaneously broken in the IR when the flavour degrees of freedom are gapped. We numerically compute the spectral function of the 2-form current and demonstrate the existence of the associated Goldstone mode. We compare to expectation…
We have modified the holographic model of Saremi and Son by using a charged black brane, instead of a neutral one, such that when the bulk pseudo scalar potential is made of quadratic and quartic terms, parity can still be broken spontaneously in the boundary theory. In our model, the 3+1 dimensional bulk has a pseudo scalar coupled to the gravitational Chern-Simons term in the anti de Sitter charged black brane back ground. Parity could be broken spontaneously in the bulk by the pseudo scalar hairy solution and give rise to non-zero Hall viscosity at the boundary theory.
We discuss the (spontaneous) chiral symmetry breaking in a strongly coupled anisotropic quark-gluon plasma (QGP) in the presence of the magnetic field, using holography. The physical quantities related to the chiral symmetry breaking (m;B_c) distinguish between the effects of the anisotropy and magnetic field on the plasma. Anisotropy affects the system similar to the temperature and for its larger values heavier quarks can live in the QGP without getting condensed. Raising the anisotropy in the system will also increase the value of the critical magnetic field, B_c, at which the spontaneous c…
The holographic dark energy model is proposed by Li as an attempt for probing the nature of dark energy within the framework of quantum gravity. The main characteristic of holographic dark energy is governed by a numerical parameter $c$ in the model. The parameter $c$ can only be determined by observations. Thus, in order to characterize the evolving feature of dark energy and to predict the fate of the universe, it is of extraordinary importance to constrain the parameter $c$ by using the currently available observational data. In this paper, we derive constraints on the holographic dark ener…
Using the holographic model for spontaneous symmetry breaking, we study some properties of the dual superfluid such as the thermodynamic exponents, Joule-Thomson coefficient, compressibility etc. Our focus is on how these properties vary with the scaling dimension and the charge of the operator that undergoes condensation.
We investigate the combined effects of anisotropy and a magnetic field in strongly interacting gauge theories by the gauge/gravity correspondence. Our main motivation is the quark-gluon plasma produced in off-central heavy-ion collisions which exhibits large anisotropy in pressure gradients as well as large external magnetic fields. We explore two different configurations, with the anisotropy either parallel or perpendicular to the magnetic field, focusing on the competition and interplay between the two. A detailed study of the RG flow in the ground state reveals a rich structure where depend…
We investigate the effects of anisotropy on the chiral condensate in a holographic model of QCD with a fully backreacted quark sector at vanishing chemical potential. The high temperature deconfined phase is a neutral and anisotropic plasma showing different pressure gradients along different spatial directions, similar to the state produced in noncentral heavy-ion collisions. We find that the chiral transition occurs at a lower temperature in the presence of anisotropy. Equivalently, we find that anisotropy acts destructively on the chiral condensate near the transition temperature. These are…
We consider the phenomenon of spontaneous pair production in presence of an external electric field for noncommutative Yang Mills theories. Using Maldacena's holographic conjecture the threshold electric field for pair production is computed from the quark-antiquark potential for noncommutative theories. As an effect of noncommutativity, the threshold electric field is seen to be smaller than its commutative counterpart. We also estimate the correction to the pair production rate of quark-antiquark pairs to the first order of the noncommutative deformation parameter. Our result bears resemblan…
We investigate non-linear extensions of the holographic soft wall model proposed by Karch, Katz, Son and Stephanov [1] including non-minimal couplings in the five-dimensional action. The non-minimal couplings bring a new parameter $a_0$ which controls the transition between spontaneous and explicit symmetry breaking near the limit of massless quarks (the chiral limit). In the physical region (positive quark mass), we show that above a critical value of the parameter $a_0$ the chiral condensate $\langle \bar{q} q \rangle$ is finite in the chiral limit, signifying spontaneous chiral symmetry bre…
We show that the field theories dual to the homogeneous holographic models with spontaneously broken translations display several distinctive properties of quasicrystals, aperiodic crystals with long-range order (e.g. incommensurate charge density waves). This interpretation suggests that the longitudinal diffusive mode, observed in the spectrum of excitations of these systems, is the diffusive Goldstone boson typical of quasicrystals -- the phason. Moreover, following this idea, and using the effective field theory formalism for Goldstone modes in dissipative systems, we are able to derive th…
We prove that the purely classical gravity dual of Fermi and non-Fermi liquids exist by employing the logarithmic behavior of entanglement entropy to probe Fermi surfaces. For isotropic systems, the logarithmic behavior originates only from UV-IR intermediate regions of the minimal area surface. For anisotropic systems, the surfaces' configuration becomes largely modified by spatial anisotropy and series of solutions exist. By imposing the null energy condition we show that the specific heat behaves as $C\propto T^α$ where $α\leq1$, in both systems. In the end, we also present an effective gra…
We study holographic isotropisation of homogeneous, strongly coupled, non-Abelian plasmas in Gauss-Bonnet gravity with a negative cosmological constant. We focus on small values of the Gauss-Bonnet coupling parameter $λ_{GB}$ and linearise the equations of motion around a time-dependent background solution with $λ_{GB}=0$. We numerically solve the linearised equations and show that the entire time evolution of the pressure anisotropy can be well approximated by the linear in $λ_{GB}$ corrections to the quasinormal mode expansion, even in the cases of high anisotropy. We finally show that, quit…
We study the spontaneous magnetization and the magnetic hysteresis using the gauge/gravity duality. We first propose a novel and general formula to compute the magnetization in a large class of holographic models. By using this formula, we compute the spontaneous magnetization in a model like a holographic superconductor. Furthermore, we turn on the external magnetic field and build the hysteresis curve of magnetization and charge density. To our knowledge, this is the first holographic model realizing the hysteresis accompanied with spontaneous symmetry breaking. By considering the Landau-Gin…
Based on the gauge-gravity duality, we study the three-dimensional QCD ($\mathrm{QCD}_{3}$) and Chern-Simons theory by constructing the anisotropic black D3-brane solution in IIB supergravity. The deformed bulk geometry is obtained by performing a double Wick rotation and dimension reduction which becomes an anisotropic bubble configuration exhibiting confinement in the dual theory. And its anisotropy also reduces to a Chern-Simons term due to the presence of the dissolved D7-branes or the axion field in bulk. Using the bubble geometry, we investigate the the ground-state energy density, quark…
We reexamine the charged AdS domain wall solution to the Einstein-Abelian-Higgs model proposed by Gubser et al as holographic superconductors at quantum critical points and comment on their statement about the uniqueness of gravity solutions. We generalize their explorations from 3+1 dimensions to arbitrary $n+1$Ds and find that the $n+1\geqslant5$D charged AdS domain walls are unstable against electric perturbations.
Implementing the Wilsonian renormalization group (RG) transformation in a nonperturbative way, we construct an effective holographic dual description with an emergent extradimension identified with an RG scale. Taking the large$-N$ limit, we obtain an equation of motion of an order-parameter field, here the chiral condensate for our explicit demonstration. In particular, an intertwined structure manifests between the first-order RG flow equations of renormalized coupling functions and the second-order differential equation of the order-parameter field, thus referred to as a nonperturbative RG-…
We consider non-linear massive gravity with two Stückelberg fields in which the diffeomorphism invariance is broken spontaneously in two of the spacetime directions. This theory admits a charged anti-de Sitter black brane solution and has recently been used in holographic context as a bulk description of a boundary field theory with momentum dissipation. Here we study the stability of the black brane solution. We identify the physical degrees of freedom and determine the healthy regions of the parameter space of the theory. We find that there is a region in parameter space, where the theory su…
