In the previous paper [arXiv:0911.0679], we showed that the Reissner-Nordstrom black hole in the 5-dimensional anti-de Sitter space coupled to the Maxwell theory with the Chern-Simons term is unstable when the Chern-Simons coupling is sufficiently large. In the dual conformal field theory, the instability suggests a spatially modulated phase transition. In this paper, we construct and analyze non-linear solutions which describe the end-point of this phase transition. In the limit where the Chern-Simons coupling is large, we find that the phase transition is of the second order with the mean field critical exponent. However, the dispersion relation with the Van Hove singularity enhances quantum corrections in the bulk, and we argue that this changes the order of the phase transition from the second to the first. We compute linear response functions in the non-linear solution and find an infinite off-diagonal DC conductivity in the new phase.
Free fermions form a Fermi surface, which results in non-zero spectral weight at low energy and finite wavevector. In this work, we find similar features in holographic phases dual to strongly coupled quantum superfluid matter. At zero temperature, the phases we consider exhibit semi-local criticality in the IR and all the charge is carried by the scalar condensate outside the black hole horizon. Depending on the value taken by the IR critical exponents, we find Fermi surfaces in the transverse sector, Fermi shells in the longitudinal sector or no spectral weight at all. When there is non-zero…
We present a string theory construction of a gravity dual of a spatially modulated phase. In our earlier work, we showed that the Chern-Simons term in the 5-dimensional Maxwell theory destabilizes the Reissner-Nordstrom black holes in anti-de Sitter space if the Chern-Simons coupling is sufficiently high. In this paper, we show that a similar instability is realized on the worldvolume of 8-branes in the Sakai-Sugimoto model in the quark-gluon plasma phase. We also construct and analyze a non-linear solution describing the end-point of the transition. Our result suggests a new spatially modulat…
We show that the five-dimensional Maxwell theory with the Chern-Simons term is tachyonic in the presence of a constant electric field. When coupled to gravity, a sufficiently large Chern-Simons coupling causes instability of the Reissner-Nordstrom black holes in anti-de Sitter space. The instability happens only at non-vanishing momenta, suggesting a spatially modulated phase in the holographically dual quantum field theory in 3+1 dimensions, with spontaneous current generation in a helical configuration. The three-charge extremal black hole in the type IIB superstring theory on AdS_5 x S^5 ba…
We performed a study of the perturbative instabilities in Einstein-Maxwell-Chern-Simons theory with a gravitational Chern-Simons term, which is dual to a strongly coupled field theory with both chiral and mixed gauge-gravitational anomaly. With an analysis of the fluctuations in the near horizon regime at zero temperature, we found that there might be two possible sources of instabilities. The first one corresponds to a real mass-squared which is below the BF bound of AdS$_2$, and it leads to the bell-curve phase diagram at finite temperature. The effect of mixed gauge-gravitational anomaly is…
We show that strongly coupled holographic matter at finite charge density can exhibit charge density wave phases which spontaneously break translation invariance while preserving time-reversal and parity invariance. We show that such phases are possible within Einstein-Maxwell-dilaton theory in general spacetime dimensions. We also discuss related spatially modulated phases when there is an additional coupling to a second vector field, possibly with non-zero mass. We discuss how these constructions, and others, should be associated with novel spatially modulated ground states.
We consider finite charge density geometries which interpolate between AdS2 x R2 in the infrared and AdS4 in the ultraviolet, while traversing an intermediate regime of anisotropic Lifshitz scaling and hyperscaling violation. We work with Einstein-Maxwell-dilaton models and only turn on a background electric field. The spatially modulated instabilities of the near-horizon AdS2 part of the geometry are used to argue that the scaling solutions themselves should be thought of as being unstable -- in the deep infrared -- to spatially modulated phases. We identify instability windows for the scalin…
Within a holographic framework we construct supersymmetric Q-lattice (`Susy Q') solutions that describe RG flows driven by supersymmetric and spatially modulated deformations of the dual CFTs. We focus on a specific $D=4$ supergravity model which arises as a consistent KK truncation of $D=11$ supergravity on the seven sphere that preserves $SO(4)\times SO(4)$ symmetry. The Susy Q solutions are dual to boomerang RG flows, flowing from ABJM theory in the UV, deformed by spatially modulated mass terms depending on one of the spatial directions, back to the ABJM vacuum in the far IR. For large eno…
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.
We investigate spatially modulated instabilities of magnetically charged $AdS_{2}\times\mathbb{R}^2$, $AdS_{3}\times\mathbb{R}^2$ and $AdS_{2}\times\mathbb{R}^3$ backgrounds in a broad class of theories, including those arising from KK reductions of ten and eleven dimensional supergravity. We show that magnetically charged black brane solutions in D=4,5 spacetime dimensions, whose zero temperature near horizon limit approach these backgrounds, can have instabilities that are dual to phases with current density waves that spontaneously break translation symmetry. Our examples include spatially …
We perform a study of possible instabilities of the infrared AdS2 x R2 region of solutions to Einstein-Maxwell-dilaton systems which exhibit an intermediate regime of hyperscaling violation and Lifshitz scaling. Focusing on solutions that are magnetically charged, we probe the response of the system to spatially modulated fluctuations, and identify regions of parameter space in which the infrared AdS2 geometry is unstable to perturbations. The conditions for the existence of instabilities translate to restrictions on the structure of the gauge kinetic function and scalar potential. In turn, th…
We study the vacuum structure of a class of Lorentz invariant field theories where the vacuum expectation values are not constant but are (phase) modulated. The vacua are classified into spatial, temporal, and light-like modulation types according to the patterns of spontaneous breaking of translational symmetry. The conditions for having temporal or light-like modulated vacua imply severe constraints on the models. We utilize the notion of generalized Nambu-Goldstone modes which appear in the modulated vacua. Finally, we examine fluctuation modes around these vacua and discuss their dynamics …
We construct supersymmetric solutions of $D=11$ supergravity, preserving 1/4 of the supersymmetry, that are holographically dual to ABJM theory which has been deformed by spatially varying mass terms depending on one of the two spatial directions. We show that the BPS equations reduce to the Helmholtz equation on the complex plane leading to rich classes of new solutions. In particular, the construction gives rise to infinite classes of supersymmetric boomerang RG flows, as well as generalising a known Janus solution.
We introduce a novel set of stability conditions for vacua with broken Lorentz symmetry. The first class of conditions require that the energy be minimized under small geometric deformations, which translates into requiring the positivity of a "stiffness" four-tensor. The second class of conditions requires that stress forces be restoring under small deformations. We then apply these conditions to examples of recently-discovered spatially modulated (or "striped") phases in holographic models of superconductors and high-density QCD. For backreacted solutions we find that the pressure condition …
We study spontaneous supersymmetry breaking in spatially modulated stable or meta-stable vacua in supersymmetric field theories. Such spatial modulation can be realized in a higher derivative chiral model for which vacuum energies are either positive, negative or zero, depending on the model parameters. There appears a Nambu-Goldstone boson associated with the spontaneously breaking of the translational and $U(1)$ symmetries without the quadratic kinetic term and with a quartic derivative term in the modulated direction, and a gapless Higgs mode. We show that there appears a Goldstino associat…
Spatial modulation has been studied for a long time in condensed matter, nuclear matter and quark matter, so far in non-relativistic field theories. In this paper, spatially modulated vacua at zero temperature and zero density are studied in relativistic field theories. We first propose an adaptation of the Nambu-Goldstone theorem to higher derivative theories under the assumption of the absence of ghosts: when a global symmetry is spontaneously broken due to vacuum expectation values of space-time derivatives of fields, a Nambu-Goldstone (NG) boson appears without a canonical kinetic (quadrat…
When Fischler and Susskind proposed a holographic prescription based on the Particle Horizon, they found that spatially closed cosmological models do not verify it due to the apparently unavoidable recontraction of the Particle Horizon area. In this article, after a short review of their original work, we expose graphically and analytically that spatially closed cosmological models can avoid this problem if they expand fast enough. It has been also shown that the Holographic Principle is saturated for a codimension one brane dominated Universe. The Fischler-Susskind prescription is used to obt…
We have considered D2-D8 model and obtain a numerical solution that exhibits spatially modulated phases corresponding to a charge density wave and a spin density wave. We have analysed behavior of the free energy density for different values of the chemical potential and the magnetic field.
We study striped phases in holographic insulator/superconductor transition by considering a spatially modulated chemical potential in AdS soliton background. Generally striped phases can develop above a critical chemical potential. When the constant leading term in the chemical potential is set to zero, a discontinuity in the plot of charge density versus chemical potential is observed in the limit of large wave vector. We explain this discontinuity using an analytical approach. When the constant leading term in the chemical potential is present, the critical chemical potential is larger than …
Modulated symmetries from generalized Lieb-Schultz-Mattis anomalies
Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS) RIKEN（理化学研究所跨学科理论与数学科学项目）；Institute for Theoretical Physics, University of Cologne（科隆大学理论物理研究所）；Center for Quantum Mathematics at IMADA, Southern Denmark University（南丹麦大学IMADA量子数学中心）；Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen（哥本哈根大学尼尔斯·玻尔研究所尼尔斯·玻尔国际学院）
We study phases of equilibrium Hawking radiation in $d$-dimensional holographic CFTs on spatially compact spacetimes with two black holes. In the particular phases chosen the dual $(d+1)$-dimensional bulk solutions describe a variety of black funnels and droplets. In the former the CFT readily conducts heat between the two black holes, but it in the latter such conduction is highly suppressed. While the generic case can be understood in certain extreme limits of parameters on general grounds, we focus on CFTs on specific geometries conformally equivalent to a pair of $d \ge 4$ AdS${}_d$-Schwar…
Anomaly-Induced Phenomena with Massive Fermions: Higher-Landau-Level Dominance from Spatially Modulated Electric Fields
Zhejiang University（浙江大学）；Research Center for Nuclear Physics, Osaka University（大阪大学核物理研究中心）；Tokyo University of Science（东京理科大学）；RIKEN iTHEMS, RIKEN（理化学研究所 iTHEMS 部门，理化学研究所）；Nagoya University（名古屋大学）；Institute of Physics, Academia Sinica（中央研究院物理学研究所）；National Center for Theoretical Sciences（国家理论科学中心）
We study mass deformations of $\mathcal{N}=4$, $d=4$ SYM theory that are spatially modulated in one spatial dimension and preserve some residual supersymmetry. We focus on generalisations of $\mathcal{N}=1^*$ theories and show that it is also possible, for suitably chosen supersymmetric masses, to preserve $d=3$ conformal symmetry associated with a co-dimension one interface. Holographic solutions can be constructed using $D=5$ theories of gravity that arise from consistent truncations of $SO(6)$ gauged supergravity and hence type IIB supergravity. For the mass deformations that preserve $d=3$…
We discuss a gravity dual of a charge density wave consisting of a U(1) gauge field and two scalar fields in the background of an AdS$_4$ Schwarzschild black hole together with an antisymmetric field (probe limit). Interactions drive the system to a phase transition below a critical temperature. We numerically compute the ground states characterized by modulated solutions for the gauge potential corresponding to a dynamically generated unidirectional charge density wave in the conformal field theory. Signatures of the holographic density waves are retrieved by studying the dynamical response t…
We investigate low-energy fluctuations in the real kink crystal phase of dense quark matter within the Nambu--Jona-Lasinio model. The modulated chiral condensate breaks both the translational symmetry and chiral symmetry spontaneously, which leads to the appearance of phonons and pions that are dominant degrees of freedom in the infrared. Using the Ginzburg-Landau expansion near the Lifshitz point, we derive elastic free energies for phonons and pions in dependence on the temperature and chemical potential. We show that the one-dimensional modulation is destroyed by thermal fluctuations of pho…
We construct cohomogeneity-three, finite temperature stationary black brane solutions dual to a field theory exhibiting checkerboard order. The checkerboards form a backreacted part of the bulk solution, and are obtained numerically from the coupled Einstein-Maxwell-scalar PDE system. They arise spontaneously and without the inclusion of an explicit lattice. The phase exhibits both charge and global U(1)-current modulation, which are periodic in two spatial directions. The current circulates within each checkerboard plaquette. We explore the competition with striped phases, finding first-order…
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 give a minimal holographic model of a disorder-driven metal-insulator transition. It consists in a CFT with a charge sector and a translation-breaking sector that interact in the most generic way allowed by the symmetries and by dynamical consistency. In the gravity dual, it reduces to a Massive Gravity-Maxwell model with new direct couplings between the Maxwell and metric that are allowed when gravity is massive. We show that, generically, the effect of disorder is to decrease the DC electrical conductivity. This happens to such an extent that the conductivity does not obey any lower bound…
We propose a new entropy construct that generalizes the Tsallis, Rényi, Sharma-Mittal, Barrow, Kaniadakis, and Loop Quantum Gravity entropies and reduces to the Bekenstein-Hawking entropy in a certain limit. This proposal is applied to the Schwarzschild black hole and to spatially homogeneous and isotropic cosmology, where it is shown that it can potentially describe inflation and/or holographic dark energy.
We propose a new model for the hemispherical power asymmetry of the CMB based on modulated reheating. Non-Gaussianity from modulated reheating can be small enough to satisfy the bound from Planck if the dominant modulation of the inflaton decay rate is linear in the modulating field $σ$. $σ$ must then acquire a spatially-modulated power spectrum with a red scale-dependence. This can be achieved if the primordial perturbation of $σ$ is generated via tachyonic growth of a complex scalar field. Modulated reheating due to $σ$ then produces a spatially modulated and scale-dependent sub-dominant con…
In condensed matter systems, zero-dimensional or one-dimensional Majorana modes can be realized respectively as the end and edge states of one-dimensional and two-dimensional topological superconductors. In this $\textit{top-down}$ approach, $(d-1)$-dimensional Majorana modes are obtained as the boundary states of a topologically nontrivial $d$-dimensional bulk. In a $\textit{bottom-up}$ approach instead, $d$-dimensional Majorana modes in a $d$-dimensional system can be realized as the continuous limit of a periodic lattice of coupled $(d-1)$-dimensional Majorana modes. We illustrate this idea…
We describe the simplest holographic model for an s-wave unbalanced superconductor in 2+1 dimensions. We study its phase diagram and linear response features with particular attention to the possibility of spatially modulated phases (LOFF) and mixed spin-electric properties. The normal phase of the model at hand allows us to analyze a strong-coupling generalization of Mott two-current model for spintronic systems; the superconducting phase features an interesting DC spin-superconductivity without spin-symmetry breaking .
We report the observation of the spatially modulated static mode in the spectrum of fluctuations around the condensed phase of the holographic d-wave superconductor. The mode involves the time component of the gauge field that is related to the charge density wave in the dual superconductor. No additional ingredients are added to the action of five dimensional theory. We speculate on the relevance of the observed mode to the formation of the pseudogap state.
We investigate an impact of the axial-vector interaction on spatial modulation of quark matter. A magnetic field coupled with baryon density leads to a topological axial current, so that the effect of the axial-vector interaction is crucially enhanced then. Using the Sakai-Sugimoto model we have found that, contrary to a naive expectation, the spatially modulated phase is less favored for a stronger magnetic field, which is realized by the presence of topological current.
We study correlation functions in an equilibrated spatially modulated phase of Einstein-Maxwell two-derivative gravity. We find that the ratio of the appropriate low frequency limit of the stress-stress two point function to the entropy density is modulated. The conductivity, the stress-current and current-stress correlation functions are also modulated. At temperatures close to the phase transition we obtain analytic expressions for some of the correlation functions.
Holographic duality is a powerful tool to investigate the far-from equilibrium dynamics of superfluids and other phases of quantum matter. For technical reasons it is usually assumed that, after a quench, the far-from equilibrium fields are still spatially uniform. Here we relax this assumption and study the time evolution of a holographic superconductor after a temperature quench but allowing spatial variations of the order parameter. Even though the initial state and the quench are spatially uniform we show the order parameter develops spatial oscillations with an amplitude that increases wi…
Using the worldline formalism we compute an effective action for fermions under a temporally modulated electric field and a spatially modulated magnetic field. It is known that the former leads to an enhanced Schwinger Mechanism, while we find that the latter can also result in enhanced particle production and even cause a reorganization of the vacuum to acquire a larger dynamical mass in equilibrium which spatially assists the Magnetic Catalysis.
Solutions of $(d+1)$-dimensional gravity coupled to a scalar field are obtained, which holographically realize interface and boundary CFTs. The solution utilizes a Janus-like $\mathrm{AdS}_d$ slicing ansatz and corresponds to a deformation of the CFT by a spatially-dependent coupling of a relevant operator. The BCFT solutions are singular in the bulk, but physical quantities such as the holographic entanglement entropy can be calculated.
We study the model for holographic dark energy in a spatially closed universe, generalizing the proposal in hep-th/0403127 for a flat universe. We provide independent arguments for the choice of the parameter $c=1$ in the holographic dark energy model. On the one hand, $c$ can not be less than 1, to avoid violating the second law of thermodynamics. On the other hand, observation suggests $c$ be very close to 1, it is hard to justify a small deviation of $c$ from 1, if $c>1$.
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 investigate the properties of the Sakai-Sugimoto model at finite magnetic field and baryon chemical potentials. We show that in a finite magnetic field, there exists a spatially homogeneous configuration carrying finite baryon number density. At low magnetic field and baryon chemical potential the equation of state of the matter coincides with that obtained from the chiral perturbation theory Lagrangian with an anomalous term. We discuss the behavior of the system at larger magnetic fields.
