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SCIENCE CHINA Physics, Mechanics & Astronomy, Volume 64 , Issue 7 : 270001(2021) https://doi.org/10.1007/s11433-021-1681-8

Holographic axion model: A simple gravitational tool for quantum matter

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  • ReceivedJan 15, 2021
  • AcceptedFeb 10, 2021
  • PublishedJun 1, 2021
PACS numbers

Abstract


Acknowledgment

Matteo Baggioli acknowledges the support of the Shanghai Municipal Science and Technology Major Project (Grant No. 2019SHZDZX01), and of the Spanish MINECO “Centro de Excelencia Severo Ochoa" Program (Grant No. SEV-2012-0249). Keun-Young Kim was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, Information and Communication Technology (ICT) Future Planning (Grant No. NRF-2017R1A2B4004810), and the Gwangju Institute of Science and Technology (GIST) Research Institute (GRI) grant funded by the GIST in 2020. Li Li is supported in part by the National Natural Science Foundation of China (Grant Nos. 12075298, 11991052, and 12047503). Wei-Jia Li is supported in part by the National Natural Science Foundation of China (Grant No. 11905024), and Dalian University of Technology (Grant No. DUT19LK20). We are grateful to the uncountable number of colleagues which participated with us in the process of understanding all the secrets of the holographic axion model. We thank Teng Ji, Giorgio Frangi, Hyun-Sik Jeong, Xi-Jing Wang and Yongjun An for useful comments and helping proof-reading an early version of this manuscript.


Supplement

Appendix

Notations and conventionsIn order to avoid confusion, in this appendix we describe in detail all the symbols and notations used in this review. Greek letters $\mu,\nu,...~$ run over spacetime indices, while Latin letters $i,~j,...$ denote spatial ones. The axion flavor indices $I,J,...$ run over the number of broken translations. In this review, we also omit the summation symbol over the axion flavor indices, and use the Einstein convention for them too. We always utilize a mostly plus metric $(-1,1,1,1)$ and we define the Fourier transform of a field $\Psi$ using the plain-wave $\text{e}^{-\text{i}~\omega~t+\text{i}~k~x}$. Finally, we indicate in Table tab1 all the symbols used.

  • Figure 1

    (Color online) A schematic illustration for the Drude model. In orange the electrons, while in green the immobile ions. The red arrows identify the direction of a constant applied electric field. The average time between collisions is given by $\tau$.

  • Figure 1

    (Color online) A schematic illustration for the Drude model. In orange the electrons, while in green the immobile ions. The red arrows identify the direction of a constant applied electric field. The average time between collisions is given by $\tau$.

  • Figure 2

    (Color online) (a) The optical conductivity in the Drude model. For simplicity we have fixed $\sigma_{\text{DC}}=1$. (b) The excellent agreement between the Drude model and the experimental data in UPD$_2$Al$_3$ at $T=2.75$ K taken from ref. [26]. Here $\sigma_1=~\mathrm{Re}\left[\sigma\right]$ and $\sigma_2=\mathrm{Im}\left[\sigma\right]$.

  • Figure 2

    (Color online) (a) The optical conductivity in the Drude model. For simplicity we have fixed $\sigma_{\text{DC}}=1$. (b) The excellent agreement between the Drude model and the experimental data in UPD$_2$Al$_3$ at $T=2.75$ K taken from ref. [26]. Here $\sigma_1=~\mathrm{Re}\left[\sigma\right]$ and $\sigma_2=\mathrm{Im}\left[\sigma\right]$.

  • Figure 3

    (Color online) A pure shear deformation and its effects on a square 2D lattice.

  • Figure 3

    (Color online) A pure shear deformation and its effects on a square 2D lattice.

  • Figure 4

    (Color online) The EFT parametrization in terms of a set of scalar fields $\Phi^I$. The equilibrium configuration is clearly $\Phi_{\text{eq}}^I=x^I$.

  • Figure 4

    (Color online) The EFT parametrization in terms of a set of scalar fields $\Phi^I$. The equilibrium configuration is clearly $\Phi_{\text{eq}}^I=x^I$.

  • Figure 5

    (Color online) A pictorial representation of the homogeneity assumption. Any system, at length-scales $\lambda~\gg\,a$ ($a$ being the characteristic microscopic scale), looks homogeneous.

  • Figure 5

    (Color online) A pictorial representation of the homogeneity assumption. Any system, at length-scales $\lambda~\gg\,a$ ($a$ being the characteristic microscopic scale), looks homogeneous.

  • Figure 6

    The action of a volume preserving diffeomorphism 37. The total volume remains unchanged.

  • Figure 6

    The action of a volume preserving diffeomorphism 37. The total volume remains unchanged.

  • Figure 7

    (Color online) An artistic representation of the Gauge-gravity duality. The bulk contains a black hole object dual to a finite temperature thermal state. The bulk spacetime terminates at the so-called boundary where the dual field theory “lives”. The bulk description contains an extra-dimension, usually denoted as radial coordinate, which describes the energy scale of the dual field theory. The dynamics of the bulk fields, including the metric $g_{\mu\nu}$ happens in a $(d+1)$-dimensional curved spacetime which is asymptotical AdS. In this picture, the boundary field theory lives on the surface of the colored sphere and the bulk region is represented by the 3D region enclosed by such a surface.

  • Figure 7

    (Color online) An artistic representation of the Gauge-gravity duality. The bulk contains a black hole object dual to a finite temperature thermal state. The bulk spacetime terminates at the so-called boundary where the dual field theory “lives”. The bulk description contains an extra-dimension, usually denoted as radial coordinate, which describes the energy scale of the dual field theory. The dynamics of the bulk fields, including the metric $g_{\mu\nu}$ happens in a $(d+1)$-dimensional curved spacetime which is asymptotical AdS. In this picture, the boundary field theory lives on the surface of the colored sphere and the bulk region is represented by the 3D region enclosed by such a surface.

  • Figure 8

    (Color online) Holography provides a geometric representation of RG flow. (a) A series of block spin transformations (coarse-graining process) labeled by the length scale $u$; (b) a cartoon of AdS space, where the radial coordinate $u$ plays the role of energy scale of the dual system. Excitations with different energy scale get put in different place in the bulk. Figures updated from ref. [57].

  • Figure 8

    (Color online) Holography provides a geometric representation of RG flow. (a) A series of block spin transformations (coarse-graining process) labeled by the length scale $u$; (b) a cartoon of AdS space, where the radial coordinate $u$ plays the role of energy scale of the dual system. Excitations with different energy scale get put in different place in the bulk. Figures updated from ref. [57].

  • Figure 9

    (Color online) A holographic example of highly inhomogeneous 2D solutions. Figure taken from ref. [75].

  • Figure 9

    (Color online) A holographic example of highly inhomogeneous 2D solutions. Figure taken from ref. [75].

  • Figure 10

    (Color online) The first holographic computation showing a finite DC electric conductivity in a inhomogeneous periodic lattice. Figure taken from ref. [14].

  • Figure 10

    (Color online) The first holographic computation showing a finite DC electric conductivity in a inhomogeneous periodic lattice. Figure taken from ref. [14].

  • Figure 11

    (Color online) The incoherent and coherent contributions to the electric current. The incoherent processes transport charge but they do not transport momentum (e.g.,particle-antiparticle pair).

  • Figure 11

    (Color online) The incoherent and coherent contributions to the electric current. The incoherent processes transport charge but they do not transport momentum (e.g.,particle-antiparticle pair).

  • Figure 12

    (Color online) The optical conductivity of the linear axion model for various values of the momentum dissipation rate $\alpha/T$. Figure taken from ref. [86].

  • Figure 12

    (Color online) The optical conductivity of the linear axion model for various values of the momentum dissipation rate $\alpha/T$. Figure taken from ref. [86].

  • Figure 13

    (Color online) The optical conductivity of the linear axion model compared with the Drude model formulas. (a) $\alpha/\mu=0.25$; (b) $\alpha/\mu=1$. Figure taken from ref. [86].

  • Figure 13

    (Color online) The optical conductivity of the linear axion model compared with the Drude model formulas. (a) $\alpha/\mu=0.25$; (b) $\alpha/\mu=1$. Figure taken from ref. [86].

  • Figure 14

    (Color online) (a) The coherent regime, where the lowest mode (in orange) is well separated from the rest of the excitations. The corresponding response function displays a nice Lorentzian peak at the position of the lowest mode. (b) The incoherent regime where the separation of scales is lost. The response function is featureless and cannot be well approximated by using only the first (orange) mode.

  • Figure 14

    (Color online) (a) The coherent regime, where the lowest mode (in orange) is well separated from the rest of the excitations. The corresponding response function displays a nice Lorentzian peak at the position of the lowest mode. (b) The incoherent regime where the separation of scales is lost. The response function is featureless and cannot be well approximated by using only the first (orange) mode.

  • Figure 15

    (Color online) The imaginary part of the lowest mode in the longitudinal sector of the linear axion model with $\alpha/T=1/2$ in the coherent regime showing the diffusive-to-propagating crossover. Figure adapted from ref. [88].

  • Figure 15

    (Color online) The imaginary part of the lowest mode in the longitudinal sector of the linear axion model with $\alpha/T=1/2$ in the coherent regime showing the diffusive-to-propagating crossover. Figure adapted from ref. [88].

  • Figure 16

    (Color online) The modes collision associated to the coherent-incoherent transition in the linear axion model. Here the wave-number $k$ is taken to be zero and the parameter $\alpha/T$ is increased from 0 to 12 in the direction of the arrows. There is a Drude-like pole near the origin at weak momentum dissipation rate. As $\alpha$ increases, it moves down the imaginary axis and collides with another purely imaginary pole at $\alpha/T~\approx~9.5$, producing two off-axis poles. Figure adapted from ref. [88].

  • Figure 16

    (Color online) The modes collision associated to the coherent-incoherent transition in the linear axion model. Here the wave-number $k$ is taken to be zero and the parameter $\alpha/T$ is increased from 0 to 12 in the direction of the arrows. There is a Drude-like pole near the origin at weak momentum dissipation rate. As $\alpha$ increases, it moves down the imaginary axis and collides with another purely imaginary pole at $\alpha/T~\approx~9.5$, producing two off-axis poles. Figure adapted from ref. [88].

  • Figure 17

    (Color online) (a) The numerical confirmation of the energy diffusion mode in the incoherent regime with $\alpha/T=100$. (b) The coherent-incoherent transition in the frequency dependent thermal conductivity for $k/T=1/2$. Values of $\alpha/T(=2,7/2,20)$ from top to bottom. Figures adapted from ref. [88].

  • Figure 17

    (Color online) (a) The numerical confirmation of the energy diffusion mode in the incoherent regime with $\alpha/T=100$. (b) The coherent-incoherent transition in the frequency dependent thermal conductivity for $k/T=1/2$. Values of $\alpha/T(=2,7/2,20)$ from top to bottom. Figures adapted from ref. [88].

  • Figure 18

    (Color online) The frequency dependent thermoelectric coefficients for the linear axion model 67with $d=3$. Figures taken from ref. [86].

  • Figure 18

    (Color online) The frequency dependent thermoelectric coefficients for the linear axion model 67with $d=3$. Figures taken from ref. [86].

  • Figure 19

    The different symmetry breaking patterns depending on the power of the potential $V(X)=X^N$.

  • Figure 19

    The different symmetry breaking patterns depending on the power of the potential $V(X)=X^N$.

  • Figure 20

    (Color online) The shear modulus $G$ normalized by its zero temperature value $G_\infty\equiv~G(m/T\rightarrow~\infty)$ as a function of the dimensionless quantity $m/T$. The dashed line comes from the analytic eq. 114for small values of $m$.

  • Figure 20

    (Color online) The shear modulus $G$ normalized by its zero temperature value $G_\infty\equiv~G(m/T\rightarrow~\infty)$ as a function of the dimensionless quantity $m/T$. The dashed line comes from the analytic eq. 114for small values of $m$.

  • Figure 21

    (Color online) The spectrum of hydrodynamic modes can be read from the QNMs of the black hole. (a) Gapless modes propagating at the sound speed $v_\text{T}$ in the transverse channel, which are related to the transverse phonons and transverse momentum in the dual field theory. (b) Gapless sound modes propagating at the speed $v_\text{L}$ in the longitudinal channel, related to the longitudinal phonons and longitudinal momentum. In addition, there is an unexpected diffusive mode (orange dots), which we will call crystal diffusion mode hereafter.

  • Figure 21

    (Color online) The spectrum of hydrodynamic modes can be read from the QNMs of the black hole. (a) Gapless modes propagating at the sound speed $v_\text{T}$ in the transverse channel, which are related to the transverse phonons and transverse momentum in the dual field theory. (b) Gapless sound modes propagating at the speed $v_\text{L}$ in the longitudinal channel, related to the longitudinal phonons and longitudinal momentum. In addition, there is an unexpected diffusive mode (orange dots), which we will call crystal diffusion mode hereafter.

  • Figure 22

    (Color online) The dispersion relation of the transverse phonons in the holographic axion model with $V(X)=X^5$. $m/T$ increases from the red line to the blue one. Figure taken from ref. [115].

  • Figure 22

    (Color online) The dispersion relation of the transverse phonons in the holographic axion model with $V(X)=X^5$. $m/T$ increases from the red line to the blue one. Figure taken from ref. [115].

  • Figure 23

    (Color online) The real part of the dispersion relation of the longitudinal phonons in the holographic axion model. Figure taken from ref. [127].

  • Figure 23

    (Color online) The real part of the dispersion relation of the longitudinal phonons in the holographic axion model. Figure taken from ref. [127].

  • Figure 24

    (Color online) $v_{\text{T}}$ extracted from QNMs (black dots) and computed by the formula $v_{\text{T}}=\sqrt{G/\chi_{\pi\pi}}$ from elasticity (solid lines) for $n\in[3,8]$ (green-orange). Figure taken from ref. [115].

  • Figure 24

    (Color online) $v_{\text{T}}$ extracted from QNMs (black dots) and computed by the formula $v_{\text{T}}=\sqrt{G/\chi_{\pi\pi}}$ from elasticity (solid lines) for $n\in[3,8]$ (green-orange). Figure taken from ref. [115].

  • Figure 25

    (Color online) The hydrodynamic diffusive modes in the transverse of the holographic fluid model 118.

  • Figure 25

    (Color online) The hydrodynamic diffusive modes in the transverse of the holographic fluid model 118.

  • Figure 26

    (Color online) The hydrodynamic limit $\lambda~\gg~a$, with $a$ being the characteristic microscopic scale and $\lambda$ the length-scale at which we are probing out system. This regime is equivalent to the standard small momentum regime $k/T~\ll~1$.

  • Figure 26

    (Color online) The hydrodynamic limit $\lambda~\gg~a$, with $a$ being the characteristic microscopic scale and $\lambda$ the length-scale at which we are probing out system. This regime is equivalent to the standard small momentum regime $k/T~\ll~1$.

  • Figure 27

    (Color online) The discrepancy between the hydrodynamic theory of ref. [133]and the holographic model of ref. [115]reported in ref. [127]. The diffusion constant of the longitudinal crystal diffusion mode is denoted as $D_\phi$.

  • Figure 27

    (Color online) The discrepancy between the hydrodynamic theory of ref. [133]and the holographic model of ref. [115]reported in ref. [127]. The diffusion constant of the longitudinal crystal diffusion mode is denoted as $D_\phi$.

  • Figure 28

    (Color online) The comparison of the crystal diffusion constant predicted by hydrodynamics 136with the holographic results. The previous discrepancy is now successfully resolved. Figure taken from ref. [117].

  • Figure 28

    (Color online) The comparison of the crystal diffusion constant predicted by hydrodynamics 136with the holographic results. The previous discrepancy is now successfully resolved. Figure taken from ref. [117].

  • Figure 29

    (Color online) The proof that the description of ref. [133]is still inaccurate even for holographic models with zero strain pressure—thermodynamically favourable. Figure taken from ref. [117].

  • Figure 29

    (Color online) The proof that the description of ref. [133]is still inaccurate even for holographic models with zero strain pressure—thermodynamically favourable. Figure taken from ref. [117].

  • Figure 30

    (Color online) The comparison between the hydrodynamic formula 140(solid lines) and the holographic results (dots). Figure taken from ref. [140].

  • Figure 30

    (Color online) The comparison between the hydrodynamic formula 140(solid lines) and the holographic results (dots). Figure taken from ref. [140].

  • Figure 31

    (Color online) The linear instability for the axions model $V(X)=X+X^2/2$ with zero strain pressure $\mathcal{P}=0$. Plot taken from ref. [117].

  • Figure 31

    (Color online) The linear instability for the axions model $V(X)=X+X^2/2$ with zero strain pressure $\mathcal{P}=0$. Plot taken from ref. [117].

  • Figure 32

    (Color online) The hydrodynamics modes in the longitudinal spectrum of one of the holographic axion models with SSB. The figure is taken from ref. [90].

  • Figure 32

    (Color online) The hydrodynamics modes in the longitudinal spectrum of one of the holographic axion models with SSB. The figure is taken from ref. [90].

  • Figure 33

    (Color online) The transverse spectrum of excitations for the linear model $V(X)=X$. $m/T$ increases from the blue line to the red one. Figures taken from ref. [167].

  • Figure 33

    (Color online) The transverse spectrum of excitations for the linear model $V(X)=X$. $m/T$ increases from the blue line to the red one. Figures taken from ref. [167].

  • Figure 34

    (Color online) (a) The numerical confirmation that $k_\text{g}\sim~1/\tau$. (b) The behaviour of the relaxation time in function of the inverse of the temperature which is in qualitative agreement with the Arrhenius law [180]. Figures taken from ref. [167].

  • Figure 34

    (Color online) (a) The numerical confirmation that $k_\text{g}\sim~1/\tau$. (b) The behaviour of the relaxation time in function of the inverse of the temperature which is in qualitative agreement with the Arrhenius law [180]. Figures taken from ref. [167].

  • Figure 35

    (Color online) The violation of the KSS bound in the linear axion model. Figures taken from ref. [6].

  • Figure 35

    (Color online) The violation of the KSS bound in the linear axion model. Figures taken from ref. [6].

  • Figure 36

    (Color online) The shear mode in the simple “linear axion model” corresponding to $V(X)=X$ is pseudo-diffusive. (a) The dispersion relation of the pseudo-diffusive mode $\omega=-\text{i}\Gamma~-\text{i}D_\pi~k^2$ for $m/T~\in~[1,6.5]$ (from black to light blue). (b) In orange the viscosity to entropy ratio $\eta/s$, in blue the dimensionless shear diffusion constant $D_\pi~T$ obtained numerically and in green the analytic eq. 152. The horizontal dashed value is $1/4\pi$. Figure taken from ref. [212].

  • Figure 36

    (Color online) The shear mode in the simple “linear axion model” corresponding to $V(X)=X$ is pseudo-diffusive. (a) The dispersion relation of the pseudo-diffusive mode $\omega=-\text{i}\Gamma~-\text{i}D_\pi~k^2$ for $m/T~\in~[1,6.5]$ (from black to light blue). (b) In orange the viscosity to entropy ratio $\eta/s$, in blue the dimensionless shear diffusion constant $D_\pi~T$ obtained numerically and in green the analytic eq. 152. The horizontal dashed value is $1/4\pi$. Figure taken from ref. [212].

  • Figure 37

    (Color online) The classical butterfly effect intended as the exponential sensitivity to the initial boundary conditions. Its quantum generalization can be formulated as a exponential growth of the OTOC 156.

  • Figure 37

    (Color online) The classical butterfly effect intended as the exponential sensitivity to the initial boundary conditions. Its quantum generalization can be formulated as a exponential growth of the OTOC 156.

  • Figure 38

    (Color online) The diffusivities and butterfly velocity can be related via the membrane paradigm.

  • Figure 38

    (Color online) The diffusivities and butterfly velocity can be related via the membrane paradigm.

  • Figure 39

    (Color online) The pole-skipping points for a scalar field in the hyperbolic space. The red line indicates the curve on which $\mathcal{G}_{AB}(\omega,k)=\infty$ and the blue one the curve on which $\mathcal{G}_{AB}(\omega,k)=0$. The white circle locates the pole-skipping point $(\omega^*,k^*)$. Here, we set $2\pi~T~=~1$. Figure taken from ref. [229].

  • Figure 39

    (Color online) The pole-skipping points for a scalar field in the hyperbolic space. The red line indicates the curve on which $\mathcal{G}_{AB}(\omega,k)=\infty$ and the blue one the curve on which $\mathcal{G}_{AB}(\omega,k)=0$. The white circle locates the pole-skipping point $(\omega^*,k^*)$. Here, we set $2\pi~T~=~1$. Figure taken from ref. [229].

  • Figure 40

    (Color online) Pole-skipping wave number ($k_*$) can be real (red), complex (blue), and pure imaginary (green) for scalar field perturbation in $\text{AdS}_4$. The momentum relaxation parameter $\bar{\alpha}=2,~2,~1$ for $\text{Im}[\Bar{\omega}]=-1,~-2,~-3$ (from top to bottom) respectively. Here, the “bar” variables denote the quantities scaled by $2\pi~T$. Figure taken from ref. [230].

  • Figure 40

    (Color online) Pole-skipping wave number ($k_*$) can be real (red), complex (blue), and pure imaginary (green) for scalar field perturbation in $\text{AdS}_4$. The momentum relaxation parameter $\bar{\alpha}=2,~2,~1$ for $\text{Im}[\Bar{\omega}]=-1,~-2,~-3$ (from top to bottom) respectively. Here, the “bar” variables denote the quantities scaled by $2\pi~T$. Figure taken from ref. [230].

  • Figure 41

    (Color online) (a) Various velocities in the holographic systems with SSB of translations. We consider $V(X)=X^N$ by fixing $N~=~3$. In orange the speed of longitudinal sound; in cyan the speed of transverse sound and in green the butterfly velocity. All the velocities are normalized by the conformal value $v_\text{c}^2~=~1/2$. (b) The dimensionless ratio $D_{\phi}T~/v_\text{B}^2$ in function of $m/T$ for various $N\in~[3,~9]$ (from black to blue). The dashed line is the AdS$_2$ value $1/2\pi$. Figure taken from ref. [212].

  • Figure 41

    (Color online) (a) Various velocities in the holographic systems with SSB of translations. We consider $V(X)=X^N$ by fixing $N~=~3$. In orange the speed of longitudinal sound; in cyan the speed of transverse sound and in green the butterfly velocity. All the velocities are normalized by the conformal value $v_\text{c}^2~=~1/2$. (b) The dimensionless ratio $D_{\phi}T~/v_\text{B}^2$ in function of $m/T$ for various $N\in~[3,~9]$ (from black to blue). The dashed line is the AdS$_2$ value $1/2\pi$. Figure taken from ref. [212].

  • Figure 42

    (Color online) In relativity, the causality allowed region is bounded by the lightcone with a slope $c$. Credits: Wikipedia, https://en.wikipedia.org/wiki/Light_cone

  • Figure 42

    (Color online) In relativity, the causality allowed region is bounded by the lightcone with a slope $c$. Credits: Wikipedia, https://en.wikipedia.org/wiki/Light_cone

  • Figure 43

    (Color online) A simple visual derivation of the upper bound on diffusion from causality. $v$ here is the lightcone speed. The region with grids is diffusion disallowed.

  • Figure 43

    (Color online) A simple visual derivation of the upper bound on diffusion from causality. $v$ here is the lightcone speed. The region with grids is diffusion disallowed.

  • Figure 44

    (Color online) The local equilibrium timescale is extracted from the imaginary part of the first non-hydrodynamic mode as in eq. 187. The arrow indicates its motion as the increase of $m/T$.

  • Figure 44

    (Color online) The local equilibrium timescale is extracted from the imaginary part of the first non-hydrodynamic mode as in eq. 187. The arrow indicates its motion as the increase of $m/T$.

  • Figure 45

    (Color online) The upper bound on diffusion. $c$ is the speed of light and $v_{\text{L}}$ is the speed of longitudinal sound. (a) The dimensionless ratio at zero density, where $D_1$ is the crystal diffusion and $D_2$ the charge diffusion. (b) The dimensionless ratio at finite density, where the blue points are for $\mu/T~=~3$ and the red for $\mu/T~=~5$.

  • Figure 45

    (Color online) The upper bound on diffusion. $c$ is the speed of light and $v_{\text{L}}$ is the speed of longitudinal sound. (a) The dimensionless ratio at zero density, where $D_1$ is the crystal diffusion and $D_2$ the charge diffusion. (b) The dimensionless ratio at finite density, where the blue points are for $\mu/T~=~3$ and the red for $\mu/T~=~5$.

  • Figure 46

    (Color online) The diffusion allowed region is bounded by quantum mechanics and causality.

  • Figure 46

    (Color online) The diffusion allowed region is bounded by quantum mechanics and causality.

  • Figure 47

    (Color online) The stiffness $\kappa$ for $N~\in~[3,9]$ (from black to yellow) in function of $m/T$. Figure taken from ref. [212].

  • Figure 47

    (Color online) The stiffness $\kappa$ for $N~\in~[3,9]$ (from black to yellow) in function of $m/T$. Figure taken from ref. [212].

  • Figure 48

    (Color online) The intuitive picture relating the spontaneous and explicit breaking of translations with the profile of the graviton mass in the holographic bulk. Picture taken from ref. [264].

  • Figure 48

    (Color online) The intuitive picture relating the spontaneous and explicit breaking of translations with the profile of the graviton mass in the holographic bulk. Picture taken from ref. [264].

  • Figure 49

    (Color online) The dynamics of the low-energy hydrodynamic modes in the pseudo-spontaneous regime. The dance of the modes is perfectly described by eq. 200. Figure adapted from ref. [140].

  • Figure 49

    (Color online) The dynamics of the low-energy hydrodynamic modes in the pseudo-spontaneous regime. The dance of the modes is perfectly described by eq. 200. Figure adapted from ref. [140].

  • Figure 50

    (Color online) The collision between the two modes by moving towards the pseudo-spontaneous regime. Figure adapted from ref. [140]. Notice how this dynamics was already present in ref. [21].

  • Figure 50

    (Color online) The collision between the two modes by moving towards the pseudo-spontaneous regime. Figure adapted from ref. [140]. Notice how this dynamics was already present in ref. [21].

  • Figure 51

    (Color online) The dispersion relation of the pseudo-phonon. Figure adapted from ref. [264].

  • Figure 51

    (Color online) The dispersion relation of the pseudo-phonon. Figure adapted from ref. [264].

  • Figure 52

    (Color online) The numerical confirmation of the GMOR relation 206. Figure adapted from ref. [140].

  • Figure 52

    (Color online) The numerical confirmation of the GMOR relation 206. Figure adapted from ref. [140].

  • Figure 53

    (Color online) Numerical verification of the universal relation 208. Figure taken from ref. [140].

  • Figure 53

    (Color online) Numerical verification of the universal relation 208. Figure taken from ref. [140].

  • Figure 54

    (Color online) (a) The shift of the Drude peak to finite intermediate frequencies as a consequence of the pseudo-spontaneous dynamics. (b) The temperature dependence of the conductivity peak. Figures taken from ref. [140].

  • Figure 54

    (Color online) (a) The shift of the Drude peak to finite intermediate frequencies as a consequence of the pseudo-spontaneous dynamics. (b) The temperature dependence of the conductivity peak. Figures taken from ref. [140].

  • Figure 55

    (Color online) Temperature dependence of the resistivity versus disorder strength at zero magnetic field for the model with $Y=1-\frac{X}{6},~V=\frac{X}{2m^2}$. (a) The metal-insulator transition driven by the disorder strength $\alpha$. (b) Scaling of resistivity with scaled temperature $T/T_0$. The collapse of data into two separated curves both in the metallic and insulating sides is manifest. Other parameters are chosen by $e=\rho=1$. Figure taken from ref. [114].

  • Figure 55

    (Color online) Temperature dependence of the resistivity versus disorder strength at zero magnetic field for the model with $Y=1-\frac{X}{6},~V=\frac{X}{2m^2}$. (a) The metal-insulator transition driven by the disorder strength $\alpha$. (b) Scaling of resistivity with scaled temperature $T/T_0$. The collapse of data into two separated curves both in the metallic and insulating sides is manifest. Other parameters are chosen by $e=\rho=1$. Figure taken from ref. [114].

  • Figure 56

    (Color online) Phases diagram for the model with $Y=1-\frac{X}{6},~V=\frac{X}{2m^2}$ in the absence of magnetic field. Four regions are denoted by (a) good metal, (b) incoherent metal, (c) bad insulator and (d) good insulator, respectively. Other parameters are chosen by $e=\rho=1$. The figure is updated from ref. [114].

  • Figure 56

    (Color online) Phases diagram for the model with $Y=1-\frac{X}{6},~V=\frac{X}{2m^2}$ in the absence of magnetic field. Four regions are denoted by (a) good metal, (b) incoherent metal, (c) bad insulator and (d) good insulator, respectively. Other parameters are chosen by $e=\rho=1$. The figure is updated from ref. [114].

  • Figure 57

    (Color online) Representative examples of the AC electric conductivity $\sigma_{xx}$ with unitary charge density $\rho=1$. There are four phases: (a) good metal (red) with $(\alpha=0.6,~T=0.5)$, (b) incoherent metal (green) with $(\alpha=1.5,~T=0.5)$, (c) bad insulator (blue) with $(\alpha=4.5,~T=0.3)$ and (d) good insulator (purple) with $(\alpha=7.8,~T=0.05)$. Figures taken from ref. [114].

  • Figure 57

    (Color online) Representative examples of the AC electric conductivity $\sigma_{xx}$ with unitary charge density $\rho=1$. There are four phases: (a) good metal (red) with $(\alpha=0.6,~T=0.5)$, (b) incoherent metal (green) with $(\alpha=1.5,~T=0.5)$, (c) bad insulator (blue) with $(\alpha=4.5,~T=0.3)$ and (d) good insulator (purple) with $(\alpha=7.8,~T=0.05)$. Figures taken from ref. [114].

  • Figure 58

    Experimental observation of the cuprates scalings for the in-plane resistivity $R_{xx}\sim~T$ and inverse Hall angle $\cot(\Theta_H)\sim~T^2$. Figures taken from ref. [285].

  • Figure 58

    Experimental observation of the cuprates scalings for the in-plane resistivity $R_{xx}\sim~T$ and inverse Hall angle $\cot(\Theta_H)\sim~T^2$. Figures taken from ref. [285].

  • Figure 59

    (Color online) $\alpha/\mu~=~1$, $T/T_{\text{c}}=~3.2,~1,0.89,0.66,~0.27$ (dotted, red, brown, green, blue). Figures adapted from ref. [301]

  • Figure 59

    (Color online) $\alpha/\mu~=~1$, $T/T_{\text{c}}=~3.2,~1,0.89,0.66,~0.27$ (dotted, red, brown, green, blue). Figures adapted from ref. [301]

  • Figure 60

    (Color online) The effects of momentum dissipation on the phase diagram of the holographic superconductors model and on the SC condensate. $\Delta$ is the conformal dimension of the complex bulk scalar and $q$ is its charge. Figures adapted from ref. [301].

  • Figure 60

    (Color online) The effects of momentum dissipation on the phase diagram of the holographic superconductors model and on the SC condensate. $\Delta$ is the conformal dimension of the complex bulk scalar and $q$ is its charge. Figures adapted from ref. [301].

  • Figure 61

    (Color online) Nernst signal for the dyonic black hole. $\alpha/T=0.5,~1,~4$ (red, green, blue). For a large $\alpha$ it is linear to magnetic field (conventional-metal-like) and for a small $\alpha$ it is bell-shaped (cuprate-like). Figure taken from ref. [106].

  • Figure 61

    (Color online) Nernst signal for the dyonic black hole. $\alpha/T=0.5,~1,~4$ (red, green, blue). For a large $\alpha$ it is linear to magnetic field (conventional-metal-like) and for a small $\alpha$ it is bell-shaped (cuprate-like). Figure taken from ref. [106].

  • Figure 62

    (Color online) The dispersion relations of the magnetophonon and magnetoplasmon in the holographic axion model with an external magnetic field. The arrow indicates the direction of growth of $B$. Figures adapted from ref. [145].

  • Figure 62

    (Color online) The dispersion relations of the magnetophonon and magnetoplasmon in the holographic axion model with an external magnetic field. The arrow indicates the direction of growth of $B$. Figures adapted from ref. [145].

  • Figure 63

    (Color online) The optical conductivity moving the magnetic field $B$ and the scaling of the peak. The position of the peak increases monotonically with the magnetic field. Figures adapted from ref. [145].

  • Figure 63

    (Color online) The optical conductivity moving the magnetic field $B$ and the scaling of the peak. The position of the peak increases monotonically with the magnetic field. Figures adapted from ref. [145].

  • Figure 64

    (Color online) The different nonlinear elastic behaviour between metals and rubbers.

  • Figure 64

    (Color online) The different nonlinear elastic behaviour between metals and rubbers.

  • Figure 65

    (Color online) (a) Shear stress strain curve for various potentials and relative (dashed) large strain scaling. (b) Shear stress strain curves for different temperatures and comparison with the analytic eq. 240(dashed lines). As expected for $T/m~\gg~1$ the formula gives a very good approximation. Figures taken from ref. [51].

  • Figure 65

    (Color online) (a) Shear stress strain curve for various potentials and relative (dashed) large strain scaling. (b) Shear stress strain curves for different temperatures and comparison with the analytic eq. 240(dashed lines). As expected for $T/m~\gg~1$ the formula gives a very good approximation. Figures taken from ref. [51].

  • Figure 66

    (Color online) The onset of nonlinear elasticity by increasing the strain amplitude. (a) The real time dynamics and the Lissajiou's figures; (b) the non-linear complex modulus. Figures updated from ref. [146].

  • Figure 66

    (Color online) The onset of nonlinear elasticity by increasing the strain amplitude. (a) The real time dynamics and the Lissajiou's figures; (b) the non-linear complex modulus. Figures updated from ref. [146].

  • Figure 67

    (Color online) The phase diagram of the holographic axion model with benchmark potential 241according to the non-linear elastic properties.

  • Figure 67

    (Color online) The phase diagram of the holographic axion model with benchmark potential 241according to the non-linear elastic properties.

  • Figure 68

    (Color online) The appearance of plasmons in the Reissner-Nordstrom background as a consequence of imposing the mixed boundary conditions 246. Figure from ref. [331].

  • Figure 68

    (Color online) The appearance of plasmons in the Reissner-Nordstrom background as a consequence of imposing the mixed boundary conditions 246. Figure from ref. [331].

  • Figure 69

    (Color online) The numerical confirmation of the inverse Matthiessen' rule for the plasmon lifetime. The dashed lines are eq. 247and the colored dots are the numerical data for various $\lambda$. Figure taken from ref. [178].

  • Figure 69

    (Color online) The numerical confirmation of the inverse Matthiessen' rule for the plasmon lifetime. The dashed lines are eq. 247and the colored dots are the numerical data for various $\lambda$. Figure taken from ref. [178].

  • Figure 70

    (Color online) (a) The fluid to solid crossover and (b) the depletion of the plasma frequency $\omega_\text{p}$. Figure taken from ref. [178].

  • Figure 70

    (Color online) (a) The fluid to solid crossover and (b) the depletion of the plasma frequency $\omega_\text{p}$. Figure taken from ref. [178].

  • Figure 71

    (Color online) Phase diagram in ($m,~p,~{\alpha}$) space for the Einstein-Maxwell-linear axion model of ref. [355]. Depending on the shape of spectral functions, one can classify different phases, such as Fermi liquid like (FL), bad metal prime (BM'), bad metal (BM), pseudogap (PG), and gapped (G). We have fixed the chemical potential $\mu$=1. See ref. [355]for more details. Figure taken from ref. [355].

  • Figure 71

    (Color online) Phase diagram in ($m,~p,~{\alpha}$) space for the Einstein-Maxwell-linear axion model of ref. [355]. Depending on the shape of spectral functions, one can classify different phases, such as Fermi liquid like (FL), bad metal prime (BM'), bad metal (BM), pseudogap (PG), and gapped (G). We have fixed the chemical potential $\mu$=1. See ref. [355]for more details. Figure taken from ref. [355].

  • Figure 72

    (Color online) Comparison with experimental data. Density plot of electric conductivity $\sigma$ and of thermal conductivity $\kappa$. Red circles are for data used in ref. [362]and black curves are for the holographic model with two currents. The regime marked in blue is for the FL that is far from the holographic theory. Figure taken from ref. [364].

  • Figure 72

    (Color online) Comparison with experimental data. Density plot of electric conductivity $\sigma$ and of thermal conductivity $\kappa$. Red circles are for data used in ref. [362]and black curves are for the holographic model with two currents. The regime marked in blue is for the FL that is far from the holographic theory. Figure taken from ref. [364].

  • Figure 73

    (Color online) Seebeck coefficient as a function of charge density $Q$. Circles are for experimental data used in ref. [366]and dashed line for hydrodynamics result. Seebeck coefficient at low temperature fits well with experiment for the holographic two currents model with $(z=3/2,~\theta=1)$. Figure updated from ref. [365].

  • Figure 73

    (Color online) Seebeck coefficient as a function of charge density $Q$. Circles are for experimental data used in ref. [366]and dashed line for hydrodynamics result. Seebeck coefficient at low temperature fits well with experiment for the holographic two currents model with $(z=3/2,~\theta=1)$. Figure updated from ref. [365].

  • Figure 74

    (Color online) Evolution for the case with top hat electric field. (a) The solid curve denotes the electric field $J$. The red dashed curve shows the quenched electric field $E(t)$ and the blue dashed curve gives the approximation to the electric conductivity. (b) The energy current. (c) The evolution of event horizon and apparent horizon. The bulk distribution of the axion field with the linear $x$-dependence subtracted is illustrated in color. The charge density has been set to one. Figure updated from ref. [381].

  • Figure 74

    (Color online) Evolution for the case with top hat electric field. (a) The solid curve denotes the electric field $J$. The red dashed curve shows the quenched electric field $E(t)$ and the blue dashed curve gives the approximation to the electric conductivity. (b) The energy current. (c) The evolution of event horizon and apparent horizon. The bulk distribution of the axion field with the linear $x$-dependence subtracted is illustrated in color. The charge density has been set to one. Figure updated from ref. [381].

  • Figure 75

    (Color online) Nonlinear electric current response as a function of effective temperature $T_\text{E}$. Curves from left to right represent runs of different initial temperatures $T_i=10^{-2},~10^{-1},~1/4,~1/2,~3/4,~1$, respectively. The black dashed line shows the DC linear response conductivity after the equilibrium temperature is promoted to $T_\text{E}$. The charge density has been set to one. Figure updated from ref. [381].

  • Figure 75

    (Color online) Nonlinear electric current response as a function of effective temperature $T_\text{E}$. Curves from left to right represent runs of different initial temperatures $T_i=10^{-2},~10^{-1},~1/4,~1/2,~3/4,~1$, respectively. The black dashed line shows the DC linear response conductivity after the equilibrium temperature is promoted to $T_\text{E}$. The charge density has been set to one. Figure updated from ref. [381].

  • Table A1  

    Table A1Notations and symbols

    Symbol Meaning Symbol Meaning
    $\eta$ shear viscosity $\mathfrak{p}$ pressure
    $G$ shear modulus $\mathcal{P}$ crystal pressure
    $\zeta$ bulk viscosity $m_\text{g}$ graviton mass
    $K$ bulk modulus $v_\text{B}$ butterfly velocity
    $E$ electric field $u$ or $r$ radial coordinate
    $B$ magnetic field $u_\text{h}$ horizon radius
    $\tau$ relaxation time $L$ AdS radius
    $\Gamma$ momentum dissipation rate $\Lambda$ cosmological constant
    $\chi_{AB}$ susceptibility $T$ temperature
    $\chi_{\pi\pi}\,\text{or}\,\chi_{pp}$ momentum susceptibility $s$ entropy density
    $\sigma_{ij}\,\text{or}\,T_{ij}$ stress $\rho$ charge density
    $\varepsilon_{ij}$ strain $\mu$ chemical potential
    $\varrho$ mass density $k_\text{g}$ $k$-gap
    $v_\text{T}\,\text{or}\,v_\perp$ shear sound speed $z$ Lifshitz exponent
    $v_\text{L}\,\text{or}\,v_\parallel$ longitudinal sound speed $\theta$ hyperscaling parameter
    $u_i$ displacements $\langle~\text{EXB}~\rangle~$ explicit breaking scale
    $\omega_0$ pinning frequency $\langle~\text{SSB}~\rangle~$ spontaneous breaking scale
    $\Omega$ phase relaxation rate $D_\pi~$ momentum diffusion constant
    $\mathcal{E}$ energy density $\omega~$ frequency
    $\Gamma_\text{L}\,\text{or}\,\Gamma_\parallel$ longitudinal sound attenuation $k~$ momentum in Fourier space or wave-number
    $\Gamma_\text{T}\,\text{or}\,\Gamma_\perp$ transverse sound attenuation $H~$ Hamiltonian
    $D_\phi$ crystal diffusion constant $\mathcal{G}_{AB}~$ Green's function
    $\xi$ Goldstone diffusion parameter $\omega_p~$ plasma frequency
  • Table 1  

    Table 1 Various coefficients in the low temperature limit, $m/T\rightarrow~\infty$, for $V(X)=X^3$ where the spatial translations are broken spontaneously. To achieve the data, we have assumed $u_\text{h}=1$

    Coefficient Value
    $\xi$ $(3-m^2)^2/324$
    $K$ $9/2$
    $G$ $3/2$
    $\mathcal{P}$ $3$
    $\chi_{\pi\pi}$ $3$
    $T$ $(3-m^2)/4\pi$
    $s'$ $8\pi^2/9$
    $v_\text{L}$ $1$
qqqq

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