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The complex life of hydrodynamic modes

Grozdanov, Sašo, Kovtun, Pavel K., Starinets, Andrei O., Tadić, Petar·2019·Journal of High Energy Physics
DOI 10.1007/jhep11(2019)097
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Abstract

A bstract: We study analytic properties of the dispersion relations in classical hydrody- namics by treating them as Puiseux series in complex momentum. The radii of convergence of the series are determined by the critical points of the associated complex spectral curves. For theories that admit a dual gravitational description through holography, the critical points correspond to level-crossings in the quasinormal spectrum of the dual black hole. We illustrate these methods in N = 4 supersymmetric Yang-Mills theory in 3+1 dimensions, in a holographic model with broken translation symmetry in 2+1 dimensions, and in con- formal field theory in 1+1 dimensions. We comment on the pole-skipping phenomenon in thermal correlation functions, and show that it is not specific to energy density correlations.

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Figure 1 Dispersion relations of the hydrodynamic modes in the strongly coupled N = 4 SYM theory, obtained using the dual holographic description. The dispersion relations are plotted in terms of dimensionless w ≡ ω/ 2 πT and q ≡ | q | / 2 πT , with complex w as functions of real positive q . The left panel shows w shear ( q ) for the shear mode, the right panel shows w + sound ( q ) for one of the two sound modes. In the left panel, the actual -Im w shear ( q ) for the shear mode is shown by the solid red curve, and the analytic hydrodynamic approximation to O ( q 8 ) (computed in section 4.1) is shown by the dashed blue curve. The blue dot indicates the pole-skipping point at q ∗ = √ 3 / 2, w ∗ = -i , discussed in section 4.7. The right panel shows Re w + sound ( q ) (solid red curve) and -Im w + sound ( q ) (dashed red curve) for the '+' sound mode. The straight dotted line indicates the light cone Re w = q .
Figure 1 Dispersion relations of the hydrodynamic modes in the strongly coupled N = 4 SYM theory, obtained using the dual holographic description. The dispersion relations are plotted in terms of dimensionless w ≡ ω/ 2 πT and q ≡ | q | / 2 πT , with complex w as functions of real positive q . The left panel shows w shear ( q ) for the shear mode, the right panel shows w + sound ( q ) for one of the two sound modes. In the left panel, the actual -Im w shear ( q ) for the shear mode is shown by the solid red curve, and the analytic hydrodynamic approximation to O ( q 8 ) (computed in section 4.1) is shown by the dashed blue curve. The blue dot indicates the pole-skipping point at q ∗ = √ 3 / 2, w ∗ = -i , discussed in section 4.7. The right panel shows Re w + sound ( q ) (solid red curve) and -Im w + sound ( q ) (dashed red curve) for the '+' sound mode. The straight dotted line indicates the light cone Re w = q .CC-BY · From the PDF
Figure 2 The analytically continued sound mode frequencies in the strongly coupled N = 4 SYM theory, obtained using the dual holographic description. The dimensionless frequencies w ± sound of the two sound modes are plotted for purely imaginary dimensionless spatial momentum q , with the '+' branch in red and the ' -' branch in blue. The frequencies w ± sound are purely imaginary at imaginary q . At small momenta, the curves are linear with slopes ± v s , with v s = 1 / √ 3. The curves pass through pole-skipping points ( q ∗ , w ∗ ) = ( ± i √ 3 / 2 , i ) indicated by the blue dots.
Figure 2 The analytically continued sound mode frequencies in the strongly coupled N = 4 SYM theory, obtained using the dual holographic description. The dimensionless frequencies w ± sound of the two sound modes are plotted for purely imaginary dimensionless spatial momentum q , with the '+' branch in red and the ' -' branch in blue. The frequencies w ± sound are purely imaginary at imaginary q . At small momenta, the curves are linear with slopes ± v s , with v s = 1 / √ 3. The curves pass through pole-skipping points ( q ∗ , w ∗ ) = ( ± i √ 3 / 2 , i ) indicated by the blue dots.CC-BY · From the PDF
Figure 3 The first two (closest to the origin) poles of the retarded function of T xy in the strongly coupled N = 4 SYM theory, obtained using the dual holographic description. The locations of the poles are plotted as functions of the dimensionless wave vector for q purely imaginary, with Re w shown in blue, and Im w shown in red. The dots indicate the points ( q ∗ , w ∗ ) = ( ± i √ 3 / 2 , -i ), where the response function of T xy exhibits pole-skipping.
Figure 3 The first two (closest to the origin) poles of the retarded function of T xy in the strongly coupled N = 4 SYM theory, obtained using the dual holographic description. The locations of the poles are plotted as functions of the dimensionless wave vector for q purely imaginary, with Re w shown in blue, and Im w shown in red. The dots indicate the points ( q ∗ , w ∗ ) = ( ± i √ 3 / 2 , -i ), where the response function of T xy exhibits pole-skipping.CC-BY · From the PDF
Figure 4 The Newton polygon for the sound mode.
Figure 4 The Newton polygon for the sound mode.CC-BY · From the PDF
Figure 5 The approximations to the exact position of the critical point | q 2 c | = 1 / 4 in the holographic model with broken translation symmetry determined from the hydrodynamic algebraic curves F k ( q 2 , w ) = 0 as a function of k .
Figure 5 The approximations to the exact position of the critical point | q 2 c | = 1 / 4 in the holographic model with broken translation symmetry determined from the hydrodynamic algebraic curves F k ( q 2 , w ) = 0 as a function of k .CC-BY · From the PDF
Figure 6 Approximations to the position of the critical point in the complex q 2 c plane in the shear channel of the N = 4 SYM theory arising from the order k hydrodynamic polynomials F k ( q 2 , w ) = 0 (circles for k = 2, triangles for k = 3, squares for k = 4). The 'exact' position q 2 c ≈ 1 . 8906469 ± 1 . 1711505 i (see section 4.2) is marked by red asterisks.
Figure 6 Approximations to the position of the critical point in the complex q 2 c plane in the shear channel of the N = 4 SYM theory arising from the order k hydrodynamic polynomials F k ( q 2 , w ) = 0 (circles for k = 2, triangles for k = 3, squares for k = 4). The 'exact' position q 2 c ≈ 1 . 8906469 ± 1 . 1711505 i (see section 4.2) is marked by red asterisks.CC-BY · From the PDF
Figure 7 Quasinormal spectrum (poles of the retarded energy-momentum tensor two-point function in the N = 4 SYM theory) in the shear channel plotted in the complex w -plane for different values of the complex momentum q 2 = | q 2 | e iθ . Large dots in all four plots correspond to the location of the poles for purely real momentum, q 2 (i.e. at θ = 0) [41]. The hydrodynamic shear pole is the blue pole closest to the real axis in the top left panel. As θ increases from 0 to 2 π , each pole moves in a counter-clockwise direction, following the trajectory of its colour. At | q 2 | = 1, all poles follow a closed orbit (top left panel). At | q 2 | = 2 . 15 (top right panel), the trajectory of the hydrodynamic pole intersects the trajectories of the two nearest gapped poles. With | q 2 | further increasing to | q 2 | = 2 . 21, the poles nearly collide at the positions marked by asterisks (bottom left panel). The actual collision occurs at the critical point with the momentum (4.20), | q 2 c | ≈ 2 . 224. At | q 2 | = 2 . 26 (bottom right panel), the orbits of the three poles closest to the origin ( w = 0) are no longer closed: the hydrodynamic pole and the two gapped poles exchange their positions cyclically as the phase θ increases from 0 to 2 π . This is a manifestation of the quasinormal mode level-crossing. The dispersion relation w ( q 2 ) therefore has branch point singularities in the complex momentum squared plane at q 2 c .
Figure 7 Quasinormal spectrum (poles of the retarded energy-momentum tensor two-point function in the N = 4 SYM theory) in the shear channel plotted in the complex w -plane for different values of the complex momentum q 2 = | q 2 | e iθ . Large dots in all four plots correspond to the location of the poles for purely real momentum, q 2 (i.e. at θ = 0) [41]. The hydrodynamic shear pole is the blue pole closest to the real axis in the top left panel. As θ increases from 0 to 2 π , each pole moves in a counter-clockwise direction, following the trajectory of its colour. At | q 2 | = 1, all poles follow a closed orbit (top left panel). At | q 2 | = 2 . 15 (top right panel), the trajectory of the hydrodynamic pole intersects the trajectories of the two nearest gapped poles. With | q 2 | further increasing to | q 2 | = 2 . 21, the poles nearly collide at the positions marked by asterisks (bottom left panel). The actual collision occurs at the critical point with the momentum (4.20), | q 2 c | ≈ 2 . 224. At | q 2 | = 2 . 26 (bottom right panel), the orbits of the three poles closest to the origin ( w = 0) are no longer closed: the hydrodynamic pole and the two gapped poles exchange their positions cyclically as the phase θ increases from 0 to 2 π . This is a manifestation of the quasinormal mode level-crossing. The dispersion relation w ( q 2 ) therefore has branch point singularities in the complex momentum squared plane at q 2 c .CC-BY · From the PDF
Figure 8 Quasinormal spectrum (poles of the retarded energy-momentum tensor two-point function in the N = 4 SYM theory) in the shear channel plotted in the complex w -plane for different values of the complex momentum q 2 = | q 2 | e iθ . Large dots in all four plots correspond to the location of the poles for purely real momentum, q 2 (i.e. at θ = 0) [41]. The second critical point (occuring at | q 2 c , 1 | ≈ 2 . 377) is marked by asterisks in the figures showing the trajectories just before and just after the pole collision (top panels). With | q 2 | further increasing, the second pair of gapped modes becomes involved in the collisions leading to the third critical point at | q 2 c , 2 | ≈ 3 . 214 (bottom panels).
Figure 8 Quasinormal spectrum (poles of the retarded energy-momentum tensor two-point function in the N = 4 SYM theory) in the shear channel plotted in the complex w -plane for different values of the complex momentum q 2 = | q 2 | e iθ . Large dots in all four plots correspond to the location of the poles for purely real momentum, q 2 (i.e. at θ = 0) [41]. The second critical point (occuring at | q 2 c , 1 | ≈ 2 . 377) is marked by asterisks in the figures showing the trajectories just before and just after the pole collision (top panels). With | q 2 | further increasing, the second pair of gapped modes becomes involved in the collisions leading to the third critical point at | q 2 c , 2 | ≈ 3 . 214 (bottom panels).CC-BY · From the PDF
Figure 9 Quasinormal spectrum level-crossing: the real (blue curves) and the imaginary (red curves) parts of the hydrodynamic shear mode and the closest gapped quasinormal mode dispersion relations plotted as functions of | q 2 | at the fixed phase θ ≈ 0 . 338858 π of the complex momentum q 2 = | q 2 | e iθ . At | q 2 | = | q 2 c | ≈ 2 . 224, the level-crossing occurs.
Figure 9 Quasinormal spectrum level-crossing: the real (blue curves) and the imaginary (red curves) parts of the hydrodynamic shear mode and the closest gapped quasinormal mode dispersion relations plotted as functions of | q 2 | at the fixed phase θ ≈ 0 . 338858 π of the complex momentum q 2 = | q 2 | e iθ . At | q 2 | = | q 2 c | ≈ 2 . 224, the level-crossing occurs.CC-BY · From the PDF
Figure 10 Quasinormal spectrum (poles of the retarded energy-momentum tensor two-point function in the N = 4 SYM theory) in the sound channel plotted in the complex w -plane for different values of the complex momentum q 2 = | q 2 | e iθ . Large dots in all four plots correspond to the location of the poles for purely real momentum, q 2 (i.e. at θ = 0) [41]. The hydrodynamic sound poles are the blue and the green poles closest to the real axis. As θ is tuned from 0 to 2 π , each pole moves in a counter-clockwise direction and follows the trajectory of its colour. At | q 2 | = 1 (top left panel), all poles follow a closed orbit. At | q 2 | = 1 . 95 (top right panel), the trajectory of the two hydrodynamic sound poles comes close to the trajectories of the nearest gapped poles. With | q 2 | further increasing to | q 2 | = 1 . 98, the poles nearly collide at the positions marked by asterisks (bottom left panel). The actual collision occurs at the critical value of the momentum (4.32), | q 2 c | = 2. At | q 2 | = 2 . 05 (bottom right panel), the orbits of the four uppermost poles are no longer closed: the hydrodynamic poles and the two gapped poles exchange their positions cyclically as the phase θ increases from 0 to 2 π - again, a manifestation of the quasinormal mode level-crossing. The dispersion relation w ( q ) therefore has branch cuts starting at q c .
Figure 10 Quasinormal spectrum (poles of the retarded energy-momentum tensor two-point function in the N = 4 SYM theory) in the sound channel plotted in the complex w -plane for different values of the complex momentum q 2 = | q 2 | e iθ . Large dots in all four plots correspond to the location of the poles for purely real momentum, q 2 (i.e. at θ = 0) [41]. The hydrodynamic sound poles are the blue and the green poles closest to the real axis. As θ is tuned from 0 to 2 π , each pole moves in a counter-clockwise direction and follows the trajectory of its colour. At | q 2 | = 1 (top left panel), all poles follow a closed orbit. At | q 2 | = 1 . 95 (top right panel), the trajectory of the two hydrodynamic sound poles comes close to the trajectories of the nearest gapped poles. With | q 2 | further increasing to | q 2 | = 1 . 98, the poles nearly collide at the positions marked by asterisks (bottom left panel). The actual collision occurs at the critical value of the momentum (4.32), | q 2 c | = 2. At | q 2 | = 2 . 05 (bottom right panel), the orbits of the four uppermost poles are no longer closed: the hydrodynamic poles and the two gapped poles exchange their positions cyclically as the phase θ increases from 0 to 2 π - again, a manifestation of the quasinormal mode level-crossing. The dispersion relation w ( q ) therefore has branch cuts starting at q c .CC-BY · From the PDF
Figure 11 Quasinormal spectrum (poles of the retarded energy-momentum tensor two-point function in the N = 4 SYM theory) in the sound channel plotted in the complex w -plane for different values of the complex momentum q 2 = | q 2 | e iθ . Large dots in all four plots correspond to the location of the poles for purely real momentum, q 2 (i.e. at θ = 0) [41]. What is shown is the level-crossing phenomenon for the critical points in eq. (4.35).
Figure 11 Quasinormal spectrum (poles of the retarded energy-momentum tensor two-point function in the N = 4 SYM theory) in the sound channel plotted in the complex w -plane for different values of the complex momentum q 2 = | q 2 | e iθ . Large dots in all four plots correspond to the location of the poles for purely real momentum, q 2 (i.e. at θ = 0) [41]. What is shown is the level-crossing phenomenon for the critical points in eq. (4.35).CC-BY · From the PDF
Figure 12 Quasinormal spectrum level-crossing in the sound channel: the real (blue curves) and the imaginary (red curves) parts of the hydrodynamic sound mode and the closest gapped quasinormal mode dispersion relations plotted as functions of | q 2 | at the fixed phase θ = π/ 2 of the complex momentum q 2 = | q 2 | e iθ . At | q 2 | = | q 2 c | = 2, the level-crossing occurs.
Figure 12 Quasinormal spectrum level-crossing in the sound channel: the real (blue curves) and the imaginary (red curves) parts of the hydrodynamic sound mode and the closest gapped quasinormal mode dispersion relations plotted as functions of | q 2 | at the fixed phase θ = π/ 2 of the complex momentum q 2 = | q 2 | e iθ . At | q 2 | = | q 2 c | = 2, the level-crossing occurs.CC-BY · From the PDF
Figure 13 Quasinormal spectrum (poles of the retarded energy-momentum tensor two-point function in the N = 4 SYM theory) in the scalar channel plotted in the complex w -plane for different values of the complex momentum q 2 = | q 2 | e iθ . Large dots in all plots correspond to the location of the poles for purely real momentum, q 2 (i.e. at θ = 0) [41]. There are no hydrodynamic modes in this channel, but the gapped modes exhibit the level-crossing phenomena at complex values of momenta given by eq. (4.38).
Figure 13 Quasinormal spectrum (poles of the retarded energy-momentum tensor two-point function in the N = 4 SYM theory) in the scalar channel plotted in the complex w -plane for different values of the complex momentum q 2 = | q 2 | e iθ . Large dots in all plots correspond to the location of the poles for purely real momentum, q 2 (i.e. at θ = 0) [41]. There are no hydrodynamic modes in this channel, but the gapped modes exhibit the level-crossing phenomena at complex values of momenta given by eq. (4.38).CC-BY · From the PDF
Figure 14 Branch point singularities, branch cuts and the domain of hydrodynamic series convergence for the shear mode in the complex q 2 -plane (left panel) and the sound mode in the complex q -plane (right panel).
Figure 14 Branch point singularities, branch cuts and the domain of hydrodynamic series convergence for the shear mode in the complex q 2 -plane (left panel) and the sound mode in the complex q -plane (right panel).CC-BY · From the PDF
Figure 15 Pole-skipping and level-crossing points in a 2 d CFT correlator of operators with conformal dimension ∆ = 2, for the smallest in magnitude poles w ± 0 and w ± 1 . The red stars at Im q = ± 1 correspond to the pole-skipping points and the red dots label the critical points of levelcrossings of w ± 0 . The blue stars at Im q = ± 2 correspond to the pole-skipping points and the blue dots label the critical points of level-crossings of w ± 1 .
Figure 15 Pole-skipping and level-crossing points in a 2 d CFT correlator of operators with conformal dimension ∆ = 2, for the smallest in magnitude poles w ± 0 and w ± 1 . The red stars at Im q = ± 1 correspond to the pole-skipping points and the red dots label the critical points of levelcrossings of w ± 0 . The blue stars at Im q = ± 2 correspond to the pole-skipping points and the blue dots label the critical points of level-crossings of w ± 1 .CC-BY · From the PDF
Figure 16 BTZ quasinormal spectrum for ∆ = 1 / 8 in the complex w -plane at complex momentum q = | q | e iθ , with | q | = 1 and θ changing from 0 to 3 π/ 4. The spectrum at θ = 0 is shown by large dots.
Figure 16 BTZ quasinormal spectrum for ∆ = 1 / 8 in the complex w -plane at complex momentum q = | q | e iθ , with | q | = 1 and θ changing from 0 to 3 π/ 4. The spectrum at θ = 0 is shown by large dots.CC-BY · From the PDF
Figure 17 Left panel: solutions ψ ( τ ) of Kepler's equation (C.4) at e = 0 . 75 for τ ∈ [0 , π ] (half a period): 'exact' numerical solution of (C.4) (solid black line), series solution (C.5) truncated at 50 terms (dashed red line) and 60 terms (dotted blue line). The rate of convergence is maximal at τ = π/ 2. Right panel: the radius of convergence e c ( τ ) of the series (C.5) as a function of τ for τ ∈ [0 , π/ 2]. The red dotted line coresponds to Laplace's value e L . For 0 < e < e L , the series converges for all τ ∈ [0 , 2 π ].
Figure 17 Left panel: solutions ψ ( τ ) of Kepler's equation (C.4) at e = 0 . 75 for τ ∈ [0 , π ] (half a period): 'exact' numerical solution of (C.4) (solid black line), series solution (C.5) truncated at 50 terms (dashed red line) and 60 terms (dotted blue line). The rate of convergence is maximal at τ = π/ 2. Right panel: the radius of convergence e c ( τ ) of the series (C.5) as a function of τ for τ ∈ [0 , π/ 2]. The red dotted line coresponds to Laplace's value e L . For 0 < e < e L , the series converges for all τ ∈ [0 , 2 π ].CC-BY · From the PDF
Figure 18 Critical points of the Kepler's curve (C.4) in the complex eccentricity plane. The blue dots are the critical points at τ = π/ 2. The points closest to the origin are located at e ≈ ± 0 . 662743 i . They determine the radius of convergence e L of the series (C.5). For τ < π/ 2, the critical points are located at a larger distance from the origin. For example, the critical points at τ = π/ 4 are shown by red asterisks. They determine the radius of convergence e c ( τ ), shown in the right panel of figure 17. With τ → 0, the three critical points merge at e = 1.
Figure 18 Critical points of the Kepler's curve (C.4) in the complex eccentricity plane. The blue dots are the critical points at τ = π/ 2. The points closest to the origin are located at e ≈ ± 0 . 662743 i . They determine the radius of convergence e L of the series (C.5). For τ < π/ 2, the critical points are located at a larger distance from the origin. For example, the critical points at τ = π/ 4 are shown by red asterisks. They determine the radius of convergence e c ( τ ), shown in the right panel of figure 17. With τ → 0, the three critical points merge at e = 1.CC-BY · From the PDF

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Metadata

Authors
Grozdanov, S., Kovtun, P. K., Starinets, A. O., & Tadić, P.
Venue
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Published
2019-11
Volume
2019
Issue
11
ISSN
1029-8479
DOI
10.1007/jhep11(2019)097
Cited by
139
References
0
Language
en
Licence

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