Quantum turbulence is a far-from-equilibrium process characterized by high nonlinearity. Holographic duality provides a systematic framework for simulating the decaying (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>3</mn><mo>+</mo><mn>1</mn></mrow></math>)-dimensional quantum turbulence by numerically solving the dual Abelian-Higgs theory in a (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>4</mn><mo>+</mo><mn>1</mn></mrow></math>)-dimensional black hole background. We reveal that different types of decay behavior of the total vortex line density <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>L</mi></math> emerge depending on the initial vortex line density, ranging from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>L</mi><mo>∼</mo><msup><mi>t</mi><mrow><mo>−</mo><mn>1.5</mn></mrow></msup></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>L</mi><mo>∼</mo><msup><mrow><mi>t</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math>, similar to the experimental observation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><mi>He</mi></mrow><mprescripts></mprescripts><none></none><mn>3</mn></mmultiscripts></mrow></math> in Bradley et al. [<a href="http://dx.doi.org/10.1103/PhysRevLett.96.035301">Phys. Rev. Lett. 96, 035301 (2006)</a>], and of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><mi>He</mi></mrow><mprescripts></mprescripts><none></none><mn>4</mn></mmultiscripts></mrow></math> in Stalp et al. [<a href="http://dx.doi.org/10.1103/PhysRevLett.82.4831">Phys. Rev. Lett. 82, 4831 (1999)</a>] and in Walmsley and Golov [<a href="http://dx.doi.org/10.1103/PhysRevLett.118.134501">Phys. Rev. Lett. 118, 134501 (2017)</a>]. Furthermore, by measuring the energy flux at the black hole horizon, we determine that the energy dissipation rate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>t</mi></math> is proportional to the square of the total vortex line density, consistent with the vortex line decay equation proposed by W. F. Vinen and also the experimental measurement in <a href="http://dx.doi.org/10.1038/nphys1963">Nat. Phys. 7, 473 (2011)</a>.

China

Hua-Bi Zeng

Chuan-Yin Xia

Japan

Wei-Can Yang

Yu Tian

Makoto Tsubota

Dissipation and Decay of Three-Dimensional Holographic Quantum Turbulence

In the framework of the AdS/CFT correspondence, we find a neutral complex scalar field dynamics in a <math><mn>2</mn><mo>+</mo><mn>1</mn></math>-dimensional black hole background which can provide a scheme for studying the pattern formation process in <math><mn>1</mn><mo>+</mo><mn>1</mn></math>-dimensional reaction-diffusion systems. The patterns include plane wave, defect turbulence, phase turbulence, spatiotemporal intermittency where defect chaos coexists with a stable plane wave, and coherent structures. A phase diagram is obtained by studying the linear instability of the plane wave solutions to determine the onset of the holographic version of the Benjamin-Feir-Newell instability. Near the critical temperature, the holographic model is dual to the one-dimensional complex Ginzburg-Landau equation (CGLE), which has been studied extensively in reaction-diffusion systems, while at low temperature the holographic theory is different from CGLE.

Pattern formation from gauge/gravity duality

We employ the Einstein-Abelian-Higgs theory to investigate the structure of vortex-antivortex lattices within a superconductor driven by spatial periodic magnetic fields. By adjusting the parameters of the external magnetic field, including the period (<math><mrow><mi>T</mi></mrow></math>) and the amplitude (<math><msub><mi>B</mi><mn>0</mn></msub></math>), various distinct vortex states emerge. These states encompass the Wigner crystallization state, the vortex cluster state, and the suppressed state. Additionally, we present a comprehensive phase diagram to demarcate the specific regions where these structures emerge, contributing to our understanding of superconductivity in complex magnetic environments.

Jia-Hao Su

Vortex-antivortex lattices in a holographic superconductor

We introduce a gravitational lattice theory defined in an <math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math> black hole background that provides a holographic dual description of the linear-to-zigzag structural phase transition, characterized by the spontaneous breaking of parity symmetry observed in, e.g., confined Coulomb crystals. The transition from the high-symmetry linear phase to the broken-symmetry doubly degenerate zigzag phase can be driven by quenching the coupling between adjacent sites through the critical point. An analysis of the equilibrium correlation length and relaxation time reveals mean-field critical exponents. We explore the nonequilibrium phase transition dynamics leading to kink formation. The kink density obeys universal scaling laws in the limit of slow quenches, described by the Kibble-Zurek mechanism (KZM), and at fast quenches, characterized by a universal breakdown of the KZM.

Taiwan

Chiang-Mei Chen

Luxembourg

Spain

Adolfo del Campo

Structural phase transition and its critical dynamics from holography

In a holographic superfluid disk, when the rotational velocity is large enough, we find that a giant vortex will form in the center of the system by merging several single charged vortices at some specific rotational velocity with a phase stratification phenomenon for the order parameter. The formation of a giant vortex can be explained, as there is not enough space for a standard vortex lattice. If we keep increasing the rotational velocity, the giant vortex will disappear and there will be an appearance of a superfluid ring. In the giant vortex region, the number of vortices measured from winding number and rotational velocity always satisfies the linear Feynman relation. However, when the superfluid ring starts to appear, the number of vortices in the disk will decrease although the rotational velocity is increasing, while most of the order parameter is suppressed.

Giant vortex in a fast rotating holographic superfluid

We study the competition between the p-wave and the p+ip superfluid solutions in a holographic model with applied magnetic field intensity H. We find that when H is turned on, both the grand potential and the critical temperature of the p+ip solution are shifted, while the p-wave solution is only slightly affected. Combining the effect of H and back reaction parameter b, we build H − T phase diagrams with a slit region of p+ip phase. The zero (or finite) value of H at the starting point of the slit region is related to second (or first) order of the p-wave phase transition at zero magnetic intensity, which should be universal in systems with degenerate critical points (spinodal points) at zero magnetic field.

Yu-Ni Yang

Zhang-Yu Nie

H − T phase diagrams of a holographic p-wave superfluid

We study the influence of nonlinear terms quartic of the condensates on the phase structure of a holographic model with a multicondensate in the probe limit. We include one s-wave order and one p-wave order charged under the same U(1) gauge field in the holographic model and study the influence of the three quartic nonlinear terms with coefficients <math><msub><mi>λ</mi><mi>s</mi></msub></math>, <math><msub><mi>λ</mi><mi>p</mi></msub></math> and <math><msub><mi>λ</mi><mrow><mi>s</mi><mi>p</mi></mrow></msub></math> on the phase structure. The quartic terms in action result in cubic nonlinear terms in equations of motion, which do not affect the critical points of a single condensate because the infinitesimal condensate value vanishes the nonlinear contributions. However, the quartic terms have a clear influence on the phase structure of systems containing multicondensates. We show the influence of each of the three parameters on the phase diagram with the other two set to zero, respectively. With these nonlinear terms, we get new power on tuning the phase structure of the holographic systems showing multicondensates, and realize a reentrant phase transition as an example.

Xing-Kun Zhang

Hui Zeng

Holographic multicondensate with nonlinear terms

The <math><mrow><mi>AdS</mi><mo>/</mo><mi>CFT</mi></mrow></math> correspondence provides a unique way to study the vortex matter phases in superconductors. We solve the dynamical evolution of a superconductor in <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> dimensions at a finite temperature subjected to a magnetic field quench in terms of a gravitational “hairy black hole” dual living in an asymptotic <math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></math> space. We exploit this to determine the nature of the equilibrium states realized at long times after the quench of this two dimensional type II superconductor in a perpendicular external uniform magnetic field <math><msub><mi>B</mi><mn>0</mn></msub></math>. This holographic superconductor exhibits the generic lower (<math><msub><mi>B</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo>(</mo><mi>T</mi><mo>)</mo></math>) and upper (<math><msub><mi>B</mi><mrow><mi>c</mi><mn>2</mn></mrow></msub><mo>(</mo><mi>T</mi><mo>)</mo></math>) critical fields. For <math><msub><mi>B</mi><mn>0</mn></msub><mo>&lt;</mo><msub><mi>B</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo>(</mo><mi>T</mi><mo>)</mo></math> the magnetic field is completely expelled revealing the Meissner phase, while the superconductivity is destroyed when <math><msub><mi>B</mi><mn>0</mn></msub><mo>&gt;</mo><msub><mi>B</mi><mrow><mi>c</mi><mn>2</mn></mrow></msub><mo>(</mo><mi>T</mi><mo>)</mo></math>. Abrikosov lattices appear in the range <math><msub><mi>B</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo>(</mo><mi>T</mi><mo>)</mo><mo>&lt;</mo><msub><mi>B</mi><mn>0</mn></msub><mo>&lt;</mo><msub><mi>B</mi><mrow><mi>c</mi><mn>2</mn></mrow></msub><mo>(</mo><mi>T</mi><mo>)</mo></math> that realize various configurations in the form of hexagonal, square and slightly irregular square lattices pending the magnetic field strength and the influence of finite size boundaries. We show this to be consistent with the expectations of Ginzburg-Landau theory where the upper and lower critical fields are associated with the inverse squares of the coherence length and magnetic penetration depth, respectively.

Netherlands

Jan Zaanen

Holographic Abrikosov lattice: Vortex matter from black hole

Gauge invariance plays an important role in forming topological defects. In this work, from the AdS/CFT correspondence, we realize the clusters of equal-sign vortices during the course of critical dynamics of a strongly coupled superconductor. This is the first time to achieve the equal-sign vortex clusters in strongly coupled systems. The appearance of clusters of equal-sign vortices is a typical character of flux trapping mechanism, distinct from Kibble-Zurek mechanism which merely presents vortex-antivortex pair distributions resulting from global symmetry breaking. Numerical results of spatial correlations and net fluxes of the equal-sign vortex clusters quantitatively support the positive correlations between vortices. The linear dependence between the vortex number and the amplitude of magnetic field at the ‘trapping’ time demonstrates the flux trapping mechanism very well.

Zhi-Hong Li

Hai-Qing Zhang

Holographic topological defects and local gauge symmetry: clusters of strongly coupled equal-sign vortices

We study the homogenous quenching processes in a holographic s + p model with reentrant phase transitions. We first realize the reentrant phase transition in the holographic model in probe limit and draw the phase diagram. Next, we compare the time evolution of the two condensates in two groups of numerical quenching experiments across the reentrant region, with different quenching speed as well as different width of the reentrant region, respectively. We also study the dynamical competition between the two orders in quenching processes from the normal phase to the superconductor phase.