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We study the anisotropic properties of dynamical quantities: direct current (DC) conductivity and butterfly velocity. The anisotropy plays a crucial role in determining the phase structure of the two-lattice system. Even a small deviation from isotropy can lead to distinct phase structures, as well as the IR fixed points of our holographic systems. In particular, for anisotropic cases, the most important property is that the IR fixed point can be non-AdS$_2 times mathbb R^2$ even for metallic phases. As that of a one-lattice system, the butterfly velocity can also diagnose the quantum phase transition (QPT) in this two-dimensional anisotropic latticed system.
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In this paper, we explore the properties of holographic entanglement entropy (HEE), mutual information (MI) and entanglement of purification (EoP) in holographic Lifshitz theory. These informational quantities exhibit some universal properties of holographic dual field theory. For most configuration parameters and temperatures, these informational quantities change monotonously with the Lifshitz dynamical critical exponent $z$. However, we also observe some non-monotonic behaviors for these informational quantities in some specific spaces of configuration parameters and temperatures. A particularly interesting phenomenon is that a dome-shaped diagram emerges in the behavior of MI vs $z$, and correspondingly a trapezoid-shaped profile appears in that of EoP vs $z$. This means that for some specific configuration parameters and temperatures, the system measured in terms of MI and EoP is entangled only in a certain intermediate range of $z$.
We initiate the investigation of the zero temperature holographic superfluids with two competing orders, where besides the vacuum phase, two one band superfluid phases, the coexistent superfluid phase has also been found in the AdS soliton background for the first time. We construct the complete phase diagram in the $e-mu$ plane by numerics, which is consistent with our qualitative analysis. Furthermore, we calculate the corresponding optical conductivity and sound speed by the linear response theory. The onset of pole of optical conductivity at $omega=0$ indicates that the spontaneous breaking phase always represents the superfluid phase, and the residue of pole is increased with the chemical potential, which is consistent with the fact that the particle density is essentially the superfluid density for zero temperature superfluids. In addition, the resulting sound speed demonstrates the non-smoothness at the critical points as the order parameter of condensate, which indicates that the phase transitions can also be identified by the behavior of sound speed. Moreover, as expected from the boundary conformal field theory, the sound speed saturates to $frac{1}{sqrt{2}}$ at the large chemical potential limit for our two band holographic superfluid model.
We analyze the holographic subregion complexity in a $3d$ black hole with the vector hair. This $3d$ black hole is dual to a $1+1$ dimensional $p$-wave superconductor. We probe the black hole by changing the size of the interval and by fixing $q$ or $T$. We show that the universal part is finite across the superconductor phase transition and has competitive behaviors different from the finite part of entanglement entropy. The behavior of the subregion complexity depends on the gravitational coupling constant divided by the gauge coupling constant. When this ratio is less than the critical value, the subregion complexity increases as temperature becomes low. This behavior is similar to the one of the holographic $1+1$ dimensional $s$-wave superconductor arXiv:1704.00557. When the ratio is larger than the critical value, the subregion complexity has a non-monotonic behavior as a function of $q$ or $T$. We also find a discontinuous jump of the subregion complexity as a function of the size of the interval. The subregion complexity has the maximum when it wraps the almost entire spatial circle. Due to competitive behaviors between normal and condensed phases, the universal term in the condensed phase becomes even smaller than that of the normal phase by probing the black hole horizon at a large interval. It implies that the formed condensate decreases the subregion complexity like the case of the entanglement entropy.
Using the volume of the space enclosed by the Ryu-Takayanagi (RT) surface, we study the complexity of the disk-shape subregion (with radius R) in various (2+1)-dimensional gapped systems with gravity dual. These systems include a class of toy models with singular IR and the bottom-up models for quantum chromodynamics and fractional quantum Hall effects. Two main results are: i) in the large-R expansion of the complexity, the R-linear term is always absent, similar to the absence of topological entanglement entropy; ii) when the entanglement entropy exhibits the classic `swallowtail phase transition, the complexity is sensitive but reacts differently.
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