We study the linear response of relativistic superfluids with a non-zero superfluid velocity. For sufficiently large superflow, an instability develops via the crossing of a pole of the retarded Green’s functions to the upper half complex frequency plane. We show that this is caused by a local thermodynamic instability, i.e. when an eigenvalue of the static susceptibility matrix (the second derivatives of the free energy) diverges and changes sign. The onset of the instability occurs when ∂  ζ (n  s  ζ) = 0, with ζ the norm of the superfluid velocity and n  s  the superfluid density. The Landau instability for non-relativistic superfluids such as Helium 4 also coincides with the non-relativistic version of this criterion. We then turn to gauge/gravity duality and show that this thermodynamic instability criterion applies equally well to strongly-coupled superfluids. In passing, we compute holographically a number of transport coefficients parametrizing deviations out-of-equilibrium in the hydrodynamic regime and demonstrate that the gapless quasinormal modes of the dual planar black hole match those predicted by superfluid hydrodynamics.

Spain

Daniel Areán

France

Blaise Goutéraux

Canada

Eric Mefford

Filippo Sottovia

Hydrodynamics and instabilities of relativistic superfluids at finite superflow

Certain holographic states of matter with a global U(1) symmetry support a sound mode at zero temperature, caused neither by spontaneous symmetry breaking of the global U(1) nor by the emergence of a Fermi surface in the infrared. In this work, we show that such a mode is also found in zero density holographic quantum critical states. We demonstrate that in these states, the appearance of a zero temperature sound mode is the consequence of a mixed ‘t Hooft anomaly between the global U(1) symmetry and an emergent higher-form symmetry. At non-zero temperatures, the presence of a black hole horizon weakly breaks the emergent symmetry and gaps the collective mode, giving rise to a sharp Drude-like peak in the electric conductivity. A similar gapped mode arises at low temperatures for non-zero densities when the state has an emergent Lorentz symmetry, also originating from an approximate anomalous higher-form symmetry. However, in this case the collective excitation does not survive at zero temperature where, instead, it dissolves into a branch cut due to strong backreaction from the infrared, critical degrees of freedom. We comment on the relation between our results and the application of the Luttinger theorem to compressible holographic states of matter.

United Kingdom

Richard Davison

Zero sound and higher-form symmetries in compressible holographic phases

In this work, we study the linear stability of superfluid phases of matter irrespective of the nature of microscopic degrees of freedom and the strength of interactions between them. Famously, assuming invariance under Galilean boosts and a phonon-roton single-particle dispersion relation, Landau predicted superfluid helium 4 would become unstable for large enough superfluid velocities. Here, we demonstrate that such instabilities generically follow from a change of sign of one of the eigenvalues of the matrix of second derivatives of the free energy. Our only assumption is the existence of static thermodynamic equilibrium, irrespective of any invariance under boosts or microscopic statistics. Turning on dissipation, we show that a linear dynamical instability also develops, leading to exponential growth in time of perturbations around equilibrium. Specializing to Galilean superfluids and assuming the existence of bosonic quasiparticles, our criterion reproduces Landau’s critical velocity for Bose-Einstein condensates. Our criterion also reduces to the well-known maximal supercurrent in weakly coupled superconductors described either by Landau-Ginzburg or Bardeen-Cooper-Schrieffer theory. Further, it correctly reproduces the onset of the instability in relativistic, strongly coupled superfluids without quasiparticles at zero as well as finite temperature, which we construct using gauge/gravity duality. As a less trivial application of our criterion, we show that in dirty superfluids the instability manifests itself first in the thermal diffusion mode instead of the superfluid sound mode. Our work provides a simple, comprehensive, and unified description of the large superflow instability of superfluids and superconductors at any temperature independent of the microscopic details of the system and the strength of interactions.

Critical superflows and thermodynamic instabilities in superfluids

Approximate symmetries abound in nature. If these symmetries are also spontaneously broken, the would-be Goldstone modes acquire a small mass, or inverse correlation length, and are referred to as pseudo-Goldstones. At nonzero temperature, the effects of dissipation can be captured by hydrodynamics at sufficiently long scales compared to the local equilibrium. Here, we show that, in the limit of weak explicit breaking, locality of hydrodynamics implies that the damping of pseudo-Goldstones is completely determined by their mass and diffusive transport coefficients. We present many applications: superfluids, QCD in the chiral limit, Wigner crystal and density wave phases in the presence of an external magnetic field or not, nematic phases, and (anti)ferromagnets. For electronic density wave phases, pseudo-Goldstone damping generates a contribution to the resistivity independent of the strength of disorder, which can have a linear temperature dependence provided the associated diffusivity saturates a bound. This is reminiscent of the phenomenology of strange metal high-<math><msub><mi>T</mi><mi>c</mi></msub></math> superconductors, where charge density waves are observed across the phase diagram.

United States

Luca V. Delacrétaz

Vaios Ziogas

Damping of Pseudo-Goldstone Fields

The low energy and finite temperature excitations of a d + 1-dimensional system exhibiting superfluidity are well described by a hydrodynamic model with two fluid flows: a normal flow and a superfluid flow. In the vicinity of a quantum critical point, thermodynamics and transport in the system are expected to be controlled by the critical exponents and by the spectrum of irrelevant deformations away from the quantum critical point. Here, using gauge-gravity duality, we present the low temperature dependence of thermodynamic and charge transport coefficients at first order in the hydrodynamic derivative expansion in terms of the critical exponents. Special attention will be paid to the behavior of the charge density of the normal flow in systems with emergent infrared conformal and Lifshitz symmetries, parameterized by a Lifshitz dynamical exponent z &gt; 1. When 1 ≤ z &lt; d + 2, we recover (z = 1) and extend (z &gt; 1) previous results obtained by relativistic effective field theory techniques. Instead, when z &gt; d + 2, we show that the normal charge density becomes non-vanishing at zero temperature. An extended appendix generalizes these results to systems that violate hyperscaling as well as systems with generalized photon masses. Our results clarify previous work in the holographic literature and have relevance to recent experimental measurements of the superfluid density on cuprate superconductors.

Non-vanishing zero-temperature normal density in holographic superfluids

The transport behavior of strongly anisotropic systems is significantly richer compared to isotropic ones. The most dramatic spatial anisotropy at a critical point occurs at a Lifshitz transition, found in systems with merging Dirac or Weyl point or near the superconductor-insulator quantum phase transition. Previous work found that in these systems a famous conjecture on the existence of a lower bound for the ratio of a shear viscosity to entropy is violated, and proposed a generalization of this bound for anisotropic systems near charge neutrality involving the electric conductivities. The present study uses scaling arguments and the gauge-gravity duality to confirm the previous analysis of universal bounds in anisotropic Dirac systems. We investigate the strongly-coupled phase of quantum Lifshitz systems in a gravitational Einstein-Maxwell-dilaton model with a linear massless scalar which breaks translations in the boundary dual field theory and sources the anisotropy. The holographic computation demonstrates that some elements of the viscosity tensor can be related to the ratio of the electric conductivities through a simple geometric ratio of elements of the bulk metric evaluated at the horizon, and thus obey a generalized bound, while others violate it. From the IR critical geometry, we express the charge diffusion constants in terms of the square butterfly velocities. The proportionality factor turns out to be direction-independent, linear in the inverse temperature, and related to the critical exponents which parametrize the anisotropic scaling of the dual field theory.

Germany

Gian Inkof

Joachim Küppers

Julia Link

Jörg Schmalian

Quantum critical scaling and holographic bound for transport coefficients near Lifshitz points

The normal density of a translation-invariant superfluid often vanishes at zero temperature, as is observed in superfluid Helium and conventional superconductors described by BCS theory. Here we show that this need not be the case. We investigate the normal density in models of quantum critical superfluids using gauge-gravity duality. Models with an emergent infrared Lorentz symmetry lead to a vanishing normal density. On the other hand, models which break the isotropy between time and space may enjoy a nonvanishing normal density, depending on the spectrum of irrelevant deformations around the underlying quantum critical ground state. Our results may shed light on recent measurements of the superfluid density and low energy spectral weight in superconducting overdoped cuprates.

Normal Charge Densities in Quantum Critical Superfluids

We study a holographic model where translations are both spontaneously and explicitly broken, leading to the presence of (pseudo)-phonons in the spectrum. The weak explicit breaking is due to two independent mechanisms: a small source for the condensate itself and additional linearly space-dependent marginal operators. The low energy dynamics of the model is described by Wigner crystal hydrodynamics. In absence of a source for the condensate, the phonons remain gapless, but momentum is relaxed. Turning on a source for the condensate damps and pins the phonons. Finally, we verify that the universal relation between the phonon damping rate, mass and diffusivity reported in [1] continues to hold in this model for weak enough explicit breaking.

Belgium

Italy

Andrea Amoretti

Daniele Musso

Gapless and gapped holographic phonons

In this Letter, we uncover a universal relaxation mechanism of pinned density waves, combining gauge-gravity duality and effective field theory techniques. Upon breaking translations spontaneously, new gapless collective modes emerge, the Nambu-Goldstone bosons of broken translations. When translations are also weakly broken (e.g., by disorder or lattice effects), these phonons are pinned with a mass <math><mi>m</mi></math> and damped at a rate <math><mi>Ω</mi></math>, which we explicitly compute. This contribution to <math><mi>Ω</mi></math> is distinct from that of topological defects. We show that <math><mi>Ω</mi><mo>≃</mo><mi>G</mi><msup><mi>m</mi><mn>2</mn></msup><mi>Ξ</mi></math>, where <math><mi>G</mi></math> is the shear modulus and <math><mi>Ξ</mi></math> is related to a diffusivity of the purely spontaneous state. This result follows from the smallness of the bulk and shear moduli, as would be the case in a phase with fluctuating translational order. At low temperatures, the collective modes relax quickly into the heat current, so that late time transport is dominated by the thermal diffusivity. In this regime, the resistivity in our model is linear in temperature and the ac conductivity displays a significant rearranging of the degrees of freedom, as spectral weight is shifted from an off-axis, pinning peak to a Drude-like peak. These results could shed light on transport properties in cuprate high <math><msub><mi>T</mi><mi>c</mi></msub></math> superconductors, where quantum critical behavior and translational order occur over large parts of the phase diagram and transport shows qualitatively similar features.

Universal Relaxation in a Holographic Metallic Density Wave Phase

We study thermodynamic and transport observables of quantum critical states that arise in the infrared limit of holographic renormalization group flows. Although these observables are expected to exhibit quantum critical scaling, there are a number of cases in which their frequency and temperature dependences are in apparent contradiction with scaling theories. We study two different classes of examples and show in both cases that the apparent breakdown of scaling is a consequence of the dependence of observables on an irrelevant deformation of the quantum critical state. By assigning scaling dimensions to the near-horizon observables, we formulate improved scaling theories that are completely consistent with all explicit holographic results once the dependence on the dangerously irrelevant coupling is properly accounted for. In addition to governing thermodynamic and transport phenomena in these states, we show that the dangerously irrelevant coupling also controls late-time equilibration, which occurs at a rate parametrically slower than the temperature <math><mn>1</mn><mo>/</mo><msub><mi>τ</mi><mrow><mi>eq</mi></mrow></msub><mo>≪</mo><mi>T</mi></math>. At very late times, transport is diffusion-dominated, with a diffusivity that can be written simply in terms of <math><msub><mi>τ</mi><mrow><mi>eq</mi></mrow></msub></math> and the butterfly velocity, <math><mi>D</mi><mo>∼</mo><msubsup><mi>v</mi><mi>B</mi><mn>2</mn></msubsup><msub><mi>τ</mi><mrow><mi>eq</mi></mrow></msub></math>. We conjecture that in such cases there exists a long-lived, propagating collective mode with velocity <math><msub><mi>v</mi><mi>s</mi></msub></math>, and in this case the relation <math><mi>D</mi><mo>=</mo><msubsup><mi>v</mi><mi>s</mi><mn>2</mn></msubsup><msub><mi>τ</mi><mrow><mi>eq</mi></mrow></msub></math> holds exactly in the limit <math><msub><mi>τ</mi><mrow><mi>eq</mi></mrow></msub><mi>T</mi><mo>≫</mo><mn>1</mn></math>.