Nonequilibrium steady states in driven holographic Weyl semi-metals
Wilczek Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院）；Shanghai Research Center for Quantum Sciences（上海量子科学研究中心）；InQubator for Quantum Simulation (IQuS), Department of Physics, University of Washington（华盛顿大学物理学部量子模拟创新中心）；Department of Mathematics, University of Michigan（密歇根大学数学系）
Matteo Baggioli, Sebastian Grieninger, James Stokes
Three-dimensional Weyl materials provide a controlled setting for exploring Floquet dynamics in open quantum systems, including nonequilibrium steady states (NESS). Motivated by the desire for a strongly-coupled description, we employ holography to analyze the formation and stability of a NESS in a Weyl semi-metal induced by an external circularly polarized electric field. A time-periodic steady-state solution is constructed and its stability is determined from the spectrum of out-of-equilibrium quasinormal modes (Floquet exponents). A stable region in the drive parameter space is identified; beyond a critical curve, the Floquet exponents enter the upper half of the complex plane, leading to a superharmonic response. At sufficiently strong driving, chaotic time evolution emerges in the fully nonlinear initial-boundary value problem. The anomaly-induced response of the NESS to an external magnetic field is also computed, and the resulting behavior is related to the previously proposed chiral pumping effect.
We present a holographic model of a Weyl semi-metal. We show that upon varying a mass parameter the model undergoes a quantum phase transition from a topologically non-trivial state to a trivial one. The order parameter for this phase transition is the anomalous Hall effect (AHE). We give an interpretation of the results in terms of a holographic RG flow and compare to a weakly coupled field theoretical model. Since there are no quasiparticle excitations in the strongly coupled holographic model the topological phase can not be bound to notions of topology in momentum space.
We study the surface states of a strongly coupled Weyl semimetal within holography. By explicit numerical computation of an inhomogeneous holographic Weyl semimetal we observe the appearance of an electric current restricted to the surface in presence of electric chemical potential. The total current is universal in the sense that it only depends on the topology of the phases showing that the bulk-boundary correspondence holds even at strong coupling. The implications of this result are subtle and may shed some light on anomalous transport at weak coupling.
We study a holographic model which exhibits a quantum phase transition from the strongly interacting Weyl semimetal phase to an insulating phase. In the holographic insulating phase there is a hard gap in the real part of frequency dependent diagonal conductivities. However, the anomalous Hall conductivity is nonzero at zero frequency, indicting that it is a Chern insulator. This holographic quantum phase transition is always of first order, signified by a discontinuous anomalous Hall conductivity at the phase transition, in contrast to the very continuous holographic Weyl semimetal/trivial se…
We study the effects of momentum relaxation on the holographic Weyl semimetal which exhibits a topological quantum phase transition between the Weyl semimetal phase and a topological trivial phase. The conservation of momentum in the field theory is broken by the axion fields in holography. The topological Weyl semimetal phase is characterized by a nontrivial anomalous Hall conductivity. We find that the critical value of the phase transition decreases when we increase the momentum relaxation strength up to a special value, above which it goes to zero. This indicates that the Weyl semimetal ph…
We study the AC electrical conductivity at zero temperature in a holographic model for a Weyl semimetal. At small frequencies we observe a linear dependence in the frequency. The model shows a quantum phase transition between a topological semimetal (Weyl semimetal phase) with a non vanishing anomalous Hall conductivity and a trivial semimetal. The AC conductivity has an intermediate scaling due to the presence of a quantum critical region in the phase diagram of the system. The phase diagram is reconstructed using the scaling properties of the conductivity. We compare with the experimental da…
We continue our recently proposed holographic description of single-particle correlation functions for four-dimensional chiral fermions with Lifshitz scaling at zero chemical potential, paying particular attention to the dynamical exponent z = 2. We present new results for the spectral densities and dispersion relations at non-zero momenta and temperature. In contrast to the relativistic case with z = 1, we find the existence of a quantum phase transition from a non-Fermi liquid into a Fermi liquid in which two Fermi surfaces spontaneously form, even at zero chemical potential. Our findings sh…
We consider possible ambiguities in the holographic Weyl anomaly that are caused by local terms in the flow equation. We point out that such ambiguities actually do not give physically meaningful contributions to the Weyl anomaly.
Topological phases of matter have recently attracted a rather notable attention in the community dealing with the holographic methods applied to strongly interacting condensed matter systems. In particular, holographic models for gapless Weyl and multi-Weyl semimetals, characterized on a lattice by the monopole-antimonopole defects of the Berry curvature in momentum space, were recently formulated. In this paper, motivated by the quest for finding topological holographic phases, we show that holographic model for multi-Weyl semimetals features a rather rich landscape of phases. In particular, …
The holographic Weyl semimetal is a model of a strongly coupled topological semi-metal. A topological quantum phase transition separates a topological phase with non-vanishing anomalous Hall conductivity from a trivial state. We investigate how this phase transition depends on the parameters of the scalar potential (mass and quartic self coupling) finding that the quantum phase transition persists for a large region in parameter space. We then compute the axial Hall conductivity. The algebraic structure of the axial anomaly predicts it to be 1/3 of the electric Hall conductivity. We find that …
We present the first exact calculations of the time dependence of causal correlations in driven nonequilibrium states in (2+1)-dimensional systems using holography. Comparing exact results with those obtained from simple prototype geometries that are parametrized only by a time dependent temperature, we find that the universal slowly varying features are controlled just by the pump duration and the initial and final temperatures only. We provide numerical evidence that the locations of the event and apparent horizons in the dual geometries can be deduced from the nonequilibrium causal correlat…
We consider holographic thermalization in the presence of a Weyl correction in five dimensional AdS space. We first obtain the Weyl corrected black brane solution perturbatively, up to first order in the coupling. The corresponding AdS-Vaidya like solution is then constructed. This is then used to numerically analyze the time dependence of the two point correlation functions and the expectation values of rectangular Wilson loops in the boundary field theory, and we discuss how the Weyl correction can modify the thermalization time scales in the dual field theory. In this context, the subtle in…
We consider the effects of momentum relaxation on the topological quantum phase transitions in holographic Weyl semimetals. The translational symmetry breaking in the field theory is realized in the framework of massive gravity. We find that the critical value of the phase transition, characterized by the anomalous Hall conductivity, decreases with the increasing of graviton mass, i.e. the momentum relaxation strength. There exists a critical value of graviton mass above which the topological phase transition disappears and therefore the Weyl points are destroyed. All these phenomena are quali…
The behavior of the divergent part of the bulk AdS/CFT effective action is considered with respect to the special finite diffeomorphism transformations acting on the boundary as a Weyl transformation of the boundary metric. The resulting 1-cocycle of the Weyl group is in full agreement with the 1-cocycle of the Weyl group obtained from the cohomological consideration of the effective action of the corresponding CFT.
We give a prescription for calculating the holographic Weyl anomaly in arbitrary dimension within the framework based on the Hamilton-Jacobi equation proposed by de Boer, Verlinde and Verlinde. A few sample calculations are made and shown to reproduce the results that are obtained to this time with a different method. We further discuss continuum limits, and argue that the holographic renormalization group may describe the renormalized trajectory in the parameter space. We also clarify the relationship of the present formalism to the analysis carried out by Henningson and Skenderis.
We study the ferminoic spectrum with Weyl correction, which exhibits the non-Fermi liquid behavior. Also, we find that both the height of the peak of the fermionic spectrum and the dispersion relation exhibit a nonlinearity with the variety of the Weyl coupling parameter $γ$, which mean that such nonlinearity maybe ascribe to the one of the Maxwell field. Another important property of this system is that for the holographic fermionic system with $γ<0$, the degree of the deviation from Fermi liquid is heavier than that for the one with $γ>0$. It indicates that there is a transition of coupling …
We analytically investigate the phase transition between the holographic insulator and superconductor with Weyl corrections by using the variational method for the Sturm-Liouville eigenvalue problem. We find that similar to the curvature corrections, in p-wave model, the higher Weyl couplings make the insulator/superconductor phase transition harder to occur. However, in s-wave case the Weyl corrections do not influence the critical chemical potential, which is in contrast to the effect caused by the curvature corrections. Moreover, we observe that the Weyl corrections will not affect the crit…
In this letter we construct the Stückelberg holographic superconductor with Weyl corrections. Under such corrections, the Weyl coupling parameter $γ$ plays an important role in the order of phase transitions and the critical exponents of second order phase transitions. So do the model parameters $c_α$, $α$ and $c_{4}$. Moreover, we show that the Weyl coupling parameter $γ$ and the model parameters $c_α$, $α$, $c_{4}$ which together control the size and strength of the conductivity coherence peak and the ratio of gap frequency over critical temperature $ω_{g}/T_{c}$.
We study odd viscosity in a holographic model of a Weyl semimetal. The model is characterised by a quantum phase transition from a topological semimetal to a trivial semimetal state. Since the model is axisymmetric in three spatial dimensions there are two independent odd viscosities. Both odd viscosity coefficients are non-vanishing in the quantum critical region and non-zero only due to the mixed axial gravitational anomaly. It is therefore a novel example in which the mixed axial gravitational anomaly gives rise to a transport coefficient at first order in derivatives at finite temperature.…
We study the effect of the Weyl corrections on the holographic p-wave dual models in the backgrounds of AdS soliton and AdS black hole via a Maxwell complex vector field model by using the numerical and analytical methods. We find that, in the soliton background, the Weyl corrections do not influence the properties of the holographic p-wave insulator/superconductor phase transition, which is different from that of the Yang-Mills theory. However, in the black hole background, we observe that similar to the Weyl correction effects in the Yang-Mills theory, the higher Weyl corrections make it eas…
We develop a general approach to simplify the derivation of the holographic Weyl anomaly. As an application, we derive the holographic Weyl anomaly from general higher derivative gravity in asymptotically $AdS_{5}$ and $AdS_{7}$. Interestingly, to derive all the central charges of 4d and 6d CFTs, we make no use of equations of motion. Following Myers' idea, we propose a formula of holographic entanglement entropy for higher derivative gravity in asymptotically $AdS_5$. Applying this formula, we obtain the correct universal term of entanglement entropy for 4d CFTs. It turns out that our formula…
Recently a generalization of the Fefferman-Graham gauge for asymptotically locally AdS spacetimes, called the Weyl-Fefferman-Graham (WFG) gauge, has been proposed. It was shown that the WFG gauge induces a Weyl geometry on the conformal boundary. The Weyl geometry consists of a metric and a Weyl connection. Thus, this is a useful setting for studying dual field theories with background Weyl symmetry. Working in the WFG formalism, we find the generalization of obstruction tensors, which are Weyl-covariant tensors that appear as poles in the Fefferman-Graham expansion of the bulk metric for even…
A quick review on the analytical aspects of holographic superconductors (HSC) with Weyl corrections has been presented. Mainly we focus on matching method and variations approaches. Different types of such HSC have been investigated, s-wave, p-wave and Stúckelberg ones. We also review the fundamental construction of a p-wave type , in which the non-Abelian gauge field is coupled to the Weyl tensor. The results are compared from numerics to analytical results.
We study the (3+1) dimensional p-wave holographic superconductors with Weyl corrections both numerically and analytically. We describe numerically the behavior of critical temperature $T_{c}$ with respect to charge density $ρ$ in a limited range of Weyl coupling parameter $γ$ and we find in general the condensation becomes harder with the increase of parameter $γ$. In strong coupling limit of Yang-Mills theory, we show that the minimum value of $T_{c}$ obtained from analytical approach is in good agreement with the numerical results, and finally show how we got remarkably a similar result in t…
We compute $\mathcal O(1)$ corrections to the holographic Weyl anomaly for six-dimensional $\mathcal N=(1,0)$ and $(2,0)$ theories using the functional Schrödinger method that is conjectured to work for supersymmetric theories on Ricci-flat backgrounds. We show that these corrections vanish for long representations of the $\mathcal N=(1,0)$ theory, and we obtain an expression for $δ(c-a)$ for short representations with maximum spin two. We also confirm that the one-loop corrections to the $\mathcal N=(2,0)$ M5-brane theory are equal and opposite to the anomaly for the free tensor multiplet. Fi…
We study the dynamics of the holographic $s$-wave superconductors described by the Einstein-Maxwell-complex scalar field theory with a negative cosmological constant. If the eigenfunction of the linearized equation of motion of the scalar field in the planar RNAdS black hole background is chosen as the initial data, the bulk system will evolve to the intermediate state that corresponds to the excited state superconductor on the boundary. The process can be regarded as the non-equilibrium condensation process of the excited state of holographic superconductor. When the linear superposition of t…
In this paper we study $3+1$ holographic superconductors with Weyl corrections. We find that the critical temperature of a superconductor with Weyl corrections increases as we amplify the Weyl coupling parameter $γ$, indicating the condensation will be harder when the parameter $γ$ decreases. We also calculate the conductivity and the ratio of gap frequency over critical temperature $ω_{g}/T_{c}$ numerically for various coupling parameters. We find that the ratio $ω_g/T_c$ becomes larger when the Weyl coupling parameter $γ$ decreases. We also notice that when $γ< 0$ there is an extra spike tha…
In this paper, we analytically compute the basic parameters of the p-wave holographic superconductors with Weyl geometrical corrections using the matching method. The explicit correspondence between the critical temperature $T_c$ and the dual charge density $ρ$ has been calculated as $T_c\proptoρ^{{1}{3}}$ and the dependence of the vacuum expectation value for the dual condensate operator $\cal{O}$ on the temperature has been found analytically in the form $<{\cal O}_{+}>\propto T_c^{{3}{2}}T^{Δ-{1}{2}}\sqrt{1-({T}{T_c})^3}$. The critical exponent ${1}{2}$ is an universal quantity according to…
We investigate analytically the properties of the Weyl holographic superconductor in the Lifshitz black hole background. We find that the critical temperature of the Weyl superconductor decreases with increasing Lifshitz dynamical exponent, $z$, indicating that condensation becomes difficult. In addition, it is found that the critical temperature and condensation operator could be affected by applying the Weyl coupling, $γ$. Moreover, we compute the critical magnetic field and investigate its dependence on the parameters $γ$ and $z$. Finally, we show numerically that the Weyl coupling paramete…
We propose a recipe - arguably the simplest - to compute the holographic type-B Weyl anomaly for general higher-derivative gravity in asymptotically AdS spacetimes. In 5 and 7 dimensions we identify a suitable basis of curvature invariants that allows to read off easily, without any further computation, the Weyl anomaly coefficients of the dual CFT. We tabulate the contributions from quadratic, cubic and quartic purely algebraic curvature invariants and also from terms involving derivatives of the curvature. We provide few examples, where the anomaly coefficients have been obtained by other me…
We use the relation between certain diffeomorphisms in the bulk and Weyl transformations on the boundary to build the conformal structure of the metric in the presence of matter in the bulk. We explicitly obtain the conformal anomaly in any spacetime dimension d in a holographic frame work in the case of gravity coupled to the scalar fields. This way we also provide a holographic construction of the bulk spacetime metric and of the matter fields in the bulk of this spacetime out of sources that are related to conformal field theory data. With the latter, we produce an asymptotic expansion of t…
We compute the O(1) contribution to holographic c-a for IIB supergravity on AdS_5 x S^5/Z_n and on AdS_5 x T^{1,1}/Z_n. In both cases, we find agreement with the dual field theory results, thus providing 1/N^2 checks of AdS/CFT with reduced supersymmetry. Since the holographic computation involves a sum over shortened multiplets in the KK tower, we provide some details on the S^5 and T^{1,1} spectra in a form that is convenient when considering their Z_n orbifolds. The computation for the even Z_n orbifolds of S^5 includes a sum over the multiplets in the twisted sector that is essential for o…
The holographic Weyl anomaly for GJMS operators (or conformal powers of the Laplacian) are obtained in four and six dimensions. In the context of AdS/CFT correspondence, free conformal scalars with higher-derivative kinetic operators are induced by an ordinary second-derivative massive bulk scalar. At one-loop quantum level, the duality dictionary for partition functions entails an equality between the functional determinants of the corresponding kinetic operators and, in particular, it provides a holographic route to their Weyl anomalies. The heat kernel of a single bulk massive scalar field …
We construct a higher derivative theory involving an axionic field and the Weyl tensor in four dimensional spacetime. Up to the first order of the coupling parameters, the charged black brane solution with momentum dissipation in a perturbative manner is constructed. Metal-insulator transitions are implemented when varying the system parameters at zero temperature. Also, we study the transports including DC conductivity and optical conductivity at zero charge density. We observe the exact particle-vortex duality for some specific momentum dissipation strength.
For conformal field theories which admit a dual gravitational description in anti-de Sitter space, electrical transport properties, such as conductivity and charge diffusion, are determined by the dynamics of a U(1) gauge field in the bulk and thus obey universality relations at the classical level due to the uniqueness of the Maxwell action. We analyze corrections to these transport parameters due to higher-dimension operators in the bulk action, beyond the leading Maxwell term, of which the most significant involves a coupling to the bulk Weyl tensor. We show that the ensuing corrections to …
The holographic Weyl anomaly associated to Chern-Simons gravity in 2n+1 dimensions is proportional to the Euler term in 2n dimensions, with no contributions from the Weyl tensor. We compute the holographic energy-momentum tensor associated to Chern-Simons gravity directly from the action, in an arbitrary odd-dimensional spacetime. We show, in particular, that the counterterms rendering the action finite contain only terms of the Lovelock type.
We present a first attempt to derive the full (type-A and type-B) Weyl anomaly of four dimensional conformal higher spin (CHS) fields in a holographic way. We obtain the type-A and type-B Weyl anomaly coefficients for the whole family of 4D CHS fields from the one-loop effective action for massless higher spin (MHS) Fronsdal fields evaluated on a 5D bulk Poincaré-Einstein metric with an Einstein metric on its conformal boundary. To gain access to the type-B anomaly coefficient we assume, for practical reasons, a Lichnerowicz-type coupling of the bulk Fronsdal fields with the bulk background We…
We employ the holographic method to study fluctuations and dissipation of an $n$-dimensional moving mirror coupled to quantum critical theories in $d$ spacetime dimensions. The bulk counterpart of the mirror with perfect reflection is a D$(n+1)$ brane in the Lifshitz geometry of $d+1$ dimensions. The motion of the mirror can be realized from the dynamics of the brane at the boundary of the bulk. The excited modes of the brane in the bulk render the mirror undergoing Brownian motion. For small displacement of the mirror, we derive the analytical results of the correlation functions and response…
We present a holographic formula relating functional determinants: the fermion determinant in the one-loop effective action of bulk spinors in an asymptotically locally AdS background, and the determinant of the two-point function of the dual operator at the conformal boundary. The formula originates from AdS/CFT heuristics that map a quantum contribution in the bulk partition function to a subleading large-N contribution in the boundary partition function. We use this holographic picture to address questions in spectral theory and conformal geometry. As an instance, we compute the type-A Weyl…
We study holographic charge transport in (2+1) dimensions at finite $N$, whose dual gravity background is given by perturbative black hole solution in Einstein theory plus cubic terms of Weyl tensor. We consider the higher derivative corrections to the standard Maxwell action, given by the interacting terms between the Weyl tensor and the field strength. We calculate the DC conductivity by using both the membrane paradigm and the Kubo's formula and find precise agreement. We compute the AC conductivity and find an analog of the crossover from `metal' to `bad metal' in the low frequency limit. …
We calculate the Weyl anomaly for conformal field theories that can be described via the adS/CFT correspondence. This entails regularizing the gravitational part of the corresponding supergravity action in a manner consistent with general covariance. Up to a constant, the anomaly only depends on the dimension d of the manifold on which the conformal field theory is defined. We present concrete expressions for the anomaly in the physically relevant cases d = 2, 4 and 6. In d = 2 we find for the central charge c = 3 l/ 2 G_N in agreement with considerations based on the asymptotic symmetry algeb…
Using the relation between diffeomorphisms in the bulk and Weyl transformations on the boundary we study the Weyl transformation properties of the bulk metric on shell and of the boundary action. We obtain a universal formula for one of the classes of trace anomalies in any even dimension in terms of the parameters of the gravity action.
The γ-trace anomaly of supersymmetry current in a supersymmetric gauge theory shares a superconformal anomaly multiplet with the chiral R-symmetry anomaly and the Weyl anomaly, and its holographic reproduction is a valuable test to the AdS/CFT correspondence conjecture. We investigate how the γ-trace anomaly of the supersymmetry current of {\cal N}=1 four-dimensional supersymmetric gauge theory in an {\cal N}=1 conformal supergravity background can be extracted out from the ${\cal N}=2$ gauged supergravity in five dimensions. It is shown that the reproduction of this super-Weyl anomaly origina…
We study the magnetic field effects on the quantum critical point (QCP) in the holographic Weyl semimetal model. We show that it increases quadratically with the magnetic field for weak field and linear with the magnetic field for strong field. Our findings are compatible with previous results in the literature from other approaches.
Floquet states can be realized in quantum systems driven by continuous time-periodic perturbations. It is known that a state known as the Floquet Weyl semimetal can be realized when free Dirac fermions are placed in a rotating electric field. What will happen if strong interaction is introduced to this system? Will the interaction wash out the characteristic features of Weyl semimetals such as the Hall response? Is there a steady state and what is its thermodynamic behavior? We answer these questions using AdS/CFT correspondence in the $\mathcal{N}=2$ supersymmetric massless QCD in a rotating …
We study the charge response in complex frequency plane and the quasi-normal modes (QNMs) of the boundary quantum field theory with momentum dissipation dual to a probe generalized Maxwell system with Weyl correction. When the strength of the momentum dissipation $\hatα$ is small, the pole structure of the conductivity is similar to the case without the momentum dissipation. The qualitative correspondence between the poles of the real part of the conductivity of the original theory and the ones of its electromagnetic (EM) dual theory approximately holds when $γ\rightarrow -γ$ with $γ$ being th…
With the aim to reveal universal features of hadronic matter and correlated Dirac insulators in strong AC-electric fields, we study the $\mathcal{N}=2$ supersymmetric QCD with a finite quark mass driven by a rotating electric field $\mathcal{E}_x+i\mathcal{E}_y= E e^{iΩt}$. The analysis is done in the holographically dual D3/D7 system in the co-rotating frame, effectively. The nonequilibrium phase diagram is determined from the threshold electric field at which the insulator phase breaks down to a conductive phase due to the AC version of the Schwinger mechanism. The external field induces a r…
We explore the connection between the holographic Weyl anomaly and the superconformal index in six-dimensional $(1,0)$ theories. Using earlier results from holographic computations of the $\mathcal O(1)$ contributions $δa$ and $δ(c-a)$ to the corresponding six-dimensional Weyl anomaly coefficients, we derive a pair of differential operators that extracts these values from the large-$N$ single-trace index. In doing so, we also highlight the structure of these corrections in terms of group theory invariants of the superconformal representations.
We consider the entropy bounds recently conjectured by Fischler, Susskind and Bousso, and proven in certain cases by Flanagan, Marolf and Wald (FMW). One of the FMW derivations supposes a covariant form of the Bekenstein entropy bound, the consequences of which we explore. The derivation also suggests that the entropy contained in a vacuum spacetime, e.g. Schwarzschild, is related to the shear on congruences of null rays. We find evidence for this intuition, but in a surprising way. We compare the covariant entropy bound to certain earlier discussions of black hole entropy, and comment on the …
Holographic Weyl Anomaly and Kounterterms in AdS gravity
Universidad Adolfo Ibañez（阿道夫·伊巴涅斯大学）；Universidad Arturo Prat（阿图罗·普拉特大学）；Pontificia Universidad Católica de Valparaíso（瓦尔帕莱索天主教 Pontificia 大学）；Politecnico di Torino（都灵理工大学）
It is a well-known property of holographic theories that diffeomorphism invariance in the bulk space-time implies Weyl invariance of the dual holographic field theory in the sense that the field theory couples to a conformal class of background metrics. The usual Fefferman-Graham formalism, which provides us with a holographic dictionary between the two theories, breaks explicitly this symmetry by choosing a specific boundary metric and a corresponding specific metric ansatz in the bulk. In this paper, we show that a simple extension of the Fefferman-Graham formalism allows us to sidestep this…
