First, we reformulate RG transformations in a recursive way with introduction of an order-parameter field. As a result, we manifest the RG flow of an effective field theory through the emergence of an extra dimensional space, where both RG $β-$functions of coupling functions and RG flow equation of the order-parameter field appear in the resulting effective action explicitly through the extra dimensional space. Second, we consider an effective dual holographic description derived recently, where the classical gravity theory of the large $N$ limit takes into account quantum corrections in the all-loop order. This non-perturbative nature turns out to originate from an intertwined renormalization structure between both RG flow equations of coupling functions and order-parameter fields in the emergent extra-dimensional space, where the IR boundary condition of the order-parameter field gives rise to a mean-field equation with fully renormalized interaction coefficients. Third, comparing the RG-reformulated effective field theory with this effective dual holographic description, we obtain term-by-term matching conditions in the level of an effective action. As a result, we express all RG coefficients of fields and interaction vertices in terms of the metric tensor of the dual holographic theory. Through this metric reformulation for the RG analysis, we propose a prescription on how to find RG $β-$functions of interaction coefficients in a non-perturbative way beyond the perturbative RG analysis. In particular, we claim that the present prescription of the RG flow generalizes the holographic RG flow of the holographic duality conjecture towards the absence of conformal symmetry, where the emergent holographic dual effective field theory has been derived from the first principle, thus being applicable even away from quantum criticality.
In holographic duality, dynamics along the emergent extra-dimensional space describes a renormalization group (RG) flow of the corresponding quantum field theory (QFT). Following this idea, we develop an emergent holographic description of a QFT, where not only the information of the RG flow is introduced into an IR holographic dual effective field theory (HDEFT), but also the UV information of the QFT is encoded in the HDEFT through the IR boundary condition. In particular, we argue that this dual holographic construction is self-consistent within the assumption of bulk locality, showing the …
We revisit the theory of strongly correlated quantum matter perturbed by Harris-marginal random-field disorder, using the simplest holographic model. We argue that for weak disorder, the ground state of the theory is not Lifshitz invariant with a non-trivial disorder-dependent dynamical exponent, as previously found. Instead, below a non-perturbatively small energy scale, we predict infrared physics becomes independent of the disorder strength.
To clarify the mathematical structure of the RG-derived holographic dual field theory, we rewrite the string-theory based conventionally utilized dual holographic effective field theory based on the ADM decomposition of the metric tensor. This comparison leads us to claim that the RG-derived emergent holographic dual field theory takes into account higher-derivative curvature terms with gauge fixing in the string-theory based conventionally utilized Einstein-Klein-Gordon theory, giving rise to the RG flow of the metric tensor beyond the AdS (anti-de Sitter space) geometry. Furthermore, we comp…
We derive an effective dual holographic Einstein-Maxwell theory, applying renormalization group transformations to interacting Dirac fermions in a recursive way. In particular, we show how both background metric tensor and U(1) gauge fields become dynamical to describe the renormalization group flows of coupling functions and order parameter fields of the corresponding quantum field theory through natural emergence of an extra-dimensional space, respectively. Finally, we propose a prescription on how to calculate correlation functions in a non-perturbative way, where quantum fluctuations of bo…
Applying recursive renormalization group transformations to a scalar field theory, we obtain an effective quantum gravity theory with an emergent extra dimension, described by a dual holographic Einstein-Klein-Gordon type action. Here, the dynamics of both the dual order-parameter field and the metric tensor field originate from density-density and energy-momentum tensor-tensor effective interactions, respectively, in the recursive renormalization group transformation, performed approximately in the Gaussian level. This linear approximation in the recursive renormalization group transformation…
Implementing the Wilsonian renormalization group (RG) transformation in a nonperturbative way, we construct an effective holographic dual description with an emergent extradimension identified with an RG scale. Taking the large$-N$ limit, we obtain an equation of motion of an order-parameter field, here the chiral condensate for our explicit demonstration. In particular, an intertwined structure manifests between the first-order RG flow equations of renormalized coupling functions and the second-order differential equation of the order-parameter field, thus referred to as a nonperturbative RG-…
Holographic duality conjecture has been proposed to be a novel non-perturbative theoretical framework for the description of strongly correlated electrons. However, the duality transformation is not specified to cause ambiguity in the application of this theoretical machinery to condensed matter physics. In this study, we propose a prescription for the holographic duality transformation. Based on recursive renormalization group (RG) transformations, we obtain an effective field theory, which manifests the RG flow of an effective action through the introduction of an extra dimension. Resorting …
In this paper, we propose an {\it exact holographic mapping} which is a unitary mapping from the Hilbert space of a lattice system in flat space (boundary) to that of another lattice system in one higher dimension (bulk). By defining the distance in the bulk system from two-point correlation functions, we obtain an emergent bulk space-time geometry that is determined by the boundary state and the mapping. As a specific example, we study the exact holographic mapping for $(1+1)$-dimensional lattice Dirac fermions and explore the emergent bulk geometry corresponding to different boundary states …
We propose a way to discover, in tabletop experiments, spacetime-emergent materials, that is, materials holographically dual to higher-dimensional quantum gravity systems under the AdS/CFT correspondence. The emergence of the holographic spacetime is verified by a mathematical imaging transform of the response function on the material. We consider theories on a 1-dimensional ring-shaped material, and compute the response to a scalar source locally put at a point on the ring. When the theory on the material has a gravity dual, the imaging in the low temperature phase exhibits a distinct differe…
We explore the phenomenon of emergent Lorentz invariance in strongly coupled theories. The strong dynamics is handled using the gauge/gravity correspondence. We analyze how the renormalization group flow towards Lorentz invariance is reflected in the two-point functions of local operators and in the dispersion relations of the bound states. The deviations of these observables from the relativistic form at low energies are found to be power-law suppressed by the ratio of the infrared and ultraviolet scales. We show that in a certain subclass of models the velocities of the light bound states st…
Strongly coupled conformal field theory appears to describe universal scaling around quantum criticality, where critical exponents reflect the nature of emergent excitations. In particular, novel symmetries can emerge from strong interactions, expected to be responsible for quantum number fractionalization. The underlying mechanism has been proposed that an emergent enhanced symmetry allows a topological term associated with anomaly, which assigns a fermion's quantum number to a topological excitation, referred as the Goldstone-Wilczek current. Although this mechanism has been verified in one …
We argue that there exist simple effective field theories describing the long-distance dynamics of holographic liquids. The degrees of freedom responsible for the transport of charge and energy-momentum are Goldstone modes. These modes are coupled to a strongly coupled infrared sector through emergent gauge and gravitational fields. The IR degrees of freedom are described holographically by the near-horizon part of the metric, while the Goldstone bosons are described by a field-theoretical Lagrangian. In the cases where the holographic dual involves a black hole, this picture allows for a dire…
Holographic Representations of Topological Quantum Criticality: Emergent Symmetry Approach around the Bott Clock
Key Laboratory of Quantum Materials and Devices of Ministry of Education, School of Physics, Southeast University（东南大学物理学院教育部量子材料与器件重点实验室）；Department of Physics and Astronomy, University of British Columbia（不列颠哥伦比亚大学物理与天文系）
We explore the far from equilibrium response of a holographic superfluid using the AdS/CFT correspondence. We establish the dynamical phase diagram corresponding to quantum quenches of the order parameter source field. We find three distinct regimes of behaviour that are related to the spectrum of black hole quasi-normal modes. These correspond to damped oscillations of the order parameter, and over-damped approaches to the superfluid and normal states. The presence of three regimes, which includes an emergent dynamical temperature scale, is argued to occur more generally in time-reversal inva…
We study renormalization group flows in strongly interacting field theories with fermions that correspond to transitions between a theory without Lorentz invariance at high energies down to a theory with approximate Lorentz symmetry in the infrared. Holographic description of the strong coupling is used. The emphasis is made on emergence of chiral fermions in the low-energy theory.
In numerical simulations of classical and quantum lattice systems, 2d corner transfer matrices (CTMs) and 3d corner tensors (CTs) are a useful tool to compute approximate contractions of infinite-size tensor networks. In this paper we show how the numerical CTMs and CTs can be used, {\it additionally\/}, to extract universal information from their spectra. We provide examples of this for classical and quantum systems, in 1d, 2d and 3d. Our results provide, in particular, practical evidence for a wide variety of models of the correspondence between $d$-dimensional quantum and $(d+1)$-dimensiona…
We consider holographic superconductors whose bulk description consists of gravity minimally coupled to a Maxwell field and charged scalar field with general potential. We give an analytic argument that there is no "hard gap": the real part of the conductivity at low frequency remains nonzero (although typically exponentially small) even at zero temperature. We also numerically construct the gravitational dual of the ground state of some holographic superconductors. Depending on the charge and dimension of the condensate, the infrared theory can have emergent conformal or just Poincare symmetr…
A holographic description on antiferromagnetic quantum phase transition (QPT) induced by magnetic field and the criticality in the vicinity of quantum critical point (QCP) have been investigated numerically recently. In this paper, we show that the properties of QPT in this holographic model are governed by a CFT dual to the emergent AdS$_2$ in the IR region, which confirms that the dual boundary theory is a strong coupling theory with dynamic exponent $z=2$ and logarithmic corrections appear. We also compare them with the results from Hertz model by solving RG equation at its upper critical d…
We discuss the time dependence of subsystem entropies in interacting quantum systems. As a model for the time dependence, we suggest that the entropy is as large as possible given two constraints: one follows from the existence of an emergent light cone, and the other is a conjecture associated to the "entanglement velocity" $v_E$. We compare this model to new holographic and spin chain computations, and to an operator growth picture. Finally, we introduce a second way of computing the emergent light cone speed in holographic theories that provides a boundary dynamics explanation for a special…
We elaborate on our earlier proposal connecting entanglement renormalization and holographic duality in which we argued that a tensor network can be reinterpreted as a kind of skeleton for an emergent holographic space. Here we address the question of the large $N$ limit where on the holographic side the gravity theory becomes classical and a non-fluctuating smooth spacetime description emerges. We show how a number of features of holographic duality in the large $N$ limit emerge naturally from entanglement renormalization, including a classical spacetime generated by entanglement, a sparse sp…
We analyze the five-dimensional AdS gravity coupled to a gauge field and a charged scalar field. Under a Scherk-Schwarz compactification, we show that the system undergoes a superconductor/insulator transition at zero temperature in 2+1 dimensions as we change the chemical potential. By taking into account a confinement/deconfinement transition, the phase diagram turns out to have a rich structure. We will observe that it has a similarity with the RVB (resonating valence bond) approach to high-Tc superconductors via an emergent gauge symmetry.
We study a conjectured connection between the AdS/CFT and a real-space quantum renormalization group scheme, the multi-scale entanglement renormalization ansatz (MERA). By making a close contact with the holographic formula of the entanglement entropy, we propose a general definition of the metric in the MERA in the extra holographic direction, which is formulated purely in terms of quantum field theoretical data. Using the continuum version of the MERA (cMERA), we calculate this emergent holographic metric explicitly for free scalar boson and free fermions theories, and check that the metric …
Capturing the interplay between electronic correlations and many-particle entanglement requires a unified framework for Hamiltonian and eigenbasis renormalization. In this work, we apply the unitary renormalization group (URG) scheme developed in a companion work (Ref.[1]) to the study of two archetypal models of stongly correlated lattice electrons, one with translation invariance and one without. We obtain detailed insight into the emergence of various gapless and gapped phases of quantum electronic matter by computing effective Hamiltonians as well as entanglement signatures through their r…
We derive a holographic dual for a gauged matrix model in general dimensions from a first-principle construction. The dual theory is shown to be a closed string field theory which includes a compact two-form gauge field coupled with closed strings in one higher dimensional space. Possible phases of the matrix model are discussed in the holographic description. Besides the confinement phase and the IR free deconfinement phase, there can be two different classes of critical states. The first class describes `holographic critical states' where strings are deconfined in the bulk. The second class …
We discuss, from a quantum information perspective, recent proposals of Maldacena, Ryu, Takayanagi, van Raamsdonk, Swingle, and Susskind that spacetime is an emergent property of the quantum entanglement of an associated boundary quantum system. We review the idea that the informational principle of minimal complexity determines a dual holographic bulk spacetime from a minimal quantum circuit U preparing a given boundary state from a trivial reference state. We describe how this idea may be extended to determine the relationship between the fluctuations of the bulk holographic geometry and the…
The holographic correspondence creates an interface between classical gravitational physics and the dynamics of strongly interacting quantum field theories. This chapter will relate the physics of charged, asymptotically Anti-de Sitter spacetimes to the phenomenology of low temperature critical phases of condensed matter. Common essential features will characterise both the gravitational and field theoretic systems. Firstly, an emergent scaling symmetry at the lowest energy scales appears as an emergent isometry in the interior, `near horizon' regime of the spacetime. Secondly, the field theor…
Superconductivity is abundant near quantum-critical points, where fluctuations suppress the formation of Fermi liquid quasiparticles and the BCS theory no longer applies. Two very distinct approaches have been developed to address this issue: quantum-critical Eliashberg theory and holographic superconductivity. The former includes a strongly retarded pairing interaction of ill-defined fermions, the latter is rooted in the duality of quantum field theory and gravity theory. We demonstrate that both are different perspectives of the same theory. We derive holographic superconductivity in form of…
Deep learning emergent spacetime from fermionic spectral functions in holography
Kyoto University（京都大学）；Asia Pacific Center for Theoretical Physics（亚太理论物理中心）；Pohang University of Science and Technology（浦项科技大学）；Gwangju Institute of Science and Technology（光州科学技术院）
Emergent Symmetry and Phase Transitions on the Domain Wall of $\mathbb{Z}_{2}$ Topological Orders
Kavli Institute for Theoretical Sciences and CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences（中国科学院理论物理研究所与拓扑量子计算卓越中心，中国科学院大学）；Graduate School of China Academy of Engineering Physics（中国工程物理研究院研究生院）
We construct numerically finite density domain-wall solutions which interpolate between two $AdS_4$ fixed points and exhibit an intermediate regime of hyperscaling violation, with or without Lifshitz scaling. Such RG flows can be realized in gravitational models containing a dilatonic scalar and a massive vector field with appropriate choices of the scalar potential and couplings. The infrared $AdS_4$ fixed point describes a new ground state for strongly coupled quantum systems realizing such scalings, thus avoiding the well-known extensive zero temperature entropy associated with $AdS_2 \time…
We take the tensor network describing explicit p-adic CFT partition functions proposed in [1], and considered boundary conditions of the network describing a deformed Bruhat-Tits (BT) tree geometry. We demonstrate that this geometry satisfies an emergent graph Einstein equation in a unique way that is consistent with the bulk effective matter action encoding the same correlation function as the tensor network, at least in the perturbative limit order by order away from the pure BT tree. Moreover, the (perturbative) definition of the graph curvature in the Mathematics literature naturally emerg…
Wavelets encode data at multiple resolutions, which in a wavelet description of a quantum field theory, allows for fields to carry, in addition to space-time coordinates, an extra dimension: scale. A recently introduced Exact Holographic Mapping [C.H. Lee and X.-L. Qi, Phys. Rev. B 93, 035112 (2016)] uses the Haar wavelet basis to represent the free Dirac fermionic quantum field theory (QFT) at multiple renormalization scales thereby inducing an emergent bulk geometry in one higher dimension. This construction is, in fact, generic and we show how higher families of Daubechies wavelet transform…
