Abstract We demonstrate the emergence of a holographic dimension in a system of 2D non-interacting Dirac fermions placed on a torus, by studying the scaling of multipartite entanglement measures under a sequence of renormalisation group (RG) transformations applied in momentum space. Geometric measures defined in this emergent space can be related to the RG beta function of the spectral gap, hence establishing a holographic connection between the spatial geometry of the emergent spatial dimension and the entanglement properties of the boundary quantum theory. We prove, analytically, that changing the boundedness of the holographic space involves a topological transition accompanied by a critical Fermi surface in the boundary theory. We go on to show that this results in the formation of a quantum wormhole geometry that connects the UV and the IR of the emergent dimension. The additional conformal symmetry at the transition also supports a relation between the emergent metric and the stress-energy tensor. In the presence of an Aharonov–Bohm flux, the entanglement gains a geometry-independent piece which is shown to be topological, sensitive to changes in boundary conditions, and related to the Luttinger volume of the system. Upon the insertion of a strong transverse magnetic field, we show that the Luttinger volume is linked to the Chern number of the occupied single-particle Landau levels.
Abstract We demonstrate the emergence of a holographic dimension in a system of 2D non-interacting Dirac fermions placed on a torus, by studying the scaling of multipartite entanglement measures under a sequence of renormalisation group (RG) transformations applied in momentum space. Geometric measures defined in this emergent space can be related to the RG beta function of the spectral gap, hence establishing a holographic connection between the spatial geometry of the emergent spatial dimension and the entanglement properties of the boundary quantum theory. We prove, analytically, that changing the boundedness of the holographic space involves a topological transition accompanied by a critical Fermi surface in the boundary theory. We go on to show that this results in the formation of a quantum wormhole geometry that connects the UV and the IR of the emergent dimension. The additional conformal symmetry at the transition also supports a relation between the emergent metric and the stress-energy tensor. In the presence of an Aharonov–Bohm flux, the entanglement gains a geometry-independent piece which is shown to be topological, sensitive to changes in boundary conditions, and related to the Luttinger volume of the system. Upon the insertion of a strong transverse magnetic field, we show that the Luttinger volume is linked to the Chern number of the occupied single-particle Landau levels.
We present a unified framework for the renormalisation of the Hamiltonian and eigenbasis of a system of correlated electrons, unveiling thereby the interplay between electronic correlations and many-particle entanglement. For this, we extend substantially the unitary renormalization group (URG) scheme introduced in Refs. [1], [2], [3]. We recast the RG as a discrete flow of the Hamiltonian tensor network, i.e., the collection of various 2n-point scattering vertex tensors comprising the Hamiltonian. The renormalisation progresses via unitary transformations that block diagonalizes the Hamiltonian iteratively via the disentanglement of single-particle eigenstates. This procedure incorporates naturally the role of quantum fluctuations. The RG flow equations possess a non-trivial structure, displaying a feedback mechanism through frequency-dependent dynamical self-energies and correlation energies. The interplay between various UV energy scales enables the coupled RG equations to flow towards a stable fixed point in the IR. The effective Hamiltonian at the IR fixed point generically has a reduced parameter space, as well as number of degrees of freedom, compared to the microscopic Hamiltonian. Importantly, the vertex RG flows are observed to govern the RG flow of the tensor network that denotes the coefficients of the many-particle eigenstates. The RG evolution of various many-particle entanglement features of the eigenbasis are, in turn, quantified through the coefficient tensor network. In this way, we show that the URG framework provides a microscopic understanding of holographic renormalisation: the RG flow of the vertex tensor network generates a eigenstate coefficient tensor network possessing a many-particle entanglement metric. We find that the eigenstate tensor network accommodates sign factors arising from fermion exchanges, and that the IR fixed point reached generically involves a trivialisation of the fermion sign factor. Several results are presented for the emergence of composite excitations in the neighbourhood of a gapless Fermi surface, as well as for the condensation phenomenon involving the gapping of the Fermi surface.
Abstract We perform a unitary renormalization group (URG) study of the 1D fermionic Hubbard model. The formalism generates a family of effective Hamiltonians and many-body eigenstates arranged holographically across the tensor network from UV to IR. The URG is realized as a quantum circuit, leading to the entanglement holographic mapping (EHM) tensor network description. A topological Θ-term of the projected Hilbert space of the degrees of freedom at the Fermi surface are shown to govern the nature of RG flow towards either the gapless Tomonaga-Luttinger liquid or gapped quantum liquid phases. This results in a nonperturbative version of the Berezenskii-Kosterlitz-Thouless (BKT) RG phase diagram, revealing a line of intermediate coupling stable fixed points, while the nature of RG flow around the critical point is identical to that obtained from the weak-coupling RG analysis. This coincides with a phase transition in the many-particle entanglement, as the entanglement entropy RG flow shows distinct features for the critical and gapped phases depending on the value of the topological Θ-term. We demonstrate the Ryu-Takyanagi entropy bound for the many-body eigenstates comprising the EHM network, concretizing the relation to the holographic duality principle. The scaling of the entropy bound also distinguishes the gapped and gapless phases, implying the generation of very different holographic spacetimes across the critical point. Finally, we treat the Fermi surface as a quantum impurity coupled to the high energy electronic states. A thought-experiment is devised in order to study entanglement entropy generated by isolating the impurity, and propose ways by which to measure it by studying the quantum noise and higher order cumulants of the full counting statistics.
The thermal dynamics of the two-dimensional Ising model and quantum dynamics of the one-dimensional transverse-field Ising model (TFIM) are mapped to one another through the transfer-matrix formalism. We show that the fermionised TFIM undergoes a Fermi-surface topology-changing Lifshitz transition at its critical point. We identify the degree of freedom which tracks the Lifshitz transition via changes in topological quantum numbers (e.g., Chern number, Berry phase etc.). An emergent SU(2) symmetry at criticality is observed to lead to a topological quantum number different from that which characterises the ordered phase. The topological transition is also understood via a spectral flow thought-experiment in a Thouless charge pump, revealing the bulk-boundary correspondence across the transition. The duality property of the phases and their entanglement content are studied, revealing a holographic relation with the entanglement at criticality. The effects of a non-zero longitudinal field and interactions that scatter across the singular Fermi surface are treated within the renormalisation group (RG) formalism. The analysis reveals that the critical point of the 1D TFIM and the 1D spin-1/2 Heisenberg chain are connected via a line of SU(2)-symmetric theories. We extend our analysis to show that the classical to quantum correspondence links the critical theories of Ising models in various dimensions holographically through the universal effective Hamiltonian that describes the Lifshitz transition of the 1D TFIM. We obtain in this way a unified perspective of transitions in Ising models that lie beyond the traditional Ginzburg-Landau-Wilson paradigm. We discuss the consequences of our results for similar topological transitions observed in classical spin models, topological insulators, superconductors and lattice gauge-field theories which are related to the Ising universality class.
Part I: Entanglement Entropy of 2d Quasi-Topological Quantum Field Theory We compute the entanglement entropy of two-dimensional quasi-topological quantum field the- ories (QTFTs). These are theories in which the correlation functions depend on the topology and on the total area of the underlying space-time, but are blind to all local details of the geometry, and include topological quantum field theory (TFT) as their limiting case. We use two complimentary methods to compute the entanglement entropy; the first method is the replica trick and the other is to devise a novel tensor network representation, or more precisely, matrix product state (MPS) representation, of the quantum states of QTFT. We demonstrate that the two calculations are in agreement. Part II: Geometry of the Exact Renormalization Group and Higher Spin Holography We consider the Wilson-Polchinski exact renormalization group (RG) applied to the generating functional of single-trace operators at a free-fixed point in d = 2 + 1 dimensions. By exploiting the rich symmetry structure of free-field theory, we study the geometric nature of the RG equations and the associated Ward identities. The geometry, as expected, is holographic, with anti-de Sitter spacetime emerging correspondent with RG fixed points. In particular, we are able to cast the renormalization group equations as Hamilton equations of radial evolution in AdSd+1. We solve these bulk equations of motion in terms of a boundary source and derive an on-shell bulk action. We demonstrate that it correctly encodes all of the correlation functions of the field theory, written as “Witten diagrams.” Going further, we show that the field theory construction gives us a par- ticular vector bundle over the (d + 1)-dimensional RG mapping space, called a jet bundle, whose structure group arises from the bilocal transformations of the bare fields in the path integral. The sources for quadratic operators constitute a connection on this bundle and a section of its endomor- phism bundle. We make comparisons to Vasiliev-type higher spin theories. Detailed calculations are carried out for the case of Majorana fermions. Results and comments are presented for complex scalars. Additional details can be found in [1, 2].
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 [1] to the study of two archetypal models of strongly 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 from numerical evaluation of the various RG equations, as well as their entanglement signatures through their respective tensor network descriptions. For the translationally invariant model of a single-band of interacting electrons, this includes results on gapless metallic phases such as the Fermi liquid and Marginal Fermi liquid, as well as gapped phases such as the reduced Bardeen-Cooper-Schrieffer, pair density-wave and Mott liquid phases. Additionally, a study of a generalised Sachdev-Ye model with disordered four-fermion interactions offers detailed results on many-body localised phases, as well as thermalised phase. We emphasise the distinctions between the various phases based on a combined analysis of their dynamical (obtained from the effective Hamiltonian) and entanglement properties. Importantly, the RG flow of the Hamiltonian vertex tensor network is shown to lead to emergent gauge theories for the gapped phases. Taken together with results on the holographic spacetime generated from the RG of the many-particle eigenstate (seen through, for instance, the holographic upper bound of the one-particle entanglement entropy), our analysis offer an ab-initio perspective of the gauge-gravity duality for quantum liquids that are emergent in systems of correlated electrons.
Modelling strongly correlated systems is an incredibly hard problem, but a very fruitful one when achieved. The main reason for its difficulty is that perturbation theory approaches do not work for certain interesting coupling regimes. However it is believed that a holographic duality would map models with strong coupling strengths into weakly coupled bulk regimes where our current tools do work. Interest in applications to condensed matter systems has been growing throughout the past decade Though questions remain about how to do it and if the duality is valid in a relevant regime.Recently, it has been proposed that the multi-scale entanglement renormalisation ansatz (MERA) m an efficient representation for ground states of local Hamiltonians on a lattice m realises at least some features of the AdS/CFT correspondence. For example, an one dimensional (1D) quantum lattice model described by the MERA has its description stored in the renormalisation group (RG) flow of the ground state, detailing the state at different length scales. When implemented, this flow is encoded in a two dimensional (2D) hyperbolic tensor network where the extra dimension is the aforementioned length scale of the lattice system. Specifically the ansatz is based on a real space RG transformation, known as entanglement renormalisation, that removes local entanglement before coarse-graining the state.To look at this I studied the MERA tensor network by adding additional degrees of freedom on each bond - a connection between separate tensors in the network - to obtain a physical geometry in 2D. The focus was to keep the bulk theory description in the tensor network regime to allow straightforward applications of numerical techniques so that this procedure can be applied to any quantum field theory regardless of coupling strength. To achieve this, myself and collaborators introduced the new degrees of freedom by inserting tensors into the original MERA network, qliftingq it to the bulk MERA.n We also extended the formalism to gauge fields by analysing the symmetric MERA - a MERA constructed out of tensors symmetric under the action of some group G.By making use of symmetric tensors and symmetric MERA the formalism naturally extends to give rise to emergent gauge fields in the bulk two dimensional state. This occurred due to symmetric tensor networks decomposing into magnetic and degeneracy components via Schur's lemma and restrictions on the magnetic components to give rise to what we call the gauge degrees of freedom. Using this it was possible to analyse gauge properties of the bulk, including: Wilson loops, entanglement between gauge and degeneracy degrees of freedom, and local gauge symmetries in the bulk. Finally we also demonstrate that it is possible to construct a Hamiltonian for the 2D bulk MERA which is local, constructed from vertex and loop operators, and symmetric under this gauge symmetry.As this project was motivated from a numerical perspective a major component of the work done was to generate a code for manipulating tensor networks. For tensor networks without symmetries this is simple, however for symmetric tensor networks this becomes quite complicated. Therefore this work also included the development of a library to numerically manipulate symmetric tensor networks, including anyonic symmetries such as the quantum SU(2) symmetry. Doing this we gained access to a complete family of unitary minimal model conformal field theories with which we could numerically test the correspondence.With the critical models tested I found that this procedure could replicate a number of properties predicted by the holographic dictionary. These included: global on-site symmetries corresponding to local on-site symmetries, the Ryu-Takayanagi formula relating entropy on the boundary to geometry in the bulk, and a correspondence between bulk and boundary operators.n