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Continuous Tensor Networks, Conformal Field Theory and Holography

Qi Ping Hu

Published Jul 23, 2020• 0 citations
OpenAlex

Abstract

Tensor network methods are playing a central role in multiple disciplines of modern quantum physics, such as condensed matter physics, quantum information and quantum gravity. Based on renormalization group (RG) ideas, they provide a large class of variational ansatz for quantum many-body systems. There are various tensor network structures which are suited for different systems based on their entanglement structures, such as matrix product states (MPS), projected entangled pair states (PEPS), multi-scale entanglement renormalization ansatz (MERA). Numerous techniques involving coarse-graining tensor networks have been developed, such as tensor renormalization group (TRG), tensor network renormalization (TNR) and its variants. \n \nFirst, we study the generalization of MERA to continuous systems, or cMERA, which is expected to become a powerful variational ansatz for the ground state of strongly interacting quantum field theories. We investigate, in the simpler context of Gaussian cMERA for free theories, the extent to which the cMERA state $\\ket{\\Psi^{\\cM}}$ with finite UV cut-off $\\Lambda$ can capture the spacetime symmetries of the ground state $\\ket{\\Psi}$. For a free boson conformal field theory (CFT) in 1+1 dimensions as a concrete example, we build a quasi-local unitary transformation $V$ that maps $\\ket{\\Psi}$ into $\\ket{\\Psi^{\\cM}}$ and show two main results. (i) Any spacetime symmetry of the ground state $\\ket{\\Psi}$ is also mapped by $V$ into a spacetime symmetry of the cMERA $\\ket{\\Psi^{\\cM}}$. However, while in the CFT the stress-energy tensor $T_{\\mu\\nu}(x)$ (in terms of which all the spacetime symmetry generators are expressed) is local, the corresponding cMERA stress-energy tensor $T_{\\mu\\nu}^{\\cM}(x) = V T_{\\mu\\nu}(x) V^{\\dagger}$ is quasi-local. (ii) From the cMERA, we can extract quasi-local scaling operators $O^{\\cM}_{\\alpha}(x)$ characterized by the exact same scaling dimensions $\\Delta_{\\alpha}$, conformal spins $s_{\\alpha}$, operator product expansion coefficients $C_{\\alpha\\beta\\gamma}$, and central charge $c$ as the original CFT. We argue that these results should also apply to interacting theories. \n \nSecond we extend TNR to field theories in the continuum. A short-distance length scale $1/\\Lambda$ is introduced in the continuum partition function by smearing the fields. The resulting object is still defined in the continuum but has no fluctuations at distances shorter than $1/\\Lambda$. An infinitesimal coarse-graining step is then generated by the combined action of a \\textit{rescaling} operator $\\mathfrak{L}$ and a \\textit{disentangling} operator $\\mathfrak{K}$ that implements a quasilocal field redefinition. As demonstrated for a free boson in two dimensions, continuous TNR exactly preserves translation and rotation symmetries and can generate a proper RG flow. Moreover, from a critical fixed point of this RG flow one can then extract the conformal data of the underlying conformal field theory. \n \nLeaving behind tensor network methods, we explore the universal structure that emerges from the eigenvectors of the reduced density matrix (RDM) for 1+1 dimensional critical lattice systems. Using the Ising model as an example, we demonstrate that the large-weight eigenvectors are related to the low-lying states of the underlying CFT with suitable boundary conditions by a conformal map, and they transform accordingly under the Virasoro generators. \n \nFinally, using the quantum renormalization group (QRG), we derive the bulk geometry that emerges in the holographic dual of the fermionic U($N$) vector model at a nonzero charge density. The obstruction that prohibits the metallic state from being smoothly deformable to the direct product state under the renormalization group flow gives rise to a horizon at a finite radial coordinate in the bulk. The region outside the horizon is described by the Lifshitz geometry with a higher-spin hair determined by microscopic details of the boundary theory. On the other hand, the interior of the horizon is not described by any Riemannian manifold, as it exhibits an algebraic non-locality. The non-local structure inside the horizon carries the information on the shape of the filled Fermi sea.

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The generalization of the multiscale entanglement renormalization ansatz (MERA) to continuous systems, or cMERA [Haegeman et al., Phys. Rev. Lett. 110, 100402 (2013)], is expected to become a powerful variational ansatz for the ground state of strongly interacting quantum field theories. In this Letter, we investigate, in the simpler context of Gaussian cMERA for free theories, the extent to which the cMERA state $|{\mathrm{\ensuremath{\Psi}}}^{\mathrm{\ensuremath{\Lambda}}}⟩$ with finite UV cutoff $\mathrm{\ensuremath{\Lambda}}$ can capture the spacetime symmetries of the ground state $|\mathrm{\ensuremath{\Psi}}⟩$. For a free boson conformal field theory (CFT) in $1+1$ dimensions, as a concrete example, we build a quasilocal unitary transformation $V$ that maps $|\mathrm{\ensuremath{\Psi}}⟩$ into $|{\mathrm{\ensuremath{\Psi}}}^{\mathrm{\ensuremath{\Lambda}}}⟩$ and show two main results. (i) Any spacetime symmetry of the ground state $|\mathrm{\ensuremath{\Psi}}⟩$ is also mapped by $V$ into a spacetime symmetry of the cMERA $|{\mathrm{\ensuremath{\Psi}}}^{\mathrm{\ensuremath{\Lambda}}}⟩$. However, while in the CFT, the stress-energy tensor ${T}_{\ensuremath{\mu}\ensuremath{\nu}}(x)$ (in terms of which all the spacetime symmetry generators are expressed) is local, and the corresponding cMERA stress-energy tensor ${T}_{\ensuremath{\mu}\ensuremath{\nu}}^{\mathrm{\ensuremath{\Lambda}}}(x)=V{T}_{\ensuremath{\mu}\ensuremath{\nu}}(x){V}^{\ifmmode\dagger\else\textdagger\fi{}}$ is quasilocal. (ii) From the cMERA, we can extract quasilocal scaling operators ${O}_{\ensuremath{\alpha}}^{\mathrm{\ensuremath{\Lambda}}}(x)$ characterized by the exact same scaling dimensions ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}}$, conformal spins ${s}_{\ensuremath{\alpha}}$, operator product expansion coefficients ${C}_{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\gamma}}$, and central charge $c$ as the original CFT. Finally, we argue that these results should also apply to interacting theories.

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Tensor network methods are playing a central role in multiple disciplines of modern quantum physics, such as condensed matter physics, quantum information and quantum gravity. Based on renormalization group (RG) ideas, they provide a large class of variational ansatz for quantum many-body systems. There are various tensor network structures which are suited for different systems based on their entanglement structures, such as matrix product states (MPS), projected entangled pair states (PEPS), multi-scale entanglement renormalization ansatz (MERA). Numerous techniques involving coarse-graining tensor networks have been developed, such as tensor renormalization group (TRG), tensor network renormalization (TNR) and its variants. \n \nFirst, we study the generalization of MERA to continuous systems, or cMERA, which is expected to become a powerful variational ansatz for the ground state of strongly interacting quantum field theories. We investigate, in the simpler context of Gaussian cMERA for free theories, the extent to which the cMERA state ketPsicM\\ket{\\Psi^{\\cM}}ketPsicM with finite UV cut-off Lambda\\LambdaLambda can capture the spacetime symmetries of the ground state ketPsi\\ket{\\Psi}ketPsi. For a free boson conformal field theory (CFT) in 1+1 dimensions as a concrete example, we build a quasi-local unitary transformation VVV that maps ketPsi\\ket{\\Psi}ketPsi into ketPsicM\\ket{\\Psi^{\\cM}}ketPsicM and show two main results. (i) Any spacetime symmetry of the ground state ketPsi\\ket{\\Psi}ketPsi is also mapped by VVV into a spacetime symmetry of the cMERA ketPsicM\\ket{\\Psi^{\\cM}}ketPsicM. However, while in the CFT the stress-energy tensor Tmunu(x)T_{\\mu\\nu}(x)Tmunu​(x) (in terms of which all the spacetime symmetry generators are expressed) is local, the corresponding cMERA stress-energy tensor TmunucM(x)=VTmunu(x)VdaggerT_{\\mu\\nu}^{\\cM}(x) = V T_{\\mu\\nu}(x) V^{\\dagger}TmunucM​(x)=VTmunu​(x)Vdagger is quasi-local. (ii) From the cMERA, we can extract quasi-local scaling operators OalphacM(x)O^{\\cM}_{\\alpha}(x)OalphacM​(x) characterized by the exact same scaling dimensions Deltaalpha\\Delta_{\\alpha}Deltaalpha​, conformal spins salphas_{\\alpha}salpha​, operator product expansion coefficients CalphabetagammaC_{\\alpha\\beta\\gamma}Calphabetagamma​, and central charge ccc as the original CFT. We argue that these results should also apply to interacting theories. \n \nSecond we extend TNR to field theories in the continuum. A short-distance length scale 1/Lambda1/\\Lambda1/Lambda is introduced in the continuum partition function by smearing the fields. The resulting object is still defined in the continuum but has no fluctuations at distances shorter than 1/Lambda1/\\Lambda1/Lambda. An infinitesimal coarse-graining step is then generated by the combined action of a \textit{rescaling} operator mathfrakL\\mathfrak{L}mathfrakL and a \textit{disentangling} operator mathfrakK\\mathfrak{K}mathfrakK that implements a quasilocal field redefinition. As demonstrated for a free boson in two dimensions, continuous TNR exactly preserves translation and rotation symmetries and can generate a proper RG flow. Moreover, from a critical fixed point of this RG flow one can then extract the conformal data of the underlying conformal field theory. \n \nLeaving behind tensor network methods, we explore the universal structure that emerges from the eigenvectors of the reduced density matrix (RDM) for 1+1 dimensional critical lattice systems. Using the Ising model as an example, we demonstrate that the large-weight eigenvectors are related to the low-lying states of the underlying CFT with suitable boundary conditions by a conformal map, and they transform accordingly under the Virasoro generators. \n \nFinally, using the quantum renormalization group (QRG), we derive the bulk geometry that emerges in the holographic dual of the fermionic U(NNN) vector model at a nonzero charge density. The obstruction that prohibits the metallic state from being smoothly deformable to the direct product state under the renormalization group flow gives rise to a horizon at a finite radial coordinate in the bulk. The region outside the horizon is described by the Lifshitz geometry with a higher-spin hair determined by microscopic details of the boundary theory. On the other hand, the interior of the horizon is not described by any Riemannian manifold, as it exhibits an algebraic non-locality. The non-local structure inside the horizon carries the information on the shape of the filled Fermi sea.
The generalization of the multiscale entanglement renormalization ansatz (MERA) to continuous systems, or cMERA [Haegeman et al., Phys. Rev. Lett. 110, 100402 (2013)], is expected to become a powerful variational ansatz for the ground state of strongly interacting quantum field theories. In this Letter, we investigate, in the simpler context of Gaussian cMERA for free theories, the extent to which the cMERA state $|{\mathrm{\ensuremath{\Psi}}}^{\mathrm{\ensuremath{\Lambda}}}⟩$ with finite UV cutoff $\mathrm{\ensuremath{\Lambda}}$ can capture the spacetime symmetries of the ground state $|\mathrm{\ensuremath{\Psi}}⟩$. For a free boson conformal field theory (CFT) in 1+11+11+1 dimensions, as a concrete example, we build a quasilocal unitary transformation VVV that maps $|\mathrm{\ensuremath{\Psi}}⟩$ into $|{\mathrm{\ensuremath{\Psi}}}^{\mathrm{\ensuremath{\Lambda}}}⟩$ and show two main results. (i) Any spacetime symmetry of the ground state $|\mathrm{\ensuremath{\Psi}}⟩$ is also mapped by VVV into a spacetime symmetry of the cMERA $|{\mathrm{\ensuremath{\Psi}}}^{\mathrm{\ensuremath{\Lambda}}}⟩$. However, while in the CFT, the stress-energy tensor ${T}_{\ensuremath{\mu}\ensuremath{\nu}}(x)$ (in terms of which all the spacetime symmetry generators are expressed) is local, and the corresponding cMERA stress-energy tensor ${T}_{\ensuremath{\mu}\ensuremath{\nu}}^{\mathrm{\ensuremath{\Lambda}}}(x)=V{T}_{\ensuremath{\mu}\ensuremath{\nu}}(x){V}^{\ifmmode\dagger\else\textdagger\fi{}}$ is quasilocal. (ii) From the cMERA, we can extract quasilocal scaling operators ${O}_{\ensuremath{\alpha}}^{\mathrm{\ensuremath{\Lambda}}}(x)$ characterized by the exact same scaling dimensions ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}}$, conformal spins ${s}_{\ensuremath{\alpha}}$, operator product expansion coefficients ${C}_{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\gamma}}$, and central charge ccc as the original CFT. Finally, we argue that these results should also apply to interacting theories.
The study of the ground states of local Hamiltonians in quantum many-body and quantum field theoretic systems is a source of many research problems of great complexity. Almost in its entirety, this thesis deals with constructions related to the field of tensor networks, which are efficient parametrizations of mathematical objects from quantum many-body theory (such as quantum states and operators) that exploit our knowledge of their entanglement structures in order to obtain improvements in our computational abilities. In the past decade, the new research program of continuous tensor networks has arisen with the goal of reproducing, in the setting of quantum field theory, the success that tensor network techniques have enjoyed on the lattice. First, we present a number of results from research performed on non-interacting examples of the continuous Multiscale Entanglement Renormalization Ansatz, or cMERA, a continuous tensor network that provides a variational wavefunctional for a quantum field theoretic ground state. Initially, we study the correlations and entanglement structure of cMERA states, providing evidence that such states can be interpreted as UV regularized versions of the physical states they approximate. After that, we study what modifications of the formalism are necessary in order to consider systems with gauge symmetry, or in the presence of boundaries and defects. We obtain prescriptions for the treatment of these cases which, we expect, may also hold in the interacting setting. We then abandon briefly the area of continuous tensor networks to present a piece of research within a different though related research program: that of the study of manifestations of universal emergent behaviour in critical lattice systems. In particular we bring attention to the existence of an approximate representation of the Virasoro algebra supported on the eigenvectors of the reduced density matrices of critical lattice systems. We call this the entanglement algebra, and study its accuracy in the particular case of the Ising model. Lastly, we present a proposal for a continuous version of the Tensor Network Renormalization (TNR) algorithm, which we dub continuous TNR, or cTNR. cTNR operates on UV regularized classical partition functions or quantum Euclidean path integrals, and generates a real-space renormalization group flow via continuous coarse-graining and rescaling operations. We show that a UV regularized version of the free boson path integral can be made a fixed point of such a renormalization flow, in a way that allows for recovery of the conformal data associated to the free boson conformal field theory.
On the lattice, a renormalization group (RG) flow for two-dimensional partition functions expressed as a tensor network can be obtained using the tensor network renormalization (TNR) algorithm [G. Evenbly, G. Vidal, Phys. Rev. Lett. 115 (18), 180405 (2015)]. In this work we explain how to extend TNR to field theories in the continuum. First, a short-distance length scale 1/Λ1/Λ1/Λ is introduced in the continuum partition function by smearing the fields. The resulting object is still defined in the continuum but has no fluctuations at distances shorter than 1/Λ1/Λ1/Λ. An infinitesimal coarse-graining step is then generated by the combined action of a rescalingrescalingrescaling operator LLL and a disentanglingdisentanglingdisentangling operator KKK that implements a quasi-local field redefinition. As demonstrated for a free boson in two dimensions, continuous TNR exactly preserves translation and rotation symmetries and can generate a proper RG flow. Moreover, from a critical fixed point of this RG flow one can then extract the conformal data of the underlying conformal field theory.
Continuous tensor networks give a variational ansatz for the ground state of the quantum field theories (QFTs). Notable examples are the continuous matrix product state (cMPS) and the continuous multiscale entanglement renormalization ansatz (cMERA). While cMPS is just adapted to the nonrelativistic QFTs, only the Gaussian cMERA is well understood, which we cannot use to approximate the ground state of the interacting relativistic QFTs. But, instead, cMERA also corresponds to a real-space renormalization group flow in the context of the wave functions. In this paper, we investigate the backward Gaussian cMERA renormalization group flow of the class of cMPS by putting the standard cMPS at the IR scale. At the UV scale, for the bosonic systems in the thermodynamic limit, we achieve the variational class of states that has been proposed recently, as the relativistic cMPS (RCMPS) is adapted to the relativistic QFTs without requiring one to introduce of any additional IR or UV cutoff. We also extend the RCMPS to fermionic systems and theories on a finite circle.
Continuous tensor networks are variational wavefunctions proposed in recent years to efficiently simulate quantum field theories (QFTs). Prominent examples include the continuous matrix product state (cMPS) and the continuous multi-scale entanglement renormalization ansatz (cMERA). While the cMPS can approximate ground states of a class of QFT Hamiltonians that are both local and interacting, cMERA is only well-understood for QFTs that are quasi-local and non-interacting. In this paper we propose the magic cMERA, a concrete realization of cMERA for a free boson QFT that simultaneously satisfies four remarkable properties: (i) it is the exact ground state of a strictly local Hamiltonian; (ii) in the massless case, its spectrum of scaling operators is exactly soluble in real space; (iii) it has the short-distance structure of a cMPS; (iv) it is generated by a quasi-local entangler that can be written as a continuous matrix product operator. None of these properties is fulfilled by previous cMERA proposals. Properties (iii)-(iv) establish a firm connection between cMERA and cMPS wavefunctionals, opening the path to applying powerful cMPS numerical techniques, valid for interacting QFTs, also to cMERA calculations.
In recent years, tensor network states have emerged as a very useful conceptual and simulation framework to study quantum many-body systems at low energies. In this paper, we describe a particular way in which any given tensor network can be viewed as a representation of two different quantum many-body states. The two quantum many-body states are said to correspond to each other by means of the tensor network. We apply this ``tensor network state correspondence''---a correspondence between quantum many-body states mediated by tensor networks as we describe---to the multi-scale entanglement renormalization ansatz (MERA) representation of ground states of one dimensional (1D) quantum many-body systems. Since the MERA is a 2D hyperbolic tensor network (the extra dimension is identified as the length scale of the 1D system), the two quantum many-body states obtained from the MERA, via tensor network state correspondence, are seen to live in the bulk and on the boundary of a discrete hyperbolic geometry. The bulk state so obtained from a MERA exhibits interesting features, some of which caricature known features of the holographic correspondence of String theory. We show how (i) the bulk state admits a description in terms of ``holographic screens'', (ii) the conformal field theory data associated with a critical ground state can be obtained from the corresponding bulk state, in particular, how pointlike boundary operators are identified with extended bulk operators. (iii) We also present numerical results to illustrate that bulk states, dual to ground states of several critical spin chains, have exponentially decaying correlations, and that the bulk correlation length generally decreases with increase in central charge for these spin chains.