Holographic entanglement renormalisation of topological order in a quantum liquid Anirban Mukherjee, Siddhartha Lald-wave superconductivity  gapped liquids  entanglement  MERG  Hubbard model  
Mar 5, 2021  JPCM. 34, 275601 arXiv:2003.06118

We introduce a novel momentum space entanglement renormalization group (MERG) scheme for the topologically ordered (T.O.) ground state of the 2D Hubbard model on a square lattice (\cite{anirbanmotti,anirbanmott2}) using a unitary quantum circuit comprised of non-local unitary gates...
We introduce a novel momentum space entanglement renormalization group (MERG) scheme for the topologically ordered (T.O.) ground state of the 2D Hubbard model on a square lattice (\cite{anirbanmotti,anirbanmott2}) using a unitary quantum circuit comprised of non-local unitary gates. At each MERG step, the unitary quantum circuit disentangles a set of electronic states, thereby transforming the tensor network representation of the many-particle state. By representing the non-local unitary gate as a product of two-qubit disentangler gates, we provide an entanglement holographic mapping (EHM) representation for MERG. Using entanglement based measures from quantum information theory and complex network theory, we study the emergence of topological order in the bulk of the EHM. The MERG reveals distinct holographic entanglement features for the normal metallic, topologically ordered insulating quantum liquid and Neél antiferromagnetic symmetry-broken ground states of the 2D Hubbard model at half-filling found in Ref.\cite{anirbanmotti}. An MERG analysis of the quantum critical point of the hole-doped 2D Hubbard model found in Ref.\cite{anirbanmott2} reveals the evolution of the many-particle entanglement of the quantum liquid ground state with hole-doping, as well as how the collapse of Mottness is responsible for the emergence of d-wave superconductivity. We perform an information theoretic analysis of the EHM network, demonstrating that the information bottleneck principle is responsible for the distillation of entanglement features in the heirarchical structure of the EHM network. As a result, we construct a deep neural network (DNN) architecture based on our EHM network, and employ it for predicting the onset of topological order. We also demonstrate that the DNN is capable of distinguishing between the topologically ordered and gapless normal metallic phases.
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The paradigm of high-temperature superconductivity displayed by the doped Mott insulating cuprate family of materials serves up several outstanding challenges to our understanding of quantum condensed matter systems. One of these pertains to the very origin of superconductivity in these materials: can superconductivity arise purely from a system that has repulsive interactions among the electrons? Recall that the pairing of electrons is essential to the phenomenon of superconductivity, and that this can only happen when electrons attract one another. Instead, in metals and insulators, the electrons suffer repulsion.

‘A cube of magnetic material levitates above a superconductor. The field of the magnet induces currents in the superconductor that generate an equal and opposite field, exactly balancing the gravitational force on the cube.’

In well-known superconductors that can be explained by the famous theory of Bardeen, Cooper and Schrieffer, quantised lattice vibrations mediate an effective attractive interaction between electrons at low temperatures [1]. This eventually destabilises the metal at the critical temperature, and the formation of a macroscopic number of electronic (Cooper) pairs at and near the Fermi surface of the metal is concomitant with the condensation into a gap protected superconducting ground state. The spatial evolution of the well-defined quantum mechanical macroscopic phase of the ground state then allows for the flow of a supercurrent in the system without any dissipation of energy.

Despite much effort, there is at present no evidence for phonons playing a similar role in the superconductivity obtained upon doping the cuprates [1]. Indeed, the Mott insulating ground state of the undoped materials is itself driven by very strong repulsive interactions among the constituent electrons. How, then, does doping holes into such a system kill antiferromagnetic ordering of the spins in the Mott insulator, and obtain a superconductor?

‘Schematic T=0 quantum phase diagram of the 2D Hubbard model on the square lattice found by us in earlier works [2,3]. QCP represents the optimal hole-doping at which a novel quantum critical point was observed, and from which the d-wave superconducting phase is emergent.’

This question has been at the heart of a long-standing debate among condensed matter physicists since 1986 [1]. Earlier work by us [2,3] had already provided some answers to this question by applying a state-of-the-art analytical Hamiltonian renormalisation group method that we developed to a prototypical model for electrons on a two-dimensional square lattice that suffer purely on-site repulsion: the famous 2D Hubbard model. The T=0 (quantum) phase diagram obtained by us (see Fig.1) is in remarkable agreement with the well-known experimental phase diagram for cuprates (see [4] for a review), except that the latter has temperature on the y-axis while ours has the energyscale for quantum fluctuations in its place. Importantly, we showed in [2,3] that d-wave superconductivity is emergent at an optimal hole doping (p c in Fig.1) from the critical fluctuations of a novel quantum critical point associated with the collapse of Mottness and pseudogap behaviours. Our work also confirmed that a Marginal Fermi liquid (a well-known phenomenology for the strange metal phase of the cuprates) is indeed the parent metallic phase of the d-wave superconducting phase. See this small video abstract we made for [2,3] for a little more background.

In a recent work [5], we take the game deeper by probing for answers by looking at the many-particle entanglement and many-body correlation content of the ground state of the 2D Hubbard model at zero (origin at left end of the x-axis in Fig.1) and optimal (p c in Fig.1) hole doping. To enable this, we formulated a numerical method called the momentum space entanglement renormalisation group (MERG) that enables the RG evolution of the ground state wavefunction of the model from low (IR) to high (UV) momenta. From the series of wavefunctions obtained, we compute various measure of many-particle entanglement as well as several experimental observables (e.g., spin correlations, off- diagonal long-ranged order to probe electron pairing correlations etc.). We find these measures clearly indicate how the properties of the Mott insulating ground state at T=0 evolve with hole doping, such that the pairing of electrons becomes the dominant phase of quantum matter at optimal hole doping. To the best of our knowledge, this is one of the first studies of its kind for a model of strongly correlated electron systems, and heralds further progress in this direction.

At the same time, we have also demonstrated (for interested readers) how the MERG approach (i) is equivalent to a holographic spatial dimension containing emergent topologically ordered quantum liquids, (ii) is connected to quantum circuits that emulate a topological quantum error-correcting code, and (iii) enables the creation of a deep neural network that verifies the information bottleneck principle for machine learning, and is very efficient at identifying the emergence of topological states of quantum matter. Details of these are provided in a Supplementary Materials writeup available online at the journal site of the work.

You can also see a seminar on this topic here.


  1. The ubiquity of superconductivity. A. J. Leggett. Annu. Rev. Condens. Matter Phys. 2, 11-30 (2011).
  2. Scaling theory for Mott-Hubbard transitions – I: T=0 phase diagram of the ½-filled Hubbard model. A. Mukherjee and S. Lal. New Journal of Physics. 22, 063007 (2020).
  3. Scaling theory for Mott-Hubbard transitions – II: Quantum Criticality of the doped Mott insulator. A. Mukherjee and S. Lal. New Journal of Physics. 22, 063008 (2020).
  4. From quantum matter to high-temperature superconductivity in copper oxides. B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida and J. Zaanen. Nature 518, 179 (2015).
  5. Superconductivity from repulsion in the doped 2D electronic Hubbard model: an entanglement perspective. A. Mukherjee and S. Lal. J. Phys.:Condens. Matter 34, 275601(2022).