Jekyll2023-11-19T18:39:31+00:00https://epqm.github.io/feed.xmlEPQM @ Siddhartha Lal**EPQM** is a condensed matter group at IISER Kolkata, led by Siddhartha Lal. We explore emergent phenomena in strongly correlated quantum matter.Siddhartha Lalslal@iiserkol.ac.inFrustrating the Kondo effect might hold the key to the Mott MIT in ∞ dimensions: our new work just got published at New Journal of Physics!2023-11-13T00:00:00+00:002023-11-13T00:00:00+00:00https://epqm.github.io/esiam<p><strong>Kondo frustration via charge fluctuations: a route to Mott localisation</strong>
<span class="pub__authors">Abhirup Mukherjee, N. S. Vidhyadhiraja, A. Taraphder, <b>Siddhartha Lal</b></span><span class="btn btn--info">impurity models</span> <span class="btn btn--info">Mott MIT</span> <span class="btn btn--info">auxiliary models</span> <span class="btn btn--info">Kondo effect</span> <span class="btn btn--info">quantum phase transition</span> <br />
Feb 5, 2023 <a href="https://iopscience.iop.org/article/10.1088/1367-2630/ad08f3" class="btn btn--danger">New J. Phys (2023)</a>
<a href="https://arxiv.org/abs/2302.02328" class="btn btn--success">arXiv:2302.02328</a></p>
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<div id="short_abstract" class="archive__item-excerpt">We propose a minimal effective impurity model that captures the phenomenology of the Mott-Hubbard metal-insulator transition (MIT) of the half-filled Hubbard model on the Bethe lattice in infinite dimensions as observed by dynamical mean field theory (DMFT)...</div>
<div id="full_abstract" class="archive__item-excerpt">We propose a minimal effective impurity model that captures the phenomenology of the Mott-Hubbard metal-insulator transition (MIT) of the half-filled Hubbard model on the Bethe lattice in infinite dimensions as observed by dynamical mean field theory (DMFT). This involves extending the standard Anderson impurity model Hamiltonian to include an explicit Kondo coupling \(J\), as well as a local on-site correlation \(U_b\) on the conduction bath site connected directly to the impurity. For the case of attractive local bath correlations (\(U_{b}<0\)), the extended Anderson impurity model (e-SIAM) sheds new light on several aspects of the DMFT phase diagram. For example, the \(T=0\) metal-to-insulator quantum phase transition (QPT) is preceded by an excited state quantum phase transition (ESQPT) where the local moment eigenstates are emergent in the low-lying spectrum. Long-ranged fluctuations are observed near both the QPT and ESQPT, suggesting that they are the origin of the quantum critical scaling observed recently at high temperatures in DMFT simulations. The \(T=0\) gapless excitations at the QCP display particle-hole interconversion processes, and exhibit power-law behaviour in self-energies and two-particle correlations. These are signatures of non-Fermi liquid behaviour that emerge from the partial breakdown of the Kondo screening.</div>
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<h2 id="some-background-on-dmft--the-mott-mit">Some background on DMFT & the Mott MIT</h2>
<p>This work is devoted towards obtaining Hamiltonian-based insight into the dynamical mean-field theory (DMFT) treatment for the half-filled Hubbard model on the Bethe lattice in infinite dimensions. Our work offers an effective auxiliary model Hamiltonian that incorporates the effects of a local interacting self-energy (as obtained from the self-consistent approach of DMFT), and provides insight on the associated Mott metal-insulator transition (MIT). We recall that the conduction bath (of the effective impurity model within DMFT) becomes correlated under the requirement of self-consistency. The numerical implementation of this process precludes a deeper understanding of the precise nature of the correlations present in the conduction bath, and its implications for the electron dynamics of the associated bulk lattice (Hubbard) model.</p>
<p class="body__img"><img src="/assets/images/esiam/dmft.gif" alt="" />
“Dynamical mean-field theory (DMFT) of correlated-electron solids replaces the full lattice of atoms and electrons with a single impurity atom imagined to exist in a bath of electrons. The approximation captures the dynamics of electrons on a central atom (in orange) as it fluctuates among different atomic configurations, shown here as snapshots in time.” <a href="https://physicstoday.scitation.org/doi/10.1063/1.1712502">[Source]</a></p>
<h2 id="questions-addressed-by-us">Questions addressed by us</h2>
<p>This leads to the following questions that we address in our work:</p>
<ul>
<li>What are the quantum fluctuations that destroy the metal at \(T=0\) and lead to the insulating phase?</li>
<li>Can we obtain a universal theory for the competing tendencies that give rise to these quantum fluctuations?</li>
<li>What leads to the coexistence of metallic and insulating phases within DMFT at \(T=0\), as indicated by the fact that the insulating solution is already present within the many-body spectrum of the metallic phase prior to the transition?</li>
<li>Is it possible to obtain a low-energy theory for the local gapless excitations precisely at the MIT, where the metal is on the brink of destruction?</li>
</ul>
<h2 id="main-results-an-extended-anderson-impurity-model-and-its-phase-transition">Main results: An extended Anderson impurity model and its phase transition</h2>
<p>In order to model the interactions within the bath, we adopt an extended single impurity Anderson model (e-SIAM) as the effective auxiliary model for the Mott MIT. In addition to the usual impurity on-site repulsion U and the single-particle hybridisation V, we introduce</p>
<ul>
<li>an additional on-site correlation (\(U_b\)) on the bath site with which the impurity couples, and</li>
<li>an antiferromagnetic Kondo coupling (\(J\)) between the impurity and the conduction bath.</li>
</ul>
<p>These additions are then justified via a \(T=0\) renormalisation group (RG) analysis of the e-SIAM that reveals a quantum phase transition (QPT) at a critical value of the ratio (\(-U_b/J\)). The transition involves a change in the ground-state from a (Kondo) spin singlet to an unscreened local moment, indicating a Mott MIT in the bulk model associated with the chosen auxiliary model similar to DMFT.</p>
<p class="body__img"><img src="/assets/images/esiam/esiam_bare.svg" alt="" />
We have extended the standard Anderson impurity model by incorporating an explicit Kondo coupling \(J\) and a local correlation \(U_b\) on the bath site coupled to the impurity. Such a model shows a local metal-insulator transition.</p>
<h2 id="main-results-preformed-gap-equals-an-excited-state-phase-transition">Main results: Preformed gap equals an excited state phase transition</h2>
<p>We use the fixed-point Hamiltonians obtained from the RG procedure to calculate quantifiers of the transition, e.g., various many-body correlation functions as well as measures of many-particle entanglement. These measures reveal that the transition occurs through a breakdown of the Kondo screening effect, owing to enhanced local pairing fluctuations in the bath proximate to the QPT. In the immediate neighbourhood of the transition, these pairing fluctuations also lead to the appearance of long-ranged correlations between the impurity and the conduction bath degrees of freedom, signalling critical behaviour. Remarkably, we find that the impurity QPT from metal to insulator is preceded by an excited state quantum phase transition (ESQPT): this involves the appearance of the local moment states in the many-body spectrum as excited states. This mechanism leads to a preformed (optical) gap in the impurity spectral function, as well as the coexistence region, in the phase diagram of the e-SIAM. This is consistent with the well-known phenomenology of DMFT.</p>
<h2 id="main-results-enhanced-pairing-fluctuations-and-non-fermi-liquid-excitations-at-the-mott-mit">Main results: Enhanced pairing fluctuations and non-Fermi liquid excitations at the Mott MIT</h2>
<p>Further, the e-SIAM provides a minimal and universal local theory for describing Mott metal-insulator transition that arise from strong local correlations, and involves a competition between Kondo screening and its frustration from a locally correlated bath. The minimal model also offers detailed insights on the nature of the gapless excitations present at transition. We find that gapless non-Fermi liquid excitations arise from the ground state degeneracy at the transition, involving an orthogonality catastrophe and the vanishing of the quasiparticle residue of the local Fermi liquid. The non-Fermi liquid is characterised by anomalous power-law behaviour in the self-energy and various two-particle correlations, as well as a fractional entanglement entropy and magnetisation arising from a scattering phase shift of π/2.</p>
<p class="body__img"><img src="/assets/images/esiam/coexistence-dmft.png" alt="" />
Qualitative structure of the finite temperature coexistence region of the \(J - U_b\) model. The dotted lines on the left and the right represent the spinodals where the insulating and metallic solutions become unstable, respectively. The solid red line represents the first-order line where the free energies and the partition functions of the two solutions become equal.</p>
<h2 id="conclusions-and-outlook">Conclusions and outlook</h2>
<p>Thus, our work offers a host of new results on multiple aspects of the auxiliary model approach to the Mott MIT. In general, the interplay of multiple correlations in the e-SIAM localisation from repulsion on the impurity site (\(U\)), delocalisation from spin and charge fluctuations (\(J\) and \(V\)), and pairing from local attractive correlations in the bath (\(U_b\)) - makes this a strong candidate for describing the emergence of a variety of novel phases of correlated quantum matter. Likely future investigations include, for instance, the effects of hole doping on the nature of the local quantum criticality observed by us, and especially whether critical pairing fluctuations can condense into a superconducting state of matter.</p>
<p>You can also watch a seminar on this work <a href="https://www.youtube.com/watch?v=JuPtebPgxtQ&ab_channel=P%C3%B3sGradua%C3%A7%C3%A3oemF%C3%ADsica-InstitutodeF%C3%ADsica">here</a>.</p>Siddhartha Lalslal@iiserkol.ac.inWe provide a Hamiltonian-based explanation of the DMFT-based phenomenology of the infinite dimensional Hubbard model, obtaining novel insights on the Mott MIT in the process.We welcome Sukalyan Deb, our new PhD student, into the group.2023-08-22T00:00:00+00:002023-08-22T00:00:00+00:00https://epqm.github.io/Sukalyan-welcome<p>We welcome our newest member, Sukalyan Deb, to the EPQM family. He is a permanent resident of the district of Alipurduar . He completed his graduation in physics from Ananda Chandra College in Jalpaiguri and his post-graduation from Assam University. He has joined IISER Kolkata for his doctoral work in August. We wish him all the success in his work.</p>
<p class="body_img"><img src="../assets/images/people/sukalyan.jpg" alt="" /></p>Siddhartha Lalslal@iiserkol.ac.inWelcome, Aashish, to EPQM@IISER Kolkata!2023-06-30T00:00:00+00:002023-06-30T00:00:00+00:00https://epqm.github.io/Aashish-arrival<p>Aashish Kumar, a member of Prof. N. S. Vidhyadhiraja’s research group at JNCASR (Bengaluru) is visiting EPQM. As part of the ongoing collaborations between the two groups, Aashish will be working with us on a research project.</p>
<p class="body_img"><img src="../assets/images/about/people/spatra.png" alt="" /></p>Siddhartha Lalslal@iiserkol.ac.inCongratulations, Abhirup, on presenting your work at PP65@IISER Kolkata!2023-06-28T00:00:00+00:002023-06-28T00:00:00+00:00https://epqm.github.io/AbhirupatPP65<p>Abhirup presented his research in a rapid talk at PP65 (the 65th birthday celebration of Prof. Prasanta Panigrahi) that was held recently at DPS, IISER Kolkata. He gave a short but insightful overview of his research on the Kondo effect and what causes its breakdown. Congratulations, Abhirup!</p>
<p class="body_img"><img src="../assets/images/about/people/abhirup.png" alt="" /></p>Siddhartha Lalslal@iiserkol.ac.inCongratulations, Siddhartha Patra, on receiving your Ph.D at the 10th convocation of IISER Kolkata!2023-06-26T00:00:00+00:002023-06-26T00:00:00+00:00https://epqm.github.io/Siddhartha-Patra-convocation<p>Siddhartha Patra received his Doctorate degree at the 10th convocation of IISER Kolkata recently. Congratulations, Siddhartha! We look forward to many more wonderful achievements from you up ahead.</p>
<p class="body_img"><img src="../assets/images/about/people/spatra.png" alt="" /></p>Siddhartha Lalslal@iiserkol.ac.inCan entanglement scaling lead to universality across various metals or insulators?2023-05-27T00:00:00+00:002023-05-27T00:00:00+00:00https://epqm.github.io/metals-EE<p><strong>Universal entanglement signatures of quantum liquids as a guide to fermionic criticality</strong>
<span class="pub__authors">Siddhartha Patra, Anirban Mukherjee, <b>Siddhartha Lal</b></span><span class="btn btn--info">metals</span> <span class="btn btn--info">gapped liquids</span> <span class="btn btn--info">entanglement</span> <span class="btn btn--info">MERG</span> <span class="btn btn--info">Hubbard-model</span> <br />
Feb 12, 2023 <a href="https://iopscience.iop.org/article/10.1088/1367-2630/acd8e8" class="btn btn--danger">New J. Phys (2023)</a>
<a href="https://arxiv.org/abs/2205.11123" class="btn btn--success">arXiv:2205.11123</a></p>
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<div id="short_abstract" class="archive__item-excerpt">An outstanding challenge involves understanding the many-particle entanglement of liquid states of quantum matter that arise in systems of interacting electrons...</div>
<div id="full_abstract" class="archive__item-excerpt">An outstanding challenge involves understanding the many-particle entanglement of liquid states of quantum matter that arise in systems of interacting electrons. The Fermi liquid (FL) in \(D\) spatial dimensions shows a violation of the area-law in real-space entanglement entropy of a subsystem (of length \(L\)), \(S_{EE} \sim L^{D-1}\ln L\), widely believed to be a hallmark signature of the ground state of a gapless quantum critical system of interacting fermions. In this work, we apply a \(T=0\) renormalisation group approach to a prototype of the FL in momentum (or, \(k\))-space, unveiling thereby the RG relevant quantum fluctuations (due to forward and tangential scattering) from which long-range entanglement arises. A similar analysis of non-Fermi liquids such as the 2D marginal Fermi liquid (MFL) and the 1D Tomonaga-Luttinger liquid (TLL) reveals a universal logarithmic violation of the area-law in gapless electronic liquids for a subsystem defined within a \(k\)-space window (of size \(Λ\)) proximate to the Fermi surface, with a proportionality constant that depends on the nature of the underlying Fermi surface. We extend this analysis to the gapped quantum liquids emergent from the destabilisation of the Fermi surface by quantum fluctuations arising from backscattering processes. Indeed, we find that the \(k\)-space entanglement signatures of gapped quantum liquids appear to be governed by the nature of the Fermi surface (e.g., nested or not) from which they emerge, as well as the nature of their parent gapless metallic liquid (e.g., FL, MFL etc.). This is confirmed by our finding an enhanced entanglement entropy for the nodal MFL present at the quantum critical point recently discovered in the 2D Hubbard model at optimal hole-doping. Our work thus paves the way for an entanglement based classification of quantum liquids emergent from the criticality of interacting fermionic matter.</div>
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<h2 id="some-background">Some background</h2>
<p>Our work is devoted towards understanding whether there exist any signatures of universality in the nature of many-particle entanglement encoded within the ground state and lowest-lying excited state wavefunctions of quantum liquids that are emergent within systems of interacting electrons. The novelty of such an investigation can be gauged by recalling that 20th century condensed matter physics was governed primarily by studies of systems that fall within the Ginzburg-Landau-Wilson (GLW) paradigm for the description of ordered states of matter. Such ground states are emergent from the spontaneous breaking of symmetries, and whose order is characterised by real space order parameters that correspond to the ground state expectation values of local operators. Importantly, the short-range entangled nature of these ground states is responsible for their satisfying the cluster decomposition property of all correlation functions. Starting with the discovery of the fractional quantised Hall effect in two-dimensional electron gases and high-temperature superconductivity in the hole-doped Mott insulating cuprates, the quantum condensed matter community has increasingly become interested in understanding the physics that can lead to long-range entangled ground states, and whether this can enable the classification of quantum matter. A variety of quantum liquids have emerged as candidate systems in which to meet these challenges.</p>
<p class="body__img"><img src="/assets/images/metals_EE/strangemetal.gif" alt="" />
Researches theorise that electrons in a strange metal are in a highly entangled quantum state. Their resistance depends on Planck’s constant, suggesting that the electrons are dissipating energy as fast as physics allows (Planckian dissipation). <a href="https://www.quantamagazine.org/universal-quantum-phenomenon-found-in-superconductors-20181119/">[Source]</a></p>
<p>Many-body quantum systems with locally interacting constituents are expected to show the area-law in the entanglement entropy of a subsystem, reflecting the minimal entanglement with the rest of the system across common boundaries. In our work, we focus on quantum liquid phases of interacting electrons that are noteworthy exceptions to this rule. Of these, gapped topologically ordered liquids (e.g., the Laughlin states of the FQHE) show long-range entanglement in the form of topological entanglement entropy, while gapless liquids (e.g., the Fermi liquid in two spatial dimensions) display scaling according to a log-modified area law in real-space calculations of entanglement entropy or even volume law scaling in some cases of quantum criticality. The long-range entanglement in both types of liquids arises from the prevalence of quantum fluctuations that defy the GLW paradigm, and we begin by identifying them in several gapless quantum liquids (e.g., the 2D Fermi liquid (FL), 2D marginal Fermi liquid (MFL), and the 1D Tomonaga-Luttinger liquid).</p>
<h2 id="main-results-unification-of-the-entanglement-signatures-of-gapless-liquids">Main results: unification of the entanglement signatures of gapless liquids</h2>
<p>Gapless quantum liquids are described by effective theories that contain only the physics of the low-energy long-wavelength degrees of freedom proximate to the Fermi surface. It is then important to investigate theoretically the momentum-space quantum fluctuations related to forward and tangential scattering in a system of interacting electrons that are resolved under the renormalisation group (RG) flow from UV to IR. Thus, by employing a state-of-the-art T=0 numerical RG method based purely on many-particle unitary transformations, we confirm that a log-modified area law for the scaling of the entanglement entropy of the 2D FL for a subsystem (defined within a momentum-space window proximate to the Fermi surface) with its size arises from precisely such quantum fluctuations. A similar analysis of non-Fermi liquids such as the 2D MFL and the 1D TLL reveals striking evidence for the universal nature of this logarithmic violation of the area-law, with a proportionality constant that depends on the nature of the underlying Fermi surface. These results suggest a unification of the Fermi liquid paradigm with non-Fermi liquid metals (that are qualitatively different in terms of their low-lying excitations) based on the signatures of many-particle entanglement they display.</p>
<h2 id="main-results-entanglement-as-an-indicator-of-the-presence-of-a-gap">Main results: entanglement as an indicator of the presence of a gap</h2>
<p>We extend this analysis in order to classify the gapped quantum liquids emergent from the destabilisation of the Fermi surface by RG relevant quantum fluctuations arising from backscattering processes. Here, we find that the momentum-space entanglement signatures of gapped quantum liquids appear to be governed by the nature of the Fermi surface from which they emerge, as well as the nature of their parent gapless metallic liquid. The importance of the nature of the Fermi surface is further confirmed by our finding an enhanced entanglement entropy for the nodal MFL present at the quantum critical point recently discovered in the 2D Hubbard model at optimal hole-doping, and lying proximate to a point-like singular Fermi surface. Finally, in a supplementary materials note, we report on various quantum information theoretic measures that display the holographic entanglement of various gapless as well as gapped quantum liquids.</p>
<p>Our analysis, thus, enables an entanglement-based classification of various quantum liquids emergent from the criticality of interacting fermionic quantum matter. To the best of our knowledge, our work is the first of its kind, and paves the way for further explorations of quantum matter based on their entanglement properties.</p>Siddhartha Lalslal@iiserkol.ac.inIn a work that appeared recently at New Journal of Physics, Patra et al classify various emergent phases of matter in terms of entanglement measures.A simple graph-theoretic construct outlines much of the physics of the multichannel Kondo effect!2023-05-05T00:00:00+00:002023-05-05T00:00:00+00:00https://epqm.github.io/mck<p><strong>Frustration shapes multi-channel Kondo physics: a star graph perspective</strong>
<span class="pub__authors">Siddhartha Patra, Abhirup Mukherjee, Anirban Mukherjee, N. S. Vidhyadhiraja, A. Taraphder, <b>Siddhartha Lal</b></span><span class="btn btn--info">Kondo effect</span> <span class="btn btn--info">impurity models</span> <span class="btn btn--info">Kondo breakdown</span> <span class="btn btn--info">non-Fermi liquid</span> <br />
Jan 14, 2023 <a href="https://doi.org/10.1088/1361-648X/acd09c" class="btn btn--danger">JPCM 35 315601</a>
<a href="https://arxiv.org/abs/2205.00790" class="btn btn--success">arXiv:2205.00790</a></p>
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<div id="short_abstract" class="archive__item-excerpt">We study the overscreened multi-channel Kondo (MCK) model using the recently developed unitary renormalization group (URG) technique...</div>
<div id="full_abstract" class="archive__item-excerpt">We study the overscreened multi-channel Kondo (MCK) model using the recently developed unitary renormalization group (URG) technique. Our results display the importance of ground state degeneracy in explaining various important properties like the breakdown of screening and the presence of local non-Fermi liquids. The impurity susceptibility of the intermediate coupling fixed point Hamiltonian in the zero-bandwidth (or star graph) limit shows a power-law divergence at low temperature, signalling its critical nature. Despite the absence of inter-channel coupling in the MCK fixed point Hamiltonian, the study of mutual information between any two channels shows non-zero correlation between them. A spectral flow analysis of the star graph reveals that the degenerate ground state manifold possesses topological quantum numbers. The low energy effective Hamiltonian obtained upon adding a finite non-zero conduction bath dispersion to the star graph Hamiltonian for both the two and three-channel cases displays the presence of local non-Fermi liquids arising from inter-channel quantum fluctuations. Discontinuous behaviour is observed in several measures of ground state entanglement, signalling the underlying orthogonality catastrophe associated with the degenerate ground state manifold. We extend our results to underscreened and perfectly screened MCK models through duality arguments. A study of channel anisotropy under renormalisation flow reveals a series of quantum phase transitions due to the change in ground state degeneracy. Our work thus presents a template for the study of how a degenerate ground state manifold arising from symmetry and duality properties in a multichannel quantum impurity model can lead to novel multicritical phases at intermediate coupling.</div>
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<h2 id="the-physics-of-frustration">The physics of frustration</h2>
<p>Frustration. We think we understand it, and how to deal with it. But what does frustration refer to in a quantum system? Consider classical spins (i.e., spins that have only two configurations, say, pointing up and pointing down) placed on a triangle and interacting with one another through a nearest neighbour antiferromagnetic Ising exchange interaction. It is easy to see that <strong>Néel (anti-parallel) ordering is no longer possible</strong>: for any two spins that are anti-aligned with one another, the third is <strong>left confused</strong> on which direction to choose. This confusion is labelled as the frustration of the classical Néel order.</p>
<p class="archive__item-excerpt"><img src="/assets/images/mck/frustration.webp" alt="" />
An example of classical frustration: three spins on a triangle, connected by antiferromagnetic Ising interaction that favours antiparallel alignment. If the blue and green spins align in opposite directions, the third gray spin experiences conflicting interactions from the blue and green spins that want to align it downwards & upwards respectively.</p>
<p>Frustration can also been seen in quantum mechanical systems: while two quantum spin 1/2s with an antiferromagnetic Heisenberg interaction will form a maximally-entangled singlet state, introducing a third spin-1/2 creates a dilemma - <strong>the two-spin singlet cannot accommodate another spin</strong> (often curiously referred to as entanglement monogamy). Since the spin-flip quantum fluctuations of the system will want to lower the energy of the system by entangling all three spins, no spin can be left free and the <strong>two-spin singlet cannot be the true ground-state</strong>.</p>
<p class="archive__item-excerpt"><img src="/assets/images/mck/qmechfrustration.webp" alt="" />
An example of quantum frustration: three spins on a triangle connected by antiferromagnetic Heisenbeg interaction. The spin-flip fluctuations favour a singlet ground-state. However, if the blue and green spins form a maximally-entangled singlet state, they can no longer bind with the third gray spin (a property of the singlet), even though there are unquenched Heisenberg interactions between the third spin and the other spins respectively.</p>
<h2 id="frustration-in-the-multichannel-kondo-problem">Frustration in the multichannel Kondo problem</h2>
<p>The multichannel Kondo problem involves a <strong>local antiferromagnetic Heisenberg interaction between a single spin-1/2 impurity and the electrons of several conduction bath channels</strong>. Had there been only one conduction bath, the impurity moment would form a singlet together with a “cloud” of electrons from the bath. We refer to this as the screening of the impurity moment (as the singlet has no magnetisation). However, in the multichannel Kondo problem described above, the <strong>formation of a singlet between the impurity and electron from one of the conduction channels is frustrated</strong>. As a result, the so-called Kondo screening of the impurity spin’s magnetic moment is hampered.</p>
<p class="archive__item-excerpt"><img src="/assets/images/mck/mckondo.webp" alt="" />
Schematic of a multichannel Kondo model. The central impurity spin interacts with two conduction bath channels ( and ) through spin-flip interactions. The two channels compete with each other in trying to form a singlet with the single impurity spin, resulting in a novel frustrated ground-state.</p>
<p>Indeed, if the total spin of the conduction bath is greater than the spin of the impurity, the multichannel Kondo problem is said to be over-screened. The screening process, as well as its breakdown, are <strong>truly many-body in nature</strong>: a macroscopic number of conduction electrons interact with a single quantum impurity, and are therefore “aware” of each other. A proper description of the physics thus requires a field-theoretic treatment of the impurity-bath interactions, and the problem has been studied using a wide variety of powerful analytic and numerical methods.</p>
<h2 id="emergence-of-a-stargraph-in-the-multichannel-kondo-problem">Emergence of a “stargraph” in the multichannel Kondo problem</h2>
<p>Our contribution in this work was to show that <strong>the fascinating properties of the N-channel Kondo problem could be linked to those of the associated skeletal problem</strong>: a central quantum spin-1/2 coupled to N quantum spin-1/2s (corresponding to the N conduction channels) through identical antiferromagnetic Heisenberg exchange couplings. Such a model is often referred to as a star graph, and it can be identified as a limit of the multichannel problem in which the kinetic energy of the itinerant electrons has been switched off.</p>
<p class="archive__item-excerpt"><img src="/assets/images/mck/stargraph.webp" alt="" />
A schematic of the stargraph model, the skeletal problem associated with any multichannel Kondo model. The graph consists of a central node (the impurity spin) connected with a number of outer nodes (the local spins of the conduction channels). The bonds connecting the nodes depict the Heisenberg interaction between the impurity spin and the conduction bath spins.</p>
<p>We show in our work that <strong>certain properties of the star graph, such as the ground-state degeneracy and the magnetisation, are linked to bulk thermodynamic properties</strong>. The star graph also sets the scattering phase shift of the conduction electrons, and the scattering phase shift then dictates how the quantum fluctuations resolve themselves in order to lead to novel features. In this way, the quantum frustration inherent in the underlying simple quantum mechanical problem is seen to <strong>offer great insights into a many-body problem</strong> which looks quite daunting otherwise. Please read our work to find out more.</p>
<h2 id="further-details-on-our-results-mostly-for-the-experts">Further details on our results (mostly for the experts)</h2>
<ul>
<li>
<p>We elucidate, within the context of the multichannel Kondo problem, the importance of the zero bandwidth limit of the RG fixed point Hamiltonian. We show that this <strong>zero bandwidth Hamiltonian directly leads to several properties of the MCK problem</strong>, like the ground-state magnetisation, scattering phase shift, Wilson loop and ‘t Hooft operators and degree of compensation.</p>
</li>
<li>
<p>The <strong>ground-state degeneracy of the zero bandwidth Hamiltonian is found to be topological in nature</strong>: the orthogonal states can be explored by the application of twist operators. Integrating out the impurity from the conduction bath states leads to the emergence of a topologically degenerate local Mott liquid.</p>
</li>
<li>
<p>We also demonstrate that the <strong>effective Hamiltonian for the gapless excitations of the fixed point Hamiltonian is of the non-Fermi liquid kind</strong>, involving scattering process that connect multiple conduction channels and leading to an orthogonality catastrophe in the ground-state. In momentum space, the self-energy resembles that of a marginal Fermi liquid localised near the impurity spin.</p>
</li>
<li>
<p>We link the non-Fermi liquid behaviour and orthogonality catastrophe with an “unrenormalised” scattering phase shift that can be obtained from the zero bandwidth problem; we show that <strong>this phase shift also leads to the well-known anomalous behaviour</strong> of quantities like the specific heat, magnetic susceptibility and thermal entropy.</p>
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<li>
<p>Various entanglement measures, e.g., impurity entanglement entropy and impurity-bath mutual information, for the overscreened case <strong>show discontinuous behaviour</strong> as the conduction bath states beyond the zero-bandwidth problem are re-introduced via single-electron hopping. These discontinuities do not exist in the single-channel problem, and <strong>arise from the orthogonality catastrophe</strong> present in the overscreened ground-state.</p>
</li>
<li>
<p>By combining the strong-weak duality of the MCK Hamiltonian with that of the RG equations, we show that <strong>the strong-coupling theory is constrained to take a simple form</strong>. We also discuss an additional duality transformation connecting underscreened and overscreened models that involves exchanging the number of channels and impurity spin: this allows us to infer the infrared scaling behaviour of one class of models from a knowledge of the other.</p>
</li>
</ul>Siddhartha Lalslal@iiserkol.ac.inIn this work, we elucidate the importance of ground-state degeneracy and frustration in determining the physics of the multichannel Kondo model.Congratulations Abhirup, on winning a Best Poster award at DPS Day 2023!2023-03-18T00:00:00+00:002023-03-18T00:00:00+00:00https://epqm.github.io/abhirup-congrats<p>Abhirup Mukherjee was awarded a Best Poster award at DPS Day 2023. Abhirup presented his recent work on understanding the Mott-Hubbard MIT on the Bethe lattice in infinite dimensions. Well done, Abhirup. We look forward to many more such awards.</p>
<p><img src="../assets/images/gallery/AbhirupDPSDay2023.jpeg.jpeg" alt="" /></p>Siddhartha Lalslal@iiserkol.ac.inCongratulations Debraj, on your first poster presentation (at DPS Day 2023) as part of EPQM!2023-03-18T00:00:00+00:002023-03-18T00:00:00+00:00https://epqm.github.io/debraj-poster<p>Congratulations Debraj, on your first poster presentation as an EPQM member. Debraj presented his ongoing investigations of how quantum fluctuations encode entanglement, symmetry and coherence in a prototypical quantum many-body system.</p>
<p class="body_img"><img src="../assets/images/about/people/debraj.png" alt="" /></p>Siddhartha Lalslal@iiserkol.ac.inCan space emerge from quantum mechanics? We demonstrate this in a simple model of fermions!2023-02-22T00:00:00+00:002023-02-22T00:00:00+00:00https://epqm.github.io/metals-holgphy<p><strong>Holographic entanglement renormalisation for fermionic quantum matter: geometrical and topological aspects</strong>
<span class="pub__authors">Abhirup Mukherjee, Siddhartha Patra, <b>Siddhartha Lal</b></span><span class="btn btn--info">metals</span> <span class="btn btn--info">entanglement</span> <span class="btn btn--info">holography</span> <span class="btn btn--info">renormalisation group</span> <br />
Feb 21, 2023
<a href="https://arxiv.org/abs/2302.10590" class="btn btn--success">arXiv:2302.10590</a></p>
<div>
<div id="short_abstract" class="archive__item-excerpt">On performing a sequence of renormalisation group (RG) transformations on a system of two-dimensional non-interacting Dirac fermions placed on a torus, we demonstrate the emergence of an additional spatial dimension arising out of the scaling of multipartite entanglement...</div>
<div id="full_abstract" class="archive__item-excerpt">On performing a sequence of renormalisation group (RG) transformations on a system of two-dimensional non-interacting Dirac fermions placed on a torus, we demonstrate the emergence of an additional spatial dimension arising out of the scaling of multipartite entanglement. The renormalisation of entanglement under this flow exhibits a hierarchy across scales as well as number of parties. Geometric measures defined in this emergent space, such as distances and curvature, can be related to the RG beta function of the coupling \(g\) responsible for the spectral gap. This establishes a holographic connection between the spatial geometry of the emergent space in the bulk and the entanglement properties of the quantum theory lying on its boundary. Depending on the anomalous dimension of the coupling \(g\), three classes of spaces (bounded, unbounded and flat) are generated from the RG. We show that changing from one class to another involves a topological transition. By minimising the central charge of the conformal field theory describing the noninteracting electrons under the RG flow, the RG transformations are shown to satisfy the \(c-\)theorem of Zamolodchikov. This is shown to possess a dual within the emergent geometric space, in the form of a convergence parameter that is minimised at large distances. 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 can be related to the Luttinger volume of the system of electrons. In the presence of a strong transverse magnetic field, the system becomes insulating and Luttinger's theorem does not hold. We show instead that the entanglement contains a term that can be related to the Chern numbers of the quantum Hall states. This yields a relation between the topological invariants of the metallic and the quantum Hall systems.</div>
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<h2 id="a-bit-of-background">A bit of background</h2>
<p>In the last few decades, quantum entanglement has become very important for studying the nature of quantum condensed matter systems. For instance, gapped interacting many-body systems typically display an area-law scaling of the subsystem entanglement entropy with subsystem size, while quantum critical systems are expected to display a volume law scaling of the same. Further, a subdominant topological term in the entanglement entropy quantifies the long-ranged nature of correlations in topologically ordered insulating states of matter. Much less is known on the entanglement features of gapless metallic systems. Further, the holographic principle posits that the renormalisation group evolution of the many-particle entanglement of an interacting quantum field theory can be visualised as the emergence of an emergent spatial dimension.</p>
<p><img src="../assets/images/holog-renorm/holography.jpg" alt="" /></p>
<p class="archive__item-excerpt"><a href="https://seereal.com/how-does-it-work-holography">[Source]</a></p>
<h2 id="questions-addressed">Questions addressed</h2>
<p>The present work is thus devoted towards addressing the following questions:</p>
<ul>
<li>Can a first principles calculation of entanglement renormalisation in a relatively simple system demonstrate the holographic emergence of a spatial geometry?</li>
<li>How do the field theoretic parameters (e.g., the RG beta function) relate to the geometry quantifiers (e.g., distances and curvature) in this emergence dimension?</li>
<li>Spectral flow operations on metallic systems lead to a flux-dependent piece in the entanglement entropy. Is this term topological in nature?</li>
<li>If so, is there a correspondence between the topological terms in metallic systems and those obtained in the above-mentioned insulating systems?</li>
</ul>
<h2 id="main-results-emergence-of-additional-spatial-dimension">Main results: Emergence of additional spatial dimension</h2>
<p>Towards achieving these goals, we study a model of two-dimensional massive non-interacting Dirac fermions placed on a torus, and apply scaling transformations on the associated Hilbert space. The resultant renormalisation of multipartite measures of entanglement exhibits a hierarchy that exists across energy scales as well as across the number of parties. As our demonstration constitutes an exact holographic mapping, we argue that the scaling of the multipartite entanglement leads to the emergence of an additional spatial dimension. Geometric measures defined in this emergent space, such as distances and curvature, can be related to the RG beta function of the coupling g responsible for the spectral mass gap. Depending on the anomalous dimension of g, three classes of spaces (bounded, unbounded and flat) are generated from the RG: changing between classes involves a topological transition.</p>
<h2 id="main-results-topological-content-of-the-entanglement">Main results: Topological content of the entanglement</h2>
<p>In order to study the boundary condition sensitivity of the multipartite entanglement, we thread an Aharonov-Bohm flux through the torus. In the presence of the flux, the entanglement gains a geometry-independent piece that is shown to be topological, and can be related to the Luttinger volume of the system of electrons. In the presence of a strong transverse magnetic field, the system attains an integer quantum Hall insulating ground state, and Luttinger’s theorem does not hold. We show instead that the entanglement contains a term that can be related to the Chern numbers of the integer quantum Hall states. This yields a relation between the topological invariants of the metallic and the integer quantum Hall systems.</p>
<p>Thus, our work offers an ab initio demonstration of the holographic principle in a prototypical condensed matter system, and sheds new light on geometrical and topological aspects of entanglement in metals.</p>Siddhartha Lalslal@iiserkol.ac.inWe show the emergence of an additional dimension arising out of the scaling of multipartite entanglement upon applying RG transformations on 2D noninteracting electrons placed on a torus.