arXiv subjects

Gautam Mandal

Publications and source records attributed to Gautam Mandal.

At least 19 recordsLinked to original sources

arXiv2026

Fate of "Space-like singularities" in $c=1$ Matrix Model

A class of time dependent backgrounds in two dimensional String Theory leads to superluminal Liouville walls on the worldsheet. In the dual double scaled $c=1$ matrix model these backgrounds involve eigenvalues leaking out to infinity, and the collective field fluctuations become strongly coupled along space-like regions, resembling singularities. We realize these backgrounds as results of quantum quenches in the matrix model, retaining non-linear terms in the matrix potential, thus departing from a double scaling limit. Working in the fermion picture in a Thomas-Fermi approximation, we show that while the early time behavior of the phase space density near the maximum of the potential agrees with that obtained in the double scaled theory, at times of the order $(\log N)$ the effect of the IR wall becomes significant. At later times, with a characteristic winding time of order $(\log N)^2$, folds on the fermi surface proliferate and eventually cover the allowed region in phase space densely. Using action-angle variables, we show that the phase space density oscillates around a time independent and angle independent value rapidly at late times. A coarse-grained density in the angle space relaxes to a time independent equilibrium value as a power law with an exponent largely independent of the details of the initial state. Thus, the appearance of a space-like singularity is an artifact of the strict double scaling limit. We comment on the interpretation of the final state in String Theory.

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arXiv2026

2D or not 2D: a "holographic dictionary" for Lowest Landau Levels

We consider 2D fermions on a plane with a perpendicular magnetic field, described by Landau levels. It is wellknown that, semiclassically, restriction to the lowest Landau levels (LLL) implies two constraints on a 4D phase space, that transforms the 2D coordinate space (x,y) into a 2D phase space, thanks to the non-zero Dirac bracket between x and y. A naive application of Dirac's prescription of quantizing LLL in terms of L2 functions of x (or of y) fails because the wavefunctions are functions of x and y. We are able, however, to construct a 1D QM, sitting differently inside the 2D QM, which describes the LLL physics. The construction includes an exact 1D-2D correspondence between the fermion density ρ(x,y) and the Wigner distribution of the 1D QM. In a suitable large N limit, (a) the Wigner distribution is upper bounded by 1, since a phase space cell can have at most one fermion (Pauli exclusion principle) and (b) the 1D-2D correspondence becomes an identity transformation. (a) and (b) imply an upper bound for the fermion density ρ(x,y). We also explore the entanglement entropy (EE) of subregions of the 2D noncommutative space. It behaves differently from conventional 2D systems as well as conventional 1D systems, falling somewhere between the two. The main new feature of the EE, directly attributable to the noncommutative space, is the absence of a logarithmic dependence on the size of the entangling region, even though there is a Fermi surface. In this paper, instead of working directly with the Landau problem, we consider a more general problem, of 2D fermions in a rotating harmonic trap, which reduces to the Landau problem in a special limit. Among other consequences of the emergent 1D physics, we find that post-quench dynamics of the (generalized) LLL system is computed more simply in 1D terms, which is described by well-developed methods of 2D phase space hydrodynamics.

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arXiv2025

Exact lattice bosonization of finite N matrix quantum mechanics and c = 1

We describe a new exact lattice bosonization of matrix quantum mechanics (equivalently of non-relativistic fermions) that is valid for arbitrary rank N of the matrix, based on an exact operator bosonization introduced earlier in [1]. The trace identities are automatically incorporated in this formalism. The finite number N of fermions is reflected in the finite number N of bosonic oscillators, or equivalently to the finite number N of lattice points. The fermion Hamiltonian is exactly mappable to a bosonic Hamiltonian. At large N, the latter becomes local and corresponds to the lattice version of a relativistic boson Hamiltonian, with a lattice spacing of order 1/N. The finite lattice spacing leads to a finite entanglement entropy (EE) of the bosonic theory, which reproduces the finite EE of the fermionic theory. Such a description is not available in the standard bosonization in terms of fermion density fluctuations on the Fermi surface, which does not have a built-in short distance cut-off (see, however, [2]). The bosonic lattice is equipped with a geometry determined by the matrix potential or equivalently by the shape of the Fermi surface. Our bosonization also works in the double scaled c=1 model, where the bosonic EE again turns out to be finite, with the short distance cut-off turning as g_s l_s, and reproduces the matrix result. Once again, such a short distance cut-off cannot appear in the conventional dual of c=1 in terms of the 2D string ``tachyon'', where the expected short distance scale is l_s. This indicates our bosonization as a possibly different dual description to the c=1 matrix model appropriate for ``local physics'' like quantum entanglement, by contrast with the conventional duality to the eigenvalue density which works well for asymptotic observables like S-matrices. We briefly discuss possible relation of our bosonization to D0 branes.

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arXiv2024

Operators in the Internal Space and Locality

Realizations of the holographic correspondence in String/M theory typically involve spacetimes of the form $AdS \times Y$ where $Y$ is some internal space which geometrizes an internal symmetry of the dual field theory, hereafter referred to as an "$R$ symmetry". It has been speculated that areas of Ryu-Takayanagi surfaces anchored on the boundary of a subregion of $Y$, and smeared over the base space of the dual field theory, quantify entanglement of internal degrees of freedom. A natural candidate for the corresponding operators are linear combinations of operators with definite $R$ charge with coefficients given by the "spherical harmonics'' of the internal space: this is natural when the product spaces appear as IR geometries of higher dimensional AdS spaces. We study clustering properties of such operators both for pure $AdS \times Y$ and for flow geometries, where $AdS \times Y$ arises in the IR from a different spacetime in the UV, for example higher dimensional AdS or asymptotically flat spacetime. We show, in complete generality, that the two point functions of such operators separated along the internal space obey clustering properties at scales larger than the $AdS$ scale. For non-compact $Y$, this provides a notion of approximate locality. When $Y$ is compact, clustering happens only when the size of $Y$ is parametrically larger than the $AdS$ scale. This latter situation is realized in flow geometries where the product spaces arise in the IR from an asymptotically AdS geometry at UV, but not typically when they arise near black hole horizons in asymptotically flat spacetimes. We discuss the significance of this result for entanglement and comment on the role of color degrees of freedom.

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arXiv2023

A Microscopic Model of Black Hole Evaporation in Two Dimensions

We present a microscopic model of black hole (BH) `evaporation' in asymptotically $AdS_2$ spacetimes dual to the low energy sector of the SYK model. To describe evaporation, the SYK model is coupled to a bath comprising of $N_f$ free scalar fields $Φ_i$. We consider a linear combination of couplings of the form $O_{SYK}(t)\sum_iΦ_i(0,t)$, where $O_{SYK}$ involves products of the Kourkoulou-Maldacena operator $i J/N\sum_{k=1}^{N/2}s'_kψ_{2k-1}(t)ψ_{2k}(t)$ specified by a spin vector $s'$. We discuss the time evolution of a product of (i) a pure state of the SYK system, namely a BH microstate characterized by a spin vector $s$ and an effective BH temperature $T_{BH}$, and (ii) a Calabrese-Cardy state of the bath characterized by an effective temperature $T_{bath}$. We take $T_{bath}\ll T_{BH}$, and $T_{BH}$ much lower than the characteristic UV scale $J$ of the SYK model, allowing a description in terms of the time reparameterization mode. Tracing over the bath degrees of freedom leads to a Feynman-Vernon type effective action for the SYK model, which we study in the low energy limit. The leading large $N$ behaviour of the time reparameterization mode is found, as well as the $O(1/\sqrt N)$ fluctuations. The latter are characterized by a non-Markovian non-linear stochastic differential equation with non-local Gaussian noise. In a restricted range of couplings, we find two classes of solutions which asymptotically approach (a) a BH at a lower temperature, and (b) a horizonless geometry. We identify these with partial and complete BH evaporation, respectively. Importantly, the asymptotic solution in both cases involves the scalar product of the spin vectors $s.s'$, which carries some information about the initial state. By repeating the dynamical process $O(N^2)$ times with different choices of the spin vector $s'$, one can in principle reconstruct the initial BH microstate.

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arXiv2023

Entanglement Entropy in Internal Spaces and Ryu-Takayanagi Surfaces

We study minimum area surfaces associated with a region, $R$, of an internal space. For example, for a warped product involving an asymptotically $AdS$ space and an internal space $K$, the region $R$ lies in $K$ and the surface ends on $\partial R$. We find that the result of Graham and Karch can be avoided in the presence of warping, and such surfaces can sometimes exist for a general region $R$. When such a warped product geometry arises in the IR from a higher dimensional asymptotic AdS, we argue that the area of the surface can be related to the entropy arising from entanglement of internal degrees of freedom of the boundary theory. We study several examples, including warped or direct products involving $AdS_2$, or higher dimensional $AdS$ spaces, with the internal space, $K=R^m, S^m$; $Dp$ brane geometries and their near horizon limits; and several geometries with a UV cut-off. We find that such RT surfaces often exist and can be useful probes of the system, revealing information about finite length correlations, thermodynamics and entanglement. We also make some preliminary observations about the role such surfaces can play in bulk reconstruction, and their relation to subalgebras of observables in the boundary theory.

hep-th↗

arXiv2021

Gauge Invariant Target Space Entanglement in D-Brane Holography

It has been suggested in arXiv:2004.00613 that in Dp-brane holography, entanglement in the target space of the D-brane Yang-Mills theory provides a precise notion of bulk entanglement in the gravity dual. We expand on this discussion by providing a gauge invariant characterization of operator sub-algebras corresponding to such entanglement. This is achieved by finding a projection operator which imposes a constraint characterizing the target space region of interest. By considering probe branes in the Coulomb branch we provide motivation for why the operator sub-algebras we consider are appropriate for describing a class of measurements carried out with low-energy probes in the corresponding bulk region of interest. We derive expressions for the corresponding Renyi entropies in terms of path integrals which can be directly used in numerical calculations.

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arXiv2020

Holographic dual to charged SYK from 3D Gravity and Chern-Simons

In this paper, we obtain a bulk dual to SYK model, including SYK model with $U(1)$ charge, by Kaluza-Klein (KK) reduction from three dimensions. We show that KK reduction of the 3D Einstein action plus its boundary term gives the Jackiw-Teitelboim (JT) model in 2D with the appropriate 1D boundary term. The size of the KK radius gets identified with the value of the dilaton in the resulting near-AdS$_2$ geometry. In presence of U(1) charge, the 3D model additionally includes a $U(1)$ Chern-Simons (CS) action. In order to describe a boundary theory with non-zero chemical potential, we also introduce a coupling between CS gauge field and bulk gravity. The 3D CS action plus the new coupling term with appropriate boundary terms reduce in two dimensions to a BF-type action plus a source term and boundary terms. The KK reduced 2D theory represents the soft sector of the charged SYK model. The pseudo-Nambu-Goldstone modes of combined $\textit{Diff} /\mathbb{SL}(2,\mathbb{R})$ and $U(1)_{\rm local}/U(1)$ transformations are represented by combined large diffeomorphisms and large gauge transformations. The effective action of the former is reproduced by the action cost of the latter in the bulk dual, after appropriate identification of parameters. We compute chaotic correlators from the bulk and reproduce the result that the contribution from the "boundary photons" corresponds to zero Liapunov exponent.

hep-th↗

arXiv2020

Bulk Entanglement Entropy and Matrices

Motivated by the Bekenstein Hawking formula and the area law behaviour of entanglement entropy, we propose that in any UV finite theory of quantum gravity with a smooth spacetime, the total entropy for a pure state in a co-dimension one spatial region, to leading order, is given by $S={A\over 4 G_N}$, where $A$ is the area of the co-dimension two boundary. In the context of $Dp$ brane holography we show that for some specially chosen regions bulk entanglement can be mapped to ``target space" entanglement in the boundary theory. Our conjecture then leads to a precise proposal for target space entanglement in the boundary theory at strong coupling and large $N$. In particular it leads to the conclusion that the target space entanglement would scale like $O(N^2)$ which is quite plausible in a system with $O(N^2)$ degrees of freedom. Recent numerical advances in studying the D0 brane system hold out the hope that this proposal can be tested in a precise way in the future.

hep-th↗

arXiv2020

Thermalization in 2D critical quench and UV/IR mixing

We consider quantum quenches in models of free scalars and fermions with a generic time-dependent mass $m(t)$ that goes from $m_0$ to zero. We prove that, as anticipated in MSS \cite{Mandal:2015jla}, the post-quench dynamics can be described in terms of a state of the generalized Calabrese-Cardy form $|ψ\rangle$= $\exp[-κ_2 H -\sum_{n>2}^\infty κ_n W_n]| \hbox{Bd} \rangle$. The $W_n$ ($n=2,3,...$, $W_2=H$) here represent the conserved $W_\infty$ charges and $| \hbox{Bd} \rangle$ represents a conformal boundary state. Our result holds irrespective of whether the pre-quench state is a ground state or a squeezed state, and is proved without recourse to perturbation expansion in the $κ_n$'s as in MSS. We compute exact time-dependent correlators for some specific quench protocols $m(t)$. The correlators explicitly show thermalization to a generalized Gibbs ensemble (GGE), with inverse temperature $β= 4κ_2$, and chemical potentials $μ_n=4κ_n$. In case the pre-quench state is a ground state, it is possible to retrieve the exact quench protocol $m(t)$ from the final GGE, by an application of inverse scattering techniques. Another notable result, which we interpret as a UV/IR mixing, is that the long distance and long time (IR) behaviour of some correlators depends crucially on all $κ_n$'s, although they are highly irrelevant couplings in the usual RG parlance. This indicates subtleties in RG arguments when applied to non-equilibrium dynamics.

hep-th↗

arXiv2019

Quantum quench and thermalization to GGE in arbitrary dimensions and the odd-even effect

In many quantum quench experiments involving cold atom systems the post-quench system can be described by a quantum field theory of free scalars or fermions, typically in a box or in an external potential. We work with free scalars in arbitrary dimensions generalizing the techniques employed in our earlier work \cite{Mandal:2015kxi} in 1+1 dimensions. In this paper, we generalize to $d$ spatial dimensions for arbitrary $d$. The system is considered in a box much larger than any other scale of interest. We start with the ground state, or a squeezed state, with a high mass and suddenly quench the system to zero mass ("critical quench"). We explicitly compute time-dependence of local correlators and show that at long times they are described by a generalized Gibbs ensemble (GGE), which, in special cases, reduce to a thermal (Gibbs) ensemble. The equilibration of {\it local} correlators can be regarded as `subsystem thermalization' which we simply call 'thermalization' here (the notion of thermalization here also includes equlibration to GGE). The rate of approach to equilibrium is exponential or power law depending on whether $d$ is odd or even respectively. As in 1+1 dimensions, details of the quench protocol affect the long time behaviour; this underlines the importance of irrelevant operators at IR in non-equilibrium situations. We also discuss quenches from a high mass to a lower non-zero mass, and find that in this case the approach to equilibrium is given by a power law in time, for all spatial dimensions $d$, even or odd.

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arXiv2019

Gravitational collapse in SYK models and Choptuik-like phenomenon

SYK model is a quantum mechanical model of fermions which is solvable at strong coupling and plays an important role as perhaps the simplest holographic model of quantum gravity and black holes. The present work considers a deformed SYK model and a sudden quantum quench in the deformation parameter. The system, as in the undeformed case, permits a low energy description in terms of pseudo Nambu Goldstone modes. The bulk dual of such a system represents a gravitational collapse, which is characterized by a bulk matter stress tensor whose value near the boundary shows a sudden jump at the time of the quench. The resulting gravitational collapse forms a black hole only if the deformation parameter $Δε$ exceeds a certain critical value $Δε_c$ and forms a horizonless geometry otherwise. In case a black hole does form, the resulting Hawking temperature is given by a fractional power $T_{bh} \propto (Δε- Δε_c)^{1/2}$, which is reminiscent of the `Choptuik phenomenon' of critical gravitational collapse.

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arXiv2018

Quantum quench and thermalization of one-dimensional Fermi gas via phase space hydrodynamics

By exploring a phase space hydrodynamics description of one-dimensional free Fermi gas, we discuss how systems settle down to steady states described by the generalized Gibbs ensembles through quantum quenches. We investigate time evolutions of the Fermions which are trapped in external potentials or a circle for a variety of initial conditions and quench protocols. We analytically compute local observables such as particle density and show that they always exhibit power law relaxation at late times. We find a simple rule which determines the power law exponent. Our findings are, in principle, observable in experiments in an one dimensional free Fermi gas or Tonk's gas (Bose gas with infinite repulsion).

cond-mat.stat-mech↗

arXiv2017

Coadjoint orbit action of Virasoro group and two-dimensional quantum gravity dual to SYK/tensor models

The Nambu-Goldstone (NG) bosons of the SYK model are described by a coset space Diff/$\mathbb{SL}(2,\mathbb{R})$, where Diff, or Virasoro group, is the group of diffeomorphisms of the time coordinate valued on the real line or a circle. It is known that the coadjoint orbit action of Diff naturally turns out to be the two-dimensional quantum gravity action of Polyakov without cosmological constant, in a certain gauge, in an asymptotically flat spacetime. Motivated by this observation, we explore Polyakov action with cosmological constant and boundary terms, and study the possibility of such a two-dimensional quantum gravity model being the AdS dual to the low energy (NG) sector of the SYK model. We find strong evidences for this duality: (a) the bulk action admits an exact family of asymptotically AdS$_2$ spacetimes, parameterized by Diff/$\mathbb{SL}(2,\mathbb{R})$, in addition to a fixed conformal factor of a simple functional form; (b) the bulk path integral reduces to a path integral over Diff/$\mathbb{SL}(2,\mathbb{R})$ with a Schwarzian action; (c) the low temperature free energy qualitatively agrees with that of the SYK model. We show, up to quadratic order, how to couple an infinite series of bulk scalars to the Polyakov model and show that it reproduces the coupling of the higher modes of the SYK model with the NG bosons.

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arXiv2016

Revisiting AdS/CFT at a finite radial cut-off

We revisit AdS/CFT at finite radial cut-off, specifically in the context of double trace perturbations, $\mathbb{O}_n$= $\mathbb{O}(x) (\partial^2)^{n} {\mathcal O}(x)$, with arbitrary $n$. As well-known, the standard GKPW prescription, applied to a finite radial cut-off, leads to contact terms in correlators. de Haro et al (arXiv:hep-th/0002230) introduced bulk counterterms to remove these. However, this prescription yields additional terms in the correlator corresponding to spurious double trace deformations. Further, if we view GKPW prescription coupled with the prescription in arXiv:hep-th/0002230, in terms of a boundary wavefunction, we find that it is incompatible with radial Schrodinger evolution (in the spirit of holographic Wilsonian RG). We consider a more general wavefunction satisfying the Schrodinger equation, and find that generically such wavefunctions generate both (a) double trace deformations and (b) contact terms. However, we find that there exist special choices of these wavefunctions, amounting to a new AdS/CFT prescription at a finite cut-off, so that both (a) and (b) are removed and we obtain a pure power law behaviour for the correlator. We compare these special wavefunctions with a specific RG scheme in field theory. We give a geometric interpretation of these wavefunctions; these correspond to some specific smearing of boundary points in the Witten diagrams. We present a comprehensive calculation of exact double-trace beta-functions for all couplings $\mathbb{O}_n$ and match with a holographic computation using the method described above. The matching works with a mapping between the field theory and bulk couplings; such a map is highly constrained because the beta-functions are quadratic and exact on both sides. We generalize standard double-trace Wilson-Fisher flow to the space of the infinite number of couplings.

hep-th↗

arXiv2016

Finite size effect on dynamical entanglement entropy: CFT and holography

Time-dependent entanglement entropy (EE) is computed for a single interval in two-dimensional conformal theories from a quenched initial state in the presence of spatial boundaries. The EE is found to be periodic in time with periodicity equal to the system size $L$. For large enough $L$, the EE shows a rise to a thermal value (characterized by a temperature $1/β$ determined by the initial state), followed by periodic returns to the original value. This works irrespective of whether the conformal field theory (CFT) is rational or irrational. For conformal field theories with a holographic dual, the large $c$ limit plays an essential role in ensuring that the EE computed from the CFT is universal (independent of the details of the CFT and of boundary conditions) and is exactly matched by the holographic EE. The dual geometry is computed and it interpolates between a BTZ black hole at large $L$ and global AdS at large $β$.

hep-th↗

arXiv2015

Thermodynamics of QCD from Sakai-Sugimoto Model

Till date, the only consistent description of the deconfinement phase of the Sakai-Sugimoto model appears to be provided by the analysis of [1] (arXiv:1107.4048). The current version of the analysis, however, has a subtlety regarding the monodromy of quarks around the Euclidean time circle. In this note, we revisit and resolve this issue by considering the effect of an imaginary baryon chemical potential on quark monodromies. With this ingredient, the proposal of [1] for investigating finite temperature QCD using holography is firmly established. Additionally, our technique allows a holographic computation of the free energy as a function of the imaginary chemical potential in the deconfinement phase; we show that our result agrees with the corresponding formula obtained from perturbative QCD, namely the Roberge-Weiss potential.

hep-th↗

arXiv2015

Thermalization with chemical potentials, and higher spin black holes

We study the long time behaviour of local observables following a quantum quench in 1+1 dimensional conformal field theories possessing additional conserved charges besides the energy. We show that the expectation value of an arbitrary string of {\it local} observables supported on a finite interval exponentially approaches an equilibrium value. The equilibrium is characterized by a temperature and chemical potentials defined in terms of the quenched state. For an infinite number of commuting conserved charges, the equilibrium ensemble is a generalized Gibbs ensemble (GGE). We compute the thermalization rate in a systematic perturbation in the chemical potentials, using a new technique to sum over an infinite number of Feynman diagrams. The above technique also allows us to compute relaxation times for thermal Green's functions in the presence of an arbitrary number of chemical potentials. In the context of a higher spin (hs[λ]) holography, the partition function of the final equilibrium GGE is known to agree with that of a higher spin black hole. The thermalization rate from the CFT computed in our paper agrees with the quasinormal frequency of a scalar field in this black hole.

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