arXiv subjects

Igor R. Klebanov

Publications and source records attributed to Igor R. Klebanov.

At least 19 recordsLinked to original sources

arXiv2026

Landau-Ginzburg description of an exceptional ${\mathcal N}=1$ minimal model

The $\mathcal N=1$ superconformal minimal model with $m=12$ and the exceptional modular invariant $(E_6,D_8)$ is the unitary minimal model of the super-$W_3$ algebra. We propose its Landau-Ginzburg description using two real scalar superfields with the cubic superpotential ${\cal W}=g_1 XY^2/2 + g_2X^3/6$. For $g_1=g_2$, this superpotential is known to describe a product of two $m=3$ $\mathcal N=1$ superconformal minimal models, which is the $m=10$ model with the $(D_6,E_6)$ modular invariant. The exceptional $m=12$ superconformal minimal model is realized at a different fixed point of the same theory. Testing this Landau-Ginzburg description requires the fusion ring of the minimal model, which we obtain from the modular data of the extended algebra. The fusion ring has a $\mathbb Z_2$ grading by chiral fermion parity that the ordinary fusion coefficients do not determine. This grading, composed with conjugation, gives the generator of the R-parity $\mathbb Z_2^{R}$ of the Landau-Ginzburg theory. We then treat the theory with superpotential $\cal W$ as a Gross-Neveu-Yukawa model in $d=4-ε$ and find a weakly coupled infrared fixed point with $g_1/g_2=3/2+\mathcal O(ε)$, at which supersymmetry emerges. We also describe the renormalization group flow from this fixed point to the decoupled fixed point with $g_1=g_2$. The operator dimensions at the coupled fixed point, continued to $d=2$, agree approximately with their values in the $m=12$ superconformal minimal model. Finally, we estimate the scaling dimensions in the new interacting $d=3$ $\mathcal N=1$ superconformal field theory.

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arXiv2026

Supercurrents and (Partial) Supersymmetry in Adjoint QCD$_2$ and Its Generalizations

$1+1$-dimensional $SU(N)$ gauge theory coupled to an adjoint Majorana fermion, also known as adjoint QCD$_2$, has the surprising feature that at fermion mass $\sqrt{\frac{g^2 N}{2 π}}$ it exhibits supersymmetry. In this paper, we obtain a deeper insight into how the supersymmetry works by constructing the gauge invariant, Lorentz covariant supercurrent $j_{μA}$. Its conservation relies crucially on the presence of a quantum anomaly. We generalize this construction to a class of models where, in addition to an adjoint Majorana fermion of an appropriate mass, the gauge theory is coupled to some collection of massless fermions ($SU(N)$ may be replaced by a more general gauge group). In general, these models have a supersymmetric massive sector and a non-supersymmetric CFT sector [arXiv:2202.04017], but there are cases in which both sectors are supersymmetric. An example of such a gapless, fully supersymmetric model is $SU(N)$ gauge theory coupled to three adjoint Majorana fermions, of which two are massless and the third has mass $\sqrt{\frac{3g^2 N}{2π}}$.

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arXiv2026

The two-flavor Schwinger model at 50: Solving Coleman's puzzles

In his 1976 paper "More about the massive Schwinger model", Coleman introduced $1+1$-dimensional Quantum Electrodynamics coupled to two charged massive fermions. By applying Abelian bosonization, he elucidated much of the physics of this two-flavor Schwinger model, but he listed three puzzles at the end of his paper. We present new analytical and numerical calculations to solve Coleman's three puzzles and thereby deepen our understanding of this model. These puzzles pertain to the theory with equal fermion masses at $θ= 0$ and at $θ= π$, as well as the size of isospin-breaking effects when the fermion masses are unequal. For the puzzle at $θ= π$, the solution is related to the structure of the zero-temperature phase diagram arXiv:2305.04437: for equal fermion masses $m$, the model exhibits spontaneous breaking of charge conjugation symmetry and absence of confinement for any value of the gauge coupling $g$, so that there is a smooth interpolation from weak to strong coupling. Using two-loop Renormalization Group and integrability methods, we show that the mass gap behaves as $\sim m e^{-0.111 g^2/m^2}$ in the strong coupling regime $m\ll g$. Our numerical results using the lattice Hamiltonian are in good agreement with this behavior. For the puzzle at $θ= 0$, the solution is related to a level crossing between two isosinglet particles with different discrete quantum numbers; we demonstrate the necessity of such a crossing by comparing integrability and weak coupling calculations, and we also exhibit the crossing numerically. Finally, we provide a new estimate for the size of isospin-breaking effects caused by different fermion masses at strong coupling.

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arXiv2026

Regge trajectories from the adjoint sector of Matrix Quantum Mechanics

We reexamine the large $N$ limit of SU$(N)$ symmetric quantum mechanics of a Hermitian matrix whose singlet sector is well known to be exactly solvable via free fermions. When the Fermi level approaches a maximum of the potential, there is critical behavior corresponding to string theory in two dimensions. We uncover new phenomena in the adjoint sector by solving the Marchesini-Onofri equation both numerically and analytically using semiclassical approximations: at criticality, the spectrum is governed by Regge trajectories with energy eigenvalues growing according to $Δ^2 \sim n/ α'$. In the dual 2D string theory, we interpret these states as oscillatory excitations of a ``short'' folded open string. Up to sub-leading corrections, this Regge behavior is essentially universal and is insensitive to the particular potential we choose to approach criticality. Slightly away from criticality, the highly excited states transition into ``long strings'' that extend far into the Liouville direction.

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arXiv2025

Yang-Lee Quantum Criticality in Various Dimensions

The Yang-Lee universality class arises when imaginary magnetic field is tuned to its critical value in the paramagnetic phase of the $d<6$ Ising model. In $d=2$, this non-unitary Conformal Field Theory (CFT) is exactly solvable via the $M(2,5)$ minimal model. As found long ago by von Gehlen using Exact Diagonalization, the corresponding real-time, quantum critical behavior arises in the periodic Ising spin chain when the imaginary longitudinal magnetic field is tuned to its critical value from below. Even though the Hamiltonian is not Hermitian, the energy levels are real due to the $PT$ symmetry. In this paper, we explore the analogous quantum critical behavior in higher dimensional non-Hermitian Hamiltonians on regularized spheres $S^{d-1}$. For $d=3$, we use the recently invented, powerful fuzzy sphere method, as well as discretization by the platonic solids cube, icosahedron and dodecaherdron. The low-lying energy levels and structure constants we find are in agreement with expectations from the conformal symmetry. The energy levels are in good quantitative agreement with the high-temperature expansions and with Padé extrapolations of the $6-ε$ expansions in Fisher's $iϕ^3$ Euclidean field theory for the Yang-Lee criticality. In the course of this work, we clarify some aspects of matching between operators in this field theory and quasiprimary fields in the $M(2,5)$ minimal model. For $d=4$, we obtain new results by replacing the $S^3$ with the self-dual polytope called the $24$-cell.

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arXiv2025

Ginzburg-Landau description of a class of non-unitary minimal models

It has been proposed that the Ginzburg-Landau description of the non-unitary conformal minimal model $M(3,8)$ is provided by the Euclidean theory of two real scalar fields with third-order interactions that have imaginary coefficients. The same lagrangian describes the non-unitary model $M(3,10)$, which is a product of two Yang-Lee theories $M(2,5)$, and the Renormalization Group flow from it to $M(3,8)$. This proposal has recently passed an important consistency check, due to Y. Nakayama and T. Tanaka, based on the anomaly matching for non-invertible topological lines. In this paper, we elaborate the earlier proposal and argue that the two-field theory describes the $D$ series modular invariants of both $M(3,8)$ and $M(3,10)$. We further propose the Ginzburg-Landau descriptions of the entire class of $D$ series minimal models $M(q, 3q-1)$ and $M(q, 3q+1)$, with odd integer $q$. They involve $PT$ symmetric theories of two scalar fields with interactions of order $q$ multiplied by imaginary coupling constants.

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arXiv2025

Towards a Quintic Ginzburg-Landau Description of the $(2,7)$ Minimal Model

We discuss dimensional continuation of the massless scalar field theory with the $iϕ^5$ interaction term. It preserves the so-called $\mathcal{PT}$ symmetry, which acts by $ϕ\rightarrow -ϕ$ accompanied by $i\rightarrow -i$. Below its upper critical dimension $10/3$, this theory has interacting infrared fixed points. We argue that the fixed point in $d=2$ describes the non-unitary minimal conformal model $M(2,7)$. We identify the operators $ϕ$ and $ϕ^2$ with the Virasoro primaries $ϕ_{1,2}$ and $ϕ_{1,3}$, respectively, and $iϕ^3$ with a quasi-primary operator, which is a Virasoro descendant of $ϕ_{1,3}$. Our identifications appear to be consistent with the operator product expansions and with considerations based on integrability. Using constrained Padé extrapolations, we provide estimates of the critical exponents in $d=3$. We also comment on possible lattice descriptions of $M(2,7)$ and discuss RG flows to and from this CFT. Finally, we conjecture that the minimal models $M(2, 2n+1)$ are described by the massless scalar field theories with the $iϕ^{2n-1}$ interaction terms.

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arXiv2025

More About the Lattice Hamiltonian for Adjoint QCD$_2$

In our earlier work arXiv:2311.09334, we introduced a lattice Hamiltonian for Adjoint QCD$_2$ using staggered Majorana fermions. We found the gauge invariant space of states explicitly for the gauge group $\text{SU}(2)$ and used them for numerical calculations of observables, such as the spectrum and the expectation value of the fermion bilinear. In this paper, we carry out a more in-depth study of our lattice model, extending it to any compact and simply-connected gauge group $G$. We show how to find the gauge invariant space of states and use it to study various observables. We also use the lattice model to calculate the mixed 't Hooft anomalies of Adjoint QCD$_2$ for arbitrary $G$. We show that the matrix elements of the lattice Hamiltonian can be expressed in terms of the Wigner 6$j$-symbols of $G$. For $G = \text{SU}(3)$, we perform exact diagonalization for lattices of up to six sites and study the low-lying spectrum, the fermion bilinear condensate, and the string tension. We also show how to write the lattice strong coupling expansion for ground state energies and operator expectation values in terms of the Wigner 6$j$-symbols. For $\text{SU}(3)$ we carry this out explicitly and find good agreement with the exact diagonalizations, and for $\text{SU}(4)$ we give expansions that can be compared with future numerical studies.

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arXiv2024

Small Circle Expansion for Adjoint QCD$_2$ with Periodic Boundary Conditions

We study $1+1$-dimensional $\text{SU}(N)$ gauge theory coupled to one adjoint multiplet of Majorana fermions on a small spatial circle of circumference $L$. Using periodic boundary conditions, we derive the effective action for the quantum mechanics of the holonomy and the fermion zero modes in perturbation theory up to order $(gL)^3$. When the adjoint fermion mass-squared is tuned to $g^2 N/(2π)$, the effective action is found to be an example of supersymmetric quantum mechanics with a nontrivial superpotential. We separate the states into the $\mathbb{Z}_N$ center symmetry sectors (universes) labeled by $p=0, \ldots, N-1$ and show that in one of the sectors the supersymmetry is unbroken, while in the others it is broken spontaneously. These results give us new insights into the $(1,1)$ supersymmetry of adjoint QCD$_2$, which has previously been established using light-cone quantization. When the adjoint mass is set to zero, our effective Hamiltonian does not depend on the fermions at all, so that there are $2^{N-1}$ degenerate sectors of the Hilbert space. This construction appears to provide an explicit realization of the extended symmetry of the massless model, where there are $2^{2N-2}$ operators that commute with the Hamiltonian. We also generalize our results to other gauge groups $G$, for which supersymmetry is found at the adjoint mass-squared $g^2 h^\vee/(2π)$, where $h^\vee$ is the dual Coxeter number of $G$.

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arXiv2024

Lattice Hamiltonian for Adjoint QCD$_2$

We introduce a Hamiltonian lattice model for the $(1+1)$-dimensional $\text{SU}(N_c)$ gauge theory coupled to one adjoint Majorana fermion of mass $m$. The discretization of the continuum theory uses staggered Majorana fermions. We analyze the symmetries of the lattice model and find lattice analogs of the anomalies of the corresponding continuum theory. An important role is played by the lattice translation by one lattice site, which in the continuum limit involves a discrete axial transformation. On a lattice with periodic boundary conditions, the Hilbert space breaks up into sectors labeled by the $N_c$-ality $p=0, \ldots N_c-1$. Our symmetry analysis implies various exact degeneracies in the spectrum of the lattice model. In particular, it shows that, for $m=0$ and even $N_c$, the sectors $p$ and $p'$ are degenerate if $|p-p'| = N_c/2$. In the $N_c = 2$ case, we explicitly construct the action of the Hamiltonian on a basis of gauge-invariant states, and we perform both a strong coupling expansion and exact diagonalization for lattices of up to $12$ lattice sites. Upon extrapolation of these results, we find good agreement with the spectrum computed previously using discretized light-cone quantization. One of our new results is the first numerical calculation of the fermion bilinear condensate.

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arXiv2023

Phase Diagram of the Two-Flavor Schwinger Model at Zero Temperature

We examine the phase structure of the two-flavor Schwinger model as a function of the $θ$-angle and the two masses, $m_1$ and $m_2$. In particular, we find interesting effects at $θ=π$: along the $SU(2)$-invariant line $m_1 = m_2 = m$, in the regime where $m$ is much smaller than the charge $g$, the theory undergoes logarithmic RG flow of the Berezinskii-Kosterlitz-Thouless type. As a result, in this regime there is a non-perturbatively small mass gap $\sim e^{- A g^2/m^2}$. The $SU(2)$-invariant line lies within a region of the phase diagram where the charge conjugation symmetry is spontaneously broken and whose boundaries we determine numerically. Our numerical results are obtained using the Hamiltonian lattice gauge formulation that includes the mass shift $m_\text{lat} = m- g^2 a/4$ dictated by the discrete chiral symmetry.

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arXiv2023

Boundaries and Interfaces with Localized Cubic Interactions in the $O(N)$ Model

We explore a new approach to boundaries and interfaces in the $O(N)$ model where we add certain localized cubic interactions. These operators are nearly marginal when the bulk dimension is $4-ε$, and they explicitly break the $O(N)$ symmetry of the bulk theory down to $O(N-1)$. We show that the one-loop beta functions of the cubic couplings are affected by the quartic bulk interactions. For the interfaces, we find real fixed points up to the critical value $N_{\rm crit}\approx 7$, while for $N> 4$ there are IR stable fixed points with purely imaginary values of the cubic couplings. For the boundaries, there are real fixed points for all $N$, but we don't find any purely imaginary fixed points. We also consider the theories of $M$ pairs of symplectic fermions and one real scalar, which have quartic $OSp(1|2M)$ invariant interactions in the bulk. We then add the $Sp(2M)$ invariant localized cubic interactions. The beta functions for these theories are related to those in the $O(N)$ model via the replacement of $N$ by $1- 2M$. In the special case $M=1$, there are boundary or interface fixed points that preserve the $OSp(1|2)$ symmetry, as well as other fixed points that break it.

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arXiv2023

Discrete Chiral Symmetry and Mass Shift in Lattice Hamiltonian Approach to Schwinger Model

We revisit the lattice formulation of the Schwinger model using the Kogut-Susskind Hamiltonian approach with staggered fermions. This model, introduced by Banks et al., contains the mass term $m_{\rm lat} \sum_{n} (-1)^{n} χ^\dagger_n χ_n$, and setting it to zero is often assumed to provide the lattice regularization of the massless Schwinger model. We instead argue that the relation between the lattice and continuum mass parameters should be taken as $m_{\rm lat}=m- \frac 18 e^2 a$. The model with $m=0$ is shown to possess a discrete chiral symmetry that is generated by the unit lattice translation accompanied by the shift of the $θ$-angle by $π$. While the mass shift vanishes as the lattice spacing $a$ approaches zero, we find that including this shift greatly improves the rate of convergence to the continuum limit. We demonstrate the faster convergence using both numerical diagonalizations of finite lattice systems, as well as extrapolations of the lattice strong coupling expansions.

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arXiv2023

Ginzburg-Landau Description and Emergent Supersymmetry of the $(3,8)$ Minimal Model

A pair of the 2D non-unitary minimal models $M(2,5)$ is known to be equivalent to a variant of the $M(3,10)$ minimal model. We discuss the RG flow from this model to another non-unitary minimal model, $M(3,8)$. This provides new evidence for its previously proposed Ginzburg-Landau description, which is a $\mathbb{Z}_2$ symmetric theory of two scalar fields with cubic interactions. We also point out that $M(3,8)$ is equivalent to the $(2,8)$ superconformal minimal model with the diagonal modular invariant. Using the 5-loop results for theories of scalar fields with cubic interactions, we exhibit the $6-ε$ expansions of the dimensions of various operators. Their extrapolations are in quite good agreement with the exact results in 2D. We also use them to approximate the scaling dimensions in $d=3,4,5$ for the theories in the $M(3,8)$ universality class.

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arXiv2022

Adjoint Majorana QCD$_2$ at Finite $N$

The mass spectrum of $1+1$-dimensional $\mathrm{SU}(N)$ gauge theory coupled to a Majorana fermion in the adjoint representation has been studied in the large $N$ limit using Light-Cone Quantization. Here we extend this approach to theories with small values of $N$, exhibiting explicit results for $N=2, 3$, and $4$. In the context of Discretized Light-Cone Quantization, we develop a procedure based on the Cayley-Hamilton theorem for determining which states of the large $N$ theory become null at finite $N$. For the low-lying bound states, we find that the squared masses divided by $g^2 N$, where $g$ is the gauge coupling, have very weak dependence on $N$. The coefficients of the $1/N^2$ corrections to their large $N$ values are surprisingly small. When the adjoint fermion is massless, we observe exact degeneracies that we explain in terms of a Kac-Moody algebra construction and charge conjugation symmetry. When the squared mass of the adjoint fermion is tuned to $g^2 N / π$, we find evidence that the spectrum exhibits boson-fermion degeneracies, in agreement with the supersymmetry of the model at any value of $N$.

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arXiv2022

Critical Field Theories with OSp$(1|2M)$ Symmetry

In the paper [L. Fei et al., JHEP 09 (2015) 076] a cubic field theory of a scalar field $σ$ and two anticommuting scalar fields, $θ$ and $\bar θ$, was formulated. In $6-ε$ dimensions it has a weakly coupled fixed point with imaginary cubic couplings where the symmetry is enhanced to the supergroup OSp$(1|2)$. This theory may be viewed as a "UV completion" in $2<d<6$ of the non-linear sigma model with hyperbolic target space H$^{0|2}$ described by a pair of intrinsic anticommuting coordinates. It also describes the $q\rightarrow 0$ limit of the critical $q$-state Potts model, which is equivalent to the statistical mechanics of spanning forests on a graph. In this letter we generalize these results to a class of OSp$(1|2M)$ symmetric field theories whose upper critical dimensions are $d_c(M) = 2 \frac{2M+1}{2M-1}$. They contain $2M$ anticommuting scalar fields, $θ^i, \bar θ^i$, and one commuting one, with interaction $g\left (σ^2+ 2θ^i \bar θ^i \right )^{(2M+1)/2}$. In $d_c(M)-ε$ dimensions, we find a weakly coupled IR fixed point at an imaginary value of $g$. We propose that these critical theories are the UV completions of the sigma models with fermionic hyperbolic target spaces H$^{0|2M}$. Of particular interest is the quintic field theory with OSp$(1|4)$ symmetry, whose upper critical dimension is $10/3$. Using this theory, we make a prediction for the critical behavior of the OSp$(1|4)$ lattice system in three dimensions.

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arXiv2021

Group theoretic approach to many-body scar states in fermionic lattice models

It has been shown [arXiv:2007.00845] that three families of highly symmetric states are many-body scars for any spin-1/2 fermionic Hamiltonian of the form $H_0+OT$, where $T$ is a generator of an appropriate Lie group. One of these families consists of the well-known $η$-pairing states. In addition to having the usual properties of scars, these families of states are insensitive to electromagnetic noise and have advantages for storing and processing quantum information. In this paper we show that a number of well-known coupling terms, such as the Hubbard and the Heisenberg interactions, and the Hamiltonians containing them, are of the required form and support these states as scars without fine-tuning. The explicit $H_0+OT$ decomposition for a number of most commonly used models, including topological ones, is provided. To facilitate possible experimental implementations, we discuss the conditions for the low-energy subspace of these models to be comprised solely of scars. Further, we write down all the generators $T$ that can be used as building blocks for designing new models with scars, most interestingly including the spin-orbit coupled hopping and superconducting pairing terms. We expand this framework to the non-Hermitian open systems and demonstrate that for them the scar subspace continues to undergo coherent time evolution and exhibit the "revivals". A full numerical study of an extended 2D $tJU$ model explicitly illustrates the novel properties of the invariant scars and supports our findings.

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arXiv2021

Exact Symmetries and Threshold States in Two-Dimensional Models for QCD

Two-dimensional SU$(N)$ gauge theory coupled to a Majorana fermion in the adjoint representation is a nice toy model for higher-dimensional gauge dynamics. It possesses a multitude of "gluinoball" bound states whose spectrum has been studied using numerical diagonalizations of the light-cone Hamiltonian. We extend this model by coupling it to $N_f$ flavors of fundamental Dirac fermions (quarks). The extended model also contains meson-like bound states, both bosonic and fermionic, which in the large-$N$ limit decouple from the gluinoballs. We study the large-$N$ meson spectrum using the Discretized Light-Cone Quantization (DLCQ). When all the fermions are massless, we exhibit an exact $\mathfrak{osp}(1|4)$ symmetry algebra that leads to an infinite number of degeneracies in the DLCQ approach. More generally, we show that many single-trace states in the theory are threshold bound states that are degenerate with multi-trace states. These exact degeneracies can be explained using the Kac-Moody algebra of the SU$(N)$ current. We also present strong numerical evidence that additional threshold states appear in the continuum limit. Finally, we make the quarks massive while keeping the adjoint fermion massless. In this case too, we observe some exact degeneracies that show that the spectrum of mesons becomes continuous above a certain threshold. This demonstrates quantitatively that the fundamental string tension vanishes in the massless adjoint QCD$_2$.

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