arXiv subjects

Grigory Tarnopolsky

Publications and source records attributed to Grigory Tarnopolsky.

At least 19 recordsLinked to original sources

arXiv2026

Quantum models with the Yang-Lee phase transition

In this article, we present four different $1+1$D quantum models that realize the Yang-Lee (YL) phase transition under a deformation that preserves $PT$ symmetry. These are the antiferromagnetic Ising spin chain in transverse and longitudinal magnetic fields, the massive Schwinger model, the Blume-Capel model, and the three-state quantum clock model. Using the state-operator correspondence, we identify the YL critical point, compute the scaling dimensions of the lowest operators in each model, and find perfect agreement with the exact results for the YL criticality in two dimensions. Using bosonization for the Schwinger model and the Polyakov-Hubbard transformation for the other models, we show that in all of these quantum models the YL critical point is described, as expected, by a massless bosonic field with an $i ϕ^3$ interaction. In the quantum clock model, this critical field interacts with a massive bosonic field, and we identify the massless and massive states in the Hamiltonian spectrum. In addition, we numerically compute the two-point function of $ϕ$ at the Yang-Lee critical point and show that it grows with distance, in agreement with theoretical expectations.

hep-th↗

arXiv2026

Four-point function of the complex Sachdev-Ye-Kitaev model at finite chemical potential

It is known that, for a range of chemical potentials, the infrared behavior of the complex Sachdev-Ye-Kitaev (cSYK) model is governed by a 1D Nearly Conformal Field Theory (NCFT$_{1}$), thereby realizing a continuous line of NCFTs. A finite chemical potential $μ$ introduces an asymmetry parameter $\mathscr{E}$ into the cSYK fermion two-point function in the conformal limit. In this work, we compute the cSYK four-point function in the conformal limit for an arbitrary value of $\mathscr{E}$ at leading order in $1/N$. We show that the result is fully consistent with the NCFT$_{1}$ structure of the cSYK model and use it to extract the structure constants for correlation functions of two complex fermions with bilinear operators.

hep-th↗

arXiv2026

Scaling limits of complex Sachdev-Ye-Kitaev models and holographic geometry

We compare different limits of the Sachdev-Ye-Kitaev model of $N$ complex fermion with $p$-fermion interactions. First, we compute the fermion Green's function and free energy in the limit of large $N$ followed subsequently by the limit of large $p$. Next, we examine the `double-scaling' limit in which the large $N,p$ limits are taken at fixed $λ= p^2/N$. Earlier results on the latter limit are resummed for small $λ$, and shown to match our results for the first limit. We also describe the holographic match of our results to two-dimensional Jackiw-Teitelboim gravity with an additional $U(1)$ gauge field.

hep-th↗

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.

hep-th↗

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.

hep-th↗

arXiv2024

Precision study of the massive Schwinger model near quantum criticality

We perform a numerical analysis of the massive Schwinger model in the presence of a background electric field. Using the Density Matrix Renormalization Group (DMRG) approach, we efficiently compute the spectrum of the Schwinger model on a staggered lattice with up to 3000 qubits. As a result, we achieve a precise computation of the critical mass of the massive Schwinger model to five digits using four different 'criticality criteria', observing perfect agreement among them. Additionally, we discuss the effect of a four-fermion operator deformation of the Schwinger model and compute the critical mass for various values of the deformation parameter.

hep-th↗

arXiv2023

Magic Angle Butterfly in Twisted Trilayer Graphene

We consider a configuration of three stacked graphene monolayers with commensurate twist angles $θ_{12}/θ_{23}=p/q$, where $p$ and $q$ are coprime integers with $0<p<|q|$ and $q$ can be positive or negative. We study this system using the continuum model in the chiral limit when interlayer coupling terms between $\textrm{AA}_{12}$ and $\textrm{AA}_{23}$ sites of the moiré patterns $12$ and $23$ are neglected. There are only three inequivalent displacements between the moiré patterns $12$ and $23$, at which the three monolayers' Dirac zero modes are protected. Remarkably, for these displacements and an arbitrary $p/q$ we discover exactly flat bands at an infinite set of twist angles (magic angles). We provide theoretical explanation and classification of all possible configurations and topologies of the flat bands.

cond-mat.str-el↗

arXiv2023

Magic Angles In Equal-Twist Trilayer Graphene

We consider a configuration of three stacked graphene monolayers with equal consecutive twist angles $θ$. Remarkably, in the chiral limit when interlayer coupling terms between $\textrm{AA}$ sites of the moiré pattern are neglected we find four perfectly flat bands (for each valley) at a sequence of magic angles which are exactly equal to the twisted bilayer graphene (TBG) magic angles divided by $\sqrt{2}$. Therefore, the first magic angle for equal-twist trilayer graphene (eTTG) in the chiral limit is $θ_{*} \approx 1.05^{\circ}/\sqrt{2} \approx 0.74^{\circ}$. We prove this relation analytically and show that the Bloch states of the eTTG's flat bands are non-linearly related to those of TBG's. Additionally, we show that at the magic angles, the upper and lower bands must touch the four exactly flat bands at the Dirac point of the middle graphene layer. Finally, we explore the eTTG's spectrum away from the chiral limit through numerical analysis.

cond-mat.str-el↗

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.

hep-th↗

arXiv2022

Precise Low-Temperature Expansions for the Sachdev-Ye-Kitaev model

We solve numerically the large $N$ Dyson-Schwinger equations for the Sachdev-Ye-Kitaev (SYK) model utilizing the Legendre polynomial decomposition and reaching $10^{-36}$ accuracy. Using this we compute the energy of the SYK model at low temperatures $T\ll J$ and obtain its series expansion up to $T^{7.54}$. While it was suggested that the expansion contains terms $T^{3.77}$ and $T^{5.68}$, we find that the first non-integer power of temperature is $T^{6.54}$, which comes from the two point function of the fermion bilinear operator $O_{h_{1}}=χ\partial_τ^{3}χ$ with scaling dimension $h_{1}\approx 3.77$. The coefficient in front of $T^{6.54}$ term agrees well with the prediction of the conformal perturbation theory. We conclude that the conformal perturbation theory appears to work even though the SYK model is not strictly conformal.

hep-th↗

arXiv2022

Notes on the complex Sachdev-Ye-Kitaev model

We describe numerous properties of the Sachdev-Ye-Kitaev model for complex fermions with $N\gg 1$ flavors and a global U(1) charge. We provide a general definition of the charge in the $(G,Σ)$ formalism, and compute its universal relation to the infrared asymmetry of the Green function. The same relation is obtained by a renormalization theory. The conserved charge contributes a compact scalar field to the effective action, from which we derive the many-body density of states and extract the charge compressibility. We compute the latter via three distinct numerical methods and obtain consistent results. Finally, we present a two dimensional bulk picture with free Dirac fermions for the zero temperature entropy.

hep-th↗

arXiv2021

Excitation spectra of quantum matter without quasiparticles I: Sachdev-Ye-Kitaev models

We study the low frequency spectra of complex Sachdev-Ye-Kitaev (SYK) models at general densities. The analysis applies also to SU($M$) magnets with random exchange at large $M$. The spectral densities are computed by numerical analysis of the saddle point equations on the real frequency ($ω$) axis at zero temperature ($T$). The asymptotic low $ω$ behaviors are found to be in excellent agreement with the scaling dimensions of irrelevant operators which perturb the conformally invariant critical states. Of possible experimental interest is our computation of the universal spin spectral weight of the SU($M$) magnets at low $ω$ and $T$: this includes a contribution from the time reparameterization mode, which is the boundary graviton of the holographic dual. This analysis is extended to a random $t$-$J$ model in a companion paper.

cond-mat.str-el↗

arXiv2020

Excitation spectra of quantum matter without quasiparticles II: random $t$-$J$ models

We present numerical solutions of the spectral functions of $t$-$J$ models with random and all-to-all exchange and global SU($M$) spin rotation symmetry. The solutions are obtained from the saddle-point equations of the large volume limit, followed by the large $M$ limit. These saddle point equations involve Green's functions for fractionalized spinons and holons carrying emergent U(1) gauge charges, obeying relations similar to those of the Sachdev-Ye-Kitaev (SYK) models. The low frequency spectral functions are compared with an analytic analysis of the operator scaling dimensions, with good agreement. We also compute the low frequency and temperature behavior of gauge-invariant observables: the electron Green's function, the local spin susceptibility and the optical conductivity; along with the temperature dependence of the d.c. resistivity. The time reparameterization soft mode (equivalent to the boundary graviton in holographically dual models of two-dimensional quantum gravity) makes important contributions to all observables, and provides a linear-in-temperature contribution to the d.c. resistivity.

cond-mat.str-el↗

arXiv2020

Spontaneous Breaking of $U(1)$ Symmetry in Coupled Complex SYK Models

As shown in [1], two copies of the large $N$ Majorana SYK model can produce spontaneous breaking of a $Z_2$ symmetry when they are coupled by appropriate quartic terms. In this paper we similarly study two copies of the complex SYK model coupled by a quartic term preserving the $U(1) \times U(1)$ symmetry. We also present a tensor counterpart of this coupled model. When the coefficient $α$ of the quartic term lies in a certain range, the coupled large $N$ theory is nearly conformal. We calculate the scaling dimensions of fermion bilinear operators as functions of $α$. We show that the operator $c_{1i}^\dagger c_{2i}$, which is charged under the axial $U(1)$ symmetry, acquires a complex dimension outside of the line of fixed points. We derive the large $N$ Dyson-Schwinger equations and show that, outside the fixed line, this $U(1)$ symmetry is spontaneously broken at low temperatures because this operator acquires an expectation value. We support these findings by exact diagonalizations extrapolated to large $N$.

hep-th↗

arXiv2020

The $O(N)$ Model in $4<d<6$: Instantons and Complex CFTs

We revisit the scalar $O(N)$ model in the dimension range $4<d<6$ and study the effects caused by its metastability. As shown in previous work, this model formally possesses a fixed point where, perturbatively in the $1/N$ expansion, the operator scaling dimensions are real and above the unitarity bound. Here, we further show that these scaling dimensions do acquire small imaginary parts due to the instanton effects. In $d$ dimensions and for large $N$, we find that they are of order $e^{-N f(d)}$, where, remarkably, the function $f(d)$ equals the sphere free energy of a conformal scalar in $d-2$ dimensions. The non-perturbatively small imaginary parts also appear in other observables, such as the sphere free energy and two and three-point function coefficients, and we present some of their calculations. Therefore, at sufficiently large $N$, the $O(N)$ models in $4<d<6$ may be thought of as complex CFTs. When $N$ is large enough for the imaginary parts to be numerically negligible, the five-dimensional $O(N)$ models may be studied using the techniques of numerical bootstrap.

hep-th↗

arXiv2020

Fractional Chern Insulator States in Twisted Bilayer Graphene: An Analytical Approach

Recent experiments on bilayer graphene twisted near the magic angle have observed spontaneous integer quantum Hall states in the presence of an aligned hexagonal boron nitride (hBN) substrate. These states arise from valley ferromagnetism, and the complete filling of Chern bands. A natural question is whether fractional filling of the same bands would lead to fractional quantum Hall states, i.e. to fractional Chern insulators (FCIs). Here, we argue that the magic angle graphene bands have favorable quantum geometry for realizing FCI phases. We show that in the tractable `chiral' limit, the flat bands wavefunctions are an analytic function of the crystal momentum. This remarkable property fixes the quantum metric up to an overall momentum dependent scale factor, the local Berry curvature, whose variation is itself small. Thus the three conditions associated with FCI stability (i) narrow bands (ii) quantum metric satisfying the isotropic ideal droplet condition and (iii) relatively uniform Berry curvature are all satisfied. Our work emphasizes continuum real space approaches to FCIs in contrast to earlier works which mostly focused on tight binding models. This enables us to construct a Laughlin wavefunction on a real-space torus that is a zero energy ground state of pseudopotential-like interactions. The latter can be realized for Coulomb interaction in the limit of very short screening length. We also map the problem onto an equivalent one of Dirac particles in an inhomogeneous but periodic magnetic field and no crystal potential. Finally we discuss evolution of the band geometry on tuning away from the chiral limit and show numerically that some of the desirable properties continue to hold at a quantitative level for realistic parameter values.

cond-mat.str-el↗

arXiv2020

Metal-insulator transition in a random Hubbard model

We examine the metal-insulator transition in a half-filled Hubbard model of electrons with random and all-to-all hopping and exchange, and an on-site non-random repulsion, the Hubbard $U$. We argue that recent numerical results of Cha et al. (arXiv:2002.07181) can be understood in terms of a deconfined critical point between a disordered Fermi liquid and an insulating spin glass. We find a deconfined critical point in a previously proposed large $M$ theory which generalizes the SU(2) spin symmetry to SU($M$), and obtain exponents for the electron and spin correlators which agree with those of Cha et al. We also present a renormalization group analysis, and argue for the presence of an additional metallic spin glass phase at half-filling and small $U$.

cond-mat.str-el↗

arXiv2020

Deconfined critical point in a doped random quantum Heisenberg magnet

We describe the phase diagram of electrons on a fully connected lattice with random hopping, subject to a random Heisenberg spin exchange interactions between any pair of sites and a constraint of no double occupancy. A perturbative renormalization group analysis yields a critical point with fractionalized excitations at a non-zero critical value $p_c$ of the hole doping $p$ away from the half-filled insulator. We compute the renormalization group to two loops, but some exponents are obtained to all loop order. We argue that the critical point $p_c$ is flanked by confining phases: a disordered Fermi liquid with carrier density $1+p$ for $p>p_c$, and a metallic spin glass with carrier density $p$ for $p<p_c$. Additional evidence for the critical behavior is obtained from a large $M$ analysis of a model which extends the SU(2) spin symmetry to SU($M$). We discuss the relationship of the vicinity of this deconfined quantum critical point to key aspects of cuprate phenomenology.

cond-mat.str-el↗