arXiv subjects

Edwin Langmann

Publications and source records attributed to Edwin Langmann.

At least 19 recordsLinked to original sources

arXiv2025

On the Hartree-Fock phase diagram for the two-dimensional Hubbard model

We propose an analytical method for the construction of Hartree-Fock phase diagrams for the (fermion) Hubbard model and various generalizations thereof. Such phase diagrams are traditionally constructed numerically, but we argue that, by using asymptotic techniques, it is possible to obtain analytic formulas approximating the curves separating the different phases to very high accuracy. To illustrate the new method, we apply it to the two-dimensional Hubbard model on the square lattice at zero temperature. This yields formulas for the Hartree-Fock phase boundaries that agree with, but also improve on, earlier numerical results. In particular, our results provide the first rigorous proof of the existence of mixed phases in this model.

math-ph↗

arXiv2025

Elliptic Calogero-Sutherland model and conformal field theory

In a project with Gordon Semenoff on 1+1 dimensional QCD many years ago (when he was my postdoc advisor), we stumbled over a method to solve Calogero-Moser-Sutherland models using gauge theories. Since then, these models have reappeared in different forms in many of my research projects. In this contribution, I describe a recent such project where a second quantization of the elliptic Calogero-Sutherland model led us to a new soliton equation and a non-relativistic variant of the Coleman correspondence. (Work with Bjorn Berntson and Jonatan Lenells.)

math-ph↗

arXiv2025

On the mean-field antiferromagnetic gap for the half-filled 2D Hubbard model at zero temperature

We consider the antiferromagnetic gap for the half-filled two-dimensional (2D) Hubbard model (on a square lattice) at zero temperature in Hartree-Fock theory. It was conjectured by Hirsch in 1985 that this gap, $\Delta$, vanishes like $\exp(-2\pi\sqrt{t/U})$ in the weak-coupling limit $U/t\downarrow 0$ ($U>0$ and $t>0$ are the usual Hubbard model parameters). We give a proof of this conjecture based on recent mathematical results about Hartree-Fock theory for the 2D Hubbard model. The key step is the exact computation of an integral involving the density of states of the 2D tight binding band relation.

math-ph↗

arXiv2024

Multi-band superconductors have enhanced critical temperatures

We introduce a multi-band BCS free energy functional and prove that for a multi-band superconductor the effect of inter-band coupling can only increase the critical temperature, irrespective of its attractive or repulsive nature and its strength. Further, for weak coupling and weaker inter-band coupling, we prove that the dependence of the increase in critical temperature on the inter-band coupling is (1) linear, if there are two or more equally strongly superconducting bands, or (2) quadratic, if there is only one dominating band.

math-ph↗

arXiv2024

Elliptic Integrable Systems and Special Functions

We discuss elliptic quantum Calogero-Moser-Sutherland models, including their relativistic generalizations due to Ruijsenaars and van Diejen, and the relations of these models to classes of special functions developed and explored in recent and on-going research efforts. We give an introduction to these models and corresponding special functions aimed at non-experts. We also describe a few open problems in the field.

math-ph↗

arXiv2024

Higher bracket structure of density operators in Weyl fermion systems and topological insulators

We study the algebraic structure of electron density operators in gapless Weyl fermion systems in $d=3,5,7,\cdots$ spatial dimensions and in topological insulators (without any protecting symmetry) in $d=4,6,8,\cdots$ spatial dimensions. These systems are closely related by the celebrated bulk-boundary correspondence. Specifically, we study the higher bracket -- a generalization of commutator for more than two operators -- of electron density operators in these systems. For topological insulators, we show that the higher-bracket algebraic structure of density operators structurally parallels with the Girvin-MacDonald-Platzman algebra (the $W_{1+\infty}$ algebra), the algebra of electron density operators projected onto the lowest Landau level in the quantum Hall effect. By the bulk-boundary correspondence, the bulk higher-bracket structure mirrors its counterparts at the boundary. Specifically, we show that the density operators of Weyl fermion systems, once normal-ordered with respect to the ground state, their higher bracket acquires a c-number part. This part is an analog of the Schwinger term in the commutator of the fermion current operators. We further identify this part with a cyclic cocycle, which is a topological invariant and an element of Connes' noncommutative geometry.

cond-mat.str-el↗

arXiv2023

Partial continuum limit of the 2D Hubbard model

An effective quantum field theory of the 2D Hubbard model on a square lattice near half-filling is presented and studied. This effective model describes so-called nodal and antinodal fermions, and it is derived from the lattice model using a certain partial continuum limit. It is shown that the nodal fermions can be bosonized, which leads to spin-charge separation and a 2D analogue of a Wess-Zumino-Witten model. A bosonization formula for the nodal fermion field operator is obtained, and an exactly solvable model of interacting 2D fermions is identified. Different ways of treating the antinodal fermions are also proposed.

cond-mat.str-el↗

arXiv2023

Universal and nonuniversal features of Bardeen-Cooper-Schrieffer theory with finite-range interactions

We study analytic solutions to the Bardeen-Cooper-Schrieffer (BCS) gap equation for isotropic superconductors with finite-range interaction potentials over the full range of temperatures from absolute zero to the superconducting critical temperature, $0\leq T\leq T_c$. Using these solutions, $\Delta(\epsilon,T)$, we provide a proof of the universality of the temperature dependence of the BCS gap ratio at the Fermi level, $\Delta(\epsilon=0,T)/T_c$. Moreover, by examining the behavior of this ratio as a function of energy, $\epsilon$, we find that non-universal features emerge away from the Fermi level, and these features take the form of a temperature independent multiplicative factor, $F(\epsilon)$, which is equal to $\Delta(\epsilon,T)/\Delta(\epsilon=0,T)$ up to exponentially small corrections, i.e., the error terms vanish like $\rm{e}^{-1/\lambda}$ in the weak-coupling limit $\lambda\rightarrow 0$. We discuss the model-dependent features of both $F(\epsilon)$ and $T_c$, and we illustrate their behavior focusing on several concrete examples of physically-relevant finite-range potentials. Comparing these cases for fixed coupling constants, we highlight the importance of the functional form of the interaction potential in determining the size of the critical temperature and provide guidelines for choosing potentials which lead to higher values of $T_c$. We also propose experimental signatures which could be used to probe the energy-dependence of the gap and potentially shed light on the underlying mechanisms giving rise to superconductivity.

cond-mat.supr-con↗

arXiv2023

Closed-form propagator of the Calogero model

We present an exact closed-form expression for the propagator of the Calogero model, i.e., for the integral kernel of the time evolution operator of the quantum many-body system on the real line with an external harmonic potential and inverse-square two-body interactions. This expression is obtained by combining two results: first, a simple formula relating this propagator to the eigenfunctions of the Calogero model without harmonic potential and second, a formula for these eigenfunctions as finite sums of products of polynomial two-body functions.

quant-ph↗

arXiv2023

Conformal field theory, solitons, and elliptic Calogero--Sutherland models

We construct a non-chiral conformal field theory (CFT) on the torus that accommodates a second quantization of the elliptic Calogero-Sutherland (eCS) model. We show that the CFT operator that provides this second quantization defines, at the same time, a quantum version of a soliton equation called the non-chiral intermediate long-wave (ncILW) equation. We also show that this CFT operator is a second quantization of a generalized eCS model which can describe arbitrary numbers of four different kinds of particles; we propose that these particles can be identified with solitons of the quantum ncILW equation.

math-ph↗

arXiv2022

Elliptic soliton solutions of the spin non-chiral intermediate long-wave equation

We construct elliptic multi-soliton solutions of the spin non-chiral intermediate long-wave (sncILW) equation with periodic boundary conditions. These solutions are obtained by a spin-pole ansatz including a dynamical background term; we show that this ansatz solves the periodic sncILW equation provided the spins and poles satisfy the elliptic $A$-type spin Calogero-Moser (sCM) system with certain constraints on the initial conditions. The key to this result is a B\"{a}cklund transformation for the elliptic sCM system which includes a non-trivial dynamical background term. We also present solutions of the sncILW equation on the real line and of the spin Benjamin-Ono equation which generalize previously obtained solutions by allowing for a non-trivial background term.

math-ph↗

arXiv2022

Obstructions to odd-frequency superconductivity in Eliashberg theory

We present a necessary condition for odd-frequency (odd-f) superconductivity (SC) to occur in a large class of materials described by Eliashberg theory. We use this condition to prove a no-go theorem ruling out the occurrence of odd-f SC in standard one-band superconductors with pairing interactions mediated by phonon exchange. We also present a corresponding no-go theorem for superconductors with interactions mediated by spin-fluctuations. Our results explain why odd-f SC is rare in conventional materials, and they open up the possibility for a search for materials with interactions designed so as to allow for odd-f SC.

cond-mat.supr-con↗

arXiv2022

Spin generalizations of the Benjamin-Ono equation

We present new soliton equations related to the $A$-type spin Calogero-Moser (CM) systems introduced by Gibbons and Hermsen. These equations are spin generalizations of the Benjamin-Ono (BO) equation and the recently introduced non-chiral intermediate long-wave (ncILW) equation. We obtain multi-soliton solutions of these spin generalizations of the BO equation and the ncILW equation via a spin-pole ansatz where the spin-pole dynamics is governed by the spin CM system in the rational and hyperbolic cases, respectively. We also propose physics applications of the new equations, and we introduce a spin generalization of the standard intermediate long-wave equation which interpolates between the matrix Korteweg-de Vries equation, the Heisenberg ferromagnet equation, and the spin BO equation.

math-ph↗

arXiv2021

The non-chiral intermediate Heisenberg ferromagnet equation

We present and solve a soliton equation which we call the non-chiral intermediate Heisenberg ferromagnet (ncIHF) equation. This equation, which depends on a parameter $\delta >0$, describes the time evolution of two coupled spin densities propagating on the real line, and in the limit $\delta \to \infty$ it reduces to two decoupled half-wave maps (HWM) equations of opposite chirality. We show that the ncIHF equation is related to the A-type hyperbolic spin Calogero-Moser (CM) system in two distinct ways: (i) it is obtained as a particular continuum limit of a Inozemtsev-type spin chain related to this CM system, (ii) it has multi-soliton solutions obtained by a spin-pole ansatz and with parameters satisfying the equations of motion of a complexified version of this CM system. The integrability of the ncIHF equation is shown by constructing a Lax pair. We also propose a periodic variant of the ncIHF equation related to the A-type elliptic spin CM system.

math-ph↗

arXiv2021

Higher order deformed elliptic Ruijsenaars operators

We present four infinite families of mutually commuting difference operators which include the deformed elliptic Ruijsenaars operators. The trigonometric limit of this kind of operators was previously introduced by Feigin and Silantyev. They provide a quantum mechanical description of two kinds of relativistic quantum mechanical particles which can be identified with particles and anti-particles in an underlying quantum field theory. We give direct proofs of the commutativity of our operators and of some other fundamental properties such as kernel function identities. In particular, we give a rigorous proof of the quantum integrability of the deformed Ruijsenaars model.

math-ph↗

arXiv2021

From Kajihara's transformation formula to deformed Macdonald-Ruijsenaars and Noumi-Sano operators

Kajihara obtained in 2004 a remarkable transformation formula connecting multiple basic hypergeometric series associated with $A$-type root systems of different ranks. By multiple principle specialisations of his formula, we deduce kernel identities for deformed Macdonald-Ruijsenaars (MR) and Noumi-Sano (NS) operators. The deformed MR operators were introduced by Sergeev and Veselov in the first order case and by Feigin and Silantyev in the higher order cases. As applications of our kernel identities, we prove that all of these operators pairwise commute and are simultaneously diagonalised by the super-Macdonald polynomials. We also provide an explicit description of the algebra generated by the deformed MR and/or NS operators by a Harish-Chandra type isomorphism and show that the deformed MR (NS) operators can be viewed as restrictions of inverse limits of ordinary MR (NS) operators.

math.QA↗

arXiv2021

Super-Macdonald polynomials: Orthogonality and Hilbert space interpretation

The super-Macdonald polynomials, introduced by Sergeev and Veselov, generalise the Macdonald polynomials to (arbitrary numbers of) two kinds of variables, and they are eigenfunctions of the deformed Macdonald-Ruijsenaars operators introduced by the same authors. We introduce a Hermitian form on the algebra spanned by the super-Macdonald polynomials, prove their orthogonality, compute their (quadratic) norms explicitly, and establish a corresponding Hilbert space interpretation of the super-Macdonald polynomials and deformed Macdonald-Ruijsenaars operators. This allows for a quantum mechanical interpretation of the models defined by the deformed Macdonald-Ruijsenaars operators. Motivated by recent results in the nonrelativistic ($q\to 1$) case, we propose that these models describe the particles and anti-particles of an underlying relativistic quantum field theory, thus providing a natural generalisation of the trigonometric Ruijsenaars model.

math.QA↗

arXiv2021

On the non-chiral intermediate long wave equation II: periodic case

We study integrability properties of the non-chiral intermediate long wave (ncILW) equation with periodic boundary conditions. The ncILW equation was recently introduced by the authors as a parity-invariant relative of the intermediate long wave equation. For this new equation we: (a) derive a Lax pair, (b) derive a Hirota bilinear form, (c) use the Hirota method to construct the periodic multi-soliton solutions, (d) derive a B\"{a}cklund transformation, (e) use the B\"{a}cklund transformation to obtain an infinite number of conservation laws.

nlin.SI↗