Systematic Low-energy Effective Field Theory Research Articles

Finite-volume energy spectrum, fractionalized strings, and low-energy effective field theory for the quantum dimer model on the square lattice

We present detailed analytic calculations of finite-volume energy spectra, mean field theory, as well as a systematic low-energy effective field theory for the square lattice quantum dimer model. The analytic considerations explain why a string connecting two external static charges in the confining columnar phase fractionalizes into eight distinct strands with electric flux $\frac{1}{4}$. An emergent approximate spontaneously broken $SO(2)$ symmetry gives rise to a pseudo-Goldstone boson. Remarkably, this soft phonon-like excitation, which is massless at the Rokhsar-Kivelson (RK) point, exists far beyond this point. The Goldstone physics is captured by a systematic low-energy effective field theory. We determine its low-energy parameters by matching the analytic effective field theory with exact diagonalization results and Monte Carlo data. This confirms that the model exists in the columnar (and not in a plaquette or mixed) phase all the way to the RK point.

Baryon chiral perturbation theory transferred to hole-doped antiferromagnets on the honeycomb lattice

A systematic low-energy effective field theory for hole-doped antiferromagnets on the honeycomb lattice is constructed. The formalism is then used to investigate spiral phases in the staggered magnetization as well as the formation of two-hole bound states.

Systematic low-energy effective field theory for magnons and holes in an antiferromagnet on the honeycomb lattice

Based on a symmetry analysis of the microscopic Hubbard and t-J models, a systematic low-energy effective field theory is constructed for hole-doped antiferromagnets on the honeycomb lattice. In the antiferromagnetic phase, doped holes are massive due to the spontaneous breakdown of the $SU(2)_s$ symmetry, just as nucleons in QCD pick up their mass from spontaneous chiral symmetry breaking. In the broken phase the effective action contains a single-derivative term, similar to the Shraiman-Siggia term in the square lattice case. Interestingly, an accidental continuous spatial rotation symmetry arises at leading order. As an application of the effective field theory we consider one-magnon exchange between two holes and the formation of two-hole bound states. As an unambiguous prediction of the effective theory, the wave function for the ground state of two holes bound by magnon exchange exhibits $f$-wave symmetry.

The confining string: Lattice results versus effective field theory

In analogy to the pions in chiral perturbation theory, the transverse fluctuations of a confining string in Yang–Mills theory are described by a systematic low-energy effective field theory. The predictions of the effective theory at the two-loop level are in quantitative agreement with high-precision Monte Carlo data obtained from lattice gauge theory. The decay and ultimate breaking of strings connecting charges in higher-dimensional representations of the gauge group are also investigated.

High-precision determination of low-energy effective parameters for a two-dimensional Heisenberg quantum antiferromagnet

The 2-d spin 1/2 Heisenberg antiferromagnet with exchange coupling $J$ is investigated on a periodic square lattice of spacing $a$ at very small temperatures using the loop-cluster algorithm. Monte Carlo data for the staggered and uniform susceptibilities are compared with analytic results obtained in the systematic low-energy effective field theory for the staggered magnetization order parameter. The low-energy parameters of the effective theory, i.e.\ the staggered magnetization density ${\cal M}_s = 0.30743(1)/a^2$, the spin stiffness $\rho_s = 0.18081(11) J$, and the spin wave velocity $c = 1.6586(3) J a$ are determined with very high precision. Our study may serve as a test case for the comparison of lattice QCD Monte Carlo data with analytic predictions of the chiral effective theory for pions and nucleons, which is vital for the quantitative understanding of the strong interaction at low energies.

Linear broadening of the confining string in Yang-Mills theory at low temperature

The logarithmic broadening predicted by the systematic low-energy effective field theory for the confining string has recently been verified in numerical simulations of (2 + 1)-d SU(2) lattice Yang-Mills theory at zero temperature. The same effective theory predicts linear broadening of the string at low non-zero temperature. In this paper, we verify this prediction by comparison with very precise Monte Carlo data. The comparison involves no additional adjustable parameters, because the low-energy constants of the effective theory have already been fixed at zero temperature. It yields very good agreement between the underlying Yang-Mills theory and the effective string theory.

The width of the color flux tube at 2-loop order

The color flux tube connecting a static quark-anti-quark pair in Yang-Mills theory supports massless transverse fluctuations, which are the Goldstone bosons of spontaneously broken translation invariance. Just as in chiral perturbation theory, the dynamics of these Goldstone bosons is described by a systematic low-energy effective field theory. We use the effective theory to calculate the width of the fluctuating string at the 2-loop level, using both cylindrical and toroidal boundary conditions. At zero temperature, the string width diverges logarithmically with the quark-anti-quark distance r. On the other hand, at low but non-zero temperature T = 1/β, for r ≫ β the string width diverges linearly.

Microscopic model versus systematic low-energy effective field theory for a doped quantum ferromagnet

We consider a microscopic model for a doped quantum ferromagnet as a test case for the systematic low-energy effective field theory for magnons and holes, which is constructed in complete analogy to the case of quantum antiferromagnets. In contrast to antiferromagnets, for which the effective field theory approach can be tested only numerically, in the ferromagnetic case, both the microscopic and the effective theory can be solved analytically. In this way, the low-energy parameters of the effective theory are determined exactly by matching to the underlying microscopic model. The low-energy behavior at half-filling as well as in the single- and two-hole sectors is described exactly by the systematic low-energy effective field theory. In particular, for weakly bound two-hole states the effective field theory even works beyond perturbation theory. This lends strong support to the quantitative success of the systematic low-energy effective field theory method not only in the ferromagnetic but also in the physically most interesting antiferromagnetic case.

Systematic effective field theory investigation of spiral phases in hole-doped antiferromagnets on the honeycomb lattice

Motivated by possible applications to the antiferromagnetic precursor of the high-temperature superconductor Na$_x$CoO$_2\cdot$yH$_2$O, we use a systematic low-energy effective field theory for magnons and holes to study different phases of doped antiferromagnets on the honeycomb lattice. The effective action contains a leading single-derivative term, similar to the Shraiman-Siggia term in the square lattice case, which gives rise to spirals in the staggered magnetization. Depending on the values of the low-energy parameters, either a homogeneous phase with four or a spiral phase with two filled hole pockets is energetically favored. Unlike in the square lattice case, at leading order the effective action has an accidental continuous spatial rotation symmetry. Consequently, the spiral may point in any direction and is not necessarily aligned with a lattice direction.

The constraint effective potential of the staggered magnetization in an antiferromagnet

We employ an improved estimator to calculate the constraint effective potential of thestaggered magnetization in the spin quantum Heisenberg model using a loop-cluster algorithm. The first and second momentof the probability distribution of the staggered magnetization are in excellent agreementwith the predictions from the systematic low-energy magnon effective field theory. We alsocompare the Monte Carlo data with the universal shape of the constraint effective potentialof the staggered magnetization and study its approach to the convex effective potentialin the infinite volume limit. In this way the higher-order low-energy parameterk0 is determined from a fit to the numerical data.

Loop-cluster simulation of the zero- and one-hole sectors of thet−Jmodel on the honeycomb lattice

Inspired by the lattice structure of the unhydrated variant of the superconducting material Na$_x$CoO$_2 \cdot$yH$_2$O at $ x = {1/3}$, we study the $t$-$J$ model on a honeycomb lattice by using an efficient loop-cluster algorithm. The low-energy physics of the undoped system and of the single hole sector is described by a systematic low-energy effective field theory. The staggered magnetization per spin $\widetilde{{\cal M}}_s = 0.2688(3)$, the spin stiffness $\rho_s = 0.102(2) J$, the spin wave velocity $c= 1.297(16) J a$, and the kinetic mass $M'$ of a hole are obtained by fitting the numerical Monte Carlo data to the effective theory predictions.

Systematic low-energy effective field theory for electron-doped antiferromagnets

In contrast to hole-doped systems which have hole pockets centered at $(\ifmmode\pm\else\textpm\fi{}\frac{\ensuremath{\pi}}{2a},\ifmmode\pm\else\textpm\fi{}\frac{\ensuremath{\pi}}{2a})$, in lightly electron-doped antiferromagnets the charged quasiparticles reside in momentum space pockets centered at $(\frac{\ensuremath{\pi}}{a},0)$ or $(0,\frac{\ensuremath{\pi}}{a})$. This has important consequences for the corresponding low-energy effective field theory of magnons and electrons which is constructed in this paper. In particular, in contrast to the hole-doped case, the magnon-mediated forces between two electrons depend on the total momentum $\stackrel{P\vec}{P}$ of the pair. For $\stackrel{P\vec}{P}=0$, the one-magnon exchange potential between two electrons at distance $r$ is proportional to $1∕{r}^{4}$, while in the hole case, it has a $1∕{r}^{2}$ dependence. The effective theory predicts that spiral phases are absent in electron-doped antiferromagnets.

From the Hubbard model to a systematic low-energy effective field theory for magnons and holes in an antiferromagnet

The low-energy physics of antiferromagnets is governed by their Goldstone bosons – the magnons – and it is described by a low-energy effective field theory. In analogy to baryon chiral perturbation theory, we construct the effective field theory for magnons and holes in an antiferromagnet. It is a systematic low-energy expansion based on symmetry considerations and on the fact that the holes are located in pockets centered at k = ( π 2 a , ± π 2 a ) . Even though the symmetries are extracted from the Hubbard model, the effective theory is universal and makes model-independent predictions about the dynamical mechanisms in the antiferromagnetic phase. The low-energy effective theory has been used to investigate one-magnon exchange which leads to a d-wave-shaped bound state of holes.

Two-hole bound states from a systematic low-energy effective field theory for magnons and holes in an antiferromagnet

Identifying the correct low-energy effective theory for magnons and holes in an antiferromagnet has remained an open problem for a long time. In analogy to the effective theory for pions and nucleons in QCD, based on a symmetry analysis of Hubbard and $t\text{\ensuremath{-}}J$-type models, we construct a systematic low-energy effective field theory for magnons and holes located inside pockets centered at lattice momenta $(\ifmmode\pm\else\textpm\fi{}\frac{\ensuremath{\pi}}{2a},\ifmmode\pm\else\textpm\fi{}\frac{\ensuremath{\pi}}{2a})$. The effective theory is based on a nonlinear realization of the spontaneously broken spin symmetry and makes model-independent universal predictions for the entire class of lightly doped antiferromagnetic precursors of high-temperature superconductors. The predictions of the effective theory are exact, order by order in a systematic low-energy expansion. We derive the one-magnon exchange potentials between two holes in an otherwise undoped system. Remarkably, in some cases the corresponding two-hole Schr\odinger equations can even be solved analytically. The resulting bound states have $d$-wave characteristics. The ground state wave function of two holes residing in different hole pockets has a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-like symmetry, while for two holes in the same pocket the symmetry resembles ${d}_{xy}$.