[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: We consider a 1-parameter family of self-adjoint extensions of the
Hamiltonian for a particle confined to a finite interval with perfectly
reflecting boundary conditions. In some cases, one obtains negative energy
states which seems to violate the Heisenberg uncertainty relation. We use this
as a motivation to derive a generalized uncertainty relation valid for an
arbitrarily shaped quantum dot with general perfectly reflecting walls in $d$
dimensions. In addition, a general uncertainty relation for non-Hermitean
operators is derived and applied to the non-Hermitean momentum operator in a
quantum dot. We also consider minimal uncertainty wave packets in this
situation, and we prove that the spectrum depends monotonically on the
self-adjoint extension parameter. In addition, we construct the most general
boundary conditions for semiconductor heterostructures such as quantum dots,
quantum wires, and quantum wells, which are characterized by a 4-parameter
family of self-adjoint extensions. Finally, we consider perfectly reflecting
boundary conditions for relativistic fermions confined to a finite volume or
localized on a domain wall, which are characterized by a 1-parameter family of
self-adjoint extensions in the $(1+1)$-d and $(2+1)$-d cases, and by a
4-parameter family in the $(3+1)$-d and $(4+1)$-d cases.
Annals of Physics 05/2011; 327(1). DOI:10.1016/j.aop.2011.05.003 · 2.10 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The two-dimensional (2D) 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 Ms=0.307 43(1)/a2, the spin stiffness ρs=0.180 81(11)J, and the spin wave velocity c=1.6586(3)Ja, are determined with very high precision. Our study may serve as a test case for the comparison of lattice quantum chromodynamics 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.
[Show abstract][Hide abstract] ABSTRACT: Using an improved estimator in the loop-cluster algorithm, we investigate the
constraint effective potential of the magnetization in the spin $\tfrac{1}{2}$
quantum XY model. The numerical results are in excellent agreement with the
predictions of the corresponding low-energy effective field theory. After its
low-energy parameters have been determined with better than permille precision,
the effective theory makes accurate predictions for the constraint effective
potential which are in excellent agreement with the Monte Carlo data. This
shows that the effective theory indeed describes the physics in the low-energy
regime quantitatively correctly.
Journal of Statistical Mechanics Theory and Experiment 02/2011; 6(06). DOI:10.1088/1742-5468/2011/06/P06002 · 2.40 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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.
Journal of High Energy Physics 10/2010; 2011(1). DOI:10.1007/JHEP01(2011)057 · 6.11 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We consider lattice field theories with topological actions, which are invariant against small deformations of the fields. Some of these actions have infinite barriers separating different topological sectors. Topological actions do not have the correct classical continuum limit and they cannot be treated using perturbation theory, but they still yield the correct quantum continuum limit. To show this, we present analytic studies of the 1-d O(2) and O(3) model, as well as Monte Carlo simulations of the 2-d O(3) model using topological lattice actions. Some topological actions obey and others violate a lattice Schwarz inequality between the action and the topological charge Q. Irrespective of this, in the 2-d O(3) model the topological susceptibility \chi_t = \l< Q^2 >/V is logarithmically divergent in the continuum limit. Still, at non-zero distance the correlator of the topological charge density has a finite continuum limit which is consistent with analytic predictions. Our study shows explicitly that some classically important features of an action are irrelevant for reaching the correct quantum continuum limit. Comment: 37 pages, 11 figures
Journal of High Energy Physics 09/2010; 2010(12). DOI:10.1007/JHEP12(2010)020 · 6.11 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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/\beta, for r >> \beta, the
string width diverges linearly.
Journal of High Energy Physics 06/2010; 2010(11). DOI:10.1007/JHEP11(2010)053 · 6.11 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: We investigate the stability of strings connecting charges Q in the representation {2Q+1} of SU(2) Yang-Mills theory in (2+1) dimensions. While the fundamental {2}-string between two charges Q=1/2 is unbreakable and stable, the string connecting static charges transforming under any other representation Q>1/2 is unstable and decays. A charge Q=1 can be completely screened by gluons and so the adjoint {3}-string ultimately breaks. A charge Q=3/2 can be only partially screened to a fundamental charge Q=1/2. Thus, stretching a {4}-string beyond a critical length, it decays into the stable {2}-string by gluon pair creation. The complete breaking of a {5}-string happens in two steps, it first decays into a {3}-string and then breaks completely. A phenomenological constituent gluon model provides a good quantitative description of the energy of the screened charges at the ends of an unstable string. Comment: 7 pages, 2 figures, contribution to The XXVII International Symposium on Lattice Field Theory, July 26-31, 2009, Peking University, Beijing, China
[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: We consider wave packets of free particles with a general energy-momentum
dispersion relation $E(p)$. The spreading of the wave packet is determined by
the velocity $v = \p_p E$. The position-velocity uncertainty relation $\Delta x
\Delta v \geq {1/2} |< \p_p^2 E >|$ is saturated by minimal uncertainty wave
packets $\Phi(p) = A \exp(- \alpha E(p) + \beta p)$. In addition to the
standard minimal Gaussian wave packets corresponding to the non-relativistic
dispersion relation $E(p) = p^2/2m$, analytic calculations are presented for
the spreading of wave packets with minimal position-velocity uncertainty
product for the lattice dispersion relation $E(p) = - \cos(p a)/m a^2$ as well
as for the relativistic dispersion relation $E(p) = \sqrt{p^2 + m^2}$. The
boost properties of moving relativistic wave packets as well as the propagation
of wave packets in an expanding Universe are also discussed.
Annals of Physics 07/2009; 324(12). DOI:10.1016/j.aop.2009.09.001 · 2.10 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: It is shown that baryon chiral perturbation theory, i.e., the low‐energy effective theory for pions and nucleons in quantum chromodynamics, has its condensed matter analog: A low‐energy effective theory describing magnons as well as holes (or electrons) doped into antiferromagnets. We briefly present a symmetry analysis of the Hubbard and t‐J‐type models, and review the construction of the leading terms in the effective Lagrangian. As a nontrivial application we study different phases of hole‐ and electron‐doped antiferromagnets—in particular, we investigate whether a so‐called spiral phase with an inhomogeneous staggered magnetization (order parameter) may be stable. We would like to emphasize that the effective theory is universal and makes model‐independent predictions for a large class of systems, whereas the material‐specific properties enter the effective theory only through the numerical values of a few low‐energy parameters.
[Show abstract][Hide abstract] ABSTRACT: We employ an improved estimator to calculate the constraint effective potential of the staggered magnetization in the spin $\tfrac{1}{2}$ quantum Heisenberg model using a loop-cluster algorithm. The first and second moment of the probability distribution of the staggered magnetization are in excellent agreement with the predictions of the systematic low-energy magnon effective field theory. We also compare the Monte Carlo data with the universal shape of the constraint effective potential of the staggered magnetization and study its approach to the convex effective potential in the infinite volume limit. In this way the higher-order low-energy parameter $k_0$ is determined from a fit to the numerical data.
Journal of Statistical Mechanics Theory and Experiment 01/2009; 2009(3). DOI:10.1088/1742-5468/2009/03/P03021 · 2.40 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: Inspired by the unhydrated variant of the superconducting material NaxCoO2⋅yH2O 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 M̃s=0.2688(3), the spin stiffness ρs=0.102(2)J, the spin-wave velocity c=1.297(16)Ja, and the kinetic mass M′ of a hole are obtained by fitting the numerical Monte Carlo data to the effective field theory predictions.
Physical Review B 07/2008; 78(21). DOI:10.1103/PhysRevB.78.214406 · 3.74 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: A classical particle in a constant magnetic field undergoes cyclotron motion
on a circular orbit. At the quantum level, the fact that all classical orbits
are closed gives rise to degeneracies in the spectrum. It is well-known that
the spectrum of a charged particle in a constant magnetic field consists of
infinitely degenerate Landau levels. Just as for the $1/r$ and $r^2$
potentials, one thus expects some hidden accidental symmetry, in this case with
infinite-dimensional representations. Indeed, the position of the center of the
cyclotron circle plays the role of a Runge-Lenz vector. After identifying the
corresponding accidental symmetry algebra, we re-analyze the system in a finite
periodic volume. Interestingly, similar to the quantum mechanical breaking of
CP invariance due to the $\theta$-vacuum angle in non-Abelian gauge theories,
quantum effects due to two self-adjoint extension parameters $\theta_x$ and
$\theta_y$ explicitly break the continuous translation invariance of the
classical theory. This reduces the symmetry to a discrete magnetic translation
group and leads to finite degeneracy. Similar to a particle moving on a cone, a
particle in a constant magnetic field shows a very peculiar realization of
accidental symmetry in quantum mechanics.
Annals of Physics 07/2008; 324(2). DOI:10.1016/j.aop.2008.07.006 · 2.10 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Using a loop-cluster algorithm we investigate the spin 1/2 Heisenberg antiferromagnet on a square lattice with exchange coupling $J$ and an additional four-spin interaction of strength $Q$. We confirm the existence of a phase transition separating antiferromagnetism at $J/Q > J_c/Q$ from a valence bond solid (VBS) state at $J/Q < J_c/Q$. Although our Monte Carlo data are consistent with those of previous studies, we do not confirm the existence of a deconfined quantum critical point. Instead, using a flowgram method on lattices as large as $80^2$, we find evidence for a weak first order phase transition. We also present a detailed study of the antiferromagnetic phase. For $J/Q > J_c/Q$ the staggered magnetization, the spin stiffness, and the spinwave velocity of the antiferromagnet are determined by fitting Monte Carlo data to analytic results from the systematic low-energy effective field theory for magnons. Finally, we also investigate the physics of the VBS state at $J/Q < J_c/Q$, and we show that long but finite antiferromagnetic correlations are still present. Comment: 21 pages, 10 figures
Journal of Statistical Mechanics Theory and Experiment 10/2007; 2008(02). DOI:10.1088/1742-5468/2008/02/P02009 · 2.40 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We consider a particle moving on a cone and bound to its tip by 1/r or harmonic oscillator potentials. When the deficit angle of the cone divided by 2π is a rational number, all bound classical orbits are closed. Correspondingly, the quantum system has accidental degeneracies in the discrete energy spectrum. An accidental SU(2) symmetry is generated by the rotations around the tip of the cone as well as by a Runge–Lenz vector. Remarkably, some of the corresponding multiplets have fractional “spin” and unusual degeneracies.
Annals of Physics 07/2007; 323(1-323):82-104. DOI:10.1016/j.aop.2007.08.004 · 2.10 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We have constructed a systematic low-energy effective theory for hole- and electron-doped antiferromagnets, where holes reside in momentum space pockets centered at $(\pm\frac{\pi}{2a},\pm\frac{\pi}{2a})$ and where electrons live in pockets centered at $(\frac{\pi}{a},0)$ or $(0,\frac{\pi}{a})$. The effective theory is used to investigate the magnon-mediated binding between two holes or two electrons in an otherwise undoped system. We derive the one-magnon exchange potential from the effective theory and then solve the corresponding two-quasiparticle Schr\"odinger equation. As a result, we find bound state wave functions that resemble $d_{x^2-y^2}$-like or $d_{xy}$-like symmetry. We also study possible ground states of lightly doped antiferromagnets.