Most Downloaded Annals of Physics Articles

The density-matrix renormalization group in the age of matrix product states

January 2011
Ulrich Schollwöck

The density-matrix renormalization group method (DMRG) has established itself over the last decade as the leading method for the simulation of the statics and dynamics of one-dimensional strongly correlated quantum lattice systems. In the further development of the method, the realization that DMRG operates on a highly interesting class of quantum states, so-called matrix product states (MPS), has allowed a much deeper understanding of the inner structure of the DMRG method, its further potential and its limitations. In this paper, I want to give a detailed exposition of current DMRG thinking in the MPS language in order to make the advisable implementation of the family of DMRG algorithms in exclusively MPS terms transparent. I then move on to discuss some directions of potentially fruitful further algorithmic development: while DMRG is a very mature method by now, I still see potential for further improvements, as exemplified by a number of recently introduced algorithms.

The role of gauge symmetry in spintronics

December 2011
R.F. Sobreiro | V.J. Vasquez Otoya

In this work we employ a field theoretical approach to explain the nature of the non-conserved spin current in spintronics. In particular, we consider the usual U(1) gauge theory for the electromagnetism at classical level in order to obtain the broken continuity equation involving the spin current and spin-transfer torque. Inspired by the recent work of A. Vernes, B. L. Gyorffy and P. Weinberger where they obtain such an equation in terms of relativistic quantum mechanics, we formalize their result in terms of the well known currents of field theory such as the Bargmann–Wigner current and the chiral current. Thus, an interpretation of spintronics is provided in terms of Noether currents (conserved or not) and symmetries of the electromagnetism. In fact, the main result of the present work is that the non-conservation of the spin current is associated with the gauge invariance of physical observables where the breaking term is proportional to the chiral current. Moreover, we generalize their result by including the electromagnetic field as a dynamical field instead of an external one.

Anyons in an exactly solved model and beyond

January 2006
Alexei Kitaev

A spin-1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are non-Abelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and non-Abelian phases of the original model correspond to ν=0 and ν=±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.

Fault-tolerant quantum computation by anyons

January 2003
A.Yu. Kitaev

A two-dimensional quantum system with anyonic excitations can be considered as a quantum computer. Unitary transformations can be performed by moving the excitations around each other. Measurements can be performed by joining excitations in pairs and observing the result of fusion. Such computation is fault-tolerant by its physical nature.

Electric dipole moments as probes of new physics

July 2005
Maxim Pospelov | Adam Ritz

We review several aspects of flavour-diagonal CP-violation, focussing on the role played by the electric dipole moments (EDMs) of leptons, nucleons, atoms, and molecules, which constitute the source of several stringent constraints on new CP-violating physics. We dwell specifically on the calculational aspects of applying the hadronic EDM constraints, reviewing in detail the application of QCD sum-rules to the calculation of nucleon EDMs and CP-odd pion–nucleon couplings. We also consider the current status of EDMs in the Standard Model, and on the ensuing constraints on the underlying sources of CP-violation in physics beyond the Standard Model, focussing on weak-scale supersymmetry.

Phase space representation of quantum dynamics

August 2010
Anatoli Polkovnikov

We discuss a phase space representation of quantum dynamics of systems with many degrees of freedom. This representation is based on a perturbative expansion in quantum fluctuations around one of the classical limits. We explicitly analyze expansions around three such limits: (i) corpuscular or Newtonian limit in the coordinate–momentum representation, (ii) wave or Gross–Pitaevskii limit for interacting bosons in the coherent state representation, and (iii) Bloch limit for the spin systems. We discuss both the semiclassical (truncated Wigner) approximation and further quantum corrections appearing in the form of either stochastic quantum jumps along the classical trajectories or the nonlinear response to such jumps. We also discuss how quantum jumps naturally emerge in the analysis of non-equal time correlation functions. This representation of quantum dynamics is closely related to the phase space methods based on the Wigner–Weyl quantization and to the Keldysh technique. We show how such concepts as the Wigner function, Weyl symbol, Moyal product, Bopp operators, and others automatically emerge from the Feynmann's path integral representation of the evolution in the Heisenberg representation. We illustrate the applicability of this expansion with various examples mostly in the context of cold atom systems including sine-Gordon model, one- and two-dimensional Bose–Hubbard model, Dicke model and others.

Effective equilibrium theory of nonequilibrium quantum transport

December 2011
Prasenjit Dutt | Jens Koch | Jong Han | Karyn Le Hur

The theoretical description of strongly correlated quantum systems out of equilibrium presents several challenges and a number of open questions persist. Here, we focus on nonlinear electronic transport through an interacting quantum dot maintained at finite bias using a concept introduced by Hershfield [S. Hershfield, Phys. Rev. Lett. 70 2134 (1993)] whereby one can express such nonequilibrium quantum impurity models in terms of the system’s Lippmann–Schwinger operators. These scattering operators allow one to reformulate the nonequilibrium problem as an effective equilibrium problem associated with a modified Hamiltonian. In this paper, we provide a pedagogical analysis of the core concepts of the effective equilibrium theory. First, we demonstrate the equivalence between observables computed using the Schwinger–Keldysh framework and the effective equilibrium approach, and relate Green’s functions in the two theoretical frameworks. Second, we expound some applications of this method in the context of interacting quantum impurity models. We introduce a novel framework to treat effects of interactions perturbatively while capturing the entire dependence on the bias voltage. For the sake of concreteness, we employ the Anderson model as a prototype for this scheme. Working at the particle–hole symmetric point, we investigate the fate of the Abrikosov–Suhl resonance as a function of bias voltage and magnetic field.

Energetics of a strongly correlated Fermi gas

December 2008
Shina Tan

The energy of the two-component Fermi gas with the s-wave contact interaction is a simple linear functional of its momentum distribution:Einternal=ℏ2ΩC/4πam+∑kσ(ℏ2k2/2m)(nkσ-C/k4)where the external potential energy is not included, a is the scattering length, Ω is the volume, nkσ is the average number of fermions with wave vector k and spin σ, and C≡limk→∞k4nk↑=limk→∞k4nk↓. This result is a universal identity. Its proof is facilitated by a novel mathematical idea, which might be of utility in dealing with ultraviolet divergences in quantum field theories. Other properties of this Fermi system, including pair correlations and the dimer–fermion scattering length, are also studied.

Quantization and instability of the damped harmonic oscillator subject to a time-dependent force

December 2011
H. Majima | A. Suzuki

We consider the one-dimensional motion of a particle immersed in a potential field U(x) under the influence of a frictional (dissipative) force linear in velocity (−γẋ) and a time-dependent external force (K(t)). The dissipative system subject to these forces is discussed by introducing the extended Bateman’s system, which is described by the Lagrangian: ℒ=mẋẏ−U(x+12y)+U(x−12y)+γ2(xẏ−yẋ)−xK(t)+yK(t), which leads to the familiar classical equations of motion for the dissipative (open) system. The equation for a variable y is the time-reversed of the x motion. We discuss the extended Bateman dual Lagrangian and Hamiltonian by setting U(x±y/2)=12k(x±y/2)2 specifically for a dual extended damped–amplified harmonic oscillator subject to the time-dependent external force. We show the method of quantizing such dissipative systems, namely the canonical quantization of the extended Bateman’s Hamiltonian ℋ. The Heisenberg equations of motion utilizing the quantized Hamiltonian ℋ̂ surely lead to the equations of motion for the dissipative dynamical quantum systems, which are the quantum analog of the corresponding classical systems. To discuss the stability of the quantum dissipative system due to the influence of an external force K(t) and the dissipative force, we derived a formula for transition amplitudes of the dissipative system with the help of the perturbation analysis. The formula is specifically applied for a damped–amplified harmonic oscillator subject to the impulsive force. This formula is used to study the influence of dissipation such as the instability due to the dissipative force and/or the applied impulsive force.

Two soluble models of an antiferromagnetic chain

December 1961
Elliott Lieb | Theodore Schultz | Daniel Mattis

Two genuinely quantum mechanical models for an antiferromagnetic linear chain with nearest neighbor interactions are constructed and solved exactly, in the sense that the ground state, all the elementary excitations and the free energy are found. A general formalism for calculating the instantaneous correlation between any two spins is developed and applied to the investigation of short- and long-range order. Both models show nonvanishing long-range order in the ground state for a range of values of a certain parameter λ which is analogous to an anisotropy parameter in the Heisenberg model. A detailed comparison with the Heisenberg model suggests that the latter has no long-range order in the isotropic case but finite long-range order for any finite amount of anisotropy. The unreliability of variational methods for determining long-range order is emphasized. It is also shown that for spin 12 systems having rather general isotropic Heisenberg interactions favoring an antiferromagnetic ordering, the ground state is nondegenerate and there is no energy gap above the ground state in the energy spectrum of the total system.

Efficient wireless non-radiative mid-range energy transfer

January 2008
Aristeidis Karalis | J.D. Joannopoulos | Marin Soljačić

We investigate whether, and to what extent, the physical phenomenon of long-lifetime resonant electromagnetic states with localized slowly-evanescent field patterns can be used to transfer energy efficiently over non-negligible distances, even in the presence of extraneous environmental objects. Via detailed theoretical and numerical analyses of typical real-world model-situations and realistic material parameters, we establish that such a non-radiative scheme can lead to “strong coupling” between two medium-range distant such states and thus could indeed be practical for efficient medium-range wireless energy transfer.

From a particle in a box to the uncertainty relation in a quantum dot and to reflecting walls for relativistic fermions

January 2012
M.H. Al-Hashimi | U.-J. Wiese

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 seem 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-Hermitian operators is derived and applied to the non-Hermitian 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.

Wireless adiabatic power transfer

March 2011
A.A. Rangelov | H. Suchowski | Y. Silberberg | N.V. Vitanov

We propose a technique for efficient mid-range wireless power transfer between two coils, by adapting the process of adiabatic passage for a coherently driven two-state quantum system to the realm of wireless energy transfer. The proposed technique is shown to be robust to noise, resonant constraints, and other interferences that exist in the neighborhood of the coils.

Energy bands: Chern numbers and symmetry

December 2011
T. Iwai | B. Zhilinskii

Energy bands formed by rotation–vibrational states of molecules in the presence of symmetry and their qualitative modifications under variation of some control parameters are studied within the semi-quantum model. Rotational variables are treated as classical whereas a finite set of vibrational states is considered as quantum. In the two-state approximation the system is described in terms of a fiber bundle with the base space being a two-dimensional sphere, the classical phase space for rotational variables. Generically this rank 2 complex vector bundle can be decomposed into two complex line bundles characterized by a topological invariant, the first Chern class. A general method of explicit calculation of Chern classes and of their possible modifications under variation of control parameters in the presence of symmetry is suggested. The construction of iso-Chern diagrams which split the space of control parameters into connected domains with fixed Chern numbers is suggested. A detailed analysis of the rovibrational model Hamiltonian for a D3 invariant molecule possessing two vibrational states transforming according to the two-dimensional irreducible representation is done to illustrate non-trivial restrictions imposed by symmetry on possible values of Chern classes.

Theory of carrier mediated ferromagnetism in dilute magnetic oxides

November 2007
M.J. Calderón | S. Das Sarma

We analyze the origin of ferromagnetism as a result of carrier mediation in diluted magnetic oxide semiconductors in the light of the experimental evidence reported in the literature. We propose that a combination of percolation of magnetic polarons at lower temperature and Ruderman–Kittel–Kasuya–Yosida ferromagnetism at higher temperature may be the reason for the very high critical temperatures measured (up to ∼700K).

The cold atom Hubbard toolbox

January 2005
D. Jaksch | P. Zoller

We review recent theoretical advances in cold atom physics concentrating on strongly correlated cold atoms in optical lattices. We discuss recently developed quantum optical tools for manipulating atoms and show how they can be used to realize a wide range of many body Hamiltonians. Then, we describe connections and differences to condensed matter physics and present applications in the fields of quantum computing and quantum simulations. Finally, we explain how defects and atomic quantum dots can be introduced in a controlled way in optical lattice systems.

Topological BF field theory description of topological insulators

June 2011
Gil Young Cho | Joel E. Moore

Topological phases of matter are described universally by topological field theories in the same way that symmetry-breaking phases of matter are described by Landau–Ginzburg field theories. We propose that topological insulators in two and three dimensions are described by a version of abelian BF theory. For the two-dimensional topological insulator or quantum spin Hall state, this description is essentially equivalent to a pair of Chern–Simons theories, consistent with the realization of this phase as paired integer quantum Hall effect states. The BF description can be motivated from the local excitations produced when a π flux is threaded through this state. For the three-dimensional topological insulator, the BF description is less obvious but quite versatile: it contains a gapless surface Dirac fermion when time-reversal-symmetry is preserved and yields “axion electrodynamics”, i.e., an electromagnetic E·B term, when time-reversal symmetry is broken and the surfaces are gapped. Just as changing the coefficients and charges of 2D Chern–Simons theory allows one to obtain fractional quantum Hall states starting from integer states, BF theory could also describe (at a macroscopic level) fractional 3D topological insulators with fractional statistics of point-like and line-like objects.

Guiding structures with multiply connected cross sections: Evolution of propagation in external fields at complex Robin parameters

Available online 17 October 2011
O. Olendski

Properties of the two-dimensional ring and three-dimensional infinitely long straight hollow waveguide with unit width and inner radius ρ0 in the superposition of the longitudinal uniform magnetic field B and the Aharonov–Bohm flux are analyzed within the framework of the scalar Helmholtz equation under the assumption that the Robin boundary conditions at the inner and outer confining walls contain extrapolation lengths Λin and Λout, respectively, with nonzero imaginary parts. As a result of this complexity, the self-adjointness of the Hamiltonian is lost, its eigenvalues E, in general, become complex too and the discrete bound states of the annulus, which are characteristic for the real Λ, turn into the corresponding quasibound states with their lifetime defined by the imaginary parts Ei of the eigenenergies while the current along the wire exponentially amplifies/attenuates with the distance depending on the sign of Ei. It is shown that, compared to the disk geometry, the annulus opens up additional possibilities of varying magnetization and currents by tuning imaginary components of the Robin parameters on each confining circumference; in particular, the possibility of restoring a lossless longitudinal flux by zeroing Ei is discussed from mathematical and physical points of view. For each ρ0, the energy E turns real under a special correlation between the imaginary parts of Λin and Λout with the opposite signs being what physically corresponds to the equal transverse fluxes through the inner and outer interfaces of the annulus. In the asymptotic case of the very large radius, ρ0→∞, simple expressions are derived and applied to the analysis of the dependence of the real energy E on Λin and Λout. New features also emerge in the magnetic field influence; for example, if, for the quantum disk, the imaginary energy Ei is quenched by the strong intensities B, then for the annulus this takes place only when the inner Robin distance Λin is real; otherwise, it almost quadratically depends on B with the corresponding enhancement of the reactive scattering. The closely related problem of the hole in the otherwise uniform medium is also addressed for the real and complex extrapolation lengths with the emphasis on the comparative analysis with its dot counterpart.

Phase space quantum mechanics

February 2012
Maciej Błaszak | Ziemowit Domański

This paper develops an alternative formulation of quantum mechanics known as the phase space quantum mechanics or deformation quantization. It is shown that the quantization naturally arises as an appropriate deformation of the classical Hamiltonian mechanics. More precisely, the deformation of the point-wise product of observables to an appropriate noncommutative ⋆-product and the deformation of the Poisson bracket to an appropriate Lie bracket are the key elements in introducing the quantization of classical Hamiltonian systems.The formalism of the phase space quantum mechanics is presented in a very systematic way for the case of any smooth Hamiltonian function and for a very wide class of deformations. The considered class of deformations and the corresponding ⋆-products contains as a special case all deformations which can be found in the literature devoted to the subject of the phase space quantum mechanics.Fundamental properties of ⋆-products of observables, associated with the considered deformations are presented as well. Moreover, a space of states containing all admissible states is introduced, where the admissible states are appropriate pseudo-probability distributions defined on the phase space. It is proved that the space of states is endowed with a structure of a Hilbert algebra with respect to the ⋆-multiplication.The most important result of the paper shows that developed formalism is more fundamental than the axiomatic ordinary quantum mechanics which appears in the presented approach as the intrinsic element of the general formalism. The equivalence of two formulations of quantum mechanics is proved by observing that the Wigner–Moyal transform has all properties of the tensor product. This observation allows writing many previous results found in the literature in a transparent way, from which the equivalence of the two formulations of quantum mechanics follows naturally.In addition, examples of a free particle and a simple harmonic oscillator illustrating the formalism of the deformation quantization and its classical limit are given.

Double-component convection due to different boundary conditions in an infinite slot diversely oriented to the gravity

August 2007
N. Tsitverblit

Onset of small-amplitude oscillatory and both small- and finite-amplitude steady double-component convection arising due to component different boundary conditions in an infinite slot is studied for various slot orientations to the gravity. The main focus is on two compensating background gradients of the components. The physical mechanisms underlying steady and oscillatory convection are analyzed from the perspective of a universally consistent understanding of the effects of different boundary conditions. In a horizontal slot with inviscid fluid addressed by Welander [P. Welander, Tellus Ser. A 41 (1989) 66], oscillatory convection sets in with the most unstable wave number and oscillation frequency being zero. Exact expressions for the critical fixed-value background gradient and the respective group velocity at zero wave number are derived from the long-wavelength expansion both for the horizontal slot with independently varying background gradients and for the inclined slot with the compensating gradients. In the horizontal slot with viscous fluid, the dissipation of along-slot perturbation-cell motion reduces efficiency of the oscillatory instability feedback and thus prevents the most unstable wavelength from being infinite. Based on this interpretation, the oscillatory instability of a three-dimensional (3D) nature is predicted for an interval of long two-dimensional (2D) wavelengths in an inclined slot, and such 3D instability is indeed shown to arise. Related general conditions for three-dimensionality of most unstable disturbances are also formulated. As the slot orientation changes from the horizontal by angle θ(⩾π/2), the oscillatory 2D marginal-stability boundaries in inviscid and viscous fluid are expected to eventually transform into respective steady ones. Oscillatory instability in the vertical slot with viscous fluid, first reported by Tsitverblit [N. Tsitverblit, Phys. Rev. E 62 (2000) R7591], is of a quasi-steady nature. Its (new) mechanism is identified. It is underlain by differential gradient diffusion. As the horizontal slot at θ=π, addressed by Tsitverblit [N. Tsitverblit, Phys. Fluids 9 (1997) 2458], changes its orientation to vertical, the wave number interval of linear steady instability shrinks to the vicinity of the most unstable zero wave number and vanishes. Consistently with the basic nature of finite-amplitude steady convection being the same in the horizontal and vertical slots, the respective convective flows are continuously transformed into each other. The dissimilarity between the nature of finite-amplitude steady convective flows in the horizontal slot with θ=0, revealed by Tsitverblit [N. Tsitverblit, Phys. Lett. A 329 (2004) 445], and that in the vertical slot is shown to eventually give rise to a region of hysteresis in θ∈(0,π/2).

Quantum tunnelling in a dissipative system

September 1983
A.O Caldeira | A.J Leggett

Efficient weakly-radiative wireless energy transfer: An EIT-like approach

August 2009
Rafif E. Hamam | Aristeidis Karalis | J.D. Joannopoulos | Marin Soljačić

Inspired by a quantum interference phenomenon known in the atomic physics community as electromagnetically induced transparency (EIT), we propose an efficient weakly radiative wireless energy transfer scheme between two identical classical resonant objects, strongly coupled to an intermediate classical resonant object of substantially different properties, but with the same resonance frequency. The transfer mechanism essentially makes use of the adiabatic evolution of an instantaneous (so called “dark”) eigenstate of the coupled 3-object system. Our analysis is based on temporal coupled mode theory (CMT), and is general enough to be valid for various possible sorts of coupling, including the resonant inductive coupling on which witricity-type wireless energy transfer is based. We show that in certain parameter regimes of interest, this scheme can be more efficient, and/or less radiative than other, more conventional approaches. A concrete example of wireless energy transfer between capacitively-loaded metallic loops is illustrated at the beginning, as a motivation for the more general case. We also explore the performance of the currently proposed EIT-like scheme, in terms of improving efficiency and reducing radiation, as the relevant parameters of the system are varied.

Anyons and the quantum Hall effect—A pedagogical review

January 2008
Ady Stern

The dichotomy between fermions and bosons is at the root of many physical phenomena, from metallic conduction of electricity to super-fluidity, and from the periodic table to coherent propagation of light. The dichotomy originates from the symmetry of the quantum mechanical wave function to the interchange of two identical particles. In systems that are confined to two spatial dimensions particles that are neither fermions nor bosons, coined “anyons”, may exist. The fractional quantum Hall effect offers an experimental system where this possibility is realized. In this paper we present the concept of anyons, we explain why the observation of the fractional quantum Hall effect almost forces the notion of anyons upon us, and we review several possible ways for a direct observation of the physics of anyons. Furthermore, we devote a large part of the paper to non-abelian anyons, motivating their existence from the point of view of trial wave functions, giving a simple exposition of their relation to conformal field theories, and reviewing several proposals for their direct observation.

Kubo formulas for relativistic fluids in strong magnetic fields

December 2011
Xu-Guang Huang | Armen Sedrakian | Dirk H. Rischke

Magnetohydrodynamics of strongly magnetized relativistic fluids is derived in the ideal and dissipative cases, taking into account the breaking of spatial symmetries by a quantizing magnetic field. A complete set of transport coefficients, consistent with the Curie and Onsager principles, is derived for thermal conduction, as well as shear and bulk viscosities. It is shown that in the most general case the dissipative function contains five shear viscosities, two bulk viscosities, and three thermal conductivity coefficients. We use Zubarev’s non-equilibrium statistical operator method to relate these transport coefficients to correlation functions of the equilibrium theory. The desired relations emerge at linear order in the expansion of the non-equilibrium statistical operator with respect to the gradients of relevant statistical parameters (temperature, chemical potential, and velocity.) The transport coefficients are cast in a form that can be conveniently computed using equilibrium (imaginary-time) infrared Green’s functions defined with respect to the equilibrium statistical operator.

Uncertainty relations for noise and disturbance in generalized quantum measurements

June 2004
Masanao Ozawa

Heisenberg’s uncertainty relation for measurement noise and disturbance is commonly understood to state that in any measurement the product of the position measurement noise and the momentum disturbance is not less than Planck’s constant divided by 4π. However, it has been shown in many ways that this relation holds only for a restricted class of measuring apparatuses in the most general formulation of measuring processes. Here, Heisenberg’s uncertainty relation is generalized to a relation that holds for all the possible quantum measurements, from which rigorous conditions are obtained for measuring apparatuses to satisfy Heisenberg’s relation. In particular, every apparatus with the noise and the disturbance statistically independent from the measured object is proven to satisfy Heisenberg’s relation. For this purpose, all the possible quantum measurements are characterized by naturally acceptable axioms. Then, a mathematical notion of the distance between probability operator valued measures and observables is introduced and the basic properties are explored. Based on this notion, the measurement noise and disturbance are naturally defined for any quantum measurements in a model independent formulation. Under this formulation, various relations for noise and disturbance are also derived for apparatuses with independent noise, independent disturbance, unbiased noise, and unbiased disturbance as well as noiseless apparatuses and nondisturbing apparatuses. Two models of position measurements are also discussed in the light of the new uncertainty relations to show that Heisenberg’s relation can be violated even by approximately repeatable position measurements.