Condensed Matter Theory
Friedrich-Hund-Platz 1 • 37077 Göttingen
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Kehrein Publications
Irreversible dynamics in quantum many-body systems
M. Schmitt, S. Kehrein, arXiv:1711.00015
Irreversibility, despite being a necessary condition for thermalization, still lacks a sound understanding in the context of quantum many-body systems. In this work we approach this question by studying the behavior of generic many-body systems under imperfect effective time reversal, where the imperfection is introduced as a perturbation of the many-body state at the point of time reversal. Based on numerical simulations of the full quantum dynamics we demonstrate that observable echos occurring in this setting decay exponentially with a rate that is intrinsic to the system meaning that the dynamics is effectively irreversible.
Analysis of the buildup of spatiotemporal correlations and their bounds outside of the light cone
N. O. Abeling, L. Cevolani, S. Kehrein, arXiv:1707.02328
In non-relativistic quantum theories the Lieb-Robinson bound defines an effective light cone with exponentially small tails outside of it. In this work we use it to derive a bound for the time evolution of the correlation function of two local disjoint observables if the initial state has a power-law decay. We show that the exponent of the power-law of the bound is identical to the initial (equilibrium) decay. We explicitly verify this result by studying the full dynamics of the susceptibilities and correlations in the exactly solvable Luttinger model after a sudden quench from the non-interacting to the interacting model.
Flow Equation Holography
S. Kehrein, arXiv:1703.03925
The Ryu-Takayanagi conjecture establishes a remarkable connection between quantum systems and geometry. Specifically, it relates the entanglement entropy to minimal surfaces within the setting of AdS/CFT correspondence. This Letter shows how this idea can be generalised to generic quantum many-body systems within a perturbative expansion where the region whose entanglement properties one is interested in is weakly coupled to the rest of the system. A simple expression is derived that relates a unitary disentangling flow in an emergent RG-like direction to the min-entropy of the region under consideration. Explicit calculations for critical free fermions in one and two dimensions illustrate this relation.
Photoexcitations in a 1D manganite model: From quasiclassical light absorption to quasiparticle relaxations
Th. Köhler, S. Rajpurohit, O. Schumann, F. R. A. Biebl, M. Sotoudeh, S. C. Kramer, P. E. Blöchl, S. Kehrein, S. R. Manmana, arXiv:1610.07246
We investigate the evolution of a photoexcitation in a correlated material over a wide range of time scales from femto- to nanoseconds. The system studied is a one-dimensional model of a manganite with correlated electron, spin, orbital, and lattice degrees of freedom. In a first step, the interaction of light with electrons considering the back-action is treated. Time-dependent matrix product states (MPS) methods are then used for the electronic quantum dynamics of the resulting photoexcitation. The emergence of quasiparticles from a fully correlated excitation is addressed. We study the relaxation of the non-equilibrium quasi-particle distribution with a linearized quantum-Boltzmann equation. Our investigation treats a model-Hamiltonian with parameters extracted from ab-initio calculations of Pr1?xCaxMnO3. This model is restricted to one dimension. The ground state phases for the entire composition range are determined and rationalized by a coarse grained polaron model. A pattern of antiferromagnetically coupled Zener polarons for the half-doped material is identified, which forms the basis for the dynamical calculations discussed above. We discuss the effect of the resulting magnetic microstructure on the relaxation times of the excitation.
Thermalization rates in the one dimensional Hubbard model with next-to-nearest neighbor hopping
Fabian R. A. Biebl, Stefan Kehrein, Phys. Rev. B 95, 104304 (2017), arXiv:1607.07115
We consider a fermionic Hubbard chain with an additional next-to-nearest neighbor hopping term. We study the thermalization rates of the quasi-momentum distribution function within a quantum Boltzmann equation approach. We find that the thermalization rates are proportional to the square of the next-to-nearest neighbor hopping: Even weak next-to-nearest neighbor hopping in addition to nearest neighbor hopping leads to thermalization in a two-particle scattering quantum Boltzmann equation in one dimension. We also investigate the temperature dependence of the thermalization rates, which away from half filling become exponentially small for small temperature of the final thermalized distribution.
Effective time reversal and echo dynamics in the transverse field Ising model
Markus Schmitt, Stefan Kehrein, Europhys. Lett. 115, 50001 (2016), arXiv:1607.02272
The question of thermalisation in closed quantum many-body systems has received a lot of attention in the past few years. An intimately related question is whether a closed quantum system shows irreversible dynamics. However, irreversibility and what we actually mean by this in a quantum many-body system with unitary dynamics has been explored very little. In this work we investigate the dynamics of the Ising model in a transverse magnetic field involving an imperfect effective time reversal. We propose a definition of irreversibility based on the echo peak decay of observables. Inducing the effective time reversal by different protocols we find algebraic decay of the echo peak heights or an ever persisting echo peak indicating that the dynamics in this model is well reversible.
Universal nonanalytic behavior of the Hall conductance in a Chern insulator at the topologically driven nonequilibrium phase transition
Pei Wang, Markus Schmitt, Stefan Kehrein, Phys. Rev. B 93, 085134 (2016), arXiv:1511.05255
 The Hall conductance as a function of (Mf,Bf) for different (Mi,Bi).
We study the Hall conductance of a Chern insulator after a global quench of the Hamiltonian. The Hall conductance in the long time limit is obtained by applying linear response theory to the diagonal ensemble. We identify a topologically driven nonequilibrium phase transition, which is indicated by the nonanalyticity of the Hall conductance as a function of the energy gap m_f in the post-quench Hamiltonian H_f. The topological invariant for the quenched state is the winding number of the Green's function W, which equals the Chern number for the ground state of H_f. In the limit that m_f goes to zero, the derivative of the Hall conductance with respect to m_f is proportional to ln(|m_f|), with the constant of proportionality being the ratio of the change of W at m_f = 0 to the energy gap in the initial state. This nonanalytic behavior is universal in two-band Chern insulators such as the Dirac model, the Haldane model, or the Kitaev honeycomb model in the fermionic basis.
Quantum Quench Dynamics in the Transverse Field Ising Model at Non-zero Temperatures
Nils Abeling, Stefan Kehrein, Phys. Rev. B 93, 104302 (2016), arXiv:1510.08728
 Rate function for a double quench from g = 0.5 to g
The recently discovered dynamical phase transition denotes non-analytic behavior in the real time evolution of quantum systems in the thermodynamic limit and has been shown to occur in different systems at zero temperature [Heyl et al., Phys. Rev. Lett. 110, 135704 (2013)]. In this paper we extend the analysis to non-zero temperature by studying a generalized form of the Loschmidt echo, the work distribution function, of a quantum quench in the transverse field Ising model. Although the quantitative behavior at non-zero temperatures still displays features derived from the zero temperature non-analyticities, it is shown that in this model dynamical phase transitions do not exist if T>0. Moreover, we elucidate how the Tasaki-Crooks-Jarzynski relation can be exploited as a symmetry relation for a global quench or to obtain the change of the equilibrium free energy density.
Representation of excited states and topological order of the toric code in MERA
Johannes Oberreuter, Stefan Kehrein, arXiv:1510.08126
 The holographic contribution to the topological entanglement entropy on different scales.
The holographic duality relates a field theory to a theory of (quantum) gravity in one dimension more. The extra dimension represents the scale of the RG transformation in the field theory. It has been conjectured that the tensor networks which arise during the real space renormalization procedure like the multi-scale entanglement renormalization ansatz (MERA) are a discretized version of the background of the gravity theory. We conside an explicit and tractable example, namely the dual network of the toric code, for which MERA can be performed analytically even for excited states. Furthermore, we show how to calculate topological entanglement entropy from the geometry of MERA.
Dynamical Quantum Phase Transitions in the Kitaev Honeycomb Model
Markus Schmitt, Stefan Kehrein, Phys. Rev. B 92, 075114 (2015), arXiv:1505.03401
 Distribution of zeros of the Loschmidt echo in the complex time plane for two different quenches. The zeros form areas in the complex plane .
The notion of a dynamical quantum phase transition (DQPT) was recently introduced as the non-analytic behavior of the Loschmidt echo at critical times in the thermodynamic limit. In this work the quench dynamics in the ground state sector of the two-dimensional Kitaev honeycomb model are studied regarding the occurrence of DQPTs. For general two-dimensional systems of BCS-type it is demonstrated how the zeros of the Loschmidt echo coalesce to areas in the thermodynamic limit, implying that DQPTs occur as discontinuities in the second derivative. In the Kitaev honeycomb model DQPTs appear after quenches across a phase boundary or within the massless phase. In the 1d limit of the Kitaev honeycomb model it becomes clear that the discontinuity in the higher derivative is intimately related to the higher dimensionality of the non-degenerate model. Moreover, there is a strong connection between the stationary value of the rate function of the Loschmidt echo after long times and the occurrence of DQPTs in this model.
Topologically driven nonequilibrium phase transitions in diagonal ensembles
P. Wang, S. Kehrein, New J. Phys. 18, 053003 (2016), arXiv:1504.05689
We identify a new class of topologically driven phase transitions when calculating the Hall conductance of the Dirac model in the long-time limit after a global quench of the Hamiltonian. The Hall conductance can be expressed as the integral of the Berry curvature in the diagonal ensemble. Even if the topological invariant of the wave function is conserved under unitary evolution, the Hall conductance derived from the diagonal ensemble displays a continuous but nonanalytic behavior, that is it has a logarithmically divergent derivative whenever the post-quench Hamiltonian changes its Chern number. This nonequilibrium phase transition reveals different nonequilibrium phases which must be characterized by their symmetry protected topological order.
Dissipation-induced first-order decoherence phase transition in a non-interacting fermionic system
M. Medvedyeva, M. Cubrovic, S. Kehrein, Phys. Rev. B 91, 205416 (2015), arXiv:1409.1625
 Exponential decay of the current along the dissipative chain. A: Logarithmic scale, different lengths of system. B: Disordered system.
We consider a dissipative tight-binding chain. The dissipation manifests as tunneling into/out of the chain from/to a memoryless environment. The evolution of the system is described by the Lindblad equation. Already infinitesimally small dissipation along the chain induces a quantum phase transition (QPT). This is a decoherence QPT: the reduced density matrix of a subsystem (far from the ends of the chain) can be represented as the tensor product of single-site density matrices. We analyze the QPT in the thermodynamic limit by looking at the entropy and the response function in the bulk. We also explore the properties of the boundaries of the chain close to the transition point and observe that the boundaries behave as if they undergo a second-order phase transition with power-law divergence of the correlation functions and response function. Disorder is known to localize one-dimensional systems, but the coupling to the memoryless environment pushes the system back into the delocalized state even in the presence of disorder.
Dynamical quantum phase transitions in the axial next-nearest-neighbour Ising chain
J. Kriel, C. Karrasch, S. Kehrein, Phys. Rev. B 90, 125106 (2014), arXiv:1407.4036
 Comparison of analytical flow equation results (crosses) and tDMRG results (blue line) for the rate function after a quench across the quantum critical point. The agreement is excellent.
We investigate sudden quenches across the critical point in the transverse field Ising chain with a perturbing non-integrable next-nearest-neighbour interaction. Analytic expressions for the return (Loschmidt) amplitude and associated rate function are derived to linear order in the next-nearest-neighbour coupling using the flow equation method. In the thermodynamic limit these quantities exhibit non-analytic behaviour at a set of critical times, a phenomenon referred to as a dynamical quantum phase transition. We quantify the effect of the integrability breaking perturbation on the location and shape of these non-analyticities. Our results agree with those of earlier numerical studies and offer further support for the assertion that the dynamical quantum phase transitions exhibited by this model are a generic feature of its post-quench dynamics and is robust with respect to the inclusion of non-integrable perturbations.
Power-law approach to steady state in open lattices of noninteracting electrons
M. Medvedyeva and S. Kehrein, Phys. Rev. B 90, 205410 (2014), arXiv:1406.1408
We address the question of how a non-equilibrium steady state (NESS) is reached in the Linbdladian dynamics of an open quantum system. We develop an expansion of the density matrix in terms of the NESS-excitations, each of which has its own (exponential) decay rate. However, when the decay rates tend to zero for many NESS-excitations (the spectral gap of the Liouvillian is closed in the thermodynamic limit), the long-time dynamics of the system can exhibit a power-law behaviour. This relaxation to NESS expectation values is determined by the density of states close to zero spectral gap and the value of the operator in these states. We illustrate this main idea on the example of the lattice of non-interacting fermions coupled to Markovian leads at infinite bias voltage. The current comes towards its NESS value starting from a typical initial state as ??3/2. This behaviour is universal and independent of the space dimension.
Entanglement propagation and typicality of measurements in the quantum Kac ring
J. Oberreuter, I. Homrighausen, S. Kehrein, Ann. Phys. 348, 324 (2014), arXiv:1312.1726
 A quantum Kac ring. Time evolution generates entanglement between the system and the environment.
We study the time evolution of entanglement in a quantum version of the Kac ring. The recurrence time of this quantum many-body system is twice the length of the chain and "thermalization" only occurs on time scales much smaller than the dimension of the Hilbert space. The model thus elucidates the relation between distribution of measurement results in quantum and classical systems: While in classical systems repeated measurements are performed over an ensemble of systems, the corresponding result is obtained by measuring the same quantum system prepared in an appropriate superposition repeatedly.
Quench dynamics in a model with tuneable integrability breaking
F.H.L. Essler, S. Kehrein, S.R. Manmana, N.J. Robinson, Phys. Rev. B 89, 165104 (2014)
 Comparison of flow equation results (CUT) with t-DMRG calculations after a quantum quench in the non-integrable model.
We consider quantum quenches in an integrable quantum chain with tuneable integrability breaking interactions. In the case where these interactions are weak, we demonstrate that at intermediate times after the quench local observables relax to a prethermalized regime, which can be described by a density matrix that can be viewed as a deformation of a generalized Gibbs ensemble. We present explicit expressions for the approximately conserved charges characterizing this ensemble. We do not find evidence for a crossover from the prethermalized to a thermalized regime on the time-scales accessible to us. Increasing the integrability-breaking interactions leads to a behaviour that is compatible with eventual thermalization.
Lindblad equation for a non-interacting fermionic system: full-counting statistics
M.V. Medvedyeva, S. Kehrein, arXiv:1310.4997
 The second generalized Fano factor depending on the hopping on both ends of the chain. One clearly see a region where the third cumulant is very small and the statistics becomes approximately Gaussian.
We develop a method of calculating the full-counting statistics for a non-interacting fermionic system coupled to the memory-less reservoirs. The evolution of the system is described by the Lindblad equation. The counting statistics from the Lindblad approach does not take into account interference in the reservoirs, which gives a decreased noise in comparison with the Green function method which describes phase coherent leads: The Fano factors are different and allow to distinguish between memory-less and phase coherent reservoirs.
Non-equilibrium dynamics of a system with quantum frustration
Heiner Kohler, Andreas Hackl, and Stefan Kehrein, Phys. Rev. B 88, 205122 (2013)
 Phase diagram: The line indicates the critical asymmetry angle, which separates the localized from the delocalized phase.
Using flow equations and equilibrium and nonequilibrium dynamics of a two-level system (TLS) is investigated, which couples Ohmically via noncommuting components to two independent oscillator baths. In equilibrium, the two-level energy splitting is protected when the TLS is coupled symmetrically to both baths. A critical asymmetry angle separates the localized from the delocalized phase. Real-time decoherence of a nonequilibrium initial state is studied as well. The short-time dynamics exhibits initial slips on the scale of the cutoff frequency of the bath modes. Moreover, whereas for a single bath decay of coherence depends crucially on the chosen initial state, for a symmetric coupling to two baths this dependence vanishes.
Spatiotemporal buildup of the Kondo screening cloud
M. Medvedyeva, A. Hoffmann, and S. Kehrein, Phys. Rev. B 88, 094306 (2013)
We investigate how the Kondo screening cloud builds up as a function of space and time. Starting from an impurity spin decoupled from the conduction band, the Kondo coupling is switched on at time t=0. We work at the Toulouse point where one can obtain exact analytical results for the ensuing spin dynamics at both zero and nonzero temperature $\mathit{T}$. For $\mathit{t}>0$ the Kondo screening cloud starts building up in the wake of the impurity spin being transported to infinity. In this buildup process the impurity spin–conduction band spin susceptibility shows a sharp light cone due to causality, while the corresponding correlation function has a tail outside the light cone. At $\mathit{T}=0$ this tail has a power-law decay as a function of distance from the impurity, which we interpret as due to initial entanglement in the Fermi sea.
Dynamical Quantum Phase Transitions in the Transverse Field Ising Model
M. Heyl, A. Polkovnikov, and S. Kehrein, Phys. Rev. Lett. 110, 135704 (2013)
A phase transition indicates a sudden change in the properties of a large system. For temperature-driven phase transitions this is related to nonanalytic behavior of the free energy density at the critical temperature: The knowledge of the free energy density in one phase is insufficient to predict the properties of the other phase. In this Letter we show that a close analogue of this behavior can occur in the real time evolution of quantum systems, namely nonanalytic behavior at a critical time. We denote such behavior a dynamical phase transition and explore its properties in the transverse-field Ising model. Specifically, we show that the equilibrium quantum phase transition and the dynamical phase transition in this model are intimately related.
Relaxation of hot electrons in a degenerate two-dimensional electron system: transition to one-dimensional scattering
D. Taubert, C. Tomaras, G. J. Schinner, H. P. Tranitz, W. Wegscheider, S. Kehrein, and S. Ludwig, Phys. Rev. B 83, 235404 (2011)
Scaling approach for the time-dependent Kondo model
C. Tomaras and S. Kehrein, Eur. Phys. Lett. 93, 47011 (2011)
We present a new nonperturbative method to deal with the time-dependent quantum many-body problem, which is an extension of Wegner's flow equations to time-dependent Hamiltonians. The formalism provides a scaling procedure for the set of time-dependent interaction constants. We apply these ideas to a Kondo model with a ferromagnetic exchange coupling switched on over a time scale $\tau$. We show that the asymptotic expectation value of the impurity spin interpolates continuously between its quenched and adiabatic value.
Scaling approach for the time-dependent Kondo model
C. Tomaras and S. Kehrein, Eur. Phys. Lett. 93, 47011 (2011)
We present a new nonperturbative method to deal with the time-dependent quantum many-body problem, which is an extension of Wegner's flow equations to time-dependent Hamiltonians. The formalism provides a scaling procedure for the set of time-dependent interaction constants. We apply these ideas to a Kondo model with a ferromagnetic exchange coupling switched on over a time scale $\tau$. We show that the asymptotic expectation value of the impurity spin interpolates continuously between its quenched and adiabatic value.
X-ray edge singularity in optical spectra of quantum dots
M. Heyl and S. Kehrein
In this work we investigate the x-ray edge singularity problem realized in noninteracting quantum dots. We analytically calculate the exponent of the singularity in the absorption spectrum near the threshold and extend known analytical results to the whole parameter regime of local level detunings. Additionally, we highlight the connections to work distributions and to the Loschmidt echo.
Flow equation calculation of transient and steady state currents in the Anderson impurity model
P. Wang and S. Kehrein, Phys. Rev. B 82, 125124 (2010)
Transient and steady-state currents through dc-biased quantum impurity models beyond the linear-response regime are of considerable interest, both from an experimental and a theoretical point of view. Here we present an analytical approach for the calculation of such currents based on the flow equation method (method of infinitesimal unitary transformations). Specifically, we analyze the Anderson impurity model in its mixed valence regime where the coupling to the leads is switched on suddenly at time $mathit{t}=0. We observe the real time buildup of the current until it reaches its steady-state limit. The Crooks relation in optical spectra - universality in work distributions for weak local quenches M. Heyl and S. Kehrein, Phys. Rev. Lett. 108, 190601 (2012)  Crooks relation in the absorption spectrum of a Kondo exciton. We show that work distributions and non-equilibrium work fluctuation theorems can be measured in optical spectra for a big class of quantum systems. For the particular case of a weak local perturbation, the Crooks relation establishes a universal relation in absorption as well as in emission spectra. Due to a direct relation between the spectra and work distribution functions this is equivalent to universal relations in work distributions for weak local quenches. As two explicit examples we treat the X-ray edge problem and the Kondo exciton. New theoretical approaches for correlated systems in nonequilibrium M. Eckstein, A. Hackl, S. Kehrein, M. Kollar, M. Moeckel, P. Werner, and F. A. Wolf, Eur. Phys. J. Special Topics 180, 217 (2010)  Fermi surface discontinuity after an interaction quench in the Hubard model. The prethermalization plateau is clearly visible. We review recent developments in the theory of interacting quantum many-particle systems that are not in equilibrium, focussing mainly on the nonequilibrium generalizations of the flow equation approach and of dynamical mean-field theory (DMFT). We discuss results from the flow equation approach for nonlinear transport in the Kondo model, and further applications of this method to the relaxation behavior in the ferromagnetic Kondo model and the Hubbard model after an interaction quench. For the interaction quench in the Hubbard model, we have also obtained numerical DMFT results using quantum Monte Carlo simulations. Non-equilibrium Scaling Properties of a Double Quantum Dot System: Comparison between Perturbative Renormalization Group and Flow Equation Approach V. Koerting, P. Fritsch, and S. Kehrein, Physica B 406, 2091 (2011)  Geometry of double dot system with four leads. In this paper the differences and commons between the flow equation method out of equilibrium and the frequency-dependent poor man's scaling approach are presented for the non-equilibrium double quantum dot system. This will turn out to be a particularly suitable testing ground while being experimentally interesting in its own right. An outlook is given on the quantum critical behavior of the double quantum dot system and its accessibility with the two methods. Effect of measurement-backaction on adiabatic coherent electron transport J. Rech and S. Kehrein, Phys. Rev. Lett. 106, 136808 (2011)  Distribution of charges through QPC during CTAP protocol as a function of coupling strength D-D We study the backaction of measurement on electrons undergoing coherent transfer via adiabatic passage (CTAP) in a triple-well system. We account for this continuous measurement by treating the whole {triple-well + detector} as a closed quantum system. This approach allows to study a single realization of the measurement process while keeping track of the detector output. As one increases the coupling between the middle potential well and the detector, one finds a substantial drop in the fidelity of the CTAP scheme. Exact results for nonlinear ac-transport through a resonant level model P. Wang, M. Heyl and S. Kehrein, J. Phys. C 22, 275604 (2010)  The steady state current in one period for intermediate fast driving. We obtain exact results for the transport through a resonant level model for rectangular voltage bias as a function of time. We study both the transient behavior after switching on the tunneling and the ensuing steady state behavior. Among other effects, we observe current ringing and PAT (photon assisted tunneling) oscillations. Electron-Plasmon scattering in chiral 1D systems with nonlinear dispersion M. Heyl, S. Kehrein, F. Marquardt, and C. Neuenhahn, Phys. Rev. B 82, 033409 (2010)  The fermionic density rho(x;t) in the co-moving frame of the high-energy electron. We investigate systems of spinless one-dimensional chiral fermions occurring, e.g., in the arms of electronic Mach-Zehnder interferometers. We take into account the curvature of the fermionic spectrum and a finite interaction range. Due to an interplay of both, an injected high-energy electron can scatter off plasmons (density excitations) leading to an exponential decay of the single particle Green's function even at zero temperature. The excitation of plasmons leads to the buildup of a coherent monochromatic sinusoidal density pattern. Electron-avalanche amplifier based on the electronic Venturi effect D. Taubert, G. J. Schinner, H. P. Tranitz, W. Wegscheider, C. Tomaras, S. Kehrein, and S. Ludwig, Phys. Rev. B 82, 161416 (2010) Ballistic transport of electrons far from equilibrium is investigated in a cold two-dimensional electron system. In a three-terminal device, we realize an electronic version of the Venturi effect that enables us to build an avalanche amplifier based on non-equilibrium electrons. This device might be developed further to create a non-invasive charge detector. A preliminary model based on numerical calculations using a random phase approximation is in agreement with our data. Interaction quench dynamics in the Kondo model in presence of a local magnetic field M. Heyl and S. Kehrein, J. Phys. C 22, 345604 (2010)  Magnetization after an interaction quench in presence of an local magnetic field. We investigate the quench dynamics in the Kondo model on the Toulouse line in presence of a local magnetic field. The transient dynamics is studied by analyzing exact analytical results for the local spin dynamics. The time scale for the relaxation of the local dynamical quantities turns out to be exclusively determined by the Kondo scale. In the transient regime, one observes damped oscillations in the local correlation functions with a frequency set by the magnetic field. Non-equilibrium steady state in a periodically driven Kondo model M. Heyl and S. Kehrein, Phys. Rev. B 81, 144301 (2010)  False color plot of the dynamical spin susceptibility for intermediate fast driving. We investigate the Kondo model with time-dependent couplings that are periodically switched on and off. On the Toulouse line we derive exact analytical results for the spin dynamics in the steady state that builds up after an infinite number of switching periods. In the limit of fast driving one can show that the steady state cannot be described by some effective equilibrium Hamiltonian since a naive implementation of the Trotter formula gives wrong results. As a consequence, the steady state in the limit of fast switching serves as an example for the emergence of new quantum states not accessible in equilibrium. Sudden interaction quench in the quantum sine-Gordon model J. Sabio and S. Kehrein, New J. Phys. 12, 055008 (2010) We study a sudden interaction quench in the weak-coupling regime of the quantum sine-Gordon model. The real time dynamics of the bosonic mode occupation numbers is calculated using the flow equation method. We establish the existence of an extended regime in time where the mode occupation numbers relax to twice their equilibrium values. This factor two indicates a non-equilibrium distribution and is a universal feature of weak interaction quenches. Nonequilibrium Kondo model with voltage bias in a magnetic field P. Fritsch and S. Kehrein, Phys. Rev. B 81, 035113 (2010) Nonequilibrium magnetization dynamics of ferromagnetically coupled Kondo spins A. Hackl, M. Vojta, and S. Kehrein, Phys. Rev. B 80, 195117 (2009) Real-time evolution for weak interaction quenches in quantum systems M. Moeckel and S. Kehrein, Ann. Phys. (NY) 324, 2146 (2009) Non-Equilibrium Spin Dynamics in the Ferromagnetic Kondo Model A. Hackl, D. Roosen, S. Kehrein, and W. Hofstetter, Phys. Rev. Lett. 102, 196601 (2009) We study the real time dynamics of the impurity spin initially prepared in a product state with the bath for the ferromagnetic Kondo model. Both numerical time-dependent NRG and analytical flow equation calculations show a non-analytic spin decay that remarkably retains the memory of the initial preparation of the system for all times. Non–Equilibrium Scaling Analysis of the Kondo Model with Voltage Bias P. Fritsch and S. Kehrein, Ann. Phys. (NY) 324, 1105 (2009) Density matrix renormalization group study of a quantum impurity model with Landau-Zener time-dependent Hamiltonian C. Guo, A. Weichselbaum, S. Kehrein, T. Xiang, and J. von Delft, Phys. Rev. B 79, 115137 (2009) Interaction Quench in the Hubbard Model M. Möckel and S. Kehrein, Phys. Rev. Lett. 100, 175702 (2008) The Flow Equation Approach to Many-Particle Systems Violation of the fluctuation-dissipation theorem and heating effects in the time-dependent Kondo model Dmitry Lobaskin and Stefan Kehrein, J. Stat. Phys. 123, 301 (2006)  Effective temperature$\mathit{T_\text{eff}}$as a function of the waiting time$\mathit{t_w}$in the Kondo limit. The line is a guide to the eye. The size of the datapoints (circles) indicates the numerical error. The datapoint for$\mathit{t_w}$/$\mathit{t_K}$= 5.5 is numerically indistinguishable from zero. The fluctuation-dissipation theorem (FDT) plays a fundamental role in understanding quantum many-body problems. However, its applicability is limited to equilibrium systems and it does in general not hold in nonequilibrium situations. In this paper we present results for the violation of the FDT in the Kondo model where the impurity spin is frozen for all negative times, and set free to relax at positive times. We derive exact analytical results at the Toulouse point, and results within a controlled approximation in the Kondo limit, which allow us to study the FDT violation on all time scales. Unusual non-fermi liquid behavior of Ce$_{1-x}$La$_{x}$Ni$_{9}$Ge$_4$analyzed in a single impurity Anderson model with crystal field effects E.-W. Scheidt, F. Mayr, U. Killer, W. Scherer, H. Michor, E. Bauer, S. Kehrein, Th. Pruschke, F. Anders, Physica B 378-380, 154 (2006) Scaling and Decoherence in the Nonequilibrium Kondo Model S. Kehrein, Phys. Rev. Lett. 95, 056602 (2005)  Left: Conventional scaling picture where states are integrated out. Right: Flow equation approach. Here all scattering processes with energy transfer$\mid\Delta \mathit{E}\mid \lesssim \mathit{B^{-1/2}}$are retained in$\mathit{H(B)}$. We study the Kondo effect in quantum dots in an out-of-equilibrium state due to an applied dc-voltage bias. Using the method of infinitesimal unitary transformations (“flow equations”), we develop a perturbative scaling picture that naturally contains both equilibrium coherence and nonequilibrium decoherence effects. This framework allows one to study the competition between Kondo effect and current-induced decoherence, and it establishes a large regime dominated by single-channel Kondo physics for asymmetrically coupled quantum dots. Crossover from nonequilibrium to equilibrium behavior in the time-dependent Kondo model D. Lobaskin and S. Kehrein, Phys. Rev. B 71, 193303 (2005)  Universal curves for the spin-spin correlation function$\mathit{C(t_w, \omega)}$at the Toulouse point for various waiting times ($\mathit{t_w} = 0, t_K / 4, t_K / 2, t_K, 2t_K, \infty$from top to bottom). The inset depicts the same curves on a linear scale. We investigate the equilibration of a Kondo model that is initially prepared in a nonequilibrium state towards its equilibrium behavior. We evaluate the nonequilibrium spin-spin correlation function at the Toulouse point of the Kondo model exactly and analyze the crossover between nonequilibrium and equilibrium behavior as the nonequilibrium initial state evolves as a function of the waiting time for the first spin measurement. Crossover from Single-Ion to Coherent Non-Fermi Liquid Behavior in$Ce_{1-x}La_{x}Ni_{9}Ge_{4}$E.-W. Scheidt, U. Killer, H. Michor, E. Bauer, C. Dusek, S. Kehrein, and W. Scherer, Physica B 359-361, 254 (2005) We report specific heat and magneto-resistance studies on the compound$Ce_{1-x}La_{x}Ni_{9}Ge_{4}$for various concentrations over the entire stoichiometric range. Our data reveal single-ion scaling with Ce-concentration between$\mathit{x}$= 0.1 and 0.95. Furthermore,$CeNi_{9}Ge_{4}$turns out to have the largest ever recorded value of the electronic specific heat$\Delta c/\mathit{T} \approx 5.5 J K^{-2} mol^{-1} $at$\mathit{T}$= 0.08 K which was found in Cerium f-electron lattice systems. Unusual Single-Ion Non-Fermi Liquid Behavior in$Ce_{1-x}La_{x}Ni_{9}Ge_{4}$U. Killer, E.-W. Scheidt, G. Eickerling, H. Michor, J. Sereni, Th. Pruschke, and S. Kehrein, Phys. Rev. Lett. 93, 216404 (2004)  The electronic specific heat divided by temperature in various magnetic fields of three particular samples: (a)$CeNi_{9}Ge_{4}$, (b)$Ce_{0.5}La{0.5}Ni_{9}Ge_{4}$, and (c)$Ce_{0.1}La_{0.9}Ni_{9}Ge_{4}$. The solid lines are logarithmic fits to the zero-field data below 1.5 K. We report on specific heat, magnetic susceptibility, and resistivity measurements on the compound$Ce_{1-x}La_{x}Ni_{9}Ge_{4}$for various concentrations ranging from the stoichiometric system with$\mathit{x} = 0$to the dilute limit$\mathit{x} = 0.95$. Our data reveal single-ion scaling with the Ce concentration and the largest ever recorded value of the electronic specific heat$\Delta c = \mathit{T} \approx$5.5 J K$^{-2}$mol$^{-1}$at$\mathit{T} = 0.08 K$for the stoichiometric compound$\mathit{x} = 0$without any trace of magnetic order. Exploiting environmental resonances to enhance qubit quality factors S. Kleff, S. Kehrein, and J. von Delft, Phys. Rev. B 70, 014516 (2004)  FER results for dephasing times ($\tau_{\Delta}$and$\tau_{\Omega}$) for the structured bath compared to results from WCC ($\tau_w$), with$\Delta(\infty)$occurring therein calculated using FER. We discuss dephasing times for a two-level system (including bias) coupled to a damped harmonic oscillator. We calculate correlation functions, dephasing rates, and qubit quality factors. We find that these depend strongly on the environmental resonance frequency$\Omega$; in particular, quality factors can be enhanced significantly by tuning$\Omega$to lie$\mathit{below}$the qubit frequency$\Delta$. Quantum phase transition of Ising-coupled Kondo impurities M. Garst, S. Kehrein, Th. Pruschke, A. Rosch, and M. Vojta, Phys. Rev. B 69, 214413 (2004)  (a) Experimental setup to measure the tunneling conductance between two capacitively coupled quantum dots. (b) Schematic plot of the zero-bias anomaly of the conductance at$\mathit{T}$= 0. In the frozen minidomain phase,$\delta < \delta_c$, the conductance diverges algebraically. At the quantum phase transition,$\delta = \delta_c$, the exponent takes the universal value -2($\sqrt{2}$- 1). In the Kondo-screened phase,$\delta > \delta_c$, the conductance is finite for$\mathit{V}\to$0. We investigate a model of two Kondo impurities coupled via an Ising interaction. Exploiting the mapping to a generalized single-impurity Anderson model, we establish that the model has a singlet and a (pseudospin) doublet phase separated by a Kosterlitz-Thouless quantum phase transition. Entropy balance analysis of the AFM-region in the phase diagram of$UCu_{5-x}Pd_x$U. Killer, E. -W. Scheidt, S. Kehrein, and W. Scherer, J. Mag. Mag. Mat. 272-276, e77 (2004) We present a systematic study of the specific heat in the antiferromagnetic (afm) region of the heavy fermion system$UCu_\text{5-x} Pd_x$for annealed and unannealed samples with$\mathit{x} = 0.5; 0.6, 0.8, 1.0$. A careful entropy analysis reveals a nearly constant value of the entropy around$4.0 J mol^{-1} K^{-1}$for all values of x, which supports the scenario of an afm quantum critical phase (QCP) transition with a vanishing transition temperature$\mathit{T}_N$at$\mathit{x} = 1$: Flow equation renormalization of a spin-boson model with a structured bath S. Kleff, S. Kehrein, and J. von Delft, Physica E 18, 343 (2003) We discuss the dynamics of a spin coupled to a damped harmonic oscillator. This system can be mapped to a spin-boson model with a structured bath, i.e. the spectral function of the bath has a resonance peak. We diagonalize the model by means of infinitesimal unitary transformations (flow equations), thereby decoupling the small quantum system from its environment, and calculate spin–spin correlation functions. Spin-spin correlation functions for a spin-boson model with a structured bath S. Kleff, S. Kehrein, and J. von Delft, J. Phys. Soc. Jpn. 72, Suppl. A 161 (2003) Semi-analytical solution of the Kondo model in a magnetic field C. Slezak, S. Kehrein, Th. Pruschke, and M. Jarrell, Phys. Rev. B 67, 184408 (2003) The single impurity Kondo model at zero temperature in a magnetic field is solved by an approximate semianalytical approach based on the flow-equation method. The resulting problem is shown to be equivalent to a resonant-level model with a nonconstant hybridization function. This nontrivial effective hybridization function encodes the quasiparticle interaction in the Kondo limit, while the magnetic field enters as the impurity orbital energy. The evaluation of static and dynamic quantities of the strong-coupling Kondo model becomes very simple in this effective model. Annealing, lattice disorder, and non-Fermi-liquid behavior in$UCu_{4}Pd$C. H. Booth, E.-W. Scheidt, U. Killer, A. Weber, and S. Kehrein, Phys. Rev. B 66, 140402(R) (2002)  The electronic part of the specific heat$\Delta\mathit{C/T}$as a function of temperature from Weber et al. Fits are with a Kondo disorder model within a tight-binding prescription, as described in the text. The magnetic and electronic properties of non-Fermi-liquid$UCu_{4}Pd$depend on annealing conditions. This study provides quantitative information on the amount of disorder in$UCu_{4}Pd$and allows an assessment of its possible importance to the observed non-Fermi-liquid behavior. Spin Reduction Transition in Spin-3/2 Random Heisenberg Chains G. Refael, S. Kehrein, and D. Fisher, Phys. Rev. B 66, 060402(R) (2002) Random spin-3/2 antiferromagnetic Heisenberg chains are investigated using an asymptotically exact renormalization group. Randomness is found to induce a quantum phase transition between two random-singlet phases. In the strong randomness phase the effective spins at low energies are S$_\text{eff}$=3/2, while in the weak randomness phase the effective spins are S$_\text{eff}$=1/2. Separating them is a quantum critical point near which there is a nontrivial mixture of spin-1/2, spin-1, and spin-3/2 effective spins at low temperatures. Quantum phase transitions: experimental facts - a challenge for theory E.-W. Scheidt, D. Maurer, A. Weber, T. Goetzfried, K. Heuser, S. Kehrein and R. Tidecks, Physica B 321, 133 (2002) Non-Fermi-liquid behavior in CePt$_{(1+x)}$Si$_{(1-x)}$and CePtSi$_{(1-y)}$Ge$_{(y)}$T. Götzfried, A. Weber, D. Maurer, E.-W. Scheidt, S. Kehrein, K. Heuser and G. R. Stewart, J. Low Temp. Phys. 127, 51 (2002) The heavy-fermion system CePtSi is located close to a magnetic instability. A scaling analysis of the thermodynamic quantities at magnetic fields up to 10T and temperatures up to 6 K points to a quantum phase transition described by some unknown non-Gaussian fixed point. Order and non-Fermi-liquid behavior in$UCu_{4}Pd$A. Weber, S. Koerner, S. Kehrein, E.-W. Scheidt, and G. R. Stewart, Phys. Rev. B 63, 205116 (2001) We have studied the role of disorder in the non-Fermi-liquid system$UCu_{4}Pd$using annealing as a control parameter.Whereas the non-Fermi-liquid behavior in the specific heat can be observed over a larger temperature range after annealing, the clear non-Fermi-liquid behavior of the resistivity of the unannealed sample below 10 K disappears. We come to the conclusion that this argues against the Kondo disorder model as an explanation for the non-Fermi-liquid properties of both as-prepared and annealed$UCu_{4}Pd$. Flow equation analysis of the anisotropic Kondo model W. Hofstetter and S. Kehrein, Phys. Rev. B 63, 140402(R) (2001)  Universal scaling forms of the dynamical impurity susceptibility for different dissipation strengths$\alpha = (1 - \mathit{J_{\parallel}}/ 4\pi)^2$in the limit of small coupling$\mathit{J_{\perp}} : \omega_0$is defined by the maximum of the curves. One finds$\omega_0 \propto \mathit{T_K}$. We use the method of infinitesimal unitary transformations to calculate zero-temperature correlation functions in the strong-coupling phase of the anisotropic Kondo model. We find the dynamics on all energy scales including the crossover behavior from weak to strong coupling. Flow equation approach to the sine-Gordon model S. Kehrein, Nucl. Phys. B[FS] 592, 512 (2001)  Universal curves for the expansion parameter$\mathit{g}(\ell)$of the flow equation approach (179). The curves are from top to bottom for$\beta_{0}^{2} = 8\pi, 7\pi, 6\pi, 5\pi, 4\pi, 7\pi/2, 3\pi, 5\pi/2$. Notice that$\mathit{g}(\ell) = 0$for$\beta_{0}^{2} = 4\pi$since then our approach is exact. A continuous sequence of infinitesimal unitary transformations is used to diagonalize the quantum sine-Gordon model for$\beta^{2} \epsilon (2\pi, \infty)$. This approach can be understood as an extension of perturbative scaling theory since it links weak- to strong-coupling behavior in a systematic expansion: a small expansion parameter is identified and this parameter remains small throughout the entire flow unlike the diverging running coupling constant of perturbative scaling.We find very accurate results for the single-particle/hole spectrum in the strong-coupling phase and can describe the full crossover from weak to strong-coupling. Nuclear Spin Relaxation and Kondo-disorder in$UCu_{3.5}Pd_{1.5}$N. Buettgen, W. Trinkl, J.-E. Weber, J. Hemberger, A. Loidl and S. Kehrein, Phys. Rev. B 62, 11545 (2000)  Magnetization$\mathit{m_{DC}/H}$vs temperature$\mathit{T}$. The$^{63}$Cu spin-lattice relaxation rate$1/\mathit{T}_{1}$in$UCu_{3.5}Pd_{1.5}$has been determined at frequencies 38 MHz$\le v_{0} \le$101 MHz and for temperatures 0.4 K$ \le \mathit{T} \le$100 K.$UCu_{3.5}Pd_{1.5}$was one of the first compounds revealing non-Fermi-liquid behavior which was explained in terms of a distribution of Kondo temperatures$\mathit{T}_{K}$. In contrast to the broad distribution of single Kondo impurity temperatures$\mathit{T}_{K}$, which accounted for the bulk susceptibility$\chi$($\mathit{H,T}$), such a distribution$\mathit{P(T_K)}$does not describe the results of our$^{63}$Cu-NMR experiments performed at high applied external fields$H_{0}\ge$51 kOe. Flow equation solution for the weak to strong-coupling crossover in the sine-Gordon model S. Kehrein, Phys. Rev. Lett. 83, 4914 (1999)  Soliton mass as a function of the coupling constant for various values of$\beta_{0}^{2}$: The full lines are constrained fits of the power law behavior$\mathit{am} \propto \mathit{u_0}^{1/(2-\beta_{0}^{2}/4\pi)}$to the flow equation results (open circles) with the proportionality constant being fitted. The dashed line is the case$\beta_{0}^{2}$= 4$\pi$where the flow equation approach agrees trivially. A continuous sequence of infinitesimal unitary transformations, combined with an operator product expansion for vertex operators, is used to diagonalize the quantum sine-Gordon model. The leading order of this approximation already gives very accurate results for the single-particle gap in the strong-coupling phase. This approach can be understood as an extension of perturbative scaling theory since it links weak- to strong-coupling behavior in a systematic expansion. Absence of hysteresis at the Mott-Hubbard metal-insulator transition in infinite dimensions J. Schlipf, M. Jarrell, P. G. J. van Dongen, N. Bluemer, S. Kehrein, Th. Pruschke and D. Vollhardt, Phys. Rev. Lett. 82, 4890 (1999) The nature of the Mott-Hubbard metal-insulator transition in the infinite-dimensional Hubbard model is investigated by quantum Monte Carlo simulations down to temperature$\mathit{T} = \mathit{W}/140$($\mathit{W}$= bandwidth). The Mott-Hubbard Metal-Insulator Transition in the Limit of Large Dimensions - Insights and Outlook Symmetric Anderson impurity model with a narrow band W. Hofstetter and S. Kehrein, Phys. Rev. B 59, R12732 (1999) The single-channel Anderson impurity model is a standard model for the description of magnetic impurities in metallic systems. Usually, the bandwidth represents the largest energy scale of the problem. In this paper, we analyze the limit of a narrow band, which is relevant for the Mott-Hubbard transition in infinite dimensions. Density of states near the Mott-Hubbard transition in the limit of large dimensions S. Kehrein, Phys. Rev. Lett. 81, 3912 (1998)  Hypothetical density of states near a metal-insulator transition with vanishing spectral weight in the vicinity of the Fermi surface. The zero temperature Mott-Hubbard transition as a function of the Coulomb repulsion U is investigated in the limit of large dimensions. The behavior of the density of states near the transition at$\mathit{U} =\mathit{U_c}$is analyzed in all orders of the skeleton expansion. Diagonalization of system plus environment Hamiltonians S. Kehrein and A. Mielke, J. Stat. Phys. 90, 889 (1998)  Schematic behaviour of the effective spectral function$\mathit{J}(\omega, \ell)$for various regimes of the flow equations. A new approach to dissipative quantum systems modeled by a system plus environment Hamiltonian is presented. Using a continuous sequence of infinitesimal unitary transformations, the small quantum system is decoupled from its thermodynamically large environment. Dissipation enters through the observation that system observables generically “decay” completely into a different structure when the Hamiltonian is transformed into diagonal form. The method is particularly suited for studying low-temperature properties. This is demonstrated explicitly for the super-Ohmic spin-boson model. High-gradient operators in the$\mathit{N}$-vector model S. E. Derkachov, S. Kehrein and A. N. Manashov, Nucl. Phys. B[FS] 493, 660 (1997) It has been shown by several authors that a certain class of composite operators with many fields and gradients endangers the stability of non-trivial fixed points in$2 + \epsilon$expansions for various models. This problem is up to now unresolved. We investigate it in the$\mathit{N}$-vector model in an$1/\mathit{N}$expansion. By establishing an asymptotic naive addition law for anomalous dimensions we demonstrate that the first orders in the$2 + \epsilon$expansion can lead to erroneous interpretations for high-gradient operators. Low temperature equilibrium correlation functions in dissipative quantum systems S. Kehrein and A. Mielke, Ann. Physik (Leipzig) 6, 90 (1997) We introduce a new theoretical approach to dissipative quantum systems. By means of a continuous sequence of infinitesimal unitary transformations, we decouple the small quantum system that one is interested in from its thermodynamically large environment. This yields a trivial final transformed Hamiltonian. Dissipation enters through the observation that generically observables “decay” completely under these unitary transformations, i.e. are completely transformed into other terms. As a nontrivial example the spin-boson model is discussed in some detail. On the spin-boson model with a sub-ohmic bath S. Kehrein and A. Mielke, Physics Letters A 219, 313 (1996)  Decoupling of modes as a function of the flow parameter$\mathit{l}$in our approach,$0 < \mathit{l_1} < \mathit{l_2} < \mathit{l_3}$,$\mathit{J}$($\Delta_{\infty}, \mathit{l}$) shows an algebraic decay. We study the spin-boson model with a sub-ohmic bath using infinitesimal unitary transformations. Contrary to some results reported in the literature we find a zero temperature transition from an untrapped state for small coupling to a trapped state for strong coupling. We obtain an explicit expression for the renormalized level spacing as a function of the bare parameters of the system. Furthermore we show that thpical dynamical equilibrium correlation functions exhibit an algebraic decay at zero temperature. Theory of the Anderson impurity model: The Schrieffer-Wolff transformation reexamined S. Kehrein and A. Mielke, Ann. Phys. (New York) 252, 1 (1996)  Induced spin spin interaction in the symmetric case. We test the method of infinitesimal unitary transformations recently introduced by Wegner on the Anderson single impurity model. It is demonstrated that infinitesimal unitary transformations in contrast to the Schrieffer Wolff transformation allow the construction of an effective Kondo Hamiltonian consistent with the established results in this well understood model. The main reason for this is the intrinsic energy scale separation of Wegner's approach with respect to arbitrary energy differences coupled by matrix elements. This allows the construction of an effective Hamiltonian without facing a vanishing energy denominator problem. Similar energy denominator problems are troublesome in many models. Infinitesimal unitary transformations have the potential to provide a general framework for the systematic derivation of effective Hamiltonians without such problems. Flow equations for the spin-boson problem S. Kehrein, A. Mielke and P. Neu, Z. Phys. B 99, 269 (1996) Using continuous unitary transformations recently introduced by Wegner, we obtain flow equations for the parameters of the spin-boson Hamiltonian. The spectrum of critical exponents in ($\phi^2)^2$-theory in$d=4-\epsilon$dimensions - Resolution of degeneracies and hierarchical structures S. Kehrein, Nucl. Phys. B[FS] 453, 777 (1995) The spectrum of critical exponents of the$\mathit{N}$-vector model in$4 - \epsilon$dimensions is investigated to the second order in$\epsilon$. A generic class of one-loop degeneracies that has been reported in a previous work is lifted in two-loop order. One- and two-loop results lead to the conjecture that the spectrum possesses a remarkable hierarchical structure: The naive sum of any two anomalous dimensions generates a limit point in the spectrum, an anomalous dimension plus a limit point generates a limit point of limit points and so on. An infinite hierarchy of such limit points can be observed in the spectrum. The structure of the spectrum of anomalous dimensions in the$\mathit{N}$-vector model in$4 - \epsilon$dimensions S. Kehrein and F. Wegner, Nucl. Phys. B[FS] 424, 521 (1994) In a recent publication we have investigated the spectrum of anomalous dimensions for arbitrary composite operators in the critical$\mathit{N}$-vector model in$4 - \epsilon$dimensions. We could establish properties like upper and lower bounds for the anomalous dimensions in one-loop order. In this paper we extend these results and explicitly derive parts of the one-loop spectrum of anomalous dimensions. This analysis becomes possible by an explicit representation of the conformal symmetry group on the operator algebra. Flow equations for the Anderson Hamiltonian S. Kehrein and A. Mielke, J. Phys. A 27, 4259 (1994), arXiv:cond-mat/9405034 Using a continuous unitary transformation recently proposed by Wegner together with an approximation that neglects irrelevant contributions, we obtain flow equations for Hamiltonians. These flow equations yield a diagonal or almost diagonal Hamiltonian. Conformal symmetry and the spectrum of anomalous dimensions in the$\mathit{N}$-vector model in$4-\epsilon$dimensions S. Kehrein, F. Wegner and Y. Pismak, Nucl. Phys. B[FS] 402, 669 (1993) The subject of this paper is to study the critical N-vector model in$4—\epsilon$dimensions in one-loop order. We analyse the spectrum of anomalous dimensions of composite operators with an arbitrary number of fields and gradients.Thus one-loop contributions generally improve the stability of the nontrivial fixed point in contrast to some$2 + \epsilon\$ expansions.