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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Renormalization group in quantum mechanics at zero and finite temperature

Pierre Gosselin, Hervé Mohrbach, Alain Bérard

Published Sep 26, 2001• 3 citations
Publisher

Abstract

We apply the renormalization group formalism, to integrate quantum fluctuations of quantum mechanical systems at zero and finite temperature. At zero temperature a nonperturbative renormalization group equation allows to compute the ground state energy whereas at finite temperature a variational renormalization group equation is proposed to compute the free energy.

DOI
10.1103/physreve.64.046129

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We apply the renormalization group formalism, to integrate quantum fluctuations of quantum mechanical systems at zero and finite temperature. At zero temperature a nonperturbative renormalization group equation allows to compute the ground state energy whereas at finite temperature a variational renormalization group equation is proposed to compute the free energy.
It is shown that some recently proposed iterative approaches to the ground state of quantum systems can be obtained as zero-temperature limits of free-energy-preserving renormalization transformations, if the perturbative expansion is handled in the appropriate way. Besides giving new insight into the ground-state approaches and their mutual relationships, these results allow us to perform consistent approximate calculations of the free energy at all temperatures and to get global descriptions of the critical properties. Free-energy and specific-heat calculations are reported for XY\mathrm{XY}XY and Heisenberg spin-\textonehalf{} chains and for the triangular XY\mathrm{XY}XY model. It is also demonstrated how this extension into finite temperatures allows us to compute in a consistent way the zzz exponent, and to obtain substantial improvement in the numerical values for the ground-state energy density of the transverse Ising model.
At temperatures low compared with characteristic atomic frequencies, classical free energy functionals where the order parameter field ψ which depends only on position are inadequate to describe fluctuation phenomena; the noncommutativity of quantum‐mechanical operators leads to a ψ which also depends on time. We show how to generalize the renormalization group to such functionals. For phase transitions at T=0 caused by variation of a coupling parameter, the boundary between classical and nonclassical exponent behavior is at 4‐z, not 4 dimensions, where z≳1 depends on the dynamics of the system: z=1 for van Vleck ferromagnets, z=2 for itinerant antiferomagnets, z=3 for clean and z=4 for dirty itinerant ferromagnets. For small Tc there is a crossover region where the renormalization group equations go from their T=0 form to a Wilson form, implying a large precritical region where classical exponents are correct and a small true critical region. One also has an explicit construction of the Wilson free energy functional valid in the critical region.
We systematically explore and show the existence of finite-temperature continuous quantum phase transition (CTQPT) at a critical point, namely, during solidification or melting such that the first-order thermal phase transition is a special case within CTQPT. Infact, CTQPT is related to chemical reaction where quantum fluctuation (due to wavefunction transformation) is caused by thermal energy and it can occur maximally for temperatures much higher than zero Kelvin. To extract the quantity related to CTQPT, we use the ionization energy theory and the energy-level spacing renormalization group method to derive the energy-level spacing entropy, renormalized Bose-Einstein distribution and the time-dependent specific heat capacity. This work unambiguously shows that the quantum phase transition applies for any finite temperatures.
We revisit the two-dimensional quantum Ising model by computing renormalization group flows close to its quantum critical point. The low but finite temperature regime in the vicinity of the quantum critical point is squashed between two distinct non-Gaussian fixed points: the classical fixed point dominated by thermal fluctuations and the quantum critical fixed point dominated by zero-point quantum fluctuations. Truncating an exact flow equation for the effective action, we derive a set of renormalization group equations and analyze how the interplay of quantum and thermal fluctuations, both non-Gaussian in nature, influences the shape of the phase boundary and the region in the phase diagram where critical fluctuations occur. The solution of the flow equations makes this interplay transparent: we detect finite temperature crossovers by computing critical exponents and we confirm that the power law describing the finite temperature phase boundary as a function of control parameter is given by the correlation length exponent at zero temperature as predicted in an ϵϵϵ-expansion with ϵ=1ϵ=1ϵ=1 by Sachdev [Phys. Rev. B 55, 142 (1997)].
The physics of interacting Fermi systems is extremely sensitive to the energy scale. Of particular interest is the low energy regime where correlation induced collective behavior emerges. The theory of interacting Fermi systems is confronted with the occurrence of very different phenomena along a continuum of scales calling for methods capable of computing physical observables as a function of energy scale. In this thesis, we perform a comprehensive renormalization group analysis of two and three-dimensional Fermi systems at low and zero temperature. We examine systems with spontaneous symmetry-breaking and quantum critical behavior by deriving and solving flow equations within the functional renormalization group framework. We extend the Hertz-Millis theory of quantumphase transitions in itinerant fermion systems to phases with discrete and continuous symmetry-breaking, and to quantum critical points where the zero temperature theory is associated with a non-Gaussian fixed point. The order parameter is implemented by a bosonic Hubbard-Stratonovich field, which - for continuous symmetry-breaking - splits into two components corresponding to longitudinal and transversal Goldstone fluctuations. We compute the finite temperature phase boundary near the quantum critical point explicitly including non-Gaussian fluctuations. We then set up a coupled fermion-boson renormalization group theory that captures the mutual interplay of gapless fermions with massless order parameter fluctuations when approaching a quantum critical point. As a first application, we compute the complete set of quantum critical exponents at the semimetal-to-superfluid quantum phase transition of attractively interacting Dirac fermions in two dimensions. Both, the order parameter propagator and the fermion propagator become nonanalytic functions of momenta destroying the Fermi liquid behavior. We finally compute the effects of quantum fluctuations in the superfluid ground state of an attractively interacting Fermi system, employing the attractive Hubbard model as a prototype. The flow equations capture the influence of longitudinal and Goldstone order parameter fluctuations on non-universal quantities such as the fermionic gap and the fermion-boson vertex, as well as the exact universal infrared asymptotics present in every fermionic superfluid.