Remark on the effective action of three-dimensional QED at finite temperature

Kim, Won‐Tae; Park, Young-Jai; Kim, Kee Yong; Kim, Yong-Duk

doi:10.1103/physrevd.46.3674

Cited by 8 publications

(3 citation statements)

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“…In the non-Abelian case, for example, general arguments imply that the coefficient of the CS term must be quantized at zero temperature [5], and also at non-zero temperature [6,7], if the action is to be invariant under topologically non-trivial ('large') gauge transformations. On the other hand, simple perturbative calculations [8][9][10][11][12][13][14][15][16][17], give the result that the effect of moving to T = 0 is simply to multiply the zerotemperature CS term by a smoothly varying function of T (typically, tanh( β|M| 2 ), where β = 1 kT and M is the mass of the fermion(s) in the theory). Plainly, the perturbative result contradicts the quantization requirement [18][19][20].…”

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Gauge invariance and effective actions inD=3at finite temperature

Aitchison

Fosco

1998

Phys. Rev. D

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For background gauge field configurations reducible to the form Aµ = (Ã3, A( x)) whereÃ3 is a constant, we provide an elementary derivation of the recently obtained result for the exact induced Chern-Simons (CS) effective action in QED3 at finite temperature. The method allows us to extend the result in several useful ways: to obtain the analogous result for the 'mixed' CS term in the Dorey-Mavromatos model of parity-conserving planar superconductivity, thereby justifying their argument for flux quantization in the model; to the induced CS term for a τ -dependent flux; and to the term of second order in A( x) (and all orders inÃ3) in the effective action.PACS numbers: 11.10.Wx 11.15 11.30.Er I. INTRODUCTION.Recently, there has been significant progress [1][2][3][4] in resolving a puzzle concerning the gauge invariance of induced Chern-Simons (CS) terms at finite temperature, T . In the non-Abelian case, for example, general arguments imply that the coefficient of the CS term must be quantized at zero temperature [5], and also at non-zero temperature [6,7], if the action is to be invariant under topologically non-trivial ('large') gauge transformations. On the other hand, simple perturbative calculations [8][9][10][11][12][13][14][15][16][17], give the result that the effect of moving to T = 0 is simply to multiply the zerotemperature CS term by a smoothly varying function of T (typically, tanh( β|M| 2 ), where β = 1 kT and M is the mass of the fermion(s) in the theory). Plainly, the perturbative result contradicts the quantization requirement [18][19][20]. A similar difficulty arises in the Euclidean case for T = 0 due to the S 1 topology of the compactified Euclidean time.Apart from its theoretical interest, the resolution of this puzzle is important in some physical applications. To give one specific example, consider the Dorey-Mavromatos (DM) model [21] of two-dimensional superconductivity without parity violation. This model employs two U (1) gauge fields, one the electromagnetic field A µ , the other a 'statistical' gauge field a µ , which is also massless.There are N f ≥ 2 flavours of four-component fermions, the mass term is parity conserving, and A µ and a µ have opposite parity. At zero temperature a 'mixed Chern-Simons (MCS) term' is generated by a fermion loop with one external A and one external a leg, the leading contribution to the action, in powers of derivatives, being * CONICET

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Gauge invariance and effective actions inD=3at finite temperature

Aitchison

Fosco

1998

Phys. Rev. D

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“…The determinant corresponding to the n-mode is again written as in eq. ( 13) and we can perform the two-dimensional chiral rotation (14). The x-dependence of the phase factor φ n produces in this case a different anomalous Jacobian,…”

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Induced Parity-Breaking Term at Finite Temperature

1997

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We compute the exact induced parity-breaking part of the effective action for 2 + 1 massive fermions in QED3 at finite temperature by calculating the fermion determinant in a particular background. The result confirms that gauge invariance of the effective action is respected even when large gauge transformations are considered.PACS numbers: 11.10.Wx 11.15 11.30.ErBecause of its relevance both in Field Theory and Condensed Matter physics, much effort has been devoted in the last decade to the study of three dimensional gauge theories coupled to matter. An important ingredient in this theories is the parity anomaly [1] which induces, through fluctuation of massive Fermi fields, a Chern-Simons term in the effective action of the gauge field [2].As originally stressed in [1], a fundamental property of the Chern-Simons (CS) action is the existence of a quantization law: due to the non-invariance of the CS term under gauge transformations of "non-zero" winding number, the coefficient which appears in front of the (non-Abelian) Chern-Simons three form S CS should be quantized so that exp(iS CS ) is singled valued. Even in the Abelian case "large" gauge transformations come into play whenever the theory is formulated in an appropriately compactified manifold [3]- [4] and in that case the quantization law also holds. Putting together all these facts one can state that in three dimensional gauge theories with Fermi fields, calculations of the effective action for the gauge field using gauge-invariant regularizations lead to a parity anomaly which manifests through the occurrence of a CS term with a quantized coefficient which depends on the number of fermion species.A natural question raised when the analysis of three dimensional gauge theories was extended to the case of finite temperature was whether quantization of the CS coefficient induced by fermion fluctuations survives the effects of temperature or it is smoothly renormalized. The question was originally discussed in [5] where it was argued that the coefficient of the CS term in the effective action for the gauge field should remain unchanged at finite temperature. Contrasting with this analysis, perturbative calculations for both relativistic and non-relativistic theories, abelian and non-abelian, have yielded induced actions with CS coefficients which are smooth functions of the temperature [6]- [15], this seeming to signal a kind of gauge anomaly at finite temperature.The problem of renormalization of the CS coefficient induced by fermions at T = 0 was revisited in refs.[16]-[17] where it was concluded that, on gauge invariance grounds and in perturbation theory, the effective action for the gauge field cannot contain a smoothly renormalized CS coefficient at non-zero temperature. More recently, the exact result for the effective action of a 0 + 1 analogue of the CS system [18] as well as a zeta-function analysis of the fermion determinant at T = 0 in the 2 + 1 model [19] have explicitly shown that although the perturbative expansion leads to a smooth temperat...

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“…The CS term at finite temperature was obtained by means of the derivative expansion [35] as well as other techniques [36][37][38][39][40][41][42]. Although there was strong evidence that the CS coefficient for a non-abelian gauge field should be quantized at finite temperature [38], it was found to depend smoothly on the temperature, leading to the conjecture that it would not depend on the temperature at all [38].…”

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confidence: 99%

Effective action for QED in2+1dimensions at finite temperature

Hott

Metikas

1999

Phys. Rev. D

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Both the parity-breaking and parity-invariant parts of the effective action for the gauge field in QED 3 with massive fermions at finite temperature are obtained exactly. This is feasible because we use a particular configuration of the background gauge field, namely a constant magnetic field and a time-dependent time component of the background gauge field. Our results allow us to compute exactly physically interesting quantities such as the induced charge density and fermion condensate whose dependence on the temperature, fermion mass and gauge field is discussed. ͓S0556-2821͑99͒03416-5͔PACS number͑s͒: 12.20. Ds, 11.10.Kk, 11.10.Wx, 11.30.Rd Many aspects of quantum electrodynamics in 2ϩ1 dimensions (QED 3 ) ͓1͔ in the presence of external fields have already been studied. The dynamical generation of a ChernSimons ͑CS͒ term for the gauge field ͓2͔, the formation of a fermion condensate in the presence of an external magnetic field ͓3,4͔ and the breaking of Lorentz symmetry ͓5,4͔ are some of the many features exhibited by QED 3 . Some of these aspects are desirable in 3ϩ1 dimensions and also very important in the study of superconductivity ͓6-9͔ and the quantum Hall effect ͓10͔ in planar systems.In order to carry the study of such phenomena further it is important to consider the effects of temperature; these might sometimes be completely different from one's expectations. Particularly, it has been shown that the development of the fermion condensate catalyzed by a magnetic field is unstable at finite temperature ͓11͔ and that the issue of gauge invariance at finite temperature is only consistently addressed if a re-summation of the graphs is carried out ͓12-14͔. This question has also been extensively reported using different techniques in the context of QED 3 ͓15-20͔ as well as by means of an analogous quantum mechanical model which allows one to perform a comparative analysis of the perturbation series with exact results ͓21,22͔-in fact this was the approach used in one of the seminal works on the subject ͓12͔.Although the exact contribution of the time component of the gauge field has been considered only for the parity violating term of the effective action, it also contributes to the parity invariant one and this may result in extra effects on quantities such as the fermion condensate and the density of charge generated dynamically. To explore such effects is the main purpose of this article. By making use of the FockSchwinger proper-time technique ͓23͔ we compute the oneloop effective action at finite temperature for a specific configuration, namely a constant magnetic field and a timedependent time-component of the gauge field. We also address the aspects of gauge invariance and temperature dependence of the thermal condensate and the charge density. With this program our results go beyond those obtained in ͓18͔, where the same technique was used.In QED the one-loop effective action for the gauge field is obtained by integrating out the fermionswhich can be calculated exactly only for some configurati...

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Remark on the effective action of three-dimensional QED at finite temperature

Abstract: From three-dimensional massive QED, we obtain the temperature-dependent Chern-Simons action by calculating the vacuum current at finite temperature.

Cited by 8 publications

References 9 publications

Gauge invariance and effective actions inD=3at finite temperature

Gauge invariance and effective actions inD=3at finite temperature

Induced Parity-Breaking Term at Finite Temperature

Effective action for QED in2+1dimensions at finite temperature

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