Matsubara summation

General formalism The trick to evaluate Matsubara frequency summation is to use a Matsubara weighting function hη(z) that has simple poles located exactly at z=i\omega_n. By deformation of the contour lines to enclose the poles of g(z) (the green cross in Fig. 2), the summation can be formally accomplished by summing the residue of g(z)hη(z) over all poles of g(z), S_\eta=-\frac 1 \beta \sum_{z_0\in g(z)\text{ poles}} \operatorname{Res} g(z_0) h_\eta(z_0). Note that a minus sign is produced, because the contour is deformed to enclose the poles in the clockwise direction, resulting in the negative residue. Choice of Matsubara weighting function To produce simple poles on boson frequencies z=i\omega_n, either of the following two types of Matsubara weighting functions can be chosen h_{\rm B}^{(1)}(z)=\frac{\beta}{1-e^{-\beta z}}=-\beta n_{\rm B}(-z)=\beta(1+n_{\rm B}(z)), h_{\rm B}^{(2)}(z)=\frac{-\beta}{1-e^{\beta z}}=\beta n_{\rm B}(z), depending on which half plane the convergence is to be controlled in. h_{\rm B}^{(1)}(z) controls the convergence in the left half plane (Re z h_{\rm B}^{(2)}(z) controls the convergence in the right half plane (Re z > 0). Here n_{\rm B}(z)=(e^{\beta z}-1)^{-1} is the Bose–Einstein distribution function. The case is similar for fermion frequencies. There are also two types of Matsubara weighting functions that produce simple poles at z=i\omega_m h_{\rm F}^{(1)}(z)=\frac{\beta}{1+e^{-\beta z}}=\beta n_{\rm F}(-z)=\beta(1-n_{\rm F}(z)), h_{\rm F}^{(2)}(z)=\frac{-\beta}{1+e^{\beta z}}=-\beta n_{\rm F}(z). h_{\rm F}^{(1)}(z) controls the convergence in the left half plane (Re z h_{\rm F}^{(2)}(z) controls the convergence in the right half plane (Re z > 0). Here n_{\rm F} (z)=(e^{\beta z}+1)^{-1} is the Fermi–Dirac distribution function. In the application to Green's function calculation, g(z) always have the structure g(z)=G(z)e^{-z\tau}, which diverges in the left half plane given 0 h_\eta(z)=h_\eta^{(1)}(z). However, there is no need to control the convergence if the Matsubara summation does not diverge. In that case, any choice of the Matsubara weighting function will lead to identical results. Table of Matsubara frequency summations The following table contains S_\eta=\frac{1}{\beta}\sum_{i\omega}g(i\omega) for some simple rational functions g(z). The symbol η = ±1 is the statistical sign, +1 for bosons and −1 for fermions. == Applications in physics ==

Zero temperature limit In this limit \beta\rightarrow\infty, the Matsubara frequency summation is equivalent to the integration of imaginary frequency over imaginary axis. \frac{1}{\beta}\sum_{i\omega}=\int_{-i\infty}^{i\infty}\frac{\mathrm{d}(i\omega)}{2\pi i}. Some of the integrals do not converge. They should be regularized by introducing the frequency cutoff \Omega, and then subtracting the divergent part (\Omega-dependent) from the integral before taking the limit of \Omega\rightarrow\infty. For example, the free energy is obtained by the integral of logarithm, \eta \lim_{\Omega\rightarrow\infty}\left[ \int_{-i\Omega}^{i\Omega}\frac{\mathrm{d}(i\omega)}{2\pi i} \left(\ln(-i\omega+\xi)-\frac{\pi\xi}{2\Omega}\right)-\frac{\Omega}{\pi}(\ln\Omega-1)\right] =\left\{ \begin{array}{cc} 0 & \xi\geq0, \\ -\eta\xi & \xi meaning that at zero temperature, the free energy simply relates to the internal energy below the chemical potential. Also the distribution function is obtained by the following integral \eta \lim_{\Omega\rightarrow\infty} \int_{-i\Omega}^{i\Omega}\frac{\mathrm{d}(i\omega)}{2\pi i} \left(\frac{1}{-i\omega+\xi}-\frac{\pi}{2\Omega}\right) =\left\{ \begin{array}{cc} 0 & \xi\geq0, \\ -\eta & \xi which shows step function behavior at zero temperature. Green's function related Time domain Consider a function G(τ) defined on the imaginary time interval (0,β). It can be given in terms of Fourier series, G(\tau)=\frac{1}{\beta}\sum_{i\omega} G(i\omega) e^{-i\omega\tau}, where the frequency only takes discrete values spaced by 2/β. The particular choice of frequency depends on the boundary condition of the function G(τ). In physics, G(τ) stands for the imaginary time representation of Green's function G(\tau)=-\langle \mathcal{T}_\tau \psi(\tau)\psi^*(0) \rangle. It satisfies the periodic boundary condition G(τ+β)=G(τ) for a boson field. While for a fermion field the boundary condition is anti-periodic G(τ + β) = −G(τ). Given the Green's function G(iω) in the frequency domain, its imaginary time representation G(τ) can be evaluated by Matsubara frequency summation. Depending on the boson or fermion frequencies that is to be summed over, the resulting G(τ) can be different. To distinguish, define G_\eta(\tau)= \begin{cases} G_{\rm B}(\tau), & \text{if } \eta = +1, \\ G_{\rm F}(\tau), & \text{if } \eta = -1, \end{cases} with G_{\rm B}(\tau)=\frac{1}{\beta}\sum_{i\omega_n}G(i\omega_n)e^{-i\omega_n\tau}, G_{\rm F}(\tau)=\frac{1}{\beta}\sum_{i\omega_m}G(i\omega_m)e^{-i\omega_m\tau}. Note that τ is restricted in the principal interval (0,β). The boundary condition can be used to extend G(τ) out of the principal interval. Some frequently used results are concluded in the following table. Operator switching effect The small imaginary time plays a critical role here. The order of the operators will change if the small imaginary time changes sign. \langle \psi\psi^*\rangle=\langle \mathcal{T}_\tau \psi(\tau=0^+) \psi^*(0)\rangle =-G_\eta(\tau=0^+)=-\frac{1}{\beta}\sum_{i\omega}G(i\omega)e^{-i\omega 0^+} \langle \psi^*\psi\rangle=\eta\langle \mathcal{T}_\tau \psi(\tau=0^-) \psi^*(0)\rangle =-\eta G_\eta(\tau=0^-)=-\frac{\eta}{\beta}\sum_{i\omega}G(i\omega)e^{i\omega 0^+} Distribution function The evaluation of distribution function becomes tricky because of the discontinuity of Green's function G(τ) at τ = 0. To evaluate the summation G(0) = \sum_{i\omega}(i\omega-\xi)^{-1}, both choices of the weighting function are acceptable, but the results are different. This can be understood if we push G(τ) away from τ = 0 a little bit, then to control the convergence, we must take h_\eta^{(1)}(z) as the weighting function for G(\tau=0^+), and h_\eta^{(2)}(z) for G(\tau=0^-). Bosons G_{\rm B}(\tau=0^-)=\frac{1}{\beta}\sum_{i\omega_n}\frac{e^{i\omega_n 0^+}}{i\omega_n-\xi}=-n_{\rm B}(\xi), G_{\rm B}(\tau=0^+)=\frac{1}{\beta}\sum_{i\omega_n}\frac{e^{-i\omega_n 0^+}}{i\omega_n-\xi}=-(n_{\rm B}(\xi)+1). Fermions G_{\rm F}(\tau=0^-)=\frac{1}{\beta}\sum_{i\omega_m}\frac{e^{i\omega_m 0^+}}{i\omega_m-\xi}=n_{\rm F}(\xi), G_{\rm F}(\tau=0^+)=\frac{1}{\beta}\sum_{i\omega_m}\frac{e^{-i\omega_m 0^+}}{i\omega_m-\xi}=n_{\rm F}(\xi)-1. Free energy Bosons \frac{1}{\beta}\sum_{i\omega_n} \ln(\beta(-i\omega_n+\xi))=\frac{1}{\beta}\ln(1-e^{-\beta\xi}), Fermions -\frac{1}{\beta}\sum_{i\omega_m} \ln(\beta(-i\omega_m+\xi))=-\frac{1}{\beta}\ln(1+e^{-\beta\xi}). Diagram evaluations Frequently encountered diagrams are evaluated here with the single mode setting. Multiple mode problems can be approached by a spectral function integral. Here \omega_m is a fermionic Matsubara frequency, while \omega_n is a bosonic Matsubara frequency. Fermion self energy \Sigma(i\omega_m)=-\frac{1}{\beta }\sum _{i \omega_n } \frac{1}{i \omega_m +i \omega_n -\varepsilon }\frac{1}{i \omega_n -\Omega }=\frac{n_{\rm F}(\varepsilon )+n_{\rm B}(\Omega )}{i \omega_m -\varepsilon +\Omega }. Particle-hole bubble \Pi (i \omega_n )=\frac{1}{\beta }\sum _{i \omega_m } \frac{1}{i \omega_m +i \omega_n -\varepsilon }\frac{1}{i \omega_m -\varepsilon '}=-\frac{n_{\rm F}(\varepsilon )-n_{\rm F} \left(\varepsilon '\right)}{i \omega_n -\varepsilon +\varepsilon'}. Particle-particle bubble \Pi (i \omega_n )=-\frac{1}{\beta }\sum _{i \omega_m } \frac{1}{i \omega_m +i \omega_n -\varepsilon }\frac{1}{-i \omega_m -\varepsilon '}=\frac{1-n_{\rm F}\left(\varepsilon '\right) - n_{\rm F}(\varepsilon )}{i \omega_n -\varepsilon -\varepsilon '}. == Appendix: Properties of distribution functions ==

Distribution functions The general notation n_\eta stands for either Bose (η = +1) or Fermi (η = −1) distribution function n_\eta(\xi)=\frac{1}{e^{\beta\xi}-\eta}. If necessary, the specific notations nB and nF are used to indicate Bose and Fermi distribution functions respectively n_\eta(\xi)= \begin{cases} n_{\rm B}(\xi), & \text{if } \eta = +1, \\ n_{\rm F}(\xi), & \text{if } \eta = -1. \end{cases} Relation to hyperbolic functions The Bose distribution function is related to hyperbolic cotangent function by n_{\rm B}(\xi)=\frac{1}{2}\left(\operatorname{coth}\frac{\beta\xi}{2}-1\right). The Fermi distribution function is related to hyperbolic tangent function by n_{\rm F}(\xi)=\frac{1}{2}\left(1-\operatorname{tanh}\frac{\beta\xi}{2}\right). Parity Both distribution functions do not have definite parity, n_\eta(-\xi)=-\eta-n_\eta(\xi). Another formula is in terms of the c_\eta function n_\eta(-\xi)=n_\eta(\xi)+2\xi c_\eta(0,\xi). However their derivatives have definite parity. Bose–Fermi transmutation Bose and Fermi distribution functions transmute under a shift of the variable by the fermionic frequency, n_\eta(i\omega_m+\xi)=-n_{-\eta}(\xi). However shifting by bosonic frequencies does not make any difference. Derivatives First order n_{\rm B}^\prime(\xi)=-\frac{\beta}{4}\mathrm{csch}^2\frac{\beta \xi}{2}, n_{\rm F}^\prime(\xi)=-\frac{\beta}{4}\mathrm{sech}^2\frac{\beta \xi}{2}. In terms of product: n_\eta^\prime(\xi)= -\beta n_\eta(\xi)(1+\eta n_\eta(\xi)). In the zero temperature limit: n_\eta^\prime(\xi)=\eta\delta(\xi) \text{ as } \beta\rightarrow\infty. Second order n_{\rm B}^{\prime\prime}(\xi)=\frac{\beta^2}{4}\operatorname{csch}^2\frac{\beta \xi}{2}\operatorname{coth}\frac{\beta \xi}{2}, n_{\rm F}^{\prime\prime}(\xi)=\frac{\beta^2}{4}\operatorname{sech}^2\frac{\beta \xi}{2}\operatorname{tanh}\frac{\beta \xi}{2}. Formula of difference n_\eta(a+b)-n_\eta(a-b)=-\frac{\mathrm{sinh}\beta b}{\mathrm{cosh}\beta a-\eta\,\mathrm{cosh}\beta b}. ==== Case a = 0 ==== n_{\rm B}(b)-n_{\rm B}(-b)=\mathrm{coth}\frac{\beta b}{2}, n_{\rm F}(b)-n_{\rm F}(-b)=-\mathrm{tanh}\frac{\beta b}{2}. Case a → 0 n_{\rm B}(a+b)-n_{\rm B}(a-b)=\operatorname{coth}\frac{\beta b}{2}+n_{\rm B}^{\prime\prime}(b)a^2+\cdots, n_{\rm F}(a+b)-n_{\rm F}(a-b)=-\operatorname{tanh}\frac{\beta b}{2}+n_{\rm F}^{\prime\prime}(b)a^2+\cdots. Case b → 0 n_{\rm B}(a+b)-n_{\rm B}(a-b)=2n_{\rm B}^\prime(a)b+\cdots, n_{\rm F}(a+b)-n_{\rm F}(a-b)=2n_{\rm F}^\prime(a)b+\cdots. The function cη Definition: c_\eta(a,b)\equiv-\frac{n_\eta(a+b)-n_\eta(a-b)}{2b}. For Bose and Fermi type: c_{\rm B}(a,b)\equiv c_+(a,b), c_{\rm F}(a,b)\equiv c_-(a,b). Relation to hyperbolic functions c_\eta(a,b)=\frac{\sinh\beta b}{2b(\cosh\beta a-\eta\cosh\beta b)}. It is obvious that c_{\rm F}(a,b) is positive definite. To avoid overflow in the numerical calculation, the tanh and coth functions are used c_{\rm B}(a,b)=\frac{1}{4b}\left(\operatorname{coth}\frac{\beta(a-b)}{2} - \operatorname{coth}\frac{\beta(a+b)}{2}\right), c_{\rm F}(a,b)=\frac{1}{4b}\left(\operatorname{tanh}\frac{\beta(a+b)}{2} - \operatorname{tanh}\frac{\beta(a-b)}{2}\right). ==== Case a = 0 ==== c_{\rm B}(0,b)=-\frac{1}{2b}\operatorname{coth}\frac{\beta b}{2}, c_{\rm F}(0,b)=\frac{1}{2b}\operatorname{tanh}\frac{\beta b}{2}. ==== Case b = 0 ==== c_{\rm B}(a,0)=\frac{\beta}{4}\operatorname{csch}^2\frac{\beta a}{2}, c_{\rm F}(a,0)=\frac{\beta}{4}\operatorname{sech}^2\frac{\beta a}{2}. Low temperature limit For a = 0: c_{\rm F}(0,b)=\frac{1}{2|b|}. For b = 0: c_{\rm F}(a,0)=\delta(a). In general, c_{\rm F}(a,b)=\begin{cases} \frac{1}{2|b|}, & \text{if } |a||b| \end{cases} == See also ==