On the Dynamics of Free-Fermionic Tau-Functions at Finite Temperature

Abstract

In this work we explore an instance of the $τ$-function of vertex type operators, specified in terms of a constant phase shift in a free-fermionic basis. From the physical point of view this $τ$-function has multiple interpretations: as a correlator of Jordan-Wigner strings, a Loschmidt Echo in the Aharonov-Bohm effect, or the generating function of the local densities in the Tonks-Girardeau gas. We present the $τ$-function as a form-factors series and tackle it from four vantage points: (i) we perform an exact summation and express it in terms of a Fredholm determinant in the thermodynamic limit, (ii) we use bosonization techniques to perform partial summations of soft modes around the Fermi surface to acquire the scaling at zero temperature, (iii) we derive large space and time asymptotic behavior for the thermal Fredholm determinant by relating it to effective form-factors with an asymptotically similar kernel, and (iv) we identify and sum the important basis elements directly through a tailor-made numerical algorithm for finite-entropy states in a free-fermionic Hilbert space. All methods confirm each other. We find that, in addition to the exponential decay in the finite-temperature case the dynamic correlation functions exhibit an extra power law in time, universal over any distribution and time scale.

Figures (21)

• Figure 1: Setup for the Aharonov-Bohm effect. A cylindrical region in the center has a constant magnetic field $\mathcal{B}$ going into the page. Its circular intersection is shaded red. Strictly outside it, our loop of circumference $L$ lies in in the horizontal plane, in the page. On the loop, a number of fermions are schematically indicated in orange.
• Figure 2: Momentum space picture of unshifted and shifted $N=9$-particle ground states $\ket{\mathbf{g}}$ and $\ket{\mathbf{h}}$, respectively. $g_a,h_a$ are the single particle momenta.
• Figure 3: Diagonal overlap scaling plotted against system size on semi-logarithmic axes. Included are the exact values for the ground state, \ref{['eq:lotsofGs']}, in black. Then for $T=\frac{1}{2}$ in blue and $T=\frac{3}{2}$ in red, the solid disks are the geometric mean of 1000 states sampled numerically from the Gibbs ensemble at that $N$. Compare them to the dotted lines, which are the simple prediction as in \ref{['eq:poop']}. It is clear that for finite temperature, scaling is essentially exponential, for zero temperature sub-exponential. $\nu=0.4, L=N$.
• Figure 4: Contour $\gamma$ from \ref{['eq:AAij']}. It may be taken counter-clockwise, a distance $\epsilon$ around the whole real line. However, to compensate for picking up the pole at $k\to g=g_a=g_b\in \frac{2\pi}{L}\mathbb{Z}$, we add the reverse contribution there. Red dots symbolize poles $k\to\frac{2\pi}{L}(n-\nu), n\in\mathbb{Z}$.
• Figure 5: Plots of the real part of object $\erf\left(\frac{(i+1)(x-gt)}{2\sqrt{t}}\right)$, at $g=3$, for different times $t$. The function approximates $2\theta(x-gt)-1$. As $t\to 0$, the oscillations condense and the limit becomes exact. The imaginary part (not shown) oscillates around zero.