Phase boundary and finite temperature crossovers of the quantum Ising model in two dimensions

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 $ϵ$-expansion with $ϵ=1$ by Sachdev [Phys. Rev. B 55, 142 (1997)].

Figure 1

(Color online) Flows for the quantum Ising model in $d=2$ as a function of logarithmic cutoff-scale $s=-\text{log}\left[\Lambda /{\Lambda }_{\text{UV}}\right]$ for various temperatures and at $T=0$. We set ${\Lambda }_{\text{UV}}=1$. The infrared (ultraviolet) is to the right (left) of the graphs. The values of the classical fixed point, attained by all finite-$T$ flows, and the quantum critical fixed point, attained by the zero-temperature flow, are marked on the vertical axis. (a): Flows of the quartic self-interaction $\tilde{u}$. (b): Corresponding flows of the rescaled minimum of the effective potential $\tilde{\rho }$. (c): Flows of the anomalous dimension $\eta$.

Figure 2

(Color online) Emergence of three different regimes in the phase diagram as illustrated with three different values of the order parameter exponent $\beta$ upon approaching the QCP at very low temperatures. Here, we set $T=1.7\times {10}^{-5}$.

Figure 3

(Color online) Crossover behavior of the anomalous dimension as a function of scale $s=-\text{log}\left[\Lambda /{\Lambda }_{\text{UV}}\right]$ for $T=1\times {10}^{-6}$. Upon increasing $T$, this curve will continuously deform toward the shape of the blue dashed-dotted line in Fig. 1c.

Figure 4

(Color online) Schematic plot of the Ginzburg-line (dashed, blue), the crossover line (dotted, black) from $3D$-Ising to $2D$-Ising behavior, and the true $T_c$-line (straight, red) for the quantum Ising model in $d=2$. Units are arbitrary and will depend on the microscopic details of the specific system under investigation. The symmetry-broken phase is the area to the right of the $T_c$-line.