High-temperature criticality in strongly constrained quantum systems

The exotic nature of many strongly correlated materials at reasonably high temperatures—for instance, cuprate superconductors in their normal state—has led to the suggestion that such behavior occurs within a quantum-critical region where the physics is controlled by the influence of a phase transition down at zero temperature. Such a scenario can be thought of as a bottom-up approach, with the zero-temperature mechanisms finding a way to manifest critical behavior at high temperatures. Here we propose an alternative, top-down, mechanism by which strong kinematic constraints that can only be broken at extremely high temperatures are responsible for critical behavior at intermediate but still high temperatures. This critical behavior may extend all the way down to zero temperature, but this outcome is not one of necessity, and the system may order at low temperatures. We provide explicit examples of such high-temperature criticality when additional strong interactions are introduced in quantum Heisenberg, transverse-field Ising, and some bosonic lattice models.

Figure 1

(Color online) Generic phase diagram for a strongly constrained quantum system that satisfies conditions (1)–(6) in Sec. 1. The parameter space encodes the competition between two energy scales, the temperature $T$, and a characteristic energy scale $g$ that selects a classical ordered phase, while ${\Gamma }_{\mathrm{eff}}$ and $U$ are held fixed. A quantum-critical scaling regime, if it exists, is restricted to a region represented by the upper half of a disk of radius $\sim \mid {\Gamma }_{\mathrm{eff}}\mid$ and centered at the origin $(g,T)=(0,0)$ of parameter space. The focus of this paper is on the constrained entropic scaling regime at temperatures intermediate between the small characteristic quantum energy scale ${\Gamma }_{\mathrm{eff}}$ and the large characteristic energy scale $U$ set by a strong constraint. At a fixed temperature, the constrained entropic scaling regime terminates in a phase transition that needs not be continuous upon increasing $\mid g\mid$. At fixed $g$, the constrained entropic scaling regime crosses over to the conventional high-temperature phase—say, a paramagnetic one for spin degrees of freedom—when $T$ is of the order of $U$. The transition from the constrained entropic scaling regime to the quantum regime upon lowering $T$ at fixed $g$ is system specific.

Figure 2

(Color online) Qualitative phase diagram of the classical constrained system assuming the existence of two phases, a classical ordered phase and a disordered one, separated by a phase boundary that extends all the way to $T=0$. Not all of the phase boundary between the ordered phase and the paramagnetic phase is shown here. The topology of the phase boundary upon approaching $T=0$ depends on the microscopics, as we imply by depicting the small-$T$ side of the phase boundary with a dotted segment. The dashed lines represent curves where $g$ and $U$ are held fixed and only the temperature is varied. These curves originate at the ordered fixed point $(\infty ,0)$ at $T=0$ and end at the disordered and unconstrained entropic fixed point $(0,\infty )$ at $T=\infty$. In the limit of $g∕U⪡1$, these curves become infinitesimally close to the $g∕T$ and $T∕U$ semiaxes, and the classification discussed in this paper applies. The universal physics occurs in regime (3.2c); therefore, in the region $T∕U⪡1$, $g∕T⪡1$ close to the origin of the coordinate system. The shaded region represents the scaling entropic region appearing in a constrained system due to the proximity to the constrained entropic critical point at the origin. (Top) This diagram encompasses scenarios I, III, and IV depending on the behavior of the system along the segment $0⩽g∕T⩽{(g∕T)}_{\mathrm{c}}$ with $T∕U=0$. (Bottom) Phase diagram corresponding to scenario II, where ${(g∕T)}_{\mathrm{c}}=0$.

Figure 3

(Color online) (Top) Behavior of the (finite-size) central charge $c_{L_i,L_{i+1}}$ as a function of $h∕T$, obtained from finite-size scaling of the free energy computed via transfer matrix. From top to bottom, the different curves correspond to increasing diameters $L_j=4,\dots ,14$ for the systems used in the scaling fit to compute the value of the central charge $c_{L_i,L_{i+1}}$. The system in presence of a field is symmetric upon the sign change $h\leftrightarrow -h$; therefore, only the $h>0$ axis is shown. Notice that the high-temperature criticality is robust with respect to a uniform field and it survives at large but finite temperatures. (Bottom) Behavior of the two smallest scaling dimensions allowed by the conformal field theory as a function of $h∕T$. The extrapolated values for $L\to \infty$ are shown here for simplicity. The fact that both (degenerate) scaling dimensions remain constant in the critical regime suggests that the whole critical phase is described by the same conformal field theory. The lines between data points are guides to the eyes.

Figure 4

(Color online) (Top) Behavior of the (finite-size) central charge $c_{L_i,L_{i+1}}$ as a function of $J∕T$, obtained from finite-size scaling of the free energy computed via transfer matrix. From top to bottom, the different curves correspond to increasing diameters $L_j=4,\dots ,14$ for the systems used in the scaling fit to compute the value of the central charge $c_{L_i,L_{i+1}}$. Positive values of $J$ correspond to a ferromagnetic (F) coupling, while negative values correspond to an antiferromagnetic (AF) coupling. Notice that the high-temperature criticality is robust with respect to uniform nearest-neighbor interactions and it survives at large but finite temperatures both for positive and negative $J$. Antiferromagnetic interactions, however, deeply affect the critical behavior, inducing what appears to be a continuously varying central charge. (Bottom) Behavior of the two smallest scaling dimensions allowed by the conformal field theory as a function of $J∕T$. The extrapolated values for $L\to \infty$ are shown here for simplicity. Contrary to the uniform field case, the scaling dimensions vary on the AF side ${(J∕T)}_{\mathrm{c}}^{\left(\mathit{AF}\right)}⩽J∕T⩽0$. Variations of the scaling dimensions on the F side $0⩽J∕T<{(J∕T)}_{\mathrm{c}}^{\left(F\right)}$ cannot be resolved numerically. The lines between data points are guides to the eyes.

Figure 5

(Color online) Illustration of the phase diagram of the constrained Ising model in the presence of a nearest-neighbor coupling $J$, discussed in Sec. 5. $U$ is held fixed in the figure, and only two out of the three parameters of the model—namely, $J$ (horizontal axis) and $T$ (vertical axis)—are shown. The effects of the quantum energy scale $\Gamma$ become dominant only at very small values of the coupling $J$, in the interval $-\mid {\Gamma }_{\mathrm{eff}}\mid \lesssim J\lesssim \mid {\Gamma }_{\mathrm{eff}}\mid$, and for very small temperatures $T\lesssim \mid {\Gamma }_{\mathrm{eff}}\mid$, giving rise to the quantum regime shown in the picture. At low temperatures, but for $\mid J\mid ⪢\mid {\Gamma }_{\mathrm{eff}}\mid$, the system exhibits two classical ordered phases, a ferromagnetic (F) one for $J>0$ and an antiferromagnetic (AF) one for $J<0$. While the transition to the AF ordered phase is continuous, the transition to the F ordered phase is strongly first order (dotted line). As discussed in Ref. 9, quantum glassiness is expected to appear in the system when the temperature is lowered across the transition to the F-ordered phase, at least as long as the transition temperature is small enough for the $U$-violating defects not to play a significant role in the equilibration process.

Figure 6

(Color online) Example of longer dimers that can be thought of as pairs of monomers chained together. When the end points of a long dimer sit at sites within the same sublattice, the chained monomers have equal charges. When the end points sit at opposite sublattices, the chained monomers have opposite charges.

Figure 7

(Color online) (Top) Behavior of the extrapolated central charge as a function of $v∕T$, obtained from finite-size scaling of the free energy computed via transfer matrix. (Bottom) Behavior of the scaling dimensions corresponding to the electric and magnetic vertex operators as a function of $v∕T$, also obtained from finite-size scaling arguments and transfer matrix calculations. Observe that the scaling dimensions of the electric and magnetic vertex operators are 1 and $1∕4$, respectively, when $v∕T=0$. The lines between data points are guides to the eyes.

Figure 8

(Color online) Generic phase diagram for a strongly constrained quantum system that satisfies conditions (1)–(6) from Sec. 1. Parameter space encodes the competition between two energy scales, the temperature $T$, and a characteristic energy scale ${\Gamma }_{\mathrm{eff}}$ given by the effective coupling of the off-diagonal term in the Hamiltonian, while $g$ and $U$ are held fixed. Here we chose to represent the case of a system exhibiting a zero-temperature phase transition at a finite value of the ratio ${\Gamma }_{\mathrm{eff}}∕g$. A quantum-critical scaling regime, if it exists, is restricted to a region represented by the upper half of a disk of radius ${\Gamma }_{\mathrm{eff}}$ and centered at the origin $(g,T)=(0,0)$ of parameter space. As in the case of Fig. 1, the system exhibits a constrained entropic scaling regime at temperatures intermediate between the small characteristic quantum energy scale ${\Gamma }_{\mathrm{eff}}$ and the large characteristic energy scale $U$ set by a strong constraint. At a fixed temperature, the constrained entropic scaling regime terminates in a phase transition that needs not be continuous upon increasing $\mid {\Gamma }_{\mathrm{eff}}\mid$. At fixed ${\Gamma }_{\mathrm{eff}}$, the constrained entropic scaling regime crosses over to the conventional high-temperature phase—say, a paramagnetic one for spin degrees of freedom—when $T$ is of the order of $U$.