Spin- and Pair-Density-Wave Glasses

Spontaneous breaking of translational symmetry, known as density-wave order, is common in nature. However, such states are strongly sensitive to impurities or other forms of frozen disorder leading to fascinating glassy phenomena. We analyze impurity effects on a particularly ubiquitous form of broken translation symmetry in solids: a spin-density wave (SDW) with spatially modulated magnetic order. Related phenomena occur in pair-density-wave (PDW) superconductors where the superconducting order is spatially modulated. For weak disorder, we find that the SDW or PDW order can generically give way to a SDW or PDW glass—new phases of matter with a number of striking properties, which we introduce and characterize here. In particular, they exhibit an interesting combination of conventional (symmetry-breaking) and spin-glass (Edwards-Anderson) order. This is reflected in the dynamic response of such a system, which—as expected for a glass—is extremely slow in certain variables, but, surprisingly, is fast in others. Our results apply to all uniaxial metallic SDW systems where the ordering vector is incommensurate with the crystalline lattice. In addition, the possibility of a PDW glass has important consequences for some recent theoretical and experimental work on ${\text{La}}_{2-x}{\text{Ba}}_x{\text{Cu}}_2{\mathrm{O}}_4$.

Popular Summary

Spontaneous breaking of translational symmetry, known as density-wave order, is observed in a large number of electronic systems. In the presence of imperfections (i.e., nonmagnetic impurities), which are inherent to any real system, the nature of these states changes dramatically, and new phases of matter arise. These new phases exhibit properties that lie midway between the possible thermal phases expected in hypothetical clean realizations of the same system. One common example is given by collinear spin-density-wave order, which has been observed in a large number of materials ranging from conventional metals to more complicated systems such as cuprates. There the magnetic moments of the electrons exhibit a periodically modulated pattern in space, oscillating between alignment and antialignment with a spontaneously chosen axis. Such states break both translation symmetry and spin-rotation symmetry, and their interplay with disorder leads to a number of striking effects that we analyze here in both two and three dimensions.

Using analytical arguments supported by numerical simulations, we uncover and characterize new phases of matter that we refer to as spin-density-wave glasses (and closely analogous pair-density-wave glasses). Crucially, the magnetic moments remain aligned or antialigned to a spontaneously chosen axis even as the periodic spatial modulations are lost. As a consequence, these systems are both spin glasses and symmetry broken phases at the same time; their measured nature depends on the experimental probe. Since any solid-state experiment necessarily includes disorder, spin-density-wave glasses are likely to be a common phenomenon. These glasses present a natural starting point for interpreting experiments in any system where spin-density-wave order is observed.

Similar effects arise in systems with spatially modulated superconductivity, which have important consequences for recent theoretical and experimental work on the cuprate ${\text{La}}_{2-x}{\text{Ba}}_x{\text{Cu}}_2{\mathrm{O}}_4$.

Figure 1

Magnetic moments in the ground state of the SDW glass. The disorder pins the domain walls into a random configuration, but the structure of “antiphase” domain walls persists. Upon crossing of each domain wall, the local magnetization changes sign. Thus, the disordered state inherits the collinear structure from the parent SDW, where the axis along which the moments point is selected spontaneously, breaking the spin-rotation symmetry.

Figure 2

Goldstone mode in the SDW glass. The configuration of the domain walls is identical to the ground state, but the orientation of the moments slowly varies in space.

Figure 3

In the SDW Bragg glass, the antiphase domain walls are pinned in a configuration without dislocations.

Figure 4

Schematic phase diagram for a disordered SDW (or PDW) in 2D as a function of disorder strength $v$ and the ratio $K_s/K_c$ of the stiffnesses associated with spin waves and phonons in the absence of disorder.

Figure 5

Summary of numerical data. Left: Spatial decay of the correlation functions $C_{\text{nematic}}$ and $C_{\mathrm{CDW}}$. The inset shows a log-log plot of the same data. Combined error due to sampling and disorder averaging is shown where it exceeds the size of the shown data points. The qualitative feature that nematic correlations decay much slower than CDW correlations is generic for low temperatures and weak disorder. However, depending on parameters, $C_{\text{nematic}}$ can either follow a power law or decay exponentially. Center: Temperature dependence of the spin-nematic correlation length ${\xi }_{\text{nematic}}$ shown for parameters corresponding to a SDW glass phase (green) or a fully disorderd phase (purple) at low temperatures. For the first set of parameters, ${\xi }_{\text{nematic}}$ grows rapidly as temperature is lowered, indicating a phase transition into the SDW glass phase. For the second set of parameters, ${\xi }_N$ shows no sign of divergence and is expected to saturate at a finite value such that the low-temperature glassy phase is smoothly connected to the high-temperature phase. Solid lines are drawn as a guide to the eye. Right: The stiffness ${\rho }_N$ for spin-nematic fluctuations, as a function of system size, shown for both the SDW glass and the fully disordered phase. In the SDW glass (${\kappa }_c=-0.1{\kappa }_s$), ${\rho }_N$ depends only weakly on system size $L$ and always takes a finite value, while for ${\kappa }_c=0.1{\kappa }_s$ it exhibits substantial sample dependence and rapidly decays with system size.