Non-Abelian Symmetries and Disorder: A Broad Nonergodic Regime and Anomalous Thermalization

Previous studies reveal a crucial effect of symmetries on the properties of a single particle moving in a disorder potential. More recently, a phenomenon of many-body localization (MBL) has been attracting much theoretical and experimental interest. MBL systems are characterized by the emergence of quasilocal integrals of motion and by the area-law entanglement entropy scaling of its eigenstates. In this paper, we investigate the effect of a non-Abelian $SU(2)$ symmetry on the dynamical properties of a disordered Heisenberg chain. While $SU(2)$ symmetry is inconsistent with conventional MBL, a new nonergodic regime is possible. In this regime, the eigenstates exhibit faster than area-law, but still strongly subthermal, scaling of the entanglement entropy. Using extensive exact diagonalization simulations, we establish that this nonergodic regime is indeed realized in the strongly disordered Heisenberg chains. We use the real-space renormalization group (RSRG) to construct approximate excited eigenstates by tree tensor networks and demonstrate the accuracy of this procedure for systems of sizes up to $L=26$. As the effective disorder strength is decreased, a crossover to the thermalizing phase occurs. To establish the ultimate fate of the nonergodic regime in the thermodynamic limit, we develop a novel approach for describing many-body processes that are usually neglected by the RSRG. This approach is capable of describing systems of size $L\approx 2000$. We characterize the resonances that arise due to such processes, finding that they involve an ever-growing number of spins as the system size is increased. Crucially, the probability of finding resonances grows with the system’s size. Even at strong disorder, we can identify a large length scale beyond which resonances proliferate. Presumably, this proliferation would eventually drive the system to a thermalizing phase. However, the extremely long thermalization timescales indicate that a broad nonergodic regime will be observable experimentally. Our study demonstrates that, similar to the case of single-particle localization, symmetries control dynamical properties of disordered, many-body systems. The approach introduced here provides a versatile tool for describing a broad range of disordered many-body systems, well beyond sizes accessible in previous studies.

Popular Summary

Recent experiments have opened the door to probing the fundamentals of quantum statistical mechanics in isolated systems, as well as its limitations. Surprisingly, researchers have discovered that strong disorder can prevent certain quantum systems from reaching thermal equilibrium, a mechanism now known as many-body localization. This absence of thermalization protects quantum coherence and enables new kinds of quantum order. Here, we introduce a new class of nonthermalizing systems, distinct from those with many-body localization.

In our investigation, we mathematically explore the effects of symmetry on quantum dynamics. Specifically, non-Abelian symmetries (such as groups of rotations) turn out to be fundamentally inconsistent with many-body localization. However, disorder can induce a new regime, distinguished by a novel pattern of quantum entanglement and distinct dynamical properties. To establish the existence of this regime and address its properties, we introduce a new “renormalization approach” to describing nonequilibrium properties of large quantum systems.

This work establishes that the family of nonthermalizing systems is larger than previously thought. Our renormalization-based method is likely to be widely applicable to dynamics of disordered, interacting systems.

Figure 1

(a) A cartoon of the ground state of a random antiferromagnetic Heisenberg chain. (b) The strong-disorder renormalization group aims to construct approximate eigenstates. It yields a tree state, characterized by its geometry and the choice of total block spins at each node (see the main text). The Heisenberg Hamiltonian, written in this basis, gives rise to processes which can change the spins along the “causal” path connecting two neighboring spins, to one of the block spins. (c) A schematic dynamical phase diagram of the random Heisenberg model. There are three regimes: (I) At short length scales $L<L_1(\alpha )$, the SDRG tree states are accurate approximations of the eigenstates; (II) at intermediate length scales $L_1(\alpha )<L<L_{\mathrm{erg}}(\alpha )$, there are resonances but the system remains nonergodic; (III) above some large length scale $L>L_{\mathrm{erg}}(\alpha )$, the resonances proliferate and the system becomes thermalizing (see Secs. 4b and 4c for a definition of these scales).

Figure 2

A multiplet of eigenstates predicted by the SDRG for a system of 12 spins $1/2$. The leaves of the tree represent elementary spins in the system. The tree describes the way the elementary spins are fused into larger block spins in the course of the SDRG. The numbers in the nodes indicate the resulting spins of the blocks. The value in the top node (marked red) is the total spin $S_0$ of the system ($S_0=1$ in the present example). $S_0$ is an exact integral of motion. $(2S_0+1)$ different states in the multiplet can be distinguished by additionally specifying the projection of the spin in the top node to the $z$ axis.

Figure 3

Statistics of ${\log }_{10}{N}_E$ for $\alpha =1$ (upper) and $\alpha =0.3$ (lower). Different curves correspond to different system sizes $L=10$, $L=16$, and $L=20$; see the legend.

Figure 4

Typical number $N_E$ of tree states participating in the eigenstate $|E⟩$ (top) and the typical value of the fraction $N_E/D_{S_0,L}$ (bottom) versus the system length for different strengths of disorder (see the legend). The dashed lines in the top represent the exponential fits $N_E\propto 2^{L/{\tilde{L}}_1(\alpha )}$.

Figure 5

Level statistics for the Heisenberg chain. Each curve in the figure is produced using at least 50 eigenstates from the middle of spectrum and 1000 realizations of disorder. The dashed lines are the Wigner-Dyson (WD) and Poisson distributions. (Top) For a fixed length $L=22$ and varying $\alpha$. (Bottom) For a fixed $\alpha =1.3$ and varying length $L$. The tendency toward the Wigner-Dyson statistics is evident at both growing $L$ and $\alpha$. However, for smaller values of $\alpha$, $P(r)$ remains close to the Poisson one up to the largest available system sizes.

Figure 6

The average $r$ parameter as a function of $L$ for different values of $\alpha$. From top to bottom (for any fixed $L$), $\alpha =1.9$, 1.6, 1.3, 1, 0.8, 0.6, 0.45, 0.3. The dashed lines at $r=0.53$ and $r=0.39$ represent the WD and Poisson values, respectively. Error bars are within the symbol.

Figure 7

(a) The average $r$ parameter as a function of $L/L^{*}$, divided by $r^{*}$, falls onto a universal curve in the vicinity of its minimum. (b) The value of the minimum $r^{*}(\alpha )$. The dashed line is a fit of the form $r^{*}(\alpha )=r_{\infty }+c_1/\alpha +c_2/{\alpha }^2$, which returns $r_{\infty }=0.53\pm 0.01$ compatible with the Gaussian orthogonal ensemble value. (c) The position of the minimum $L^{*}(\alpha )$ for $\alpha =1.9$, 1.6, 1.3, 1, 0.8, 0.6, 0.45. The dashed line is a fit of the form $L^{*}(\alpha )=c{\alpha }^{-\nu }$ with $\nu =1.4\pm 0.13$ for the first four points.

Figure 8

Heat maps for the distributions of $⟨a|{\widehat{O}}_{\max }|a⟩$ (left) and $⟨a|{\widehat{O}}_{\mathrm{rand}}|a⟩$ (right) over several ($\gtrsim 25000$) eigenstates. Here, $S_0=0$, $L=22$, and $\alpha \in \{0.3,0.6,1.3\}$ (top to bottom). The concentration of the $y$ marginals around 0 denotes increasingly ergodic behavior (see the comments in the text).

Figure 9

Fraction of eigenstates with a value of $⟨{\widehat{O}}_{\max /\mathrm{rand}}⟩$ between $-1/8$ and $1/8$, for $\alpha \in \{0.6,0.8,1.0\}$ and $S_0=0$. The ETH predicts this fraction to become 1 in the infinite-size limit.

Figure 10

Median value of the half-chain entanglement entropy, $S_{\mathrm{ent}}(L/2)$, for the $S_0=0$ sector and $\alpha \in \{0.3,0.45,0.6,0.8,1.0,1.3,1.6\}$ (bottom to top). Linear fits, performed on the $L\ge 14$ points, are shown when their slope is close to that expected for an ergodic phase. Entanglement entropy for the higher disorder values is strongly subthermal.

Figure 11

Typical connectivity of the Heisenberg Hamiltonian in the tree basis obtained from the SDRG, as a function of the system size, for different values of disorder $\alpha =0.3$, 0.5, 0.6, 0.8, 1, 1.2 (see the legend). The dashed line represents the fit $K=(L/L_0{)}^{\kappa }$. The extracted value of $\kappa =2.75$ is in good agreement with the analytic result (18): $\kappa =\ln 3/\ln (3/2)\approx 2.71$. The horizontal dotted line shows the maximal sampling rate ($2\times {10}^7$ matrix elements) used in the search of resonances (see Secs. 4b and 4c).

Figure 12

The average number of resonant neighbors for an SDRG tree state in short systems. The solid lines of different colors correspond to different values of the exponent $\alpha$. Dashed lines represent linear fits, $⟨K_{\mathrm{res}}⟩=aL-b$. For each value of disorder, the legend also indicates the scale $L_1(\alpha )$ defined by $⟨K_{\mathrm{res}}⟩=1$ (see the main text).

Figure 13

(Top) The probability $P(K_{\mathrm{res}}=0)$ for an SDRG tree state to have no resonant neighbors, as a function of the system size, and for different values of disorder (see the legend). Note the logarithmic scale along the horizontal axis. While short systems are essentially free of resonances even for a relatively weak disorder, the probability $P(K_{\mathrm{res}}=0)$ becomes vanishingly small in long systems. (Bottom) Typical number of resonant neighbors for an SDRG tree state in long systems $L\le 2^{12}$. The three solid lines of different colors correspond to the two different values of the exponent $\alpha =0.3$, 0.6. The dashed lines show the power-law fits, $K_{\mathrm{res}}={[L/L_1(\alpha )]}^{1+\mu }$. The scale $L_1(\alpha )$ where the average number of resonant neighbors for an SDRG tree state equals 1 is defined in the previous subsection.

Figure 14

(Top) Average number of the block spins changing their value in a single resonant transition. (Bottom) Typical number of adjacent physical spins involved in a resonance (see the main text).

Figure 15

Characteristic energy scale for typical resonances, as a function of the system size, shown for different disorder strengths.

Figure 16

Evolution of the number (top) and the density (bottom) of the resonant blocks spins in the course of the SDRG. The horizontal axes shows the ratio of the current system size to the initial length of the system.

Figure 17

(a) Maximum density ${\rho }_{\max }$ of the resonant spins developed in the course of the SDRG (cf. Fig. 16) as a function of the system size. The condition ${\rho }_{\max }\sim {\rho }_{\max ,c}$ defines the ergodization length scale $L_{\mathrm{erg}}(\alpha )$ and the corresponding energy scale $E_{\mathrm{erg}}(\alpha )=V_{\mathrm{typ}}[L_{\mathrm{erg}}(\alpha )]$. (b), (c) Estimates for the ergodization length and energy scales obtained by fixing the critical density to ${\rho }_{\max ,c}=0.25$.

Figure 18

Transformation of spins under the SDRG. $i_0$ denotes the strongest links to be removed. After the SDRG transformation, the spins ${\tilde{S}}_{i_0-1}$ and ${\tilde{S}}_{i_0+1}$ are regenerated iff they are no longer consistent with the rules of angular momentum addition.

Figure 19

SDRG trees representing states for a system of two (a) and three (b) spins. Explicit wave functions corresponding to these trees are give by Eqs. (a17) and (a18).

Figure 20

Tree state generated by the SDRG and its entanglement properties. A Schmidt cut at the middle of the system gives rise to a cut of the tree into a forest and prescribes a view of the chain as a collection of clusters lying to the left (${\mathcal{L}}_i$) and to the right (${\mathcal{R}}_i$) of the cut. With the full chain in the quantum state described by the tree, each cluster has the projection of the total momentum as the only d.o.f.

Figure 21

The probability distribution $P({\log }_{10}{K}_{\mathrm{res}})$ at $\alpha =0.3$.