Signatures of rare states and thermalization in a theory with confinement

There is a dichotomy in the nonequilibrium dynamics of quantum many-body systems. In the presence of integrability, expectation values of local operators equilibrate to values described by a generalized Gibbs ensemble, which retains extensive memory about the initial state of the system. On the other hand, in generic systems such expectation values relax to stationary values described by the thermal ensemble, fixed solely by the energy of the state. At the heart of understanding, this dichotomy is the eigenstate thermalization hypothesis (ETH): individual eigenstates in nonintegrable systems are thermal, in the sense that expectation values agree with the thermal prediction at a temperature set by the energy of the eigenstate. In systems where ETH is violated, thermalization can be avoided. Thus, establishing the range of validity of ETH is crucial in understanding whether a given quantum system thermalizes. Here, we study a simple model with confinement, the quantum Ising chain with a longitudinal field, in which ETH is violated. Despite an absence of integrability, there exist rare (nonthermal) states that persist far into the spectrum. These arise as a direct consequence of confinement: pairs of particles are confined, forming new “meson” excitations whose energy can be extensive in the system size. We show that such states are nonthermal in both the continuum and in the low-energy spectrum of the corresponding lattice model. We highlight that the presence of such states within the spectrum has important consequences, with certain quenches leading to an absence of thermalization and local observables evolving anomalously.

Figure 1

Schematic of the low-energy spectrum of the Ising field theory (2). The longitudinal field $g$ leads to a complete restructuring of the low-energy continuum (shaded) and the appearance of confined “meson” states (solid lines).

Figure 2

Comparison of the eigenstate expectation values (EEV, black) of the local magnetization $\sigma$ with the microcanonical ensemble prediction (MCE, orange) for eigenstates at energy density $E/R$ in the Ising field theory (2) with $m=1, g=0.1$ on the ring of size (a) $R=25$; (b) $R=35$; (c) $R=45$. Eigenstates are constructed via the TSM with an energy cutoff of (a) $E_{\mathrm{\Lambda }}=13.5$; (b) $E_{\mathrm{\Lambda }}=10.5$; (c) $E_{\mathrm{\Lambda }}=9$. States containing up to 10 particles are considered in the truncated Hilbert space. The MCE is constructed by uniformly averaging over an energy window of size $\mathrm{\Delta }E=0.1$, with error bars denoting the standard deviation. The red arrows highlight the nonthermal states considered further in Fig. 3.

Figure 3

(a)–(e) The finite-size scaling of the total magnetization $R\langle \sigma \rangle$ of the $n=11–15$ rare states (EEV), compared to the MCE prediction. (f) The average magnetization of the $n=11–15$ rare states compared to that predicted by the average of the MCE results. Clearly, the rare states and their proximate thermal states have different magnetization, and this persists to the $R\to \infty$ limit. In all cases, error bars show the standard deviation of the data averaged over. Note that $R({\langle \sigma \rangle }_n-{\langle \sigma \rangle }_0)$ measures the number of overturned spins in the $n\mathrm{th}$ meson as compared to the (ordered) ground state.

Figure 4

Off-diagonal matrix elements of the local spin operator $|\langle E_{\beta }|\sigma \left(0\right)|E_{\alpha }\rangle |$ ($\alpha \ne \beta$) between the first 500 eigenstates constructed with the TSM for $m=1, g=0.1$, and $R=35$ [cf. Fig. 2]. Parameters are $m=1, g=0.1$. Matrix elements satisfying $|{\sigma }_{\alpha \beta }|<{10}^{-7}$ are plotted as white. Note the visible vertical/horizontal lines are not remnants of the plotting, but show eigenstates that are only coupled very weakly to other states by the spin operator $\sigma \left(0\right)$.

Figure 5

The running average of the absolute value of off-diagonal matrix elements $|{\sigma }_{\alpha \beta }|$ between eigenstates with energies satisfying $(E_{\alpha }+E_{\beta })<8$. For the largest system $R=50$, this condition restricts to the lowest (in energy) $\sim 1000$ eigenstates (and fewer at smaller volumes). The running average is computed over 300 data points.

Figure 6

The two-particle weight $w_2$ (20) and the eigenstate expectation values of the local magnetization operator $\langle \sigma \rangle$ in the field theory (2) with $m=1, g=0.1$ on the ring of size $R=35$. Eigenstates are constructed with the TSM for an energy cutoff of $E_{\mathrm{\Lambda }}=10.5$ and a maximum of 10 particles in the basis states. Arrows (upper and lower) are drawn at the energies of the first 40 confined states, $M_n$, as computed in Appendix pp3. Higher particle weights, as well as histograms of their distributions, are provided in Appendix pp2.

Figure 7

The second-order correction to the energy (measured relative to the ground state) of the first 19 meson masses due to mixing with zero and four domain-wall fermion states. This should be compared to the correction to the mass of the spin-flip excitation in the disordered phase ($m<0$), shown in blue, which is between two and four orders of magnitude larger. In the two cases we consider (a) $g=0.1$ (reproduced from [81]); (b) $g=0.2$. Error bars denote uncertainty in the extrapolation to infinite volume.

Figure 8

The wave functions in momentum space for the first 10 odd-numbered bound states for $g=0.1$. We see that the maxima in the wave functions do not occur at momenta that are increasing in a uniform fashion.

Figure 9

The second-order correction to the energy (measured relative to the ground state) as a function of longitudinal field strength $g$ for four mesons (the first, third, fifth, and ninth ordered by energy) arising from mixing with zero and four domain-wall fermion states. Error bars denote uncertainties in extrapolating to the infinite volume.

Figure 10

The energies as a function of longitudinal field strength $g$ for four mesons (the first, third, fifth, and ninth ordered by energy) whose energy corrections are given in Fig. 9. The dashed line marks the $4m$ threshold. Once the energy of the meson crosses this threshold, we no longer expect the energy of hybridization with the vacuum and four-particle states to increase monotonically with increasing $g$.

Figure 11

We compare the results of a full TSM procedure with $E_{\mathrm{\Lambda }}=10.5$ (for a system with $g=0.1, m=1$ of size $R=35$) to an approximate block diagonalization by fermion number. Two-particle states ($2p$) are considered up to cutoff $E_{\mathrm{\Lambda }}=200$; four-particle states ($4p$) are considered up to cutoff $E_{\mathrm{\Lambda }}=16$. Dashed lines show the threshold energies for eigenstates containing $1–3$ mesons (i.e., up to six fermions).

Figure 12

The diagonal ensemble (DE) result for long-time expectation values and the microcanonical ensemble (MCE) prediction following the sudden quench $(m=1,g=0.1)\to (m=1,g=0.2)$ in the field theory (2). Each data point corresponds to starting from a different low-energy eigenstate of the initial Hamiltonian $m=1,g=0.1$. In all cases, results are on a system of size $R=25$ and with energy cutoff $E_{\mathrm{\Lambda }}$. We note that $E_{\mathrm{\Lambda }}=13$ ($E_{\mathrm{\Lambda }}=13.5$) corresponds to 23,238 (32,149) computational basis states. The MCE is computed by averaging an energy window of size $\mathrm{\Delta }E=0.1$, and error bars denote the standard deviation of data averaged over.

Figure 13

(a) Time evolution of the local magnetization $\langle \sigma \left(0\right)\rangle$ following the quench $(m=1,g=0.2)\to (m=1,g=0.1)$ in (2) with $R=25$. Time evolution is computed via TSM+CHEB with cutoff $E_{\mathrm{\Lambda }}=9,10$ and the expansion evaluated to order 2000. The magnetization is oscillating about its diagonal ensemble (DE) value, computed via the TSM, which also coincides with the microcanonical ensemble prediction. (b) The power spectrum of the time evolution. We note a few prominent frequencies in terms of the post-quench confined state energies: $\mathrm{\Delta }{M}_{ij}=M_i-M_j$ and $\mathrm{\Delta }{M}_j=M_j$.

Figure 14

(a) Time evolution of the local magnetization $\langle \sigma \left(0\right)\rangle$ when starting from the first excited state of (2) for the same quench described in Fig. 13. (b) The associated power spectrum. We note a few prominent frequencies in terms of the post-quench confined state energies: $\mathrm{\Delta }{M}_{ij}=M_i-M_j$ and $\mathrm{\Delta }{M}_j=M_j$.

Figure 15

Time evolution of the local spin operator $\sigma \left(0\right)$ following a quench $H_i=H(1,0.1)\to H_f=H(1,0.2)$ computed via TSM+CHEB. Each data set starts from a different eigenstate $n$ of the initial Hamiltonian. Dashed lines denote the diagonal ensemble prediction for the long-time limit, while the shaded region shows the thermal result from the microcanonical ensemble (with the uncertainty representing the standard deviation of the data averaged over).

Figure 16

Power spectrum ${\left|\sigma \left(\omega \right)\right|}^2$ for (a) the $n=100$ rare state; (b) the $n=333,352$ thermalizing states as computed from data in Fig. 15.

Figure 17

The eigenstate expectation values for the chain-averaged magnetization $\frac{1}{N}{\sum }_{l=1}^N\langle {\sigma }_l^z\rangle$ in the lowest $\sim 150$ states of the lattice model (33), with $J=-1, h^x=-0.5$, and $h^z=0.05$. Eigenstates were computed with DMRG for a chain of (a) $N=20$; (b) $N=30$; (c) $N=40$ sites. We encourage the reader to compare this to the analogous plot in the scaling limit, Fig. 2, and note the striking similarity. In plotting (a) and (b), we exclude the false vacuum state. The labels in (c) indicate the states shown in Fig. 18.

Figure 18

The spin expectation value $\langle {\sigma }_{\ell }^z\rangle$, on each site $\ell =1,2,\cdots ,40$, for the 0th (ground), 1st, 44th, 48th, and 51st states in the spectrum as found using DMRG. These states are marked in Fig. 17. The 1st and 48th states are examples of mesons localized on the boundary.

Figure 19

The EEV spectrum obtained by DMRG on the finite lattice of $N$ sites for the Hamiltonian (33) with $J=-1, h^x=-0.5$, and $h^z=0.05$. Here, we consider the total magnetization of the eigenstates relative to the ground state, as a function of energy (relative to the ground-state energy $E_0$). See Fig. 17 for further details.

Figure 20

The EEV spectrum for $m=1, g=0.1$ in systems of sizes $R=25–45$ (see Figs. 2 for details). Here, we plot the total magnetization (relative to the ground state) as a function of energy (relative to the ground state). We see that the meson states are well converged as a function of system size $R$, while the thermal continuum is qualitatively converged (in the sense that the features are correct, but the density of state increases with increasing $R$). Note the striking similarity to Fig. 19.

Figure 21

(a) The entanglement entropy averaged over bipartitions of the system ${\sum }_n{S}_E\left(n\right)$ in each eigenstate of energy $E$ relative to the ground state, for the lattice model (33) with $J=-1, h^x=-0.5$, and $h^z=0.05$. Data are presented for three different system sizes $N$. We see that there are two classes of states: those that have $N$-independent entanglement (i.e., area law states) and those with volume law entanglement (linear in $N$). The presence of volume law entanglement states is shown more clearly in (b), where we scale the bipartition averaged entanglement by dividing through by the volume $N$. The nonthermal states have area law entanglement, while the thermal states have volume law.

Figure 22

The eigenstate expectation value spectrum of the local spin operator $\sigma \left(0\right)$ in (2) with $m=1, g=0.1, R=35$, for four values of the cutoff $E_{\mathrm{\Lambda }}$.

Figure 23

The diagonal ensemble (DE) result for expectation values following a quench of the longitudinal field $g_i=0.1\to g_f=0.2$ in the field theory (2) with $m=1$ and $R=25$ for four values of the energy cutoff. Free-fermion basis states containing up to 10 particles are considered.

Figure 24

The MCE prediction for $\langle \sigma \left(0\right)\rangle$ at a given energy density in the field theory (2) with $m=1, g=0.2$, and $R=25$. Error bars denote the standard deviation of the averaged data for the window $\mathrm{\Delta }E=0.1$.

Figure 25

The weights (a) $w_0$; (b) $w_4$; (c) $w_6$; (d) $w_8$. Eigenstates were constructed with the TSM on the length $R=35$ chain with $m=1, g=0.1$ and an energy cutoff $E_{\mathrm{\Lambda }}=10.5$, with up to 10 free fermions in the basis states. These plots illustrate the approximate low-energy U(1) symmetry discussed in the main text: below an energy density of $E/R\sim 0.15$, the weight for each state is either $w_N\approx 1$ or $w_N\approx 0$. This is emphasized in (e), where we show the distribution of the weights for eigenstates with energy density $E/R<0.15$, where bins are of width $\mathrm{\Delta }{w}_N=0.02$. We see almost all states have weights close to zero or close to one, with the histograms being bimodal (note the logarithmic $y$-axis scale). The issue of where this approximate U(1) symmetry breaks is tough to address, with cutoff effects dominating results at higher energies.

Figure 26

(a) Time evolution via the $N_{\max }$-order Chebyshev expansion for the quantum quench $(m,g)=(1,0.1)\to (1,0.2)$ of (2) for $R=25$. We start from the first excited state of the initial Hamiltonian, computed via the TSM with an energy cutoff $E_{\mathrm{\Lambda }}=9$. The breakdown time $t_{\text{br}}$ is seen by the divergence of the result from higher-order predictions; see the purple line at $mt\sim 35$, the green line at $mt\sim 70$, and the cyan line at $mt\sim 110$. (b) Time evolution shown for three values of $E_{\mathrm{\Lambda }}$ with $N_{\max }=2000$. We see that the energy cutoff plays only a small role in the convergence of the presented results.

Figure 27

Time evolution via the $N_{\max }=2000$-order Chebyshev expansion for the quantum quench $(m,g)=(1,0.1)\to (1,0.2)$ of (2) for $R=25,40$. We start from the (a) ground state; (b) first excited state of the initial Hamiltonian, computed via the TSM with an energy cutoff $E_{\mathrm{\Lambda }}=9$. Good agreement of the results is seen until $t\sim 145$ [not shown in (a)], where the finite order of the Chebyshev expansion causes the results for the large system size to break down.