Eigenstate thermalization in systems with spontaneously broken symmetry

Abstract

A strongly non-integrable system is expected to satisfy the eigenstate thermalization hypothesis, which states that the expectation value of an observable in an energy eigenstate is the same as the thermal value. This must be revised if the observable is an order parameter for a spontaneously broken symmetry, which has multiple thermal values. We propose that in this case the system is unstable towards forming nearby eigenstates which yield each of the allowed thermal values. We provide strong evidence for this from a numerical study of the 2D transverse-field quantum Ising model.The eigenstate thermalization hypothesis (ETH) can explain how an isolated, quantum many-body system in an initial pure state can come to thermal equilibrium (as determined by measurements of a specified set of observables) in finite time [1][2][3]. ETH is expected to hold in systems without disorder that are sufficiently far (in parameter space) from points of integrability, for observables that are sufficiently simple (e.g. local) functions of the fundamental degrees of freedom. In recent years ETH has been the subject of intensive analytic and numerical investigations, e.g. [3][4][5][6][7][8][9][10][11][12][13][14]; see [15] for an overview including the connection to experimental results in cold atoms and other systems.The key statement of ETH is that expectation values of a relevant observable M in an energy eigenstate |α (of the full many-body hamiltonian H) take the formwhere M(E) is a smooth function of E and E α is the energy eigenvalue. In a system with N 1 degrees of freedom, this is enough information to show that M(E) is equal, up to O(N −1/2 ) corrections, to the canonical thermal average of the operator M ,where the temperature T is implicitly determined as a function of energy E by the usual relationA second key statement of ETH is that the off-diagonal matrix elements of M in the energy basis, α|M |β with α = β, are exponentially small in N . This is needed to explain why the diagonal matrix elements of eq. (1) dominate the instantaneous expectation value of M (in a generic time-dependent state) at almost all times, which in turn is necessary for thermal equilibrium to be maintained once it has been achieved. However this aspect of ETH will not be our focus.The ETH paradigm must be revisited for a system that is capable of exhibiting spontaneous symmetry breaking (SSB). Suppose that the observable M is an order parameter for a global symmetry. At energies corresponding to the broken-symmetry phase, and in the infinite-volume limit, we expect the system to have states with the same energy but with different values of M (that are related by the symmetry). In this case, eq. (1) cannot hold as written. We conjecture that, instead, the single smooth function M(E) is replaced by a multivalued function, with one branch for each allowed value of the order parameter.We note that the compatibility of ETH and SSB was assumed to hold in [16], in which the tunnelling dynamics of the order parameter were studied in "Schrodinger cat" ...

Discrete Time Crystals

“…(Or rather, its unrotated version U approx F = U † U approx F U, but one can check that conjugation by local unitaries does not affect the time crystal definition). Moreover, if D spontaneously breaks the symmetry generated by X at finite temperature, then the corresponding finite-energy eigenstates of D come in pairs |↑ and |↓ with opposite magnetization[84][85][86]. Hence we conclude that the corresponding eigenstates of U approx F are |↑ ± |↓ ), separated in quasi-energy by Ω/2, which indeed satisfy the definition of a discrete time crystal from Section IV D.…”

“…As motivation, let us first recall how spontaneous symmetry breaking works at zero temperature; that is, in the ground state of a static Hamiltonian H. The classic example is the ground state subspace of an Ising ferromagnet (which has a Z 2 Ising spin-flip symmetry) is degenerate and spanned by a pair of spin-polarized states in which the spins have a net magnetization in the up direction or the down direction (in the limit of vanishing transverse field, they are fully-polarized in the up or down direction); we call these states |↑ and |↓ , and they are related by the Ising symmetry. On any finite system, however, there is some tunneling amplitude between these two states, as a consequence of which the true eigenstates are the symmetric and anti-symmetric combinations |± = 1 Although this eigenstate multiplet structure is most familiar in ground states, the same structure is found in highly excited states for systems that exhibit spontaneous symmetry breaking at finite energy density; this is true both in systems that obey ETH [84][85][86] Here by "approximately the same energy", we mean that the energy difference is exponentially small in the system size.…”

Initial-state-independent equilibration at the breakdown of the eigenstate thermalization hypothesis

“…Next we address the existence of an integrable regime in the model at hand and study its relevance for the emergence of ISI for the MOD states. Again, similar to the discussion of ETH and ISI, a range of papers more or less explicitly states that non-integrability is imperative for ISI [14], whereas other works analyze ISI without even mentioning integrability. Also different features of "statistical relaxation" ( other than ISI) are addressed; examples exist in which the occurrence of statistical relaxation does not depend on integrability [19].…”

Eigenstate thermalization and quantum chaos in the Holstein polaron model

“…So far, the ETH has been verified for a wide number of lattice models such as nonintegrable spin-1/2 chains [23][24][25][26][27][28][29][30][31][32][33], ladders [26,[34][35][36] and square lattices [37][38][39], interacting spinless fermions [40,41], Bose-Hubbard [26,42] and Fermi-Hubbard chains [43], dipolar hard-core bosons [44], quantum dimer models [45] and Fibonacci anyons [46]. In these examples, mostly, direct two-body interactions in systems of either spins, fermions or bosons are responsible for rendering the system ergodic.…”