Universality in Non-Equilibrium Quantum Systems

Conformal symmetry is invariance under conformal transformations, or transformations that preserve angles. More specifically, a conformal transformation is one that rescales the metric $g_{\mu\nu}\to\Omega(r)g_{\mu\nu}$. In two dimensions, we can move to complex coordinates by defining the variables $z=x+i\tau$, $\bar{z}=x-i\tau$, which are treated as independent of one another. We can at this stage remain agnostic as to the physical significance of $x$ and $\tau$, and simply treat them as Euclidean coordinates, with the theory defined on this Euclidean space. In practice, a spin chain or other quantum system tends to map onto such a statistical mechanics model only through the use of a Wick rotation to imaginary time, $\tau=it$ with $t$ the physical time,³⁵³⁵35This comes with its own host of problems; one must make sure that, when the imaginary time answer is analytically continued back to real time, no divergences or non-analyticities are encountered. A simple failure mode is the analytic continuation of a sinusoid: perhaps some answer gives $e^{-\omega\tau}$, which is decaying and negligible for large $\tau$, but is simply oscillating with no decay in real time, as $e^{-i\omega t}$. Often issues arise when there are poles in the imaginary time answer, preventing a smooth analytic continuation to real time. Worries about analytic continuation can generally be put to rest with numerical evidence, in the case that the analytic continuation cannot be proven to be safe. or when $\tau$ is treated as an inverse temperature $\tau=\beta$. Once we’ve made the move to the complex plane, it can be shown that any analytic function is a conformal transformation. This amazing fact is what empowers CFT in two dimensions, as first elucidated by Belavin, Polyakov and A. B. Zamolodchikov ^{belavin_infinite_1984}.³⁶³⁶36Note that this is Alexander Borisovich Zamolodchikov, not his twin brother Alexei Borisovich Zamolodchikov – also a noted theoretical physicist in his own right. Having two prominent A. B. Zamolodchikov’s in the same field at the same time must have made for a publisher’s nightmare (and it’s still difficult to tell sometimes which work is by whom).

A central object of interest in CFT is what is known as a primary operator. Note that operators can be viewed either as functions of position $r=(x,\tau)$ as $\phi(r)$, or in complex notation as $\phi(z,\bar{z})$. Primary operators transform simply under conformal transformations; each primary operator $\phi$ has conformal weights $h_{\phi}$ and $\bar{h}_{\phi}$, which are generally distinct real numbers, and in a unitary CFT, $h,\bar{h}\geq 0$. These are usually combined to form the quantities $\Delta=h+\bar{h}$, called the conformal dimension, and $s=h-\bar{h}$, called the spin. Under a conformal transformation $z\to w(z)$, $\phi(z,\bar{z})$ transforms as

$$\phi(z,\bar{z})=\left(\frac{\partial w}{\partial z}\right)^{h_{\phi}}\left(\frac{\partial\bar{w}}{\partial\bar{z}}\right)^{\bar{h}_{\phi}}\phi(w(z),\bar{w}(\bar{z})).$$

(12)

This expression is valid when $\phi$ is inside of an expectation, $\langle\phi\ldots\rangle$, and the factors out front are just the Jacobian of the transformation raised to conformal dimension of the operator. This little formula has extremely broad implications; often in CFT we don’t know how to compute correlation functions of an operator in some complicated geometry, like a multi-sheeted Riemann surface with cuts³⁷³⁷37As is common in entanglement entropy calculations with the replica trick; see Cardy and Calabrese ^{1751-8121-42-50-504005, calabrese_entanglement_2004}. or even a simple disk, but we can use this relation to conformally map onto a surface where we do know how to compute correlation functions. That said, let us examine correlation functions on the plane, with the understanding that this gives us all correlation functions on geometries with the same topology.

Conformal invariance completely fixes the form of the two-point function on the plane:³⁸³⁸38The one-point function vanishes by symmetry.

$$\langle\phi_{i}(r_{1})\phi_{j}(r_{2})\rangle=\frac{\delta_{ij}C_{i}}{|z_{1}-z_{2}|^{2h_{i}}|\bar{z}_{1}-\bar{z}_{2}|^{2\bar{h}_{i}}}=\frac{\delta_{ij}C_{i}}{|z_{1}-z_{2}|^{2(h_{i}+\bar{h}_{i})}}=\frac{\delta_{ij}C_{i}}{|r_{1}-r_{2}|^{2\Delta_{i}}}.$$

(13)

The $\delta_{ij}$ is because any two-point function of operators with different weights must vanish.³⁹³⁹39A simple argument goes as follows: make a conformal transformation on the plane, and consider the two quantities $\langle\phi_{i}(z_{1},\bar{z}_{1})\phi_{j}(z_{2},\bar{z}_{2})\rangle$ and $\langle\phi_{j}(z_{1},\bar{z}_{1})\phi_{i}(z_{2},\bar{z}_{2})\rangle$, switching the operators. In fact, these must be equal (!), since the two point function has just been rotated by $\pi$. After the transformation, the only way they may be equal is if their Jacobian prefactors $a^{h_{i}}b^{h_{j}}c^{\bar{h}_{i}}d^{\bar{h}_{j}}$ and $b^{h_{i}}a^{h_{j}}d^{\bar{h}_{i}}c^{\bar{h}_{j}}$ are equal, which entails either $h_{i}=h_{j}$ and $\bar{h}_{i}=\bar{h}_{j}$, meaning that two operators are really the same; or the two-point function is 0. ∎ $C_{i}$ is a constant that depends on the operator in question; in general, we cannot deduce its value from CFT alone, and it needs to be determined from a microscopic calculation in the particular model (it is not a universal number). Similarly, the form of the three-point function is also fixed by conformal symmetry alone, namely

$$\langle\phi_{i}(r_{1})\phi_{j}(r_{2})\phi_{k}(r_{3})\rangle=\frac{C_{ijk}}{|r_{1}-r_{2}|^{\Delta_{i}+\Delta_{j}-\Delta_{k}}|r_{2}-r_{3}|^{\Delta_{j}+\Delta_{k}-\Delta_{i}}|r_{3}-r_{1}|^{\Delta_{k}+\Delta_{i}-\Delta_{j}}},$$

(14)

which is quite elegant. Our luck runs out for the four point function, though, due to the presence of the cross-ratios or anharmonic ratios, like $\frac{|r_{1}-r_{2}||r_{3}-r_{4}|}{|r_{1}-r_{3}||r_{2}-r_{4}|}$, which are invariant under a conformal transformation. Since these ratios are invariant, the 4-point function could be any function of these ratios and still satisfy conformal symmetry, so we cannot fix the form; obviously, higher point functions have this issue as well. Nonetheless, knowing the 2- and 3-point functions is quite powerful, and in some cases, such as the free fermion $\psi$, we know the full $N$-point function due to Wick’s theorem (which relates $N$-point functions to sums of products of 2-point functions). The Coulomb gas formalism can give $N$-point functions for some other operators (such as the $\sigma$ field in the Ising model with $\Delta=1/8$), but generic $N$-point functions are not known. Finally, a central concept in CFT is of the operator product expansion or OPE, which states that as we bring two operators close together, we may replace them by a sum of operators (rather than a product). The intuition for this comes from, essentially, counting poles; bringing two poles close together creates a single pole of higher order, in the same sense that bringing two operators of some dimension close together creates a new operator of a different conformal dimension, if we don’t care about the non-singular part of the function. In math, the OPE is

$$\phi_{i}(z_{1},\bar{z}_{1})\phi_{j}(z_{2},\bar{z}_{2})=\sum_{k}C_{ij}^{k}\phi_{k}(z_{1},\bar{z}_{1}).$$

(15)

This is again only valid inside correlation functions $\langle\phi_{i}(z_{1},\bar{z}_{1})\phi_{j}(z_{2},\bar{z}_{2})\ldots\rangle$ in the limit $z_{1}\approx z_{2}$, $\bar{z}_{1}\approx\bar{z}_{2}$. The $C_{ij}^{k}$ are known as structure constants, with the above formula also called a fusion rule.⁴⁰⁴⁰40There is a deep connection between CFTs and the braiding of anyons, hence the similarity in terminology. OPEs are not restricted to primary operators, and exist for all pairs of operators; but, the primary fields being the atoms of any CFT, we are generally most interested in their OPEs.⁴¹⁴¹41With the exception of OPEs of primaries against the stress-energy tensor $T$.

The last bit of CFT we will need is the concept of a central charge, $c$. Earlier, we introduced primary operators, which transform simply under conformal maps. Luckily, many physical operators are primary, such as the Ising field $\psi$ in the Ising CFT. However, it is easily shown that their derivatives (also called descendants) like $\partial_{\mu}\phi$ are not primary, transforming in more complicated ways. The most notable non-primary field is the stress-energy tensor $T_{\mu\nu}$. Now, the stress energy tensor may be expanded in Laurent series as

$$T(z)=\sum_{m=-\infty}^{\infty}\frac{L_{m}}{z^{m+2}}\Leftrightarrow L_{m}=\frac{1}{2\pi i}\oint dz\ z^{m+1}T(z),$$

(16)

with similar expressions for $\bar{T}(\bar{z})$. The significance of the $L_{m}$’s (and their cousins, the $\bar{L}_{m}$’s) are that they generate the group of conformal transformations, in the sense of Lie algebras; alternatively, they are the Noether charges associated to the symmetry transformations $\delta z=z^{n+1}$. In particular, $L_{-1}$ generates translations, and $L_{0}$ generates dilations and rotations. These $L_{m}$ operators form the Virasoro algebra, perhaps the most fundamental aspect of CFT.⁴²⁴²42This is reminiscent of the algebra of angular momentum operators, but now with an infinite tower of $L$’s and a term with $c$ in it. In particular,

$$[L_{m},L_{n}]=(m-n)L_{m+n}+\frac{c}{12}m(m^{2}-1)\delta_{n+m,0}.$$

(17)

The quantity $c$ is the central charge. (This is the formal definition, and is pretty dry.) Note that we have two copies of this algebra, one for $L_{m}$’s and one for $\bar{L}_{m}$’s; the $\bar{L}_{m}$ have their own central charge $\bar{c}$, which may differ from $c$.

The central charge is the defining characteristic of a CFT.⁴³⁴³43We’ll concern ourselves here with only the “minimal models”, with a finite number of primary fields, plus the free boson at $c=1$ with an infinite number. The Ising universality class is defined by $c=1/2$, the free boson by $c=1$, the tricritical Ising model by $c=7/10$, and the 3-state Potts model by $c=4/5$, to name a few. Unitary theories have $c>0$, and minimal models (i.e. the models we generally care about in condensed matter physics) have $c\leq 1$.⁴⁴⁴⁴44In this range, $c$ may only take discrete values $c=1-6/m(m+1)$, where $m=3,4,5\ldots$. The central charge is universal, in some sense the ur-universal object, and appears extensively.

The modern interpretation of $c$ is as the prefactor of the growth of entanglement; without getting unnecessarily derailed, since CFTs are gapless, the entanglement entropy, defined as $S_{L}=-\operatorname{Tr}\rho_{L}\log\rho_{L}$ for a subsystem of size $L$, grows logarithmically⁴⁵⁴⁵45In a gapped theory, we expect an area law for entanglement in the ground state, $S_{L}$ $\sim$ $\partial L$, i.e. a constant in 1+1 dimensions, and a volume law for generic states with $S_{L}\sim L$. The area law is a remarkable conjecture that has only been proven in the case of one-dimensional gapped Hamiltonians, by Hastings ^{Hastings_2007}. as

$$S_{L}\sim\frac{c+\bar{c}}{6}\log L.$$

(18)

The central charge also appears in the specific heat, in Cardy’s formula $C\sim c\pi k_{B}/3\beta$, and pops up in many places where $T_{\mu\nu}$ makes an appearance. Finally, $c$ satisfies a remarkable theorem known, straightforwardly, as the $c$-theorem ^{zamo_cTheorem}, that states that $c$ is monotonically decreasing along RG flows.⁴⁶⁴⁶46More accurately, there is a function $C(H,\lambda)$, with $H$ the Hamiltonian and $\lambda$ the RG scale, that is monotonically decreasing along flow, and is fixed only at the RG fixed points, where it equals the central charge $c$. We remark that in higher dimensions, while there is no $c$, there are analogues of the $c$-theorem, including the $A$-theorem in 3+1 dimensions ^{CARDY1988749, OSBORN198997} and the $F$-“theorem” in 2+1 dimensions ^{Klebanov:2011aa}.⁴⁷⁴⁷47Conjectured theorem, i.e. a conjecture. Special cases have been, to my understanding, proven only using supersymmetry in particular theories like $\mathcal{N}$=2 supersymmetric Yang-Mills.

We should finally spend a few words on boundary CFT, which is usually defined on the half-plane $\mathrm{Im}\ {z}\geq 0$. Boundary CFT is actually easier than bulk CFT; there is only one copy of the Virasoro algebra, which we can think of as being $z$ for $\mathrm{Im}\ {z}\geq 0$ and $\bar{z}$ for $\mathrm{Im}\ z\leq 0$. Consequently there is only one central charge $c$, and, when the region $L$ is taken from the boundary inward, we have $S_{L}\sim(c/6)\log L$.⁴⁸⁴⁸48This is the formula for open boundary conditions; for periodic boundary conditions or for regions in the bulk of an OBC system, we again use $(c/3)\log L$, assuming $c=\bar{c}$ and no finite-size effects. Incidentally, finite size effects can be exactly corrected for, as shown by Cardy and Calabrese, by mapping the strip to a cylinder with radius $L$. Boundary CFT has its own peculiarities, however.

There are, for any CFT, only a finite number of boundary conditions that are consistent with conformal symmetry.⁴⁹⁴⁹49The seminal papers here are Cardy 1984 ^{cardy_conformal_1984} and 1989 ^{cardy_boundary_1989}, and Ishibashi 1989 ^{doi:10.1142/S0217732389000320}. In the case of the Ising model, calling the boundary field $h_{B}$,⁵⁰⁵⁰50More correctly, the value of the field $\psi$ is what is fixed to have expectation 0 or $\pm 1$, but this corresponds to the stated values of the coupling $h_{B}$. these are $h_{B}=0$ (“free”, which is always conformal for any theory) and $h_{B}=\pm\infty$ (“fixed”); any other values would change upon rescaling. We can thus flow boundary conditions under the RG, leading to some boundary conditions being stable, and some unstable, fixed points. For Ising, $h_{B}=0$ is unstable, and arbitrarily small $h_{B}$ will flow off to infinity. Whenever there is sharp change in boundary conditions, correlation functions in this geometry can be related to those in the geometry with a free boundary via the insertion of a boundary-condition-changing operator, or BCC operator.⁵¹⁵¹51The reason why this works is deep, and related to the state-operator correspondence. In general in CFT, there is actually an isomorphism between states and operators – which is certainly not true for general quantum field theories! The hand-waving way to see this correspondence is to imagine the far past state at the end of a cylinder, which under a conformal transformation gets mapped to a disturbance at the origin – an operator. Similarly, boundary conditions, viewed as states, can be mapped to operators, and the mixed boundary condition case is a BCC operator. That is, if we have a sharp change from boundary condition $a$ to boundary condition $b$ at a point $x$ along the boundary, then $\langle\ldots\rangle_{\text{ab}}=\langle\phi_{ab}(x)\ldots\rangle_{\text{free}}$. Finally, even the bulk is slightly different: due to the presence of the boundary, any $N$-point function in the half-plane is related to a $2N$-point function in the full plane, via the method of images: $\langle\phi(x,\tau)\ldots\rangle_{\text{half plane}}=\langle\phi(-x,\tau)\phi(x,\tau)\ldots\rangle_{\text{full plane}}$.⁵²⁵²52The reason why this works is interesting. When I first learned it, I wondered whether there was an equivalent to Laplace’s equation, as for the usual method of images: solutions are unique given fixed boundary conditions, so we arrange charges so as to mimic a particular set of isocontours of the field (such as a plane conductor or spherical conductor). The reason the method of images works here is because we can “unfold” the CFT from the half plane to the full plane by mapping the left-moving sector to $x<0$ and the right-moving sector to $x>0$. So long as we are not on the boundary of $x=0$, we get two copies of the operator in the plane after we do this, at $x$ and $-x$. This means that the one-point function with a boundary does not generally vanish (and its form is Eq. 13), while the two-point function is not fixed by symmetry due to the cross-ratios.

This concludes our lightning review of the key concepts of CFT; we now move on to apply these techniques to a critical quantum spin chain driven at its boundary.

Recent years have witnessed substantial progress in understanding the dynamics of periodically driven (Floquet) systems. Such driving has traditionally been used for engineering non-trivial effective Hamiltonians ^{doi:10.1080/00018732.2015.1055918, PhysRevLett.116.205301, 0953-4075-49-1-013001, PhysRevX.4.031027}, but recent research has shown that these dynamics can differ drastically from their static counterparts. Examples include the recently observed Floquet time crystals ^{khemani_prl_2016, pi-spin-glass, else_floquet_2016, PhysRevX.7.011026, Zhang:2017aa}, the emergence of topological quasiparticles protected by driving ^{PhysRevLett.106.220402, PhysRevB.94.045127}, Floquet topological insulators ^{PhysRevB.79.081406, PhysRevB.82.235114, Lindner:2011aa, rechtsman_photonic_2013, cayssol_floquet_2013, titum_disorder-induced_2015}, and Floquet symmetry-protected topological phases ^{PhysRevB.92.125107, PhysRevB.93.245145, PhysRevB.93.201103, potter_classification_2016}. More broadly, periodically driven systems touch on fundamental issues in statistical and condensed-matter physics such as thermalization ^{lazarides_periodic_2014, abanin_effective_2015, PhysRevLett.115.256803, kuwahara_floquetmagnus_2016, mori_rigorous_2016} and phase structure ^{khemani_prl_2016}.

However, relatively little attention has been paid to driven systems at criticality, whose low-energy dynamics are often described by a conformal field theory (CFT). Such quantum critical systems are a natural setting in which to study Floquet dynamics, as many insights into the non-equilibrium dynamics of many-body systems have come from the study of CFTs in 1+1d ^{PhysRevLett.96.136801, 1742-5468-2005-04-P04010, 1742-5468-2007-10-P10004, 1742-5468-2016-6-064003}. A naïve expectation is that such a driven critical system would simply heat up. However, in the presence of a boundary drive, the energy injected per cycle is not extensive in system size, and there are multiple possible behaviors in an arbitrarily long period prior to thermalization. Moreover, as CFTs are integrable, it is natural to expect they can escape heating even at low frequencies provided the scaling limit is taken before the long time limit. This opens the door to using scaling theory combined with the analytical toolkit of boundary CFT ^{cardy_boundary_2006, cardy_conformal_1984, cardy_boundary_1989, Affleck:1996mm} to characterize multiple regimes of universal dynamics in such boundary driven quantum critical points.

In this Letter, we study the dynamics of entanglement entropy $S_{l}(t)$ and Loschmidt echo ${\cal L}(t)=\left|\left\langle\psi(0)\middle|\psi(t)\right\rangle\right|^{2}$ in conformally-invariant quantum critical systems subject to a periodic boundary drive. We find two distinct regimes in which boundary conformal field theory provides an excellent description of the dynamics. For suitably slow drives, the system behaves almost as though subject to a series of independent quantum quenches but with amplitude corrections related to multiple-point correlation functions, while for fast drives, the boundary drive can be averaged out, and the system responds as though subject to a single quench at an averaged value of the field. For intermediate driving frequency, we find universal heating which crosses over from a perturbative regime at weak drive to non-perturbative boundary CFT regime at strong drive. The dynamics in all driving regimes are universal and can be described using field-theoretic tools. We numerically confirm that the dynamics remain robust against adding integrability-breaking interactions up to the finite times that may be simulated.

While our results apply to arbitrary boundary-driven CFTs, for concreteness we will focus on the archetypical transverse-field Ising (TFI) model on the half-line with a time-dependent symmetry-breaking boundary field

$$H=-\sum_{i\geq 0}\left(J\sigma_{i}^{z}\sigma_{i+1}^{z}+h\sigma_{i}^{x}+\Gamma\sigma_{i}^{x}\sigma_{i+1}^{x}\right)-h_{b}(t)\sigma_{0}^{z},$$

(19)

with $\Gamma$ an integrability-breaking perturbation and $h\sim J$ tuned to the critical point. This model has a convenient description in terms of free fermions when $\Gamma=0$, seen by performing a Jordan-Wigner transformation ^{sachdev2011quantum} and is thus an ideal numerical test-bed for our model-independent analytical arguments. We initially prepare the system in the ground state at fixed boundary field $h_{b}(t<0)$ then quench on a periodic boundary drive, $h_{b}(t+T)=h_{b}(t)$, for $t\geq 0$. In equilibrium, the low energy description of this spin chain at criticality is well-understood in terms of gapless left- and right-moving Majorana fields satisfying $\{\eta_{R/L}(x),\eta_{R/L}(y)\}=\delta(x-y)$, with Hamiltonian

$$H=-\frac{iv}{2}\int_{0}^{\infty}dx\left(\eta_{R}\partial_{x}\eta_{R}-\eta_{L}\partial_{x}\eta_{L}\right)-\lambda(t)\sigma_{b}(0),$$

(20)

where we dropped irrelevant terms. Here, $v$ is a non-universal velocity ($v=2J$ for $\Gamma=0$) and $\lambda\propto h_{b}$. In this Majorana formulation, the boundary spin can be represented as $\sigma_{b}(0)=i(\eta_{R}+\eta_{L})\gamma$ ^{doi:10.1142/S0217751X94001552}, where $\gamma^{\dagger}=\gamma$ is an ancilla Majorana satisfying $\gamma^{2}=1$ that anticommutes with all fields. In the following, we will assume that the drive is characterized by a single scale $\|h_{b}(t)\|\sim h_{b}$ which we take to be much smaller than the single particle bandwidth $h_{b}\ll\Lambda\equiv 2J=2$ (setting $J=1$), for which field theory is a good equilibrium description. The boundary field is a relevant perturbation with scaling dimension $\Delta=\frac{1}{2}<1$ in the renormalization group (RG) language, with characteristic time scale $t_{b}\sim|h_{b}|^{-\nu_{b}}$, $\nu_{b}=(1-\Delta)^{-1}=2$.

There are three energy scales in this problem: the driving frequency $\omega=2\pi/T$, the bandwidth $\Lambda$, and the scale of the boundary perturbation $t_{b}^{-1}\sim h_{b}^{\nu_{b}}\ll\Lambda$. We will now consider various orderings of these scales and argue that essentially all regimes can be understood using a combination of field theory and scaling arguments, even though the drive is continuously injecting energy into the system. While the Hamiltonian (19) for $\Gamma=0$ can be mapped onto free fermions for numerical convenience (see Sec. Numerical methods: free fermions and interactions), we note that our main conclusions follow from general field theory arguments and therefore continue to hold in the non-integrable case. We emphasize that although we choose to focus on the Ising field theory (20) as an example, our field-theoretic arguments are model-independent, so our results carry over immediately to any boundary driven CFT, such as a driven quantum impurity problem with $t_{b}^{-1}\to T_{K}$, the Kondo temperature.

We start by considering the slow driving regime $\omega\ll t_{b}^{-1}\ll\Lambda$ for a step drive starting from the initial field $h_{b}(t<0)=-h_{b}$ with $h_{b}(t)=+h_{b}$ for $0\leq t\leq T/2$ (Hamiltonian $H_{1}$) and $h_{b}(t)=-h_{b}$ if $T/2\leq t\leq T$ (Hamiltonian $H_{0}$) for $t\geq 0$. Intuitively, this drive looks like independent local quenches. Focusing on the Loschmidt echo (return probability) ${\cal L}=\left|\langle\psi_{0}\left|\psi(t)\right\rangle\right|^{2}$ ^Goussev:2012, this behavior is best understood by Wick rotating to imaginary time $\tau=it$, where the spin-chain Loschmidt echo can be mapped onto a CFT correlation function. After computing this correlation function, we Wick rotate back to real time to obtain the dynamical echo. In imaginary time, the initial state can be generated by an infinite imaginary time evolution $\lim_{\tau\to\infty}e^{-\tau H_{0}}\left|0\right\rangle\propto\left|\psi_{0}\right\rangle$ from arbitrary initial state $\left|0\right\rangle$. In imaginary time, $\exp(-\tau H)$ acts as a projector onto the ground state of $H$, so for large $T\gg t_{b}$ we essentially oscillate between the ground states of $H_{0}$ and $H_{1}$, for which $\sigma_{0}^{z}$ is locked in the direction of the boundary field $\pm h_{b}$. In the CFT language, a sharp change in boundary conditions can be treated by inserting a boundary-condition changing (BCC) operator ^{cardy_boundary_2006}, as diagrammed in Fig. 3. This means that the Loschmidt echo $\mathcal{L}(NT)$ after $N$ periods of drive corresponds to the $2N$-point correlation function of a BCC operator $\phi_{\rm BCC}$ changing the boundary condition from fixed $\sigma_{0}^{z}=\pm 1$ to $\sigma_{0}^{z}=\mp 1$.

Analytically continuing to real time, we expect the Loschmidt echo to be a universal function ${\cal L}(T/t_{b},N)$ in the field theory regime. In the limit $T\gg t_{b}$, this reduces to the $2N$-point function

$${\cal L}(NT)\underset{T\gg t_{b}}{\sim}\left|\left\langle\prod_{n=0}^{2N-1}\phi_{{\rm BCC}}(nT/2)\right\rangle\right|^{2}=c_{N}\left(\frac{T}{t_{b}}\right)^{-\gamma N},$$

(21)

whose form is fixed by scale invariance. The universal exponent $\gamma=4h_{+-}=2$ is given by the scaling dimension $h_{+-}=\frac{1}{2}$ of the BCC operator $\phi_{\rm BCC}$ ^{cardy_conformal_1984, cardy_boundary_1989}. Other step drives can be dealt with in a similar fashion; for example, a step drive from $h_{b}=0$ to $h_{b}\neq 0$ corresponds to the insertion of a BCC field with scaling dimension $h_{\rm BCC}=\frac{1}{16}$. We emphasize that Eq. (21) holds for arbitrary boundary step drives in more general CFTs with the appropriate choice of BCC operator.

Note that although the Loschmidt echo decays exponentially with $N$, consistent with the independent quenches picture, the fact that the quenches are not fully independent is encoded in the non-trivial $N$ dependence of the coefficients $c_{N}$. The ratio $c_{N}/(c_{1})^{N}$ is universal and can be computed exactly for this specific drive, since the BCC operator $\phi_{\rm BCC}$ corresponds to a chiral fermionic field $\psi$ in the Ising field theory with $2N$-point correlator given by a Pfaffian: ${\cal L}(NT)\sim\left|\left\langle\psi(0)\psi(T/2)\psi(T)\dots\right\rangle\right|^{2}\sim\left|{\rm Pf}(1/(t_{i}-t_{j}))\right|^{2}$ with $t_{i}=0,T/2,T,\dots,(N-\frac{1}{2})T$. For step drives in general CFTs, such universal ratios can be computed within the Coulomb gas (bosonization) framework. These analytical expressions are in excellent agreement with numerical simulations for $\Gamma=0$ (Fig. 3), where the only non-universal fit parameter is $c_{1}$. Since these predictions rely solely on field theory, they apply equally well to the non-integrable case $\Gamma\neq 0$; the interactions $\Gamma\sigma_{i}^{x}\sigma_{i+1}^{x}$ are irrelevant in the RG sense and therefore do not change the universality class. We confirm this numerically by locating the new critical point for $\Gamma\neq 0$ using exact diagonalization, obtaining the ground state using standard density matrix renormalization group (DMRG) techniques ^{PhysRevLett.69.2863, Schollwock201196}, and simulating the dynamics of this driven interacting chain using time-evolving block decimation (TEBD) ^{PhysRevLett.93.040502}. We find excellent agreement with our field-theoretic argument, as shown in Sec. Interactions.

Consider now a more general drive such as $h_{b}(t>0)=-h_{b}\cos(\pi t/T)$ with $h_{b}(t<0)=-h_{b}$. In the large $T$ limit $h_{b}(t)$ crosses the critical value slowly rather than suddenly, yet the BCC picture suggests that the field should quickly flow to infinity. We find, however, that the vanishing (but finite) crossing speed is strongly relevant, changing the power law entirely (Fig. 4). To understand this difference, we use the concept of Kibble-Zurek (KZ) scaling, which is frequently applied to bulk drives crossing a bulk quantum critical point ^{PhysRevLett.95.105701, PhysRevB.72.161201, PhysRevLett.95.245701, 2016arXiv161202259L} but has not been studied for such boundary drives to our knowledge.

Let us imagine that the drive crosses $h_{b}=0$ as a power-law $h_{b}(t)=h_{b}|\frac{t}{T}|^{r}\mathrm{sgn}(t)$ with $r=1$ in the cosine drive considered above and $r=0$ for a quench ^{PhysRevB.81.012303}. The effective time scale $t_{b}(t)\sim\left[h_{b}(t)\right]^{-\nu_{b}}$ now becomes time-dependent, and we expect the dynamics to be controlled by an emergent time scale

$$t_{\rm KZ}\sim T^{\frac{r\nu_{b}}{1+r\nu_{b}}}h_{b}^{-\frac{\nu_{b}}{1+r\nu_{b}}},$$

(22)

given by $t_{\rm KZ}\sim t_{b}(t_{\rm KZ})$. Though our system is always gapless so that there is no adiabatic limit, it is straightforward to show that this dynamical scale emerges directly from the equations of motion of Eq. (20) ^mike_prl. It is natural to expect that the slow driving limit $T\gg t_{\rm KZ}$ should still be described by boundary CFT, suggesting that the Loschmidt echo would scale as (21) with $t_{b}$ replaced $t_{\rm KZ}$. We therefore see that the effect of the slow driving amounts to renormalizing the dimension $h_{\rm BCC}$ of the BCC operator by a factor $\alpha_{\rm KZ}=1/(1+r\nu_{b})$ with $\nu_{b}=2$ in our case. More generally, for a drive where $h_{b}(t)$ crosses or touches the critical value $n$ times within a single cycle, we predict that the universal exponent $\gamma$ controlling the exponential decay of the Loschmidt echo is given by

$$\gamma=2\sum_{i=1}^{n}\frac{h^{i}_{\rm BCC}}{1+r_{i}\nu_{b}},$$

(23)

where $r_{i}$ is the power of $|h_{b}(t)|\sim|t-t^{i}_{c}|^{r_{i}}$ near the critical time $t^{i}_{c}$. For our model, $h^{i}_{\rm BCC}=\frac{1}{2}$ if $h_{b}(t)$ crosses zero and $h^{i}_{\rm BCC}=\frac{1}{16}$ if it touches zero without changing sign ^{cardy_conformal_1984, cardy_boundary_1989, cardy_boundary_2006}. For example, a cosine or triangle drive oscillating between $\pm h_{b}$ has $n=2$, $r_{1}=r_{2}=1$ so that $\gamma=2/3$, while a sawtooth drive combines slow ($r_{1}=1$) and fast ($r_{2}=0$) crossings to give $\gamma=4/3$.

These predictions give good agreement with numerics (Fig. 4).⁵³⁵³53We note that the agreement is systematically worse for larger $r$ due to finite system size and Trotter step errors. Furthermore, the only effect of the slow driving is to renormalize the scaling dimensions of the BCC operators while keeping the structure of the $2N$-point function unchanged. In particular, we find that the universal numbers $c_{N}/(c_{1})^{N}$ in Eq. (21) are still given by the boundary CFT predictions for a step drive (see Sec. Structure of the $N$-point correlation functions and Kibble-Zurek Scaling).

We now consider the high-frequency regime $t_{b}^{-1}\ll\Lambda\ll\omega$. This is naïvely outside the regime where field theory results should apply, but we can take advantage of standard Floquet machinery to write a Floquet-Magnus high-frequency expansion for the Floquet Hamiltonian $H_{F}$ defined by $U(T)={\cal T}{\rm e}^{-i\int_{0}^{T}dtH(t)}={\rm e}^{-iTH_{F}}$ ^{CPA:CPA3160070404}. For example, $H_{F}=\frac{1}{2}(H_{0}+H_{1})-\frac{i}{4\omega}[H_{0},H_{1}]+\mathcal{O}(\omega^{-2})$ for a step drive. While higher order terms in this expansion are suppressed by powers of $\omega^{-1}$ as for any high-frequency Floquet system, we note here that the Floquet Hamiltonian $H_{F}$ itself corresponds to a CFT subject to an effective boundary field $\overline{h}_{b}=(1/T)\int_{0}^{T}h_{b}(t)dt$ with higher order terms in the high frequency expansion being RG irrelevant. This is most easily seen using the field theory Hamiltonian (20) where the small parameter controlling the expansion is $v/\omega\ll 1$ with $v\sim\Lambda=2J$. While the first boundary term has scaling dimension $\Delta=1/2$ and corresponds to the averaged field $\overline{h}_{b}$, dimensional analysis immediately implies that terms of order $\omega^{-n}$ have scaling dimension of at least $n+1/2$ due to terms such as $\partial^{n}\eta(0)$ and are thus irrelevant for $n>0$ (see Sec. RG analysis of the High Frequency Expansion). Therefore at late times, the system behaves as though subject to a single local quantum quench with effective boundary field $\overline{h}_{b}$ (Fig. 5), a problem whose universal dynamics has been studied extensively ^{PhysRevLett.110.240601, PhysRevX.4.041007}. We remark that though RG techniques may be in general ill-defined in a Floquet system which, for instance, lacks a notion of ground state, in this high-frequency limit the Floquet evolution is well-controlled by an effective static Hamiltonian. Since our initial state is a conformally invariant ground state and the effective Hamiltonian implements a local quench, the notion of RG flow is well-defined ^{PhysRevLett.110.240601} and provides a powerful tool of analysis. Additionally, for the non-interacting (free fermions) case with $\Gamma=0$ in Eq. (19), one may prove that the high-frequency expansion is convergent for $\omega\lower 2.0pt\hbox{$\,\buildrel\scriptstyle>\over{\scriptstyle\sim}\,$}\Lambda$ by bounding the spectral width of the single-particle Hamiltonian (see Sec. Convergence of the HFE and heating in the high-frequency limit). More generally, this effective single quench picture will survive even in the presence of integrability-breaking interactions controlled by $\Gamma$ up to exponentially long time scales $\tau_{\rm th}\sim{\rm e}^{C\omega/\Lambda}$ ^{abanin_effective_2015, PhysRevLett.115.256803, mori_rigorous_2016, kuwahara_floquetmagnus_2016}. We simulated the dynamics of this interacting chain subject to the same drive using TEBD and found excellent agreement with the single effective quench picture even at moderate frequencies (Fig. 5).

Finally, we discuss the intermediate crossover regime $t_{b}^{-1}\sim\omega\ll\Lambda$. We focus on a free-to-fixed step drive from $h_{b}=0$ to $h_{b}\not=0$ with $\Gamma=0$ for simplicity. In this regime, we expect the system to absorb energy (“heat”) via resonant processes within the single-particle bandwidth. This leads to exponential decay of the Loschmidt echo,

$${\cal L}(NT)\underset{t_{b}^{-1},\omega\ll\Lambda}{\sim}{\rm e}^{-N/N_{\star}(\omega t_{b})},$$

(24)

with $N_{\star}(\omega t_{b})$ a universal function (Fig. 6a). For weak drive ($\omega t_{b}\gg 1$), resonant heating occurs with a rate $\tau^{-1}\sim h_{b}^{2}/J$ given by Fermi’s golden rule, so that $N_{\star}\sim\tau/T\sim\omega t_{b}$. For strong drive ($\omega t_{b}\ll 1$), we recover the boundary CFT prediction $N_{\star}\sim-1/(\gamma\log\omega t_{b})$. We also find that entanglement entropy of boundary intervals of size $\ell$, relative to the ground state entropy, saturates to a volume law behavior $S_{\ell}\sim\ell$ at long times in the regime $\omega t_{b}\gg 1$, consistent with heating.⁵⁴⁵⁴54We note that by heating we simply mean absorption of energy, not thermalization. Since this is an integrable model, the reduced density matrix should more accurately be described by a generalized Gibbs ensemble ^{PhysRevLett.98.050405} which in some observables can look highly athermal At low frequencies, the entropy simply oscillates between ground state values⁵⁵⁵⁵55In the absence of boundary field, at large distances the ground state entanglement entropy is given by the well-known result $S_{l}(h_{b}=0)=\frac{c}{3}\ln\frac{l}{\tilde{a}}$^{1751-8121-42-50-504005}, where $c=1/2$ is the central charge of the CFT and $\tilde{a}$ is a non-universal constant. Pinning the boundary spin causes a universal reduction of the entanglement entropy for $h_{b}\neq 0$ of $\Delta S=-\log\sqrt{2}$, known as the Affleck-Ludwig entropy ^{affleck_universal_1991}. though it may become extensive at much later times. We leave a detailed analysis of the role of interactions in this intermediate regime for future work.