Boundaries and unphysical fixed points in Dynamical Quantum Phase Transitions

TL;DR

This paper investigates how boundary conditions influence dynamical quantum phase transitions (DQPT) and reveals the presence of unphysical fixed points affecting the nature of these transitions in quantum spin models.

Contribution

It demonstrates that boundary effects can suppress bulk DQPTs and identifies unphysical RG fixed points through exact analysis of quantum Ising and Potts models.

Findings

Boundary conditions can completely suppress bulk DQPTs.

Unphysical fixed points are involved in DQPTs.

DQPTs in the Potts chain relate to period-2 fixed points.

Abstract

We show that dynamic quantum phase transitions (DQPT) in many situations involve renormalization group (RG) fixed points that are unphysical in the context of thermal phase transitions. In such cases, boundary conditions are shown to become relevant to the extent of even completely suppressing the bulk transitions. We establish these by performing exact RG analysis of the quantum Ising model on scale-invariant lattices of different dimensions, and by analyzing the zeros of the Loschmidt amplitude. Further corroboration of boundaries affecting the bulk transition comes from the three-state quantum Potts chain, for which we also show that the DQPT corresponds to a pair of period-2 fixed points.

Figures

Equations

$$L(t)=⟨{\psi }_0|e^{-itH}|{\psi }_0⟩\sim e^{-Nf(t)},(\hslash =1),$$

$$H=-J\sum _{⟨jk⟩}{\sigma }_j^{\mathrm{z}}{\sigma }_k^{\mathrm{z}},H_{\Gamma }=-\Gamma \sum _j{\sigma }_j^{\mathrm{x}},(J,\Gamma >0),$$

$$L(y)=2^{-N}\sum _C{y}^{-E_C/2J},f(y)=-N_B^{-1}\ln L(y),$$

$$Z_n(y)$$

$$f_n(y)$$

$$y_1=2^{-b}(y+y^{-1}{)}^b.$$

$$\displaystyle Z_{1}=\left\{\begin{array}[]{ll}2(y^{1/2}+y^{-1/2}),&{\rm(Open~{}BC)}\\ y^{1/2},&{\rm(Fixed~{}BC)}(\uparrow\uparrow),\end{array}\right.$$

$$f_{+}(y)=-\frac{1}{2}\ln {\cos }^2\frac{\theta }{2},\mathrm{and}{f}_{-}(y)=-\frac{1}{2}\ln {\sin }^2\frac{\theta }{2},$$

$$L(t,h)=(\cos Jt{)}^{N-1}{\cos }^{2}ht+(i\sin Jt{)}^{N-1}{\sin }^{2}ht.$$

$$H=-J\sum _j({\Omega }_j^{†}{\Omega }_{j+1}+H.c.),$$

$$y_1=R(y)\equiv (y^2+2)/(2y+1),$$

$$d=\frac{\ln 2b}{\ln 2},$$

$$N_n=b{B}_{n-1}+N_{n-1}=2+b\frac{(2b{)}^n-1}{2b-1}.$$

$$Z=\sum _{C\in 2^N\mathrm{states}}{e}^{-\beta E_C},$$

$$E_C=-J\sum _{⟨jk⟩}{s}_j{s}_k,$$

$$Z_1{|}_{++}$$

$$Z_1{|}_{+-}$$

$$y_1$$

$$\zeta (y)$$

$$Z_{N}=\left\{\begin{array}[]{ll}{\rm Tr}~{}{\sf T}^{N},&{\rm(periodic\ BC),}\\ \sum_{j,\;k=1,\;2}~{}[{\sf T}^{N-1}]_{jk},&{\rm(free\ BC),}\\ (e^{zh}\,e^{-zh}){\sf T}^{N-1}\left({{\rm e}^{zh}}\atop{e^{-zh}}\right),&{\rm(with\ boundary\ fields),}\end{array}\right.$$

$$Z_N={\Lambda }_{+}^N+{\Lambda }_{-}^N,$$

$$Z_{N}=\left\{\begin{array}[]{ll}\Lambda_{+}^{N},&{\rm for}\ \theta\ {\rm near}\ 0,\ |\Lambda_{+}|>|\Lambda_{-}|\\ \Lambda_{-}^{N},&{\rm for}\ \theta\ {\rm near}\ \pi,\ |\Lambda_{-}|>|\Lambda_{+}|\end{array}\right.$$

$$f_{\pm }(\theta )\equiv -\mathrm{Re}\underset{N\to \infty }{\lim }\ln (Z_N/2^N)=-\mathrm{Re}\ln ({\Lambda }_{\pm }/2),$$

$$f_{+}=-\frac{1}{2}\ln {\cos }^2\frac{\theta }{2},\mathrm{and}{f}_{-}=-\frac{1}{2}\ln {\sin }^2\frac{\theta }{2},$$

$$y=i\cot \frac{(2n+1)\pi }{N},n=-N,...,N-1,$$

$$Z_{N}=\left\{\begin{array}[]{ll}\Lambda_{1}^{N}+2\Lambda_{2}^{N},&\quad(\rm periodic~{}BC),\\ \Lambda_{1}^{N},&\quad(\rm Open~{}BC),\end{array}\right.$$

$$y=-\frac{1}{2}+i\frac{3}{2}\cot \frac{\pi (2n+1)}{N},(\mathrm{PBC}).$$

$$Z_n(y)$$

$$\mathrm{and}{f}_n(y)$$

$$f_{n+1}(y)=\sum _{j=1}^n\frac{1}{(2b{)}^j}g(y_j)+\frac{1}{(2b{)}^n}{f}_1(y_n),$$

$$y^2-2y^{(b+1)/b}+1=0.$$

$$f_{\infty }(1)=\frac{b}{2b-1}\ln 2,$$

$$f_c=\frac{1}{4b}\ln \frac{4^b}{y_c}+\frac{1}{4b(2b-1)}\ln (4^b{y}_c)-\frac{b}{2b-1}\ln 2,$$

Full text

Dynamical quantum phase transitions (DQPT), a recent discovery of phase transitions, often periodic, in large quantum systems during time evolution heylprl ; heyl2 ; comm1 , have generated a lot of interest because here time itself acts as the parameter inducing the transitions. Also, to be at a transition point, only time needs to be chosen properly without any requirement of fine-tuning of system parameters, unlike thermal transitions comm11 . The signature of DQPT is the nonanalytic behaviour of various quantities in time around critical times $t_{c}$’s. These transitions have now been shown in many models, like the transverse-field Ising model (TFIM), spin chains, quantum Potts models, the Kitaev model, and many othersheylprl ; heyl2 ; Andras ; ks17 ; zvyagin ; pozsgay ; heylreview , and also observed experimentallyexpt2 ; experiment . In spite of being a zero-temperature quantum phenomenon, DQPT is not determined by the quantum phase transitions of the system but rather seems related to the classical thermal criticalities of an associated systemheylreview . However, despite the use of many techniques so far, very few exact results are known on the scaling and universality in DQPT heylreview . Moreover, the natures of the possible phases and the transitions remain to be properly classified, e. g., whether only equilibrium phases and transitions would suffice or there can be specialities of its ownheylreview .

A general approach for phase transitions is the renormalization group (RG) framework cardy in terms of length-dependent effective parameters and their flows to the fixed points (FP), with the stable FPs determining the allowed phases, and the unstable ones (or separatrices) the phase transitions. In this Letter, we adopt an exact RG scheme for TFIM and the three-state quantum Potts chain (3QPC). Our exact results establish that there are DQPTs involving FPs that are unphysical in traditional thermal transitions. Second, we show that, for those unphysical FPs, boundary conditions (BC) are relevant and can even lead to a suppression of the transitions completely, unlike thermal cases where BCs do not affect the bulk transitions. Another surprising result is the emergence of a pair of period-2 FPs, never seen in the thermal context, that controls the DQPT in 3QPC, in contrast to the zero-temperature FP heyl2 for the Ising DQPT case. In short, our exact results bring out several distinctive features of DQPT, not to be found in equilibrium transitions.

If a quantum system, with Hamiltonian $H$, is prepared in a noneigenstate $|\psi_{0}\rangle$ and suddenly allowed to evolve, then the probability for the system to be in state $|\psi_{0}\rangle$ after time $t$ is given by $P(t)=|L(t)|^{2}\sim e^{-N\lambda(t)},$ where

[TABLE]

is the Loschmidt amplitude with $f(t)$ and $\lambda(t)=2{\rm Re}f(t)$ as the large-deviation rate functionstouchet for a large system of $N(\to\infty)$ degrees of freedom. Often, $\lambda(t)$ and $f(t)$ show phase-transition-like nonanalyticities at time $t=t_{c}$. These phase transitions in time are the DQPTszvyagin ; heylreview .

TFIM is defined on a lattice as $H_{\rm I}=H+H_{\Gamma}$, where

[TABLE]

$\sigma^{\alpha}_{j}$ being the Pauli matrices ($\alpha=$ x, y, z) at lattice site $j$, and $\langle jk\rangle$ denoting nearest neighboursbkc . The interaction favours an aligned state in the z direction spectrum , and $H_{\Gamma}$ is the transverse field term that aligns the spins in the x direction. We may add a boundary term given by $H_{\rm B}=-h(\sigma_{1}^{\rm z}+\sigma_{N}^{\rm z}),$ where the boundary field $h$ acts only on the first and the $N$th spins. Two special cases are $h=0$ and $h\to\infty$ corresponding to open BC and fixed BC (both up in the z-direction) respectively. For periodic BC in one dimension, $H_{B}=-J\sigma_{1}^{\rm z}\;\sigma_{N}^{\rm z}.$

The TFIM is prepared in a product state $|\psi_{0}\rangle$ commx with all spins aligned in the x direction, e.g., by $\Gamma\to\infty.$ At time $t=0$, we set $\Gamma=0$. So, the magnet evolves with $H$ of Eq. (2) and any boundary term mentioned above. This is the particular sudden quench we use in this Letter. By expressing $|\psi_{0}\rangle$ in terms of the eigenstates of $H$, the Loschmidt amplitude and the rate function per bond can be expressed as heylreview ; spectrum ,

[TABLE]

respectively, where $y=e^{2zJ},N_{B}$ is the number of bonds, and, for generality, $z$ is taken as a complex number. $L(y)$ is an analytic continuation of the partition function of the traditional nearest neighbour Ising modelhuang defined for $1\leq y<\infty$ on the real positive axis ($z=\beta$ being the inverse temperature). The quantum time evolution in Eq. (1) is given by the unit circle $|y|=1$ ($y=e^{i2Jt}$) in the complex $y$ plane. A phase transition—defined as the point of nonanalyticity of $f$—is expected along the unit circle if there are zeros or limit points of zeros of $L(y)$ on the pathhuang . An isolated zero on the circle, in contrast, just indicates orthogonality of the evolved and the initial states. The $S^{1}$ (circle) topology guarantees (via winding numbers) that, if there are zeros on the circle, there will be periodic transitions in time.

In one-dimensional TFIM and 3QPC, similar DQPT occurs, viz., linear kinks in $f(t)$, despite the absence of any thermal transitionsheylprl ; ks17 . For the two-dimensional TFIM, DQPT was found to be the same as the two dimensional Ising critical pointheyl2 . However, the generality of these results has not yet been established. In this context, we focus on a class of exactly solvable models that would help us in alienating the specialities of DQPT.

We choose scale-invariant lattices for which the real space renormalization group (RSRG) can be implemented exactly. The lattices are constructed hierarchically by replacing a bond iteratively by a diamondlike motif of $b$ brancheshier ; smb as shown in Fig. 1. Such lattices appear naturally in approximate RSRG for usual lattices. Three cases are considered here, (i) $b=1$ corresponding to a one-dimensional lattice, (ii) $b=2$ which is two dimensional but not a Bravais lattice, and (iii) $b=3$ as a fractal-type lattice.

The hierarchical structure of the lattice allows us to calculate $L(y)$ via a real space renormalization group approach, by decimating spins on individual motifshier ; derrida . Let us define $Z_{n}=2^{N}L_{n}$ and $f_{n}=(2b)^{-n}\ln Z_{n}$ for the $n$th generation. Note that $f_{n}(y)$ is related to $f(y)$ of Eq. (3) by $f=f_{n}(1)-f_{n}(y).$ $Z_{n},$ and $f_{n}$ satisfy the following recursion relations (see Supplemental Material suppl )

[TABLE]

with $f_{1}=\ln Z_{1}$.

Equation (4c) has FPs at $y=1$ (infinite-temperature FP, paramagnetic phase), $y=\infty$ (zero-temperature FP, ordered phase), and a $b$-dependent unstable FP at $y=y_{c}$ (for $b>1$) representing the critical point. For any odd $b>1$, there are additional “unphysical” FPs at $y=-1,-y_{c}$ ($\pm\infty$ to be identified). There is no $y_{c}$ for $b=1$, as there is no thermal phase transition for the one-dimensional Ising model. The zeros of $L_{n}(y)$ can be determined from those of $L_{n-1}$ via Eqs.(4a) and (4c), starting from the BC-dependent roots of $L_{1}(y)=0$. In the $n\to\infty$ limit, the zeros then belong to the set of points that do not flow to infinity, thereby constituting the Julia set of the transformationderrida . These sets, obtained by MATHEMATICA, are shown for $b=1,2,$ and $3$ in Figs. 2 and 3.

The $y,y^{-1}$ symmetry in Eq. (4c) ensures that if $y^{*}$ is a FP, then $1/y^{*}$ flows to $y^{*}$. Therefore, there are four special points on the unit circle which flow to the nontrivial FPs, and are, necessarily, members of the Julia set. These four points on the unit circle in Figs. 2(iii), 2(iv), and Fig. 3 are the four critical points in time for $b>1$. Incidentally, Eq. (4c) also ensures that any point on the unit circle, $y=e^{i\theta}$, under iteration, first flows to the real axis to $\cos\theta$ and then remains real afterwards. Consequently, complex RG fixed points for $b>1$ are not important.

DQPT has been studied for $b=1$ under periodic BCheylprl . The surprising result we find here is that, unlike the thermal case, boundary conditions may even suppress the bulk transition. The transfer matrix solution of the 1D Ising model describes the partition function by the two eigenvalues $\Lambda_{\pm}=y^{1/2}\pm y^{-1/2}$, with the larger one determining the $N\to\infty$ behaviourhuang . For $y$ flowing to $y^{*}=+1~{}(y^{*}=-1)$, the larger eigenvalue in magnitude is $\Lambda_{+}~{}(\Lambda_{-})$, so that, with $y=e^{i\theta}$, the rate functions for the two regions ($f_{\pm}\sim\ln\Lambda_{\pm}/2$) are (see Supplemental material suppl )

[TABLE]

respectively. As characteristics of the high-temperature phases, $f_{\pm}$ should be independent of dimensions, remaining valid for all $b$. Open BC yields only one zero at $y=-1$ [Fig. 2(i)], and therefore no DQPT. On the other hand, periodic and fixed BC give zeros on the imaginary-$y$ axis [Fig. 2(ii)]. Two zeros $y=\pm i$ on the unit circle, demarcating the RG flows of the points on the unit circle to $y=\pm 1$, are the known transition pointsheylprl ; heyl2 . The transitions are from a paramagnetic (described by FP at $y=1$, and $f_{+}$) to another paramagnetic phase, which we call para*′*, described by FP $y=-1$, and rate function $f_{-}$]. [Fig. 4(i)].

Now consider an open chain with the boundary term $H_{\rm B}=-h(\sigma_{1}^{\rm z}+\sigma_{N}^{\rm z})$. For a finite chain, there will be contributions from both the FPs $y=\pm 1$, so that for an $N$-site chain (see Supplemental Meterial suppl )

[TABLE]

DQPT with $f_{\pm}(t)$ is recovered in the $N\to\infty$ limit only if $h\neq 0$. See Fig. 4(i). For an open chain, $L(t)\equiv L(t,{h=0})=(\cos Jt)^{N-1}.$ Hence, there is no transition [Fig. 4(ii)], consistent with one single zero [Fig. 2(i)]. There are four sectors of possible configurations of the two boundary spins, viz., $(\pm,\pm)$. Each of these four sectors individually shows DQPT. However, for the zero-field open chain, requiring superposition of the four sectors, there is a perfect cancellation of the $y=-1$ contributions. Thus, only $f_{+}$ survives [Fig. 4(ii)]. When the subtle cancellation of the four sectors is disturbed by the small boundary fields, the transitions appear, as shown in Fig. 4(i). We see that boundary conditions (like open-chain) become relevant only at the unphysical fixed point.

For any odd $b>1$, there are four critical points on the unit circle, Figs. 2, and 3. These are A1, $\theta_{A}=2J\tau_{1}=\arccos{y_{c}}^{-1/b}$, and A2, $2J\tau_{2}=2\pi-\theta_{A}$ on the right half-plane, flowing to $y_{c}>0$, and K1, $2J\kappa_{1}=\pi-\theta_{A}=\arccos({-y_{c}})^{-1/b}$, and K2, $2J\kappa_{2}=\pi+\theta_{A}$, on the left half-plane, flowing to the unphysical FP at $-y_{c}<0$, via $y=-1/y_{c}$. These transition points ($l\pi\pm\theta_{A}$, for any integer $l$) are determined exactly. In this particular case, the nature of the singularity happens to be the same for all, as for the thermal case [a diverging third derivative of $f$, Fig.  4(iii)]. The flows of the four arcs of the unit circle are shown in Fig. 2(vi). K1K2, being characterized by FP $y=-1$, is expected to be sensitive to any constraint on the boundary spins. For, say, fixed boundary spins, a sequence of phases occurs in time, para-ferro-para*′-ferro-para, separated by the four critical points. The two para phases with FP $y=\pm 1$ have ferromagnetic phases in between. However, in the unbiased case, the algebraic sum of the contributions of the four boundary sectors may lead to cancellation as in the $b=1$ case. A signature of the cancellation in the K1K2 region is the failure of Eq. (4b) for $f$ as $y\to-1$ on renormalization. This stability problem is also seen in the one-dimensional case, Fig. 4(i) vis-à-vis Fig. 4(ii) [the recursion relation, Eq. (4b), fails for $\pi/2<2Jt<3\pi/2$]. We, therefore, conjecture that for the open BC case (free boundary spins) there is no intermediate para′* phase, but instead the whole arc A1K1K2A2 represents the ferro phase—a major boundary effect on bulk DQPT.

For even $b$, there are again four points on the unit circle [Figs. 2(iii) and 3(i)], A${}_{i},(i=1,4),$ which have identical angular relations as the four points for odd $b$, except that here all flow to $y_{c}$ in two steps via $y=1/y_{c}$. All points in arcs A1A4 and A2A3 flow to $\infty$ implying an ordered state, while the remaining two arcs, A1A2 and A3A4, flow to $1$, the disordered phase. Therefore, there is an oscillation between ordered (broken-symmetry) phase and the standard disordered phase with critical points at four different times. The nonanalytic features at the four critical times are the same as for the temperature-driven critical point at $y_{c}$. In essence, DQPT here follows closely the thermal transition.

To show the generality of the boundary effect, let us consider the three state Potts chain of $N$ sites (3QPC) involving $3\times 3$ matrices ks17 . The interaction term is

[TABLE]

where $\Omega={\rm diag}(1,e^{i2\pi/3},e^{i4\pi/3})$. Analogous to the transverse field of Eq. (2), the spin flipping term for Potts spin is $H_{\Gamma}=-\Gamma\sum_{j}T_{j}$, where the elements of the $3\times 3$ matrix $T$ are given by $T^{\alpha\beta}=1-\delta_{\alpha\beta},(\alpha,\beta=1,2,3).$ $\Gamma$ can be used to prepare the chain in a product state of equal-amplitude superpositions of the three states of each spin. The chain evolves in time with $H$ of Eq. (7), once $\Gamma$ is switched off. Two boundary conditions are considered here, viz., periodic and open BCs. These two differ by an interaction term connecting the first and the $N$th sites. (See Supplemental Material suppl .)

In the Potts model the basic energy scale for a bond is the gap $3J$, and so define $y=\exp(3\beta J)$. The RG equation for $y$ isderrida ; comm2

[TABLE]

whose fixed points are $y_{p}=1$, $y_{u}=-2$ (“unphysical”), and $y=\pm\infty$. The DQPT involves the transition between the two stable phases described by $y_{p}$ and $y_{u}$, (analogous to Fig. 2(iv)), with the critical times at the points of intersection of the unit circle and the line of zeros of $L(y)=0$. These intersectioons are A1, $y_{A1}=e^{i2\pi/3}$, and A2, $y_{A2}=e^{i4\pi/3}$, which flow into each other under the RG transformation. In other words, A1,i and A2 are period-2 FPs of RG, i.e., the fixed points of $R^{(2)}=R(R(y))$. By linearizing $R^{(2)}$ around A1 or A2, the thermal eigenvalue is found to be 1, which leads to a kink in ${\rm Re}\,f$ at the transition points ks17 . The emergence of these novel “unphysical” fixed points distinguishes the DQPT from thermal transitions in general, and, in particular, the 3-state Potts chain from TFIM, though both show similar nonanalyticity comm20 .

Now consider a free (i.e., open) chain. There is only one zero at $y=-2$ outside the unit circle [compare with Fig.2(i)]. There cannot be any DQPT. A direct computation of the rate function $f(y)$ by the transfer matrix methodks17 ; huang shows that $f(y)\sim-\ln[(y+2)/3],(y=e^{i\theta}),$ and no DQPT, in agreement with the zeros. From the RG point of view, we see that the phase described by the unphysical fixed point at $y=-2$ does not occur for the open chain case, though it exists for the periodic case. This exact result provides yet another example of boundary conditions affecting the bulk transition in the quantum case, when an unphysical FP is involved.

Our results are summarized at the beginning, and, to that, we add the following details. The four transition points (critical times) for dynamical quantum phase transitions are determined exactly for the Ising model on hierarchical lattices of any $b>1$. The ordered (broken-symmetry) state appears as a phase only in higher dimensions ($b>1$). For all odd $b$, the phase transitions involve one phase characterized by a stable, but unphysical, fixed point. There are no such unphysical fixed points for even $b$, and, therefore, no sensitivity to boundary conditions. We anticipate that our results would lead to explorations of other unphysical fixed points in various quantum systems to look for new phases, criticality, and boundary effects, not found in thermal phase transitionscomm3 .

Note added.—Recently, Ref. comm3 appeared, which dis- cussed the role of complex fixed points for Potts chains.

S1 I. On hierarchical latices

For a hierarchical lattice (diamond type) of $b$ branches, the dimensionSsmb is given by

[TABLE]

so that $d=1,~{}2$ for $b=1,~{}2$, while $d=\ln 6/\ln 2=2.58...$ for $b=3$. For the $n$th generation, the number of bonds is $B_{n}=(2b)^{n}$ and the number of sites is

[TABLE]

In the $n\to\infty$ limit, $\frac{N_{n}}{B_{n}}\to\frac{b}{2b-1}.$

S2 II. Partition function

The partition function of the traditional nearest neighbour Ising model is given byShuang

[TABLE]

with

[TABLE]

as the energy of configuration C. Here, $s_{j}=\pm 1$, $\beta=1/k_{B}T$, $T$ being the temperature and $k_{B}$ the Boltzmann constant. The free energy is given by $-k_{B}T\ln Z$.

We take $y=e^{2\beta J}$ as the variable because $2J$ is the energy gap for a single bond.

The existence of the thermodynamic limit ($N\to\infty$) for the Ising model ensures the large deviation form $L~{}=~{}\exp(-Nf(y))$, with $f$ as the analog of the free energy extended to the complex plane.

S3 RG relations for the Ising model

We derive the RG relations for the Ising model on a hierarchical lattice of $b$ branches. The decimation of a motif to a bond is shown in Fig. S1

Denoting the partition function of a bond of two spins $s_{1},s_{2}=\pm 1$ by $Z_{s_{1}s_{2}}(y)$, the summation over the internal spins (green disks) yields the relation between the partition function $Z_{1}$ of the larger cell (generation $n=1$) as a function of $y$ and the partition function of a bond ($n=0$) as a function of the renormalized parameter $y_{1}$. Explicitly,

[TABLE]

exploiting the fact that the $b$ branches are independent. Only two relations are sufficient, thanks to the symmetry that $Z_{++}(y)=Z_{--}(y),$ and $Z_{+-}(y)=Z_{-+}(y).$ Noting that $Z_{++}(y)=y^{1/2},$ and $Z_{+-}(y)=y^{-1/2}$, one gets

[TABLE]

S4 III. Derivation of $f_{\pm}$ and Zeros of the one-dimensional Ising chain

The two high temperature fixed points of Eq. S5a, $y^{*}=\pm 1$ characterize the two possible phases of the one dimensional Ising model. The Ising partition function can be determined by a transfer matrix approach Shuang .

If ${\sf T}$ is the $2\times 2$ transfer matrix, then the partition functions for a chain of $N$ sites under different BCs are given by

[TABLE]

For periodic BC

[TABLE]

where $\Lambda_{\pm}=y^{1/2}\pm y^{-1/2}$ are the two eigenvalues of ${\sf T}$. Eq. 6 of the text follows from Eq. (S6)

For large $N$, and $y=e^{i\theta}$

[TABLE]

The rate functions

[TABLE]

are then given by

[TABLE]

as quoted in Eq. (5) of the text.

Zeros of the one-dimensional Ising chain

The partition function for an $N$-site Ising chain with periodic boundary condition is given in Eq. (S7). The zeros of $L_{N}(y)=Z_{N}/2^{N}$ are then given by

[TABLE]

which lie along the imaginary axis. This is shown in Fig 2(b) in the text.

For open BC, the partition function is given by $Z_{N}=\Lambda_{+}^{N}$. There is therefore only one zero at $\Lambda=0$. Therefore, $L_{N}(y)$ has only one zero at $y=-1$, as shown in Fig 2(a) in the text.

S5 V. Zeros of the one-dimensional Potts chain

For the Potts model $y=\exp(3\beta J)$. The partition functions for the Potts chain are given by

[TABLE]

where $\Lambda_{1}\propto(y+2),$ and $\Lambda_{2}\propto(y-1)$ are the two eigen values of the transfer matrix given in Ref. sks17 . The zeros for the periodic BC case are given by $\Lambda_{1}/\Lambda_{2}=e^{i(2n+1)\pi/N}$, so that

[TABLE]

For an open chain (free BC), it follows from Eq. (S10) that there is only one zero at $\Lambda_{1}=0$, i.e., at $y=-2$.

S6 V. On $L_{n}$ and $f_{n}$

Under the decimation transformation of Fig. S1 and Eq. (S4a), the partition function for generation $n$ with parameter $y$, $Z_{n}(y)$, can be related to that of the $(n-1)$th generation with parameter $y_{1}$. The recursion relations for $Z_{n}$, and $f_{n}=(2b)^{-n}\ln Z_{n}$ can be written asSderrida

[TABLE]

where $g(x)=\ln\zeta(x)=2^{-1}\ln(4^{b}x)$, and $y_{1}$ is given by Eq. (S5a). Note that $f_{n}$ is defined without the normalization factor $2^{N}$ (Eq. (4a) in text). With successive transformation $y\to y_{1}\to...\to y_{n}$ following the RG flow equation, the Loschmidt amplitude is given by a rapidly convergent sum for large $n$ as

[TABLE]

provided the functions remain well-defined at the transformed arguments.

The fixed points of Eq.(S5a) satisfy the equation

[TABLE]

The nontrivial fixed points are $y_{c}=3.38298....$ for $b=2$, and $y_{c}=2.05817...$ for b = 3.

In the limit $n\to\infty$, one gets

[TABLE]

which is the infinite temperature entropy per bond Eq. (S2)). The rate function $f$ as defined in Eq.  (4a) in the text, is given by $f(y)\equiv f_{n}(1)-f_{n}(y)$, where $f_{n}(1)$ takes care of the normalization in $L$. Fig. 4 in the text shows the plots of $f(y)$.

For points $A_{i}$’s and $K_{i}$’s that flow to $\pm y_{c}$ in two steps, we have the same value of $f=f_{c}$ for all of them with

[TABLE]

which evaluates to

[TABLE]

The intersection of the $f=f_{c}$ line with the $f$-vs-$t$ curve gives the critical points $A_{i}$’s and $K_{i}$’s.

For $b=3$, the transition points are

[TABLE]

For $b=2$, the transition points are

[TABLE]