--- title: Three-State Quantum Clock Model url: https://www.emergentmind.com/topics/three-state-quantum-clock-model type: topic --- # Three-State Quantum Clock Model Searching arXiv for the cited clock-model papers to verify metadata and grounding. The three-state quantum clock model is a one-dimensional $\mathbb Z_3$ lattice model whose local degrees of freedom are three-level “clock” variables acted on by clock and shift operators obeying a discrete Weyl algebra. In its standard quantum clock-chain form, it interpolates between a $\mathbb Z_3$-broken ordered phase and a disordered phase, with a second-order quantum critical point at self-duality. In related chiral and antiferromagnetic realizations, the same three-state structure supports nonconformal critical dynamics, Kosterlitz–Thouless transitions, and a massless incommensurate phase. Recent work has emphasized not only equilibrium scaling but also dynamical diagnostics based on the Loschmidt echo, rate functions, and finite-time scaling under driven protocols [2301.08198]. ## 1. Algebraic structure and canonical Hamiltonians The standard $q$-state quantum clock chain on an $N$-site ring with periodic boundary conditions is defined by \[ H \;=\; -\,J\sum_{j=1}^N\bigl(U_{j+1}^\dagger\,U_j + U_j^\dagger\,U_{j+1}\bigr) \;-\;h\sum_{j=1}^N\bigl(V_j + V_j^\dagger\bigr), \qquad U_{N+1}\equiv U_1, \] where the local Hilbert space at each site is spanned by $\{|0\rangle,|1\rangle,\dots,|q-1\rangle\}$, and \[ U_j\,|m\rangle_j=\omega^m\,|m\rangle_j,\qquad V_j\,|m\rangle_j=|m+1\rangle_j,\qquad \omega=e^{2\pi i/q}. \] These operators satisfy \[ V_jU_j=\omega\,U_jV_j,\qquad U_j^q=V_j^q=1. \] For the three-state case, $\omega=e^{2\pi i/3}$ and, in the basis $\{|0\rangle,|1\rangle,|2\rangle\}$, \[ U_j= \begin{pmatrix} 1 & & \\ & \omega & \\ & & \omega^2 \end{pmatrix}, \qquad V_j= \begin{pmatrix} 0&1&0\\ 0&0&1\\ 1&0&0 \end{pmatrix}. \] The Hamiltonian becomes \[ H_{q=3}= -J\sum_{j=1}^N\bigl(U_{j+1}^\dagger U_j + U_j^\dagger U_{j+1}\bigr) -h\sum_{j=1}^N\bigl(V_j + V_j^\dagger\bigr) \] [2301.08198]. A closely related formulation is the three-state quantum chiral clock model, written in terms of operators $\sigma_i$ and $\tau_i$ satisfying \[ \sigma_i^3=\tau_i^3=\mathbb I,\qquad \sigma_i\tau_i=\omega\,\tau_i\sigma_i,\qquad \omega=e^{2\pi i/3}, \] with Hamiltonian \[ \mathcal H = -\,f\sum_i\bigl(\tau_i^\dagger e^{-i\phi}+\tau_i e^{i\phi}\bigr) -\sum_i\bigl(\sigma_i^\dagger\sigma_{i+1}\,e^{-i\theta}+\mathrm{h.c.}\bigr) -h\sum_i\bigl(\sigma_i+\sigma_i^\dagger\bigr). \] In the time-reversal-symmetric case considered numerically, $\phi=0$ and the chirality parameter is $\theta\neq 0$, with most calculations using $\theta=\pi/8$ [1901.04818]. In the antiferromagnetic regime, the three-state chiral clock model can also be written as \[ H_{\rm clock} = -\sum_{j=1}^\infty \Bigl[ \cos\bigl(a\,(p_j-\Delta_p)\bigr) + \beta\,\cos\bigl(a\,(\theta_{j+1}-\theta_j-\Delta_\theta)\bigr) \Bigr], \qquad a=\frac{2\pi}{3}, \] or, at $\Delta=0$, as the mixed ferro-antiferromagnetic Potts chain \[ H_{\rm Potts} = \sum_{j=-\infty}^\infty \Bigl[ J\bigl(X_jX_{j+1}^2 + X_j^2X_{j+1}\bigr) -h\,R_j \Bigr], \qquad \beta=-\frac{J}{h},\quad J>0, \] with \[ R_j=Z_j+Z_j^{-1}=\mathrm{diag}(2,-1,-1). \] These forms make explicit that the three-state clock, Potts, and chiral-clock descriptions share the same local $\mathbb Z_3$ operator algebra while differing in coupling structure and phase organization [1608.04960]. ## 2. Equilibrium criticality of the standard three-state clock chain For $q\leq 4$, the quantum $q$-state clock model has a single second-order $(\mathbb Z_q)$ quantum critical point at the self-dual coupling $(h/J)_c=1$, with $J=1$ throughout. In the three-state case, the ordered phase at $hh_c$ is dominated by the transverse-field term and has a unique ground state given by the uniform superposition in the $U$-basis [2301.08198]. At the critical point $h_c=1$, the correlation-length exponent is $\nu_3=5/6\approx 0.833$, while the numerical extraction reported from dynamical scaling gives $\nu_3=0.842\pm0.006$. The agreement between these values identifies the three-state clock chain with the same equilibrium critical behavior as the exact three-state Potts result quoted in the dynamical study [2301.08198]. A common simplification is to treat all three-state clock models as if they shared the same critical structure. The available results distinguish several cases. The standard self-dual chain has a single second-order critical point at $(h/J)_c=1$ for $q=3$ [2301.08198]. By contrast, in the chiral model with $\theta=0$ one has the ordinary quantum three-state Potts chain with a conformal critical point of central charge $c=4/5$, while for $\theta>0$ the critical point is no longer conformal and the hallmark is a noninteger dynamical exponent $z>1$ [1901.04818]. In the antiferromagnetic region of the chiral model, the phase diagram contains commensurate, incommensurate “floating,” and disordered phases, separated by Kosterlitz–Thouless transitions [1608.04960]. This suggests that the phrase “three-state quantum clock model” names a family of closely related $\mathbb Z_3$ chains rather than a single universal phase diagram. ## 3. Loschmidt echo and small-quench dynamical scaling A central dynamical result for the standard three-state clock chain is that phase transitions can be characterized by an enhanced decay behavior of the Loschmidt echo under a small quench. The protocol prepares the system in the ground state $\ket{\psi_0}$ at $h_i=h_c$ and suddenly quenches to $h_f=h_c+g$ with small $g=0.01$. The Loschmidt echo is \[ L(t)=\Bigl|\langle\psi_0|e^{-iH(h_f)t}|\psi_0\rangle\Bigr|^2. \] The first minimum $L_{\min}(N)$ at time $t_{{\rm min},1}$ obeys the finite-size scaling law \[ 1-L_{\min}(N)\propto g^2\,N^{2/\nu_3}. \] Equivalently, \[ L_{\min}(N)\sim N^{-\mu_3},\qquad \mu_3=\frac{2}{\nu_3}\approx \frac{2}{0.842}\approx 2.38 \] [2301.08198]. The same exponent can be extracted from the short-time-averaged rate function. Defining \[ r(t)=-\frac{1}{N}\ln|L(t)|^2 \] and \[ \bar r(N,h,g) = -\frac{1}{N\,g^2}\,\ln\!\Bigl[\frac1T\!\int_0^T L(t)\,dt\Bigr], \] one locates the peak of $\bar r$ in $h$ for each $N$, defining a pseudo-critical $h^*(N)$, and then measures the corresponding $L_{\min}(N)$. This again yields \[ 1-L_{\min}(N)\sim g^2N^{2/\nu_3}, \qquad \nu_3=\frac{2}{\text{slope}}, \] with numerical estimate $\nu_3=0.842\pm0.006$ [2301.08198]. These results are significant because they show that equilibrium critical exponents can be recovered from nonequilibrium observables defined through wave-function overlaps. In the three-state case, the dynamical exponent $\mu_3$ is not introduced as an independent quantity but as a rewriting of the correlation-length scaling through $\mu_3=2/\nu_3$. The method therefore ties critical spectroscopy in finite systems to Loschmidt-echo minima rather than to conventional static susceptibilities. ## 4. Dynamical quantum phase transitions in the three-state chain For the three-state clock chain, dynamical quantum phase transitions are analyzed through a large quench from $h_i=\infty$ to $h_f=0$. In this case the Loschmidt amplitude can be written in transfer-matrix form, \[ G(t)=\mathrm{Tr}\bigl[\mathbf T(t)\bigr]^N, \qquad \mathbf T_{mn}(t)=\frac1{3}e^{\,2iJt\cos\!\tfrac{2\pi(m-n)}3}, \] and admits the closed expression \[ G(t) = 2\Bigl[\frac13\,e^{-iJt}\!(e^{3iJt}-1)\Bigr]^N + \Bigl[\frac13\,e^{-iJt}\!(e^{3iJt}+2)\Bigr]^N. \] The associated rate function, \[ r(t)=-\,\frac1N\ln|G(t)|^2, \] has its first non-analytic cusp at \[ t_{c1}=\frac{2\pi}{9J}, \qquad r(t_{c1})\xrightarrow[N\to\infty]{}\ln 3 \] [2301.08198]. A time-dependent $\mathbb Z_3$ order parameter is defined as \[ W(t) = \frac1N\sum_j\langle\psi(t)|V_j+V_j^\dagger|\psi(t)\rangle, \] which for $q=3$ reduces exactly to \[ W(t)=\frac{2}{9}\,\bigl[\,2\cos(3Jt)+1\bigr]^2. \] Its first zero occurs at the same critical time, \[ t_{c1}=\frac{2\pi}{9J}, \] demonstrating a one-to-one correspondence between Loschmidt-rate cusps and order-parameter zeros for $q\leq4$ [2301.08198]. Within the same comparison across $q=2,3,4$, the first DQPT cusp obeys \[ r(t_{c1})=\ln q\qquad (q\leq 4), \] so that the first-cusp rate function grows logarithmically with $q$. For $q>4$, the correspondence between singularities of the Loschmidt echo and zeros of the order parameter no longer exists; the Loschmidt echo near its first minimum converges, while the order parameter at its first zero increases linearly with $q$ [2301.08198]. In the three-state case, however, the DQPT structure remains in the regime where cusp times and order-parameter zeros coincide exactly. ## 5. Chiral deformation and nonequilibrium finite-time scaling The three-state quantum chiral clock model generalizes the standard chain by introducing chirality through the phase $e^{-i\theta}$ on bonds. When $\theta=0$, the model is the ordinary quantum three-state Potts chain and exhibits a conformal critical point with central charge $c=4/5$. For $\theta>0$, space-inversion and time-reversal are broken, and the critical point is no longer conformal. Field-theory and density-matrix-renormalization-group studies summarized in the finite-time-scaling work show that, for small $\theta$, the incommensurate phase shrinks to a point and one obtains a direct, continuous transition between an ordered $\mathbb Z_3$-broken phase and a disordered phase. The hallmark is a noninteger dynamical exponent $z>1$ [1901.04818]. The order parameter is \[ m=\frac12\,\langle \sigma_i+\sigma_i^\dagger\rangle, \] and the distance to criticality is defined by $g\equiv f-f_c$. Two driven protocols are considered. For a transverse-field quench, \[ f(t)=f_i+Rt, \] with $f_i\ll f_c$, the finite-time-scaling ansatz is \[ m(g,R)=R^{\beta/(\nu r)}\,F_1\!\Bigl(g\,R^{-1/(\nu r)}\Bigr), \qquad r=\frac1\nu+z. \] The zero-crossing and amplitude scale as \[ f_0(R)=f_c+c\,R^{1/(\nu r)}, \qquad m_0(R)\propto R^{\beta/(\nu r)}. \] For a longitudinal-field quench, \[ h(t)=h_i+R_h t\qquad \text{at fixed } g=0, \] the scaling form becomes \[ m(R_h,h)=R_h^{\beta/(\nu r_h)}\, F_2\!\Bigl(h\,R_h^{-\beta\delta/(\nu r_h)}\Bigr), \qquad r_h=\frac{\beta\delta}{\nu}+z, \] with \[ h_0(R_h)\propto R_h^{\beta\delta/(\nu r_h)}, \qquad m_0(R_h)\propto R_h^{\beta/(\nu r_h)} \] [1901.04818]. All simulations use infinite-MPS time evolution (iTEBD), with bond dimension up to $D=300$. For $\theta=\pi/8$, the extracted critical parameters are \[ f_c = 0.8609(2),\qquad z = 1.22(8),\qquad \nu = 0.81(5),\qquad \beta = 0.063(3),\qquad \delta = 22.8(10). \] The best collapse of the two-point correlation function \[ G(x)=\bigl|\langle\sigma_i\sigma_{i+x}\rangle -\langle\sigma_i\rangle\langle\sigma_{i+x}\rangle\bigr| \sim x^{-(d+z-2+\eta)}\,\mathcal G\!\bigl(x\,R^{1/r}\bigr), \] with \[ \eta=2-\beta(\delta-1)/\nu, \] occurs at $z\approx 1.22$ [1901.04818]. The same work extends finite-time scaling to thermal initial states. For the transverse protocol, \[ m(R,g_i,g,T) = R^{\beta/(\nu r)} F_3\!\bigl(g_iR^{-1/(\nu r)},\,gR^{-1/(\nu r)},\,TR^{-z/r}\bigr), \] and analogously for the longitudinal protocol, \[ m(R_h,h_i,h,T) = R_h^{\beta/(\nu r_h)} F_4\!\bigl(h_iR_h^{-\beta\delta/(\nu r_h)},\,hR_h^{-\beta\delta/(\nu r_h)},\,TR_h^{-z/r_h}\bigr). \] Perfect collapse is reported even when starting from a thermal state close to criticality, confirming that finite temperature can be treated as an additional scaling field of dimension $z$ [1901.04818]. ## 6. Antiferromagnetic regime, massless phase, and entanglement diagnostics In the antiferromagnetic region of the three-state quantum chiral clock model, the $(\Delta,\beta)$ phase diagram contains three phases: a commensurate ordered phase, an incommensurate floating phase, and a disordered phase. Two Kosterlitz–Thouless transitions occur for $\beta<0$. At $\Delta=0$, the transition between disordered and incommensurate phases is \[ \beta_c(0)\approx -7.0(1), \] inferred from the Potts-chain critical field \[ h_c/J\simeq 0.143(3). \] At $\Delta=\tfrac12$, the exact duality $\beta\leftrightarrow 1/\beta$, $\Delta\leftrightarrow \tfrac12-\Delta$ gives \[ \beta_c\!\left(\tfrac12\right)\approx -0.143(3) \] [1608.04960]. The critical point $h_c/J\simeq 0.143(3)$ is obtained from the bipartite von Neumann entropy of an infinite matrix-product state with bond dimension $\chi$, \[ S(\chi)= -\sum_{\alpha=1}^{\chi}\lambda_\alpha^2\log_2(\lambda_\alpha^2), \] where $\{\lambda_\alpha\}$ are Schmidt coefficients. The entropy exhibits a pronounced peak at a pseudo-critical field $h_c(\chi)$, and extrapolating \[ h_c(\chi)/J=(h_c/J)+a\,\chi^b \] gives the quoted critical value [1608.04960]. Within the massless phase $h\le h_c$, finite-entanglement scaling yields \[ S(\chi)\simeq \frac{c\kappa}{6}\log_2\chi+\mathrm{const.}, \qquad \xi(\chi)\simeq a_\xi \chi^\kappa, \] and the central charge is estimated as \[ c\simeq 1.0(1). \] At the special point $h/J=-1$, corresponding to the purely antiferromagnetic three-state Potts model, one recovers exactly $c=1$. The equivalent finite-block formula is \[ S_{vN}(\ell)=\frac{c}{3}\log\!\Bigl[\frac{L}{\pi}\sin\!\frac{\pi\ell}{L}\Bigr]+\mathrm{const.} \] [1608.04960]. The spin-spin correlator \[ \mathcal C_{12}(r)=\langle X_iX_{i+r}^{\,2}\rangle \] decays algebraically in the massless phase, \[ \mathcal C_{12}(r)\simeq a_0\,r^{-\eta}, \] with continuously varying exponent $\eta$. Numerically, one finds $\eta_\infty=0.3346(9)\simeq 1/3$ at $h/J=-1$, while as $h$ approaches $h_c\approx 0.143J$, $\eta_\infty$ decreases continuously and reaches approximately $0.248(1)$ at $h_c$. For $h>h_c$, the correlation decays exponentially, $\sim e^{-r/\xi_0}$ [1608.04960]. These results distinguish the antiferromagnetic chiral-clock regime from the direct ordered-to-disordered transition emphasized in the small-chirality nonequilibrium study. ## 7. Representative results and conceptual distinctions The following values summarize the principal three-state results reported in the cited studies. | Regime | Quantity | Value | |---|---|---| | Standard clock chain | Critical point | $(h/J)_c=1$ | | Standard clock chain | Correlation-length exponent | $\nu_3=0.842\pm0.006$ | | Standard clock chain | Small-quench exponent | $\mu_3=2/\nu_3\approx 2.38$ | | Standard clock chain | First DQPT time | $t_{c1}=2\pi/(9J)$ | | Standard clock chain | First-cusp rate value | $r(t_{c1})\to \ln 3$ | | Chiral clock model, $\theta=\pi/8$ | Critical point | $f_c=0.8609(2)$ | | Chiral clock model, $\theta=\pi/8$ | Dynamical exponent | $z=1.22(8)$ | | Antiferromagnetic Potts line | Critical field | $h_c/J\simeq 0.143(3)$ | | Massless antiferromagnetic phase | Central charge | $c\simeq 1.0(1)$ | Several conceptual distinctions follow directly from these results. First, the standard three-state clock chain is self-dual and exhibits a single second-order critical point at $(h/J)_c=1$ [2301.08198]. Second, adding chirality changes the critical dynamics qualitatively: for small $\theta$, the transition remains direct and continuous but becomes nonconformal, with noninteger $z$ [1901.04818]. Third, in the antiferromagnetic region the phase diagram is not reduced to a single critical point; instead it includes a massless incommensurate phase bounded by Kosterlitz–Thouless transitions and characterized by $c\simeq 1$ and continuously varying $\eta$ [1608.04960]. A common misconception is that dynamical signatures in $\mathbb Z_3$ chains are universally equivalent across all parameter regimes. The available evidence supports a more specific statement: in the standard $q=3$ clock chain, large quenches produce a one-to-one correspondence between Loschmidt-rate cusps and zeros of the order parameter, and the first cusp satisfies $r(t_{c1})\to \ln 3$ [2301.08198]. This correspondence was established for the standard clock chain for $q\le 4$; it should not be transferred without qualification to the chiral or antiferromagnetic regimes, where the cited works focus instead on finite-time scaling, entanglement entropy, and massless-phase diagnostics. Taken together, these results place the three-state quantum clock model at the intersection of discrete-symmetry breaking, nonconformal quantum criticality, and nonequilibrium scaling. The standard chain provides a controlled setting in which equilibrium exponents can be extracted from Loschmidt-echo scaling and DQPT cusps, while the chiral and antiferromagnetic variants demonstrate that the same local $\mathbb Z_3$ algebra supports qualitatively different critical structures, including direct continuous transitions with noninteger $z$ and extended $c=1$ massless phases [2301.08198]. Source: https://www.emergentmind.com/topics/three-state-quantum-clock-model