Universality of Shallow Global Quenches in Critical Spin Chains.

Measuring universal data in the strongly correlated regime of quantum critical points remains a fundamental objective for quantum simulators. In foundational work, Calabrese and Cardy demonstrated how these data govern the dynamics of certain global quenches to 1+1-dimensional conformal field theories. While the quasiparticle picture they introduce has been widely successful in both theory and experiment, their seminal prediction that the critical exponents are simply encoded in the relaxation rates of local observables is challenging to investigate experimentally. In this Letter, we examine the critical quench dynamics of local observables from two types of readily accessible initial conditions: ground states and finite-temperature ensembles. We identify universal scaling collapses and scaling functions, utilizing a combination of conformal perturbation theory and tensor network numerics. For the finite-temperature quenches, we determine a regime in which the conformal field theory results are recovered, thereby allowing universal quantum critical data to be extracted from realistic quenches.Measuring universal data in the strongly correlated regime of quantum critical points remains a fundamental objective for quantum simulators. In foundational work, Calabrese and Cardy demonstrated how these data govern the dynamics of certain global quenches to 1+1-dimensional conformal field theories. While the quasiparticle picture they introduce has been widely successful in both theory and experiment, their seminal prediction that the critical exponents are simply encoded in the relaxation rates of local observables is challenging to investigate experimentally. In this Letter, we examine the critical quench dynamics of local observables from two types of readily accessible initial conditions: ground states and finite-temperature ensembles. We identify universal scaling collapses and scaling functions, utilizing a combination of conformal perturbation theory and tensor network numerics. For the finite-temperature quenches, we determine a regime in which the conformal field theory results are recovered, thereby allowing universal quantum critical data to be extracted from realistic quenches.

arXiv:2509.22773Sep 26, 20250 citations

Tensor networks, quantum spin chains, and quantum field theory

Yijian Zou

Understanding the universality class of continuous phase transition is of central importance in condensed matter physics. In one spatial dimension, the universal properties are encoded in the conformal field theory (CFT), which is in turn specified by the conformal data. In this thesis, we propose a systematic method to extract complete and accurate conformal data from the critical quantum spin chain based on the operator-state correspondence, thus completing the project initiated by Cardy and others in the 80's. \n \nOur method is based on the low-energy eigenstates of the critical quantum spin chain with periodic boundary conditions. First, scaling dimensions and conformal spins are extracted by the energies and momenta. Second, the primary states and conformal towers are identified by using the Koo-Saleur lattice Virasoro generators. Third, we propose a systematic way of identifying lattice operators with CFT operators, which enables us to compute operator product expansion coefficients from the low-energy eigenstates. Finally, the whole approach is generalized to critical quantum spin chains with antiperiodic boundary condition. In order to reduce finite-size corrections, we put forward the periodic uniform matrix product state (puMPS) algorithm which enables us to compute low-energy eigenstates of a critical quantum spin chain up to several hundreds of spins. Our method also enables us to study nonperturbatively the renormalization group flow between two CFTs as well as explore the emergence of extended symmetries beyond conformal symmetry. \n \nWe test our method with the Ising model and its generalization due to O'Brien and Fendley. The latter model is featured by a tricritical point described by the tricritical Ising CFT and a line of critical points interpolating between the Ising CFT and the tricritical Ising CFT. We extract complete conformal data from the two models and find excellent agreement with analytical results. Furthermore, we study the spectral renormalization group flow between the two CFTs nonperturbatively. At the tricritical point, the underlying CFT has an extended symmetry, the superconformal symmetry. We propose lattice operators that correspond to supervirasoro generators and verify their action on low-energy eigenstates. In this way we study the emergence of superconformal symmetry in critical quantum spin chains.Understanding the universality class of continuous phase transition is of central importance in condensed matter physics. In one spatial dimension, the universal properties are encoded in the conformal field theory (CFT), which is in turn specified by the conformal data. In this thesis, we propose a systematic method to extract complete and accurate conformal data from the critical quantum spin chain based on the operator-state correspondence, thus completing the project initiated by Cardy and others in the 80's. \n \nOur method is based on the low-energy eigenstates of the critical quantum spin chain with periodic boundary conditions. First, scaling dimensions and conformal spins are extracted by the energies and momenta. Second, the primary states and conformal towers are identified by using the Koo-Saleur lattice Virasoro generators. Third, we propose a systematic way of identifying lattice operators with CFT operators, which enables us to compute operator product expansion coefficients from the low-energy eigenstates. Finally, the whole approach is generalized to critical quantum spin chains with antiperiodic boundary condition. In order to reduce finite-size corrections, we put forward the periodic uniform matrix product state (puMPS) algorithm which enables us to compute low-energy eigenstates of a critical quantum spin chain up to several hundreds of spins. Our method also enables us to study nonperturbatively the renormalization group flow between two CFTs as well as explore the emergence of extended symmetries beyond conformal symmetry. \n \nWe test our method with the Ising model and its generalization due to O'Brien and Fendley. The latter model is featured by a tricritical point described by the tricritical Ising CFT and a line of critical points interpolating between the Ising CFT and the tricritical Ising CFT. We extract complete conformal data from the two models and find excellent agreement with analytical results. Furthermore, we study the spectral renormalization group flow between the two CFTs nonperturbatively. At the tricritical point, the underlying CFT has an extended symmetry, the superconformal symmetry. We propose lattice operators that correspond to supervirasoro generators and verify their action on low-energy eigenstates. In this way we study the emergence of superconformal symmetry in critical quantum spin chains.

Jul 7, 20200 citations

Universal scaling of work statistics in conformal field theory models

Zhaoyu Fei, C. P. Sun

We systematically study the work statistics for quantum phase transition. For a quantum system approached by an anisotropic conformal field theory near the critical point, the driving protocol is divided into three different regimes for different quench rates, which reflects the competition between the frozen time and the quench timescale. In each regime, we find universal scaling behaviors in work statistics (after renormalization). It is shown that the critical exponents are determined by the space-time dimension $d$ , the dynamical critical exponent $z$ , the correlation-length exponent $\ensuremath{\nu}$, and the power-law protocols. These universal scalings in nonequilibrium processes may be found in quantum phase transition by measuring the Loschmidt echo or the Ramsey interferometry.We systematically study the work statistics for quantum phase transition. For a quantum system approached by an anisotropic conformal field theory near the critical point, the driving protocol is divided into three different regimes for different quench rates, which reflects the competition between the frozen time and the quench timescale. In each regime, we find universal scaling behaviors in work statistics (after renormalization). It is shown that the critical exponents are determined by the space-time dimension $d$, the dynamical critical exponent $z$, the correlation-length exponent $\ensuremath{\nu}$, and the power-law protocols. These universal scalings in nonequilibrium processes may be found in quantum phase transition by measuring the Loschmidt echo or the Ramsey interferometry.

arXiv:2012.15501Physical review. B./Physical review. BApr 27, 20214 citations

Quantum quenches to a critical point in one dimension: some further results

John Cardy

Abstract. We describe several results concerning global quantum quenches from states with short-range correlations to quantum critical points whose low-energy properties are described by a 1 + 1-dimensional conformal field theory (CFT), extending the work of Calabrese and Cardy (2006 Phys. Rev. Lett . 96 136801 ): (a) for the special class of initial states discussed in that paper we show that, once a finite region falls inside the horizon, its reduced density matrix is exponentially close in L 2 norm to that of a thermal Gibbs state; (b) small deformations of this initial state in general lead to a (non-Abelian) generalized Gibbs distribution with, however, the possibility of parafermionic conserved charges; (c) small deformations of the CFT, corresponding to curvature of the dispersion relation and (non-integrable) left-right scattering, lead to a dependence of the speed of propagation on the initial state, as well as diffusive broadening of the horizon.Abstract. We describe several results concerning global quantum quenches from states with short-range correlations to quantum critical points whose low-energy properties are described by a 1 + 1-dimensional conformal field theory (CFT), extending the work of Calabrese and Cardy (2006 Phys. Rev. Lett . 96 136801 ): (a) for the special class of initial states discussed in that paper we show that, once a finite region falls inside the horizon, its reduced density matrix is exponentially close in L 2 norm to that of a thermal Gibbs state; (b) small deformations of this initial state in general lead to a (non-Abelian) generalized Gibbs distribution with, however, the possibility of parafermionic conserved charges; (c) small deformations of the CFT, corresponding to curvature of the dispersion relation and (non-integrable) left-right scattering, lead to a dependence of the speed of propagation on the initial state, as well as diffusive broadening of the horizon.

arXiv:1507.07266Journal of Statistical Mechanics Theory and ExperimentFeb 1, 2016135 citations

Conformal Fields and Operator Product Expansion in Critical Quantum Spin Chains

Yijian Zou, Ashley Milsted, Guifré Vidal

At a quantum critical point, the low-energy physics of a quantum spin chain is described by conformal field theory (CFT). Given the Hamiltonian of a critical spin chain, we propose a variational method to build an approximate lattice representation ϕ_{α} of the corresponding primary CFT operators ϕ_{α}^{CFT}. We then show how to numerically compute the operator product expansion coefficients C_{αβγ}^{CFT} governing the fusion of two primary fields. In this way, we complete the implementation of Cardy's program, outlined in the 1980s, which aims to extract the universality class of a phase transition, as encoded in the so-called conformal data of the underlying CFT, starting from a microscopic description. Our approach, demonstrated here for the critical quantum Ising model, only requires a generic (i.e., in general, nonintegrable) critical lattice Hamiltonian as its input.At a quantum critical point, the low-energy physics of a quantum spin chain is described by conformal field theory (CFT). Given the Hamiltonian of a critical spin chain, we propose a variational method to build an approximate lattice representation ϕ_{α} of the corresponding primary CFT operators ϕ_{α}^{CFT}. We then show how to numerically compute the operator product expansion coefficients C_{αβγ}^{CFT} governing the fusion of two primary fields. In this way, we complete the implementation of Cardy's program, outlined in the 1980s, which aims to extract the universality class of a phase transition, as encoded in the so-called conformal data of the underlying CFT, starting from a microscopic description. Our approach, demonstrated here for the critical quantum Ising model, only requires a generic (i.e., in general, nonintegrable) critical lattice Hamiltonian as its input.

arXiv:1901.06439Physical Review LettersJan 28, 202045 citations

Non-equilibrium quantum dynamics after local quenches

Uma Divakaran, Ferenc Iglói, Heiko Rieger

We study the quantum dynamics resulting from preparing a one-dimensional quantum system in the ground state of initially two decoupled parts which are then joined together (local quench). Specifically we focus on the transverse Ising chain and compute the time-dependence of the magnetization profile, m_l(t), and correlation functions at the critical point, in the ferromagnetically ordered phase and in the paramagnetic phase. At the critical point we find finite size scaling forms for the nonequilibrium magnetization and compare predictions of conformal field theory with our numerical results. In the ferromagnetic phase the magnetization profiles are well matched by our predictions from a quasi-classical calculation.We study the quantum dynamics resulting from preparing a one-dimensional quantum system in the ground state of initially two decoupled parts which are then joined together (local quench). Specifically we focus on the transverse Ising chain and compute the time-dependence of the magnetization profile, m_l(t), and correlation functions at the critical point, in the ferromagnetically ordered phase and in the paramagnetic phase. At the critical point we find finite size scaling forms for the nonequilibrium magnetization and compare predictions of conformal field theory with our numerical results. In the ferromagnetic phase the magnetization profiles are well matched by our predictions from a quasi-classical calculation.

arXiv:1105.5317Journal of Statistical Mechanics Theory and ExperimentOct 24, 201136 citations

d

, the dynamical critical exponent

z

, the correlation-length exponent $\ensuremath{\nu}$, and the power-law protocols. These universal scalings in nonequilibrium processes may be found in quantum phase transition by measuring the Loschmidt echo or the Ramsey interferometry.