We study semidefinite programming (SDP) methods to analyze spin chain Hamiltonians. We examine the ground state energy, the first excited charged states and ground state correlators in two simple models: the Ising model in a transverse magnetic field and the closely related 3-state Potts model. Our goal is to understand precisely what the SDP program is doing and when it works well, why it does so. We focus on the following novel ingredients: using charge constraints to obtain excited states and to see if additional constraints from integrable models are effective at improving the method. At criticality we also explore to what extent we can use approximate Virasoro correlators to extract conformal data: the central charge and some critical exponents of charged states. We also use these to identify the location of the phase transition. In the special case where the system is made of free fermions we prove that the SDP finds the exact energy of the ground state and produces the correct two point functions of the fermions. Away from free fermion theories, the SDP gets progressively worse at estimating data beyond the value of the ground state energy (like correlation functions), although it qualitatively matches these. In order to be effective, the SDP seems to run into scaling issues where the amount of input needed scales poorly with the lattice volume.
We study semidefinite programming (SDP) methods to analyze spin chain Hamiltonians. We examine the ground state energy, the first excited charged states and ground state correlators in two simple models: the Ising model in a transverse magnetic field and the closely related 3-state Potts model. Our goal is to understand precisely what the SDP program is doing and when it works well, why it does so. We focus on the following novel ingredients: using charge constraints to obtain excited states and to see if additional constraints from integrable models are effective at improving the method. At criticality we also explore to what extent we can use approximate Virasoro correlators to extract conformal data: the central charge and some critical exponents of charged states. We also use these to identify the location of the phase transition. In the special case where the system is made of free fermions we prove that the SDP finds the exact energy of the ground state and produces the correct two point functions of the fermions. Away from free fermion theories, the SDP gets progressively worse at estimating data beyond the value of the ground state energy (like correlation functions), although it qualitatively matches these. In order to be effective, the SDP seems to run into scaling issues where the amount of input needed scales poorly with the lattice volume.
We analyze the bootstrap approach (a dual optimization method to the variational approach) to one-dimensional spin chains, leveraging semidefinite programming to extract numerical results. We study how correlation functions in the ground state converge to their true values at and away from criticality and at relaxed optimality. We consider the transverse Ising model, the three state Potts model, and other non-integrable spin chains and investigate to what extent semidefinite methods can reliably extract numerical emergent physical data, including conformal central charges, correlation lengths and scaling dimensions. We demonstrate procedures to extract these data and show preliminary results in the various models considered. We compare to exact analytical results and to exact diagonalization when the system volume is small enough. When we attempt to go to the thermodynamic limit, the semidefinite numerical method with translation invariance imposed as a constraint finds the solution with periodic boundary conditions even if these have not been specified. This implies that the determination of all conformal data in correlators has to be handled at finite volume. Our investigation reveals that the approach has practical challenges. In particular, the correlation functions extracted from the optimal solution, which function as slack variables in the optimization, have convergence issues that suggest an underlying exponential complexity in the system size.
We present a generic approach to the condensed-matter ground-state problem which is complementary to variational techniques and works directly in the thermodynamic limit. Relaxing the ground-state problem, we obtain semidefinite programs (SDP). These can be solved efficiently, yielding strict lower bounds to the ground-state energy and approximations to the few-particle Green's functions. As the method is applicable for all particle statistics, it represents, in particular, a novel route for the study of strongly correlated fermionic and frustrated spin systems in D>1 spatial dimensions. It is demonstrated for the XXZ model and the Hubbard model of spinless fermions. The results are compared against exact solutions, quantum Monte Carlo calculations, and Anderson bounds, showing the competitiveness of the SDP method.
We study the emergence of 2D conformal symmetry in critical quantum spin chains on the finite circle. Our goal is to characterize the conformal field theory (CFT) describing the universality class of the corresponding quantum phase transition. Using only the lattice Hamiltonian H=∑jhj as an input, we construct operators Hn (Fourier modes of the local Hamiltonian terms hj) that transform the low-energy eigenstates of H in the same way as certain combinations of the Virasoro generators do in the CFT. These operators were introduced for certain integrable lattice models by Koo and Saleur [Nucl. Phys. B 426, 459 (1994)]. In this paper we propose and demonstrate the use of operators Hn to extract conformal data in nonintegrable quantum spin chains. In particular, we explain how to estimate the central charge from a simple ground-state expectation value and to systematically identify the low-energy states of H corresponding to Virasoro primary operators, as must be done in order to identify the CFT.
This thesis is concerned with the finite lattice study of spin models. The underlying theme in Parts I and II is the exploitation of Bethe ansatz equations to provide finite lattice data well beyond the range normally available. In Part I the Bethe ansatz equations for eigenvalues of the eight-vertex model are solved numerically to yield mass gap data on infinitely long strips of up to 512 sites in width. The finite-size corrections, at criticality, to the free energy per site and polarization gap are found to agree with recent studies of the XXZ spin chain. The leading corrections to the finite-size scaling estimates of the critical line and thermal exponent are also found, providing an explanation of the poor convergence seen in earlier studies. Away from criticality, the linear scaling fields are derived exactly in the full parameter space of the spin system, allowing a thorough test of a recently proposed method of extracting linear scaling fields and related exponents from finite lattice data. In Part II, the numerical solutions of the Bethe ansatz equations for the eigen-energies of the XXZ Hamiltonian on very large chains are used to identify, via conformal invariance, the scaling dimensions of various two-dimensional models. With periodic boundary conditions, eight-vertex and Gaussian model operators are found. The scaling dimensions of the Ashkm-Teller and Potts models are obtained by the exact relating of eigenstates of their quantum Hamiltonians to those of the XXZ chain with modified boundary conditions. Eigenstates of the Ashkin-Teller and Potts models with free boundaries are also obtained, allowing an examination of their critical surface properties. In Part HI the critical behaviour of an Ising model with competing firstand third- nearest neighbour interactions on the square lattice is investigated using the finite lattice method. In the ferromagnetic region, the phase boundary is located with an accuracy at least equal to that of alternative methods. In the antiphase region, distinctive structure in the finite lattice estimators is found over an extended temperature range. However, the nature of the transition remains unclear.
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 and approximating extremal energy states of local Hamiltonians is a central problem in quantum physics and complexity theory. Recent work has focused on developing approximation algorithms for local Hamiltonians, and in particular the ``Quantum Max Cut'' (QMax-Cut) problem, which is closely related to the antiferromagnetic Heisenberg model. In this work, we introduce a family of semidefinite programming (SDP) relaxations based on the Navascues-Pironio-Acin (NPA) hierarchy which is tailored for QMaxCut by taking into account its SU(2) symmetry. We show that the hierarchy converges to the optimal QMaxCut value at a finite level, which is based on a new characterization of the algebra of SWAP operators. We give several analytic proofs and computational results showing exactness/inexactness of our hierarchy at the lowest level on several important families of graphs. We also discuss relationships between SDP approaches for QMaxCut and frustration-freeness in condensed matter physics and numerically demonstrate that the SDP-solvability practically becomes an efficiently-computable generalization of frustration-freeness. Furthermore, by numerical demonstration we show the potential of SDP algorithms to perform as an approximate method to compute physical quantities and capture physical features of some Heisenberg-type statistical mechanics models even away from the frustration-free regions.