Geometry of Quantum Phase Transitions Angelo Carolloa,b, Davide Valentia, Bernardo Spagnoloa,b,c aDepartment of Physics and Chemistry, University of Palermo, Viale delle Scienze, Ed. 18, I-90128 Palermo, Italy bRadiophysics Department, Lobachevsky State University of Nizhni Novgorod, 23 Gagarin Avenue, Nizhni Novgorod 603950, Russia cIstituto Nazionale di Fisica Nucleare, Sezione di Catania, Via S. Sofia 64, I-90123 Catania, Italy Abstract In this article we provide a review of geometrical methods employed in the analysis of quantum phase transitions and non-equilibrium dissipative phase transitions. After a pedagogical introduction to geometric phases and geomet- ric information in the characterisation of quantum phase transitions, we describe recent developments of geometrical approaches based on mixed-state generalisation of the Berry-phase, i.e. the Uhlmann geometric phase, for the inves- tigation of non-equilibrium steady-state quantum phase transitions (NESS-QPTs ). Equilibrium phase transitions fall invariably into two markedly non-overlapping categories: classical phase transitions and quantum phase transitions, whereas in NESS-QPTs this distinction may fade off. The approach described in this review, among other things, can quantitatively assess the quantum character of such critical phenomena. This framework is applied to a paradig- matic class of lattice Fermion systems with local reservoirs, characterised by Gaussian non-equilibrium steady states. The relations between the behaviour of the geometric phase curvature, the divergence of the correlation length, the character of the criticality and the gap - either Hamiltonian or dissipative - are reviewed. Keywords: Quantum geometric information, Geometric phase, Quantum phase transitions, Dissipative phase transitions, Quantum metrology Contents 1 Introduction 2 2 Continuous Phase Transition and Universality 5 3 Quantum vs Classical Phase Transitions 7 4 Quantum Phase Transitions 7 5 Geometric Phase and Criticality 8 5.1 Berry’s phase . 9 arXiv:1911.10196v1 [quant-ph] 19 Nov 2019 5.2 A Simple Two-Level System . 12 5.3 The XY Model and Its Criticalities . 13 5.4 Geometric Phases and XY Criticalities . 14 5.5 General Considerations . 16 5.6 Dicke model and Geometric Phases . 18 6 Quantum Phase Transition and Information Geometry 23 6.1 Metric on the Hilbert Space: the Fubini-Study metric . 23 6.2 Fubini-Study distance as a statistical distance . 25 6.3 Fubini-Study metric and quantum phase transitions . 25 6.4 Quantum Phase Transition and Super-Extensitivity of the Quantum Geometric Tensor . 27 6.5 XY Model and Information Geometry . 28 Preprint submitted to Physics Reports November 25, 2019 7 Mixed States and Phase Transitions. 29 7.1 Definition of Mixed Geometric Phase via State Purification . 29 7.2 Purification . 30 7.3 Parallel Transport of Density Matrices and Uhlmann Geometric Phase . 31 8 Fidelity and Bures Metric 33 8.1 The Bures Metric . 35 9 Uhlmann Connection and Uhlmann Curvature 37 10 Multi-parameter quantum state estimation 39 10.1 Formulation of the problem . 40 10.2 Multi-parameter Incompatibility: a Measure of Quantumness . 41 11 Thermal States and Classical Phase Transitions. 43 12 Susceptibility and mean Uhlmann curvature 43 13 Dissipative Non-Equilibrium Phase Transitions 44 14 Gaussian Fermionic States 46 15 Dissipative Markovian Quadratic Models 48 15.1 Diagonalisation of the Lindblad Equation . 50 15.2 Liouvillian Spectrum . 51 16 Geometric Properties of the Steady States 52 16.1 Mean Uhlmann Curvature and Bures Metric of Gaussian Fermionic States . 53 16.2 Super-Extensivity of the (Generalised) Quantum Geometric Tensor . 56 17 Translationally Invariant Models 58 17.1 Mean Uhlmann Curvature and Criticality in Translationally Invariant Models . 59 18 Applications 62 18.1 Vanishing dissipative gap without criticality . 62 18.2 Rotated XY-model with Local Dissipation . 63 18.3 Boundary driven XY-model . 63 19 Conclusions 67 Appendix A Fermionic Gaussian States 69 Appendix A.1 Symmetric Logarithmic derivative for Fermionic Gaussian States . 70 Appendix B Spectral Properties of Quadratic Liouvillians 72 1. Introduction Several systems manifest phase transitions as the temperature or other external parameters are modified, and are characterised by a qualitative changes in the system properties. Examples of phase transitions range from our mundane experience of ice melting or the loss of ferromagnetism in iron to the far more exotic superfluid Mott-insulator phase transitions in optical lattices [1]. 2 Thermal phase transitions, i.e. phase transitions that occur at finite temperature, are characterised by a macroscopic order, such as magnetisation, or crystal structure, which gets destroyed by the thermal fluctuations [2, 3, 4, 5, 6]. At zero temperature, a different type of phase transitions can take place as a non-thermal parameter, such as magnetic field, pressure or chemical composition, is changed. The result is an order in the ground state properties of the system which is destroyed exclusively by quantum fluctuations. Competition between different terms of the system’s Hamiltonian, and the ordering thereof, are responsible for such fluctuations, which are a macroscopic manifestation of the Heisenberg uncertainty principle [7, 8, 9, 10, 11, 12]. In the past few decades, the studies of phase transitions at zero and finite temperature has been fuelled by the formidable success of Ginzburg-Landau-Wilson theories, local ordering, spontaneous symmetry breaking and the renormalisation group [13, 14, 15, 16] in spelling out the nature of a wide variety of critical behaviours. Such a standard approach has been recently paralleled by the development of a framework emerging from the cross-pollination of quantum information theory, information geometry and metrology. This approach is differential- geometric in essence and relies on two intertwined concepts: geometric phase and fidelity. The idea is to explore the critical features of many-body systems through the geometrical properties of its equilibrium state. The intuition behind it is relatively straightforward. Phase transitions are characterised by dramatic differences in some system’s observable under infinitesimal variation of a control parameter, which are embodied in major structural changes of the state of the system. One could compare the states associated to infinitesimally close values of the parameters by some “similarity” function, or distance function. One would then be able to reveal these major structural changes in the state through the singular behaviour of such a distance function. The Uhlmann fidelity [17, 18, 19, 20, 21, 22, 23, 24] provides a quantitative measure of such a distance, called the Bures metric [25]. Alternatively, in a quantum phase transitions (QPT), one may look at these dramatic changes as the manifes- tation of level crossings, occurring at the thermodynamic limit, which involve ground state and low lying part of the energy spectrum. In much the same way as mass-energy singularity bends the geometry of the space-time, or a Dirac monopole twists the topology of the field configuration around it [26], a level crossing bends the geometry of the ground-state manifold in a singular way. A quantitative account of this effect is provided by the geometric phase [27, 28, 29, 30], or more precisely, its infinitesimal counterpart, the Berry curvature. Driving the system close to or around these singularities results in dramatic effects on the state geometry, picked up by a quantum state in the form of geometric phase instabilities [31, 32, 33]. Geometric phases are at the core of the characterisation of topological phase transitions [34, 35, 36], and have been employed in the description and detection of QPT, both theoretically [31, 37, 38, 32, 39, 40] and experimentally [41]. Singular curvature and singular metric are two complementary manifestations of the same exceptional behaviour of the quantum state arising across phase transitions [42]. These quantities are a direct expression of the geometry of the state manifold and require no a-priori notions of order parameters, or symmetry breaking. They do not even require a detailed knowledge on the dynamics of the physical model, (e.g. the Hamiltonian), but only on its kinematics (i.e. the state of the system, and its dependence on parameters). This means that this conceptual framework can be freed from the boundaries of the standard Ginzburg-Landau-Wilson paradigm, and can be used to study a wide range of critical behaviours where a notion of local order does not hold, or cannot be easily identified. This motivates the employment of geometric phases in the investigation of critical properties in a plethora of differ- ent scenarios [31, 32, 39, 37, 38, 43, 44, 45, 42, 40, 46, 47, 48, 49, 50, 33, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 58, 61, 62, 63, 64, 65, 66, 41, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 83, 85, 86, 87, 88, 86, 89, 90, 91, 92, 85, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115]. The idea that QPTs could be explored through the Berry phase properties was first proposed and applied in the pro- totypical XY spin-1/2 chain [31, 32, 37, 39, 40, 57, 65, 76] and extended to many other many-body systems, such as the Dicke model [38, 44], the Lipkin-Meshkov-Glick model [43, 60, 72], Yang-Baxter spin-1/2 model [49, 58], quasi free-Fermion systems [47, 56, 53, 85, 102], interacting Fermion models [51, 116, 68, 63, 77, 79, 98], in ultracold atoms [73, 74,