Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1808.09547

Spontaneous Symmetry Breaking in Finite Quantum Systems: a decoherent-histories approach

David Wallace

Published Aug 28, 2018• 7 citations
arXivHTMLPDF

Abstract

Spontaneous symmetry breaking (SSB) in quantum systems, such as ferromagnets, is normally described as (or as arising from) degeneracy of the ground state; however, it is well established that this degeneracy only occurs in spatially infinite systems, and even better established that ferromagnets are not spatially infinite. I review this well-known paradox, and consider a popular solution where the symmetry is explicitly broken by some external field which goes to zero in the infinite-volume limit; although this is formally satisfactory, I argue that it must be rejected as a physical explanation of SSB since it fails to reproduce some important features of the phenomenology. Motivated by considerations from the analogous classical system, I argue that SSB in finite systems should be understood in terms of the approximate decoupling of the system's state space into dynamically-isolated sectors, related by a symmetry transformation; I use the formalism of decoherent histories to make this more precise and to quantify the effect, showing that it is more than sufficient to explain SSB in realistic systems and that it goes over in a smooth and natural way to the infinite limit.

DOI
10.48550/arxiv.1808.09547

Related papers

Spontaneous symmetry breaking in quantum systems: Emergence or reduction?

Nicolaas P. Landsman

Beginning with Anderson (1972) , spontaneous symmetry breaking ( ssb ) in infinite quantum systems is often put forward as an example of (asymptotic) emergence in physics , since in theory no finite system should display it. Even the correspondence between theory and reality is at stake here, since numerous real materials show ssb in their ground states (or equilibrium states at low temperature), although they are finite. Thus against what is sometimes called ‘Earman's Principle’, a genuine physical effect (viz. ssb ) seems theoretically recovered only in some idealisation (namely the thermodynamic limit), disappearing as soon as the idealisation is removed. We review the well-known arguments that (at first sight) no finite system can exhibit ssb , using the formalism of algebraic quantum theory in order to control the thermodynamic limit and unify the description of finite- and infinite-volume systems. Using the striking mathematical analogy between the thermodynamic limit and the classical limit, we show that a similar situation obtains in quantum mechanics (which typically forbids ssb ) versus classical mechanics (which allows it). This discrepancy between formalism and reality is quite similar to the measurement problem (now regarded as an instance of asymptotic emergence), and hence we address it in the same way, adapting an argument of the Landsman and Reuvers (2013) that was originally intended to explain the collapse of the wave-function within conventional quantum mechanics. Namely, exponential sensitivity to (asymmetric) perturbations of the (symmetric) dynamics as the system size increases causes symmetry breaking already in finite but very large quantum systems. This provides continuity between finite- and infinite-volume descriptions of quantum systems featuring ssb and hence restores Earman's Principle (at least in this particularly threatening case).

arXiv:1305.4473•Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern Physics•Sep 13, 2013•54 citations

Symmetry Breaking through Superselection by Boundary Conditions

Silvester G. A. Borsboom, José P. Dupont

Spontaneous symmetry breaking (SSB) is central to modern physics but is conventionally defined only for infinite systems, raising challenges for its interpretation in finite, real-world setups. This paper argues that the key to resolving this issue lies in the underappreciated role of boundary conditions in quantum systems. Inspired by both the relational approach to symmetries and the physical mechanism behind symmetry breaking, we formulate a relational interpretation of SSB: a finite system exhibits SSB relative to a reference environment which can induce perturbations across the boundary. This eliminates the need for the thermodynamic limit, offering a more physical picture of SSB that emphasizes the observable consequences of the interactions that real-life systems inevitably have with their environment. We show how, in this relational interpretation, SSB for both lattice systems and (gauge) field theories should be understood as subtle, rather than spontaneous, symmetry breaking, still in contrast to explicit symmetry breaking. We also explain how algebraic definitions of SSB for infinite systems relate to the intuitive picture of SSB in finite systems and illustrate how asymptotic boundary conditions push the environment "to infinity". In this way, our relational interpretation of SSB provides a unified conceptual framework applicable to symmetry-breaking in systems of any size.

physics.hist-phCross-listedarXiv:2606.15272•Jun 16, 2026•0 citations

Philosophical Aspects of Spontaneous Symmetry Breaking

Giacomo Schwarz

This essay expounds the algebraic framework describing general physical the-ories, within which the phenomenon of spontaneous symmetry breaking (SSB) makes its appearance in infinite quantum systems. This is in contrast with the fact that a large class of theories- both classical and quantum, finite and infinite- are termed, in the conventional account of classical and quantum mechanics, as exhibiting SSB. This discrepancy will be understood in the light of an interpretation that finds the symmetry breaking to be in some respects stronger in the algebraic account than is generally the case in the conventional picture. The case of SSB in the standard account of quantum field theory (QFT) will then be discussed, and it will be argued that, although one would expect a connection with the algebraic account to be possible, this turns out to be problematic. Finally the role of the idealisation of infinite systems, crucial to algebraic SSB, will be dis-cussed.

•May 4, 2012•0 citations

Giant Number-Parity Effect Leading to Spontaneous Symmetry Breaking in Finite-Size Quantum Spin Models

Filippo Caleca, Saverio Bocini, Fabio Mezzacapo +1 more

Spontaneous symmetry breaking (SSB) occurs when a many-body system governed by a symmetric Hamiltonian, and prepared in a symmetry-broken state by the application of a field coupling to its order parameter O, retains a finite O value even after the field is switched off. SSB is generally thought to occur only in the thermodynamic limit N→∞ (for N degrees of freedom). In this limit, the time to restore the symmetry once the field is turned off, either via thermal or quantum fluctuations, is expected to diverge. Here, we show that SSB can also be observed in finite-size quantum spin systems, provided that the spin is half-integer and three conditions are met: (1) the ground state of the system has long-range correlations, (2) the Hamiltonian conserves the (spin) parity of the order parameter, and (3) N is odd. Using a combination of analytical arguments and numerical results (based on time-dependent variational Monte Carlo and rotor + spin-wave theory), we show that SSB on finite-size systems can be achieved via a quasiadiabatic preparation of the ground state-which, in U(1)-symmetric systems, is shown to require a symmetry-breaking field vanishing over timescales τ∼O(N). In these systems, the symmetry-broken state exhibits spin squeezing with Heisenberg scaling.

arXiv:2412.15493•Physical Review Letters•Nov 10, 2025•1 citation

Spontaneous Symmetry Breaking

Tomáš Brauner

The roots of the modern understanding of symmetries in physics can be traced to the work of Sophus Lie on transformations of differential equations in the nineteenth century. In this chapter, the history of the subject is used to emphasize the distinction between the symmetry of the dynamics of a physical system and the symmetry of its specific state. The central concept of spontaneous symmetry breaking is then introduced as the discrepancy between the two notions of symmetry. It is shown how the presence of spontaneous symmetry breaking can be detected through the sensitivity of measurements on the physical state to external perturbations. This naturally leads to the concepts of the order parameter and the vacuum manifold. The rest of the chapter is devoted to highlighting some subtle features of spontaneous symmetry breaking. These include the properties of the ground state and the tower of low-lying excited states in a finite volume, and the absence of well-defined unitary operators representing broken symmetry in the thermodynamic limit.

Lecture notes in physics•Jan 1, 2024•0 citations

Spontaneous Symmetry Breaking: Quantum Statistical Mechanics versus Quantum Field Theory

Doreen Fraser

Philosophical analysis of spontaneous symmetry breaking (SSB) in particle physics has been hindered by the unavailability of rigorous formulations of models in quantum field theory (QFT). A strategy for addressing this problem is to use the rigorous models that have been constructed for SSB in quantum statistical mechanics (QSM) systems as a basis for drawing analogous conclusions about SSB in QFT. On the basis of an analysis of this strategy as an instance of the application of the same mathematical formalism to different domains and as an instance of drawing analogies between domains, I conclude that certain structural explanations can be exported from QSM to QFT but that causal explanations cannot.

Philosophy of Science•Nov 19, 2012•2 citations
Spontaneous symmetry breaking (SSB) in quantum systems, such as ferromagnets, is normally described as (or as arising from) degeneracy of the ground state; however, it is well established that this degeneracy only occurs in spatially infinite systems, and even better established that ferromagnets are not spatially infinite. I review this well-known paradox, and consider a popular solution where the symmetry is explicitly broken by some external field which goes to zero in the infinite-volume limit; although this is formally satisfactory, I argue that it must be rejected as a physical explanation of SSB since it fails to reproduce some important features of the phenomenology. Motivated by considerations from the analogous classical system, I argue that SSB in finite systems should be understood in terms of the approximate decoupling of the system's state space into dynamically-isolated sectors, related by a symmetry transformation; I use the formalism of decoherent histories to make this more precise and to quantify the effect, showing that it is more than sufficient to explain SSB in realistic systems and that it goes over in a smooth and natural way to the infinite limit.
Beginning with Anderson (1972) , spontaneous symmetry breaking ( ssb ) in infinite quantum systems is often put forward as an example of (asymptotic) emergence in physics , since in theory no finite system should display it. Even the correspondence between theory and reality is at stake here, since numerous real materials show ssb in their ground states (or equilibrium states at low temperature), although they are finite. Thus against what is sometimes called ‘Earman's Principle’, a genuine physical effect (viz. ssb ) seems theoretically recovered only in some idealisation (namely the thermodynamic limit), disappearing as soon as the idealisation is removed. We review the well-known arguments that (at first sight) no finite system can exhibit ssb , using the formalism of algebraic quantum theory in order to control the thermodynamic limit and unify the description of finite- and infinite-volume systems. Using the striking mathematical analogy between the thermodynamic limit and the classical limit, we show that a similar situation obtains in quantum mechanics (which typically forbids ssb ) versus classical mechanics (which allows it). This discrepancy between formalism and reality is quite similar to the measurement problem (now regarded as an instance of asymptotic emergence), and hence we address it in the same way, adapting an argument of the Landsman and Reuvers (2013) that was originally intended to explain the collapse of the wave-function within conventional quantum mechanics. Namely, exponential sensitivity to (asymmetric) perturbations of the (symmetric) dynamics as the system size increases causes symmetry breaking already in finite but very large quantum systems. This provides continuity between finite- and infinite-volume descriptions of quantum systems featuring ssb and hence restores Earman's Principle (at least in this particularly threatening case).
Spontaneous symmetry breaking (SSB) is central to modern physics but is conventionally defined only for infinite systems, raising challenges for its interpretation in finite, real-world setups. This paper argues that the key to resolving this issue lies in the underappreciated role of boundary conditions in quantum systems. Inspired by both the relational approach to symmetries and the physical mechanism behind symmetry breaking, we formulate a relational interpretation of SSB: a finite system exhibits SSB relative to a reference environment which can induce perturbations across the boundary. This eliminates the need for the thermodynamic limit, offering a more physical picture of SSB that emphasizes the observable consequences of the interactions that real-life systems inevitably have with their environment. We show how, in this relational interpretation, SSB for both lattice systems and (gauge) field theories should be understood as subtle, rather than spontaneous, symmetry breaking, still in contrast to explicit symmetry breaking. We also explain how algebraic definitions of SSB for infinite systems relate to the intuitive picture of SSB in finite systems and illustrate how asymptotic boundary conditions push the environment "to infinity". In this way, our relational interpretation of SSB provides a unified conceptual framework applicable to symmetry-breaking in systems of any size.
This essay expounds the algebraic framework describing general physical the-ories, within which the phenomenon of spontaneous symmetry breaking (SSB) makes its appearance in infinite quantum systems. This is in contrast with the fact that a large class of theories- both classical and quantum, finite and infinite- are termed, in the conventional account of classical and quantum mechanics, as exhibiting SSB. This discrepancy will be understood in the light of an interpretation that finds the symmetry breaking to be in some respects stronger in the algebraic account than is generally the case in the conventional picture. The case of SSB in the standard account of quantum field theory (QFT) will then be discussed, and it will be argued that, although one would expect a connection with the algebraic account to be possible, this turns out to be problematic. Finally the role of the idealisation of infinite systems, crucial to algebraic SSB, will be dis-cussed.
Spontaneous symmetry breaking (SSB) occurs when a many-body system governed by a symmetric Hamiltonian, and prepared in a symmetry-broken state by the application of a field coupling to its order parameter O, retains a finite O value even after the field is switched off. SSB is generally thought to occur only in the thermodynamic limit N→∞ (for N degrees of freedom). In this limit, the time to restore the symmetry once the field is turned off, either via thermal or quantum fluctuations, is expected to diverge. Here, we show that SSB can also be observed in finite-size quantum spin systems, provided that the spin is half-integer and three conditions are met: (1) the ground state of the system has long-range correlations, (2) the Hamiltonian conserves the (spin) parity of the order parameter, and (3) N is odd. Using a combination of analytical arguments and numerical results (based on time-dependent variational Monte Carlo and rotor + spin-wave theory), we show that SSB on finite-size systems can be achieved via a quasiadiabatic preparation of the ground state-which, in U(1)-symmetric systems, is shown to require a symmetry-breaking field vanishing over timescales τ∼O(N). In these systems, the symmetry-broken state exhibits spin squeezing with Heisenberg scaling.
The roots of the modern understanding of symmetries in physics can be traced to the work of Sophus Lie on transformations of differential equations in the nineteenth century. In this chapter, the history of the subject is used to emphasize the distinction between the symmetry of the dynamics of a physical system and the symmetry of its specific state. The central concept of spontaneous symmetry breaking is then introduced as the discrepancy between the two notions of symmetry. It is shown how the presence of spontaneous symmetry breaking can be detected through the sensitivity of measurements on the physical state to external perturbations. This naturally leads to the concepts of the order parameter and the vacuum manifold. The rest of the chapter is devoted to highlighting some subtle features of spontaneous symmetry breaking. These include the properties of the ground state and the tower of low-lying excited states in a finite volume, and the absence of well-defined unitary operators representing broken symmetry in the thermodynamic limit.
Philosophical analysis of spontaneous symmetry breaking (SSB) in particle physics has been hindered by the unavailability of rigorous formulations of models in quantum field theory (QFT). A strategy for addressing this problem is to use the rigorous models that have been constructed for SSB in quantum statistical mechanics (QSM) systems as a basis for drawing analogous conclusions about SSB in QFT. On the basis of an analysis of this strategy as an instance of the application of the same mathematical formalism to different domains and as an instance of drawing analogies between domains, I conclude that certain structural explanations can be exported from QSM to QFT but that causal explanations cannot.