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Universal Scaling Properties After Quantum Quenches

Damián A. Galante

Published Jan 1, 2016• 0 citations
OpenAlex

Abstract

In this Thesis, the problem of a quantum quench in quantum field theories is analyzed. This involves studying the real time evolution of observables in a theory that undergoes a change in one of its couplings. These quenches are then characterized by two parameters: $\\delta \\lambda$, the magnitude of the quench and most importantly, $\\delta t$, the quench duration. In contrast to previous studies of abrupt quenches in the condensed matter theory community, we will be interested in smooth quenches with a finite $\\delta t$.\nMotivated by existing results in holographic theories, we studied the problem of a fast smooth quench in free field theories by quenching the mass of a free scalar and a free Dirac fermion. For specific mass profiles, exact analytic answers were found. We found that expectation values for the quenched operators obey universal scaling properties. We provide both numerical and analytic evidence for this scaling to hold in free field theories and argue that the same scaling properties should hold independently of the theory, the coupling or the quench profile. In fact, we show that the renormalized expectation value of an operator ${\\cal{O}}_\\Delta$ of dimension $\\Delta$ should scale as $\\vev{{\\cal{O}}_\\Delta}_{ren} \\sim \\delta \\lambda/(\\delta t)^{2\\Delta -d}$, when the quench rate is fast compared to the quench amplitude. This growth is further enhanced by a logarithmic factor in even dimensions.\nThis result suggests that, for $\\Delta>d/2$, expectation values will diverge in the limit of $\\delta t \\to 0$, which seem to contradict previous studies of abrupt quenches. In this Thesis, we carefully analzye the relation between the two approaches and establish restrictions to the kind of objects that can be studied using the abrupt approximation.\nApart from the fast smooth quench, another regime of interest is the slow quench through a critical point. In free field theories we found that adiabatic behaviour breaks down when the system is close enough to the critical point and renormalized expectation values scale different, following expectations from the Kibble-Zurek argument. Given any finite fixed time, we are able to show how expectation values scale at any value of the quench duration $\\delta t$.

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Global quantum quench with a finite quench rate which crosses critical points is known to lead to universal scaling of correlation functions as functions of the quench rate. In this work, we explore scaling properties of the entanglement entropy of a subsystem in a harmonic chain during a mass quench which asymptotes to finite constant values at early and late times and for which the dynamics is exactly solvable. When the initial state is the ground state, we find that for large enough subsystem sizes the entanglement entropy becomes independent of size. This is consistent with Kibble–Zurek scaling for slow quenches, and with recently discussed “fast quench scaling” for quenches fast compared to physical scales, but slow compared to UV cutoff scales.

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In this Thesis, the problem of a quantum quench in quantum field theories is analyzed. This involves studying the real time evolution of observables in a theory that undergoes a change in one of its couplings. These quenches are then characterized by two parameters: deltalambda\\delta \\lambdadeltalambda, the magnitude of the quench and most importantly, deltat\\delta tdeltat, the quench duration. In contrast to previous studies of abrupt quenches in the condensed matter theory community, we will be interested in smooth quenches with a finite deltat\\delta tdeltat.\nMotivated by existing results in holographic theories, we studied the problem of a fast smooth quench in free field theories by quenching the mass of a free scalar and a free Dirac fermion. For specific mass profiles, exact analytic answers were found. We found that expectation values for the quenched operators obey universal scaling properties. We provide both numerical and analytic evidence for this scaling to hold in free field theories and argue that the same scaling properties should hold independently of the theory, the coupling or the quench profile. In fact, we show that the renormalized expectation value of an operator {\\cal{O}}_\\Delta of dimension Delta\\DeltaDelta should scale as \\vev{{\\cal{O}}_\\Delta}_{ren} \\sim \\delta \\lambda/(\\delta t)^{2\\Delta -d}, when the quench rate is fast compared to the quench amplitude. This growth is further enhanced by a logarithmic factor in even dimensions.\nThis result suggests that, for Delta>d/2\\Delta>d/2Delta>d/2, expectation values will diverge in the limit of deltatto0\\delta t \\to 0deltatto0, which seem to contradict previous studies of abrupt quenches. In this Thesis, we carefully analzye the relation between the two approaches and establish restrictions to the kind of objects that can be studied using the abrupt approximation.\nApart from the fast smooth quench, another regime of interest is the slow quench through a critical point. In free field theories we found that adiabatic behaviour breaks down when the system is close enough to the critical point and renormalized expectation values scale different, following expectations from the Kibble-Zurek argument. Given any finite fixed time, we are able to show how expectation values scale at any value of the quench duration deltat\\delta tdeltat.
We expand on the investigation of the universal scaling properties in the early time behaviour of fast but smooth quantum quenches in a general d-dimensional conformal field theory deformed by a relevant operator of dimension Δ with a time-dependent coupling. The quench consists of changing the coupling from an initial constant value λ 1 by an amount of the order of δλ to some other final value λ 2, over a time scale δt. In the fast quench limit where δt is smaller than all other length scales in the problem, δt ≪ λ 1 1/() , λ 2 1/() , δλ 1/(Δ − d), the energy (density) injected into the system scales as δℰ ∼ (δλ)2(δt) d− 2Δ . Similarly, the change in the expectation value of the quenched operator at times earlier than the endpoint of the quench scales as ⟨OΔ⟩∼δλ(δt)d−2Δ \left\langle {\mathcal{O}}_{\varDelta}\right\rangle \sim \delta \lambda {{\left(\delta t\right)}^d}^{-2\varDelta } ⟨OΔ​⟩∼δλ(δt)d−2Δ , with further logarithmic enhancements in certain cases. While these results were first found in holographic studies, we recently demonstrated that precisely the same scaling appears in fast mass quenches of free scalar and free fermionic field theories. As we describe in detail, the universal scaling refers to renormalized quantities, in which the UV divergent pieces are consistently renormalized away by subtracting counterterms derived with an adiabatic expansion. We argue that this scaling law is a property of the conformal field theory at the UV fixed point, valid for arbitrary relevant deformations and insensitive to the details of the quench protocol. Our results highlight the difference between smooth fast quenches and instantaneous quenches where the Hamiltonian abruptly changes at some time.
We study the time evolution of a conformal field theory deformed by a relevant operator under a smooth but fast quantum quench which brings it to the conformal point. We argue that when the quench time scale δt\delta tδt is small compared to the scale set by the relevant coupling, the expectation value of the quenched operator scales universally as δg/δt2Δ−d\delta g/ \delta t ^{2\Delta-d}δg/δt2Δ−d where δg\delta gδg is the quench amplitude. This growth is further enhanced by a logarithmic factor in even dimensions. We present explicit results for free scalar and fermionic field theories, supported by an analytic understanding of the leading contribution for fast quenches. Results from this Letter suggest that this scaling result, first found in holography, is in fact universal to quantum quenches. Our considerations also show that this limit of fast smooth quenches is quite different from an instantaneous quench from one time-independent Hamiltonian to another, where the Schrodinger picture state at the time of the quench simply serves as an initial condition for subsequent evolution with the final Hamiltonian.
We study the time evolution of a conformal field theory deformed by a relevant operator under a smooth but fast quantum quench which brings it to the conformal point. We argue that when the quench time scale δt is small compared to the scale set by the relevant coupling, the expectation value of the quenched operator scales universally as δλ/δt(2Δ-d), where δλ is the quench amplitude. This growth is further enhanced by a logarithmic factor in even dimensions. We present explicit results for free scalar and fermionic field theories, supported by an analytic understanding of the leading contribution for fast quenches. Our results suggest that this scaling result, first found in holography, is in fact quite general. Our considerations also show that this limit of fast smooth quenches is quite different from an instantaneous quench from one time-independent Hamiltonian to another, where the state at the time of the quench serves as an initial condition for subsequent evolution with the final Hamiltonian.
Quantum quenches display universal scaling in several regimes. For quenches which start from a gapped phase and cross a critical point, with a rate slow compared to the initial gap, many systems obey Kibble-Zurek scaling. More recently, a different scaling behaviour has been shown to occur when the quench rate is fast compared to all other physical scales, but still slow compared to the UV cutoff. We investigate the passage from fast to slow quenches in scalar and fermionic free field theories with time dependent masses for which the dynamics can be solved exactly for all quench rates. We find that renormalized one point functions smoothly cross over between the regimes.
We examine in detail the relationship between smooth fast quantum quenches, characterized by a time scale δt, and instantaneous quenches, within the framework of exactly solvable mass quenches in free scalar field theory. Our earlier studies [1, 2] highlighted that the two protocols remain distinct in the limit δt → 0 because of the relation of the quench rate to the UV cut-off, i.e., 1/δt ≪ Λ always holds in the fast smooth quenches while 1/δt ∼ Λ for instantaneous quenches. Here we study UV finite quantities like correlators at finite spatial distances and the excess energy produced above the final ground state energy. We show that at late times and large distances (compared to the quench time scale) the smooth quench correlator approaches that for the instantaneous quench. At early times, we find that for small spatial separation and small δt, the correlator scales universally with δt, exactly as in the scaling of renormalized one point functions found in earlier work. At larger separation, the dependence on δt drops out. The excess energy density is finite (for finite mδt) and scales in a universal fashion for all d. However, the scaling behaviour produces a divergent result in the limit mδt → 0 for d ≥ 4, just as in an instantaneous quench, where it is UV divergent for d ≥ 4. We argue that similar results hold for arbitrary interacting theories: the excess energy density produced is expected to diverge for scaling dimensions Δ > d/2.
Global quantum quench with a finite quench rate which crosses critical points is known to lead to universal scaling of correlation functions as functions of the quench rate. In this work, we explore scaling properties of the entanglement entropy of a subsystem in a harmonic chain during a mass quench which asymptotes to finite constant values at early and late times and for which the dynamics is exactly solvable. When the initial state is the ground state, we find that for large enough subsystem sizes the entanglement entropy becomes independent of size. This is consistent with Kibble–Zurek scaling for slow quenches, and with recently discussed “fast quench scaling” for quenches fast compared to physical scales, but slow compared to UV cutoff scales.